Abstract
We are concerned with the existence and regularity of the solutions to the Dirichlet problem, for a class of quasilinear elliptic equations driven by a general differential operator, depending on \((x,u,\nabla u)\), and with a convective term f. The assumptions on the members of the equation are formulated in terms of Young’s functions, therefore we work in the Orlicz-Sobolev spaces. After establishing some auxiliary properties, that seem new in our context, we present two existence and two regularity results. We conclude with several examples.
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1 Introduction
In the present paper we deal with the existence and regularity of solutions to the the following nonlinear elliptic problem
\(\Omega \) is a subset of \({{{\mathbb {R}}}^n}\) having finite measure, \({\mathcal {A}}:\Omega \times {\mathbb {R}}\times {{{\mathbb {R}}}^n}\rightarrow {{{\mathbb {R}}}^n}\) and \(f:\Omega \times {\mathbb {R}}\times {{{\mathbb {R}}}^n}\rightarrow {\mathbb {R}}\) are Carathéodory functions.
In spite of the many boundedness (see, for instance, [9, 22, 37, 41] and the references therein) or regularity (see [12, 16, 23, 24, 38, 39]) results, in which the existence of a solution is assumed a priori, there are fewer results about the existence of solutions for (1.1), unless the principal part is the \(p-\)Laplacian or growths like the \(p-\)Laplacian, or the function f depends only on x.
This is due to the fact that when the operator \({\mathcal {A}}\) in (1.1) depends also from the unknown u, technical difficulties arise. First of all, variational methods are not directly applicable (and this holds also for problems with a convective term, namely with a nonlinearity f, depending also on \(\nabla u\)), and the properties of \({\mathcal {A}}\) are actually well known in standard contexts (see [17, 50]). This means, in particular, that in most situations, the abstract framework for the study of (1.1) is the classical Sobolev space \(W_0^{1,p}(\Omega )\). The paper [42] deserves particular attention, because the authors study problems of higher order than ours and do not require the \(\Delta _2\) condition on the Young function involved. Comparing their result (for elliptic operators) with our we can only state that the growth conditions on the s variable are more restrictive than ours. This is quite natural, due to their conditions on the Orlicz space used (see Remark 3.5).
In this regard, we note that there are many papers dealing with the existence of solutions to (1.1), whenever the principal part does not depend explicitely on u, and the tecniques used are the most diverse. This also testifies to the growing interest in recent years for the study of these problems.
The two main motivations of the paper are to investigate problems whose differential part depends explicitely also on the unknown, and to remove restrictions of power type growth in the study of (1.1). To do this, we have to establish some properties of \({\mathcal {A}}\), which are of independent interest (see Proposition 2.17). Just to give an idea, in our contest we can manage problems driven by an operator having a power-times-logarithmic type growth, like (see Example 2.16)
To the best of our knowledge, very few results are actually available for problems driven by such operators (see, for instance [42]). Conversely, many existence and regularity results are available for (1.1), when \({\mathcal {A}}\) is the p-Laplacian, or the (p, q)-Laplacian, namely when \({\mathcal {A}}(\nabla u)=|\nabla u|^{p-2}\nabla u\) or \({\mathcal {A}}(\nabla u)=|\nabla u|^{p-2}\nabla u+ |\nabla u|^{q-2}\nabla u\) (see [1, 4, 7, 26, 40, 43, 44, 46, 51]). There is also an extensive literature concerning problems in which the structure of \({\mathcal {A}}\) allows to tackle (1.1) with the same techniques adopted for the \(p-\)Laplacian. We refer to [28, 45] for problems with \({\mathcal {A}}(\nabla u)\), to [13,14,15, 35, 43, 44, 49] for problems with \({\mathcal {A}}(x,\nabla u)\) and finally, for problems with \({\mathcal {A}}(x,u,\nabla u)\) in \(W_0^{1,p}(\Omega )\), we cite [27, 31].
Much less is available regarding more general operators, built via a function radial with respect to \(\xi \), but not necesseraly a polinomyal. This new situation requires the use of Young functions and Orlicz spaces. Existence (and regularity) results for problems with \({\mathcal {A}}(\nabla u)=a(|\nabla u|)\nabla u\) can be found in [5, 6, 10, 11, 18, 23, 29]. In [25] the authors deal with an operator depending on the three variables via Young functions of a real variable.
In this paper, the growth conditions on the terms appearing in (1.1) require to replace the customary Sobolev space with an Orlicz space (see (1.2)). These conditions cover several instances already studied in the papers cited above. We stress that Young’s functions are also involved in the growth of the convective term f. Similar hypotheses can be found in [10, 11, 25].
Given the non-variational nature of the problem, we use the method of sub and super solutions, togheter with truncation techniques and the theorem of existence of zeros for monotone operators. For the regularity results, a main tool is a Theorem due to Lieberman (see [39, Theorem 1.7] and Proposition 4.1).
The first step necessary to obtain our results consists in establishing some properties of the operator \({\mathcal {A}}\) in Orlicz spaces. This properties, as well as basic definitions and some other auxiliary results, are collected in Sect. 2. The main existence theorems (and a general example of function \({\mathcal {A}}\) satisfying our assumptions) are presented in Sect. 3: beside to a general existence result, Theorem 3.4, in Theorem 3.9 we consider a special instance of (1.1), where a subsolution and a supersolution are obtained via variational methods. In Sect. 4 we prove our regularity results. Theorems 3.9 and 4.4 have among the main hypothesis, the existence of a subsolution and a supersolution. Starting from them, we construct a suitable functional, in which appropriate truncations of \({\mathcal {A}}\) and of the convective term f are involved.
Finally, Sect. 5 is devoted to some examples where we highlight how the method of sub and super solutions works well in all situations where the convective term f has two zeros of opposite sign, namely \(f(x,s_1,0)=f(x,s_2,0)=0\) for all \(x\in \Omega \) and for some \(s_1,s_2\in {\mathbb {R}}\), enjoying the condition \(s_1\cdot s_2<0\).
In this regard, it should be noted that the use of the method of sub and super solutions, combined with truncation techniques, is extremely valid when a mix of conditions related to both \({\mathcal {A}}\) and f occur. This is due to the fact that once we have a sub and a super solution, we restrict our attention to a suitable truncation of \({\mathcal {A}}(x,u,\nabla u)\). If the structure of f allow for sub and super solutions having good properties, such as, for instance, boundedness, then the truncated operator satisfies the hypotheses of the abstract result, even if the growth of \({\mathcal {A}}\) is more general than one would expect (see Examples 5.1 and 5.3).
2 Preliminaries
In this Section we give the main definitions on Young functions and introduce the Orlicz Sobolev spaces that we use in the sequel. We also collect some new results, auxiliary for the proof of the main theorems. Classical results concerning Young functions and Orlicz spaces can be found in [19, 36, 47, 48].
Definition 2.1
A function \(A: [0, \infty ) \rightarrow [0, \infty ]\) is called a Young function if it is convex, vanishes at 0, and is neither identically equal to 0, nor to infinity.
It is not restrictive to assume that any finite valued Young function is continuous. For Young functions
Definition 2.2
The Young conjugate of a Young function A is the Young function \({{\widetilde{A}}}\) defined as
The following inequalities are a consequence of Definition 2.2
The inverse function in (2.2) is the generalized right continuous inverse. In general a Young function is left continuous (in effect it is continuous, unless it takes the value \(+\infty \)), and when we deal with the inverse we consider the right continuous one.
Definition 2.3
A Young function A is said to dominate another Young function B near infinity, if there exist constants \(k>0\) and \(M\ge 0\) such that
If (2.4) holds with \(M=0\), then we say that A dominates B Two Young functions A and B are called equivalent near infinity (globally) if they dominate each other near infinity (globally) and w e briefly write \(A\approx B\) near infinity (globally).
Definition 2.4
Two functions \(f,g: (0, \infty )\rightarrow [0, \infty )\), are equivalent near infinity (briefly \(f \approx g \) near infinity) if and only if there exist suitable positive constants \(c_1\), \(c_2\) and \(s_0\) such that
Definition 2.5
A Young function B is said to increase essentially more slowly than A near infinity (briefly \(B\ll A\)), if B is finite valued and
Condition (2.6) is equivalent to
where \(A^{-1}\) denotes the right continuous inverse of A. Note that if \(A(t)=+\infty \) for \(t>t_0\), then \(A^{-1}(s)\) is constant for s sufficiently large, whatever inverse we take.
Proposition 2.6
Let A and B be two Young functions, such that \(B\ll A\). Then there exists a Young function C such that \(B\ll C\ll A\).
Proof
Let us define \(C^{-1}(t)=\sqrt{A^{-1}(t)B^{-1}(t)}\) for \(t\ge 0\). One can easily check that \(C^{-1}\) is concave, strictly increasing (inasmuch A and B are strictly increasing), and
Thus C satisfies \(B\ll C\ll A\). \(\square \)
Definition 2.7
A Young function A is said to satisfy the \(\Delta _2\)-condition near infinity (briefly \(A\in \Delta _2\) near infinity) if it is finite valued and there exist two constants \(K\ge 2\) and \(M\ge 0\) such that
Definition 2.8
The function A is said to satisfy the \(\nabla _2\)-condition near infinity (briefly \(A\in \nabla _2\) near infinity) if there exist two constants \(K>2\) and \(M\ge 0\) such that
If (2.9) or (2.10) holds with \(M=0\), then A is said to satisfy the \(\Delta _2\)-condition (globally), or the \(\nabla _2\)-condition (globally), respectively. If (2.9) or (2.10) holds for \(0\le t\le M\), then A is said to satisfy the \(\Delta _2\)-condition near zero, or the \(\nabla _2\)-condition near zero, respectively. If \(A\in \Delta _2\cap \nabla _2\) near infinity then \({\widetilde{A}}\in \Delta _2\cap \nabla _2\) near infinity (see [8], Proposition 6.6).
We give basic definitions and the main properties on the Orlicz spaces. Let \(\Omega \) be a measurable set in \({{{\mathbb {R}}}^n}\), with \(n\ge 1\). Given a Young function A, the Orlicz space \(L^A(\Omega )\) is the set of all measurable functions \(u:\Omega \rightarrow {\mathbb {R}}\) such that the Luxemburg norm
is finite. The functional \(\Vert \cdot \Vert _{L^A(\Omega )}\) is a norm on \(L^A(\Omega )\), and it is a Banach space (see [3]). If A and B are Young functions, \(A\approx B\) near infinity and \(|\Omega |<\infty \) then \(L^A(\Omega )=L^B(\Omega )\), and there exist \(k_1(,t_0,|\Omega |), k_2(t_0,|\Omega |)\) such that
If A is a Young function and \({{\widetilde{A}}}\) denotes its conjugate, then a generalized Hölder inequality
holds for every \(u\in L^A (\Omega )\) and \(v\in L^{{{\widetilde{A}}}}(\Omega )\).
If \(A\in \Delta _2\) globally (or \(A\in \Delta _2\) near infinity and \(\Omega \) has finite measure) then
Also, if \(B\ll A\) near infinity and \(\Omega \) has finite measure, then
Given a Young function \(A \in C^1([0,+\infty ))\), define the quantities
The conditions
are equivalent to the fact that \(A\in \nabla _2\cap \Delta _2\) globally (near infinity). The following result extends Lemma A.1 of [20] to Young functions \(A\in \nabla _2\) near infinity.
Lemma 2.9
Let \(\Omega \) be a subset of \({{{\mathbb {R}}}^n}\), with finite measure. Let A be a Young function. Assume that \(i_A^\infty >1\). Then there exists \(k_3(i_A^\infty ,\,|\Omega |)\ge 0\) such that
Here \(k_1\) is that of (2.11).
Proof
Since \(i_A^\infty >1\), corresponding to \(\varepsilon \in (0,i_A^\infty -1) \) there exists \(t_0>0\) such that \(\displaystyle {\frac{t\cdot A'(t)}{A(t)}>i_A^\infty -\varepsilon }\) for \(t\ge t_0\). Fix \(\alpha >i_A^\infty \) and consider the Young function
where \(c_1\) and \(c_2\) are chosen in such a way that \(A_1\in C^1([0,+\infty [)\). Then \(A_1\approx A\) near infinity and \(i_{A_1}^\infty =i_{A}^\infty \). Let \(v\in L^A(\Omega )\) be such that \(\Vert v\Vert _{L^A(\Omega )}\ge \frac{1}{k_1}. \)
Using Lemma A.1 of [20] and (2.11), with \(B=A_1\)
where \(m_0=\min _{0\le t\le t_0}(A(t)-A_1(t))\le 0\). Thus (2.16) follows with \(k_3= -m_0|\Omega |\ge 0\). \(\square \)
The Lemma below is a consequence of the Theorems of Vitali and De la Vallée-Poussin (see [5, Theorems 2.6 and 2.8]). It will be used several times in the paper.
Lemma 2.10
Let \(\Omega \subseteq {{{\mathbb {R}}}^n}\) be such that \(|\Omega |<\infty \) and let A be a Young function. If \(\{u_k\}_{k\in N}\subseteq L^A(\Omega )\) satisfies
-
\(u_k(x)\rightarrow u(x) \) a.e. in \(\Omega \) ,
-
\(\{u_k\}_{k\in N}\) is bounded in \(L^A(\Omega )\) ,
then \(u_k\rightarrow u\) in \(L^B(\Omega )\) for all Young functions \(B\ll A\) near infinity.
If \(L^A(\Omega )\) is reflexive, then \(|u_k-u|\rightharpoonup 0\) in \(L^A(\Omega )\).
Proof
Let \(B\ll A\) near infinity. Then B is finite valued and we may assume, without loss of generality, that B is continuous. We must prove that \(\lim _{k\rightarrow +\infty }\int _\Omega B\left( \frac{|u_k-u|}{\lambda }\right) dx=0\), for all \(\lambda >0\).
If \(A(t)\equiv +\infty \) for \(t>t_0>0\), then \(L^A(\Omega )=L^\infty (\Omega )\) and the conclusion follows via the dominated convergence theorem.
If A is finite valued, then we may assume that it is continuous. Fix \(\lambda >0\) and let \(M>0\) be such that \(\Vert u_k\Vert _{L^A(\Omega )}\le M\) for all \(n\in N\). Then, the continuity of A and the Fatou’s lemma guarantee that \(u\in L^A(\Omega )\) and \(\Vert u\Vert _{L^A(\Omega )}\le M\). Let \(\Psi (t)= A\big (\frac{\lambda }{2\,M}B^{-1}(t)\big )\), for \(t\ge 0\). Then \(\Psi \) is increasing and the condition \(B\ll A\) guarantees that
Note that
Thus the family \(\left\{ B\left( \frac{|u_k-u|}{\lambda }\right) \right\} _{k\in N}\) is equintegrable and we can apply the Vitali’s Theorem. This proves that \(u_k\rightarrow u\) in \(L^B(\Omega )\).
For the last part of the proof we observe that \(|u_k-u|\rightharpoonup v\) in \(L^A(\Omega )\), up to a subsequence. From the proof above we know that \(|u_k-u|\rightarrow 0\) in \(L^B(\Omega )\). Thus \(v\equiv 0\). This apply to all the subsequences, thus it holds for the whole sequence. \(\square \)
From now and throughout the paper we assume that \(\Omega \) is an open set in \({{{\mathbb {R}}}^n}\) with \(|\Omega |<\infty \). The isotropic Orlicz-Sobolev spaces \(W^{1,A}(\Omega )\) and \(W^{1,A}_0(\Omega )\) are defined as
and
The spaces \(W^{1,A}(\Omega )\) and \({W_0^{1,A}(\Omega )}\) equipped with the norms
are Banach spaces. The norm on \({W_0^{1,A}(\Omega )}\) is equivalent to the standard one
If \(A\in \Delta _2\cap \nabla _2\) near infinity, then the Orlicz-Sobolev space \(W^{1,A}_0(\Omega )\) is reflexive [8, Proposition 3.1].
Definition 2.11
The optimal Sobolev conjugate of A is defined by \(A_n:[0,\infty )\rightarrow [0,\infty ]\)
where \({ H}:[0,\infty ) \rightarrow [0, \infty )\) is given by
provided that the integral is convergent. Here, \({ H}^{-1}\) denotes the generalized left-continuous inverse of H.
If
then (see [21, Theorem 1])
and there exists a constant \(C=C(n)\) such that
Thus from [21, Theorem 3] (or [32, Theorem 3.1]), the embedding \({W_0^{1,A}(\Omega )}\rightarrow L^{E}(\Omega )\) is compact for every Young function \(E\ll A_n\) near infinity.
If
then \(A_n\) assumes only finite values, while when
then \(A_n(t)=\infty \) for t large and (2.26) yields
When (2.27) holds then (see [8, (3.13)])
Example 2.12
A general Young function
Let \(1<p<+\infty \), \(p_1\in {\mathbb {R}}\). Consider a Young function \(B:[0,+\infty [\rightarrow [0,+\infty [\), such that
Then
and \(W_0^{1,B}(\Omega )\) is reflexive. Also
Thus, given any Young function A such that
and a set \(\Omega \) with finite measure, it holds \({W_0^{1,A}(\Omega )}=W_0^{1,B}(\Omega )\), up to equivalent norms. The conjugate \({\widetilde{A}}\) and the optimal Sobolev conjugate of A satisfy
and
If \(p+p_1-1\ge 0\) then the function B satisfying (2.30) for all \(t>0\) is in fact a Young function. When \(p+p_1-1>0\) the function \(B\in \nabla _2\) globally. All the examples of the paper will involve functions A complying with (2.33).
We recall now some definitions and the theorem on pseudomonotone operators. Then we introduce the conditions on the function \({\mathcal {A}}\) in (1.1).
Definition 2.13
Let X be a real reflexive Banach space. A mapping \({{{\mathcal {B}}}}: X\rightarrow X^*\) is called
-
(i)
coercive if \(\lim _{\Vert u\Vert \rightarrow \infty }\frac{\langle {{{\mathcal {B}}}}u,u\rangle }{\Vert u\Vert }=+\infty \);
-
(ii)
bounded if it maps bounded sets into bounded sets;
-
(iii)
pseudomonotone if \(u_n\rightharpoonup u\) and \(\limsup _{k\rightarrow +\infty }\langle {{{\mathcal {B}}}}u_k,u_k-u\rangle \le 0\) imply that \({{{\mathcal {B}}}}u_k\rightharpoonup {{{\mathcal {B}}}}u\) and \(\langle {{{\mathcal {B}}}}u_k,u_k\rangle \rightarrow \langle {{{\mathcal {B}}}}u,u\rangle \).
-
(iv)
satisfies the \((S)_+\) property if \(u_k\rightharpoonup u\) and \(\limsup _{k\rightarrow +\infty }\langle {{{\mathcal {B}}}}u_k,u_k-u\rangle \le 0\) imply that \(u_k\rightarrow u\) (strongly) in X.
Theorem 2.14
(see [17, Theorem 2.99]) Let X be a real reflexive Banach space and let \({{{\mathcal {B}}}}:X\rightarrow X^*\) be a bounded, coercive and pseudomonotone operator. Then, for every \(b\in X^*\) the equation \({{{\mathcal {B}}}}x=b\) has at least solution \(x\in X\).
Let us now introduce the hypotheses on the function \( {\mathcal {A}}\) operator and verify the properties of the integral operator associated with it, in a standard way. We note that this is a Leray Lions type operator. The properties of the auxiliary truncated operator will easily follow from those of the non-truncated operator (see Proposition 2.17). We extend the results in [17, 50] in several directions: Orlicz spaces are considered, and even in the case of Lebesgue spaces, the hypotheses are slightly more general.
Let \(\Omega \subset {{{\mathbb {R}}}^n}\) be a set with finite measure and let A be a Young function, \(A\in \Delta _2\cap \nabla _2\) near infinity. Consider the vector valued function \({\mathcal {A}}:\Omega \times {\mathbb {R}}\times {{{\mathbb {R}}}^n}\rightarrow {{{\mathbb {R}}}^n}\), \({\mathcal {A}}=(a_1,\ldots a_n)\), enjoying with the properties that each \(a_i(x,s,\xi )\) is a Carathéodory function, and
Here \(q\in L^{{{\widetilde{A}}}} (\Omega )\), \(b>0\), and F is a Young function, \(F\ll A_n\) near infinity.
Here \(c,d>0\), G is a Young function, \(G\ll A_n\) near infinity, and \(r\in L^{1}(\Omega )\).
Remark 2.15
Using the condition \(A\in \Delta _2\) near infinity, and \(|\Omega |<\infty \), it can be shown that (2.38) is equivalent to
We will use (2.39) in the sequel.
Example 2.16
We present some examples of functions \({\mathcal {A}}(x,s,\xi )\) satisfying (2.36), (2.37) and (2.38).
The conditions below are used in all the examples presented.
Let \(p,\,p_1\in {\mathbb {R}}\), satisfying
Consider also a Young function D,
Let A(t) be defined as in (2.33). Note that \({\widetilde{A}}^{-1}(A(|\xi |))\approx |\xi |^{p-1}\lg ^{p_1}(1+|\xi |)\) near infinity.
Case 1: \(\mathbf{p<n}\). We choose three numbers \(\delta ,\,\beta , \, \beta _1\) satisfying
We assume also that the number r in (2.41) fulfills \(r>\frac{np}{n\delta -\beta (n-p)}\). We take a positive function \(a\in L^D(\Omega )\), and define
It holds
for all \((x,s,\xi )\in \Omega \times {\mathbb {R}}\times {{{\mathbb {R}}}^n}\).
Now we use twice formula (2.3) in (2.44): first with \(B(t)=t^{\frac{p-1}{\delta }}\) and then with \(B(t)=t^{\frac{r\delta }{p}}\).
Due to (2.41) and to \(a\in L^D(\Omega )\), it holds \(a(x)^{\frac{r(p-1)}{\delta }}\in L^{{\widetilde{A}}}(\Omega )\). Let F be a Young function, satisfying \(F(s)\approx |s|^{\frac{\beta rp}{r\delta -p}}\lg ^{\frac{\beta _1rp}{r\delta -p}-\frac{q}{p-1}}(1+|s|)\) near infinity. Then \(F\ll A_n\) near infinity, and \({\widetilde{A}}^{-1}(F(|s|))\) is equivalent, near infinity, to the function in the s variable, appearing in (2.45). Then (2.36) holds with \(q(x)=a(x)+k\), for some \(k>0\). Condition (2.37) is satisfied, because (2.40) guarantees that the function \(B(t)=t^{p-1-\delta }\lg ^{p_1\left( 1-\frac{\delta }{p-1}\right) }(1+t)\) is increasing, for \(t\ge 0\). Condition (2.38) clearly holds true.
Case 2: \(\mathbf{p=n,\ p_1<n-1}\). We choose \(\delta ,\,\beta \) satisfying
We assume also that the number r in (2.41) fulfills \( r>\frac{n}{\delta }\). We take a positive function \(a\in L^D(\Omega )\) and define
It holds
for all \((x,s,\xi )\in \Omega \times {\mathbb {R}}\times {{{\mathbb {R}}}^n}\).
Now we use twice formula (2.3) in (2.48), with the same functions of previous case.
For the function a it holds \(a(x)^{\frac{r(p-1)}{\delta }}\in L^{{\widetilde{A}}}(\Omega )\). Let F be a Young function, satisfying \(F(s)\approx e^{s^{\beta }}\) near infinity. Then \(F\ll A_n\) near infinity, and \({\widetilde{A}}^{-1}(F(|s|))\) is equivalent, near infinity, to the function in the s variable, appearing in (2.49). Then (2.36) holds with \(q(x)=a(x)+k\), for some \(k>0\).
Condition (2.37) is satisfied, because (2.40) guarantees that the function \(B(t)=t^{n-1-\delta }\lg ^{n-1-\delta }(1+t)\) is increasing, for \(t\ge 0\). Condition (2.38) clearly holds true.
Case 3: \(\mathbf{p=n,\ p_1=n-1}\). We choose \(\delta ,\,\beta \) satisfying
We assume that the number r in (2.41) fulfills \( r>\frac{n}{\delta }\) and we take a positive function \(a\in L^D(\Omega )\). Finally, we define
It holds
for all \((x,s,\xi )\in \Omega \times {\mathbb {R}}\times {{{\mathbb {R}}}^n}\).
Now we use twice (2.3) in (2.52), with the same functions of the first case.
For the function a it holds \(a(x)^{\frac{r(n-1)}{n}}\in L^{{\widetilde{A}}}(\Omega )\). Let F be a Young function satisfying \(F(s)\approx e^{e^{s^{\beta }}}\) near infinity. Then \(F\ll A_n\) near infinity and \({\widetilde{A}}^{-1}(F(|s|))\) is equivalent, near infinity, to the function in the s variable, appearing (2.53). Then (2.36) holds with \(q(x)=a(x)+k\), for some \(k>0\). Conditions (2.37) and (2.38) are satisfied, in view of previous case.
We now consider the borderline instances concernig \(\beta \). When \(\beta =0\), then we can take \(r=\frac{p}{\delta }\). When \(\beta =\frac{n\delta }{n-p}\) in (2.432.47), then in addition to \(\beta _1>-\frac{n\delta }{n-p}\), it is necessary \(\beta _1<\frac{p_1\delta (n-1)}{(n-p)(p-1)}\) and \(a\in L^{\infty }(\Omega )\).
Define now the operator \({{{\mathcal {S}}}}: {W_0^{1,A}(\Omega )}\rightarrow ({W_0^{1,A}(\Omega )})^*\) as
The properties of \({{{\mathcal {S}}}}\) are listed in the next proposition.
Proposition 2.17
Let A be a Young function, \(A\in \nabla _2\cap \Delta _2\) near infinity. Assume that \({\mathcal {A}}\) satisfies (2.36), (2.37), (2.39). Then the operator \({{{\mathcal {S}}}}\), introduced in (2.54), is well defined, bounded, continuous, and satisfies the \((S_+)\) property.
Proof
Let \(u\in {W_0^{1,A}(\Omega )}\). From (2.36), (2.29) and Propositon 6.6 of [8] we can find \(c_1>0\) such that
Thus \(|{\mathcal {A}}(x,u,\nabla u)|\in L^{{{\widetilde{A}}}}(\Omega )\) and \(\Vert {\mathcal {A}}(x,u,\nabla u) \Vert _{L^{{{\widetilde{A}}}}(\Omega )}\le \max \{1,\,M\}\). From (2.12)
Similarly, if \({{{\mathcal {C}}}}\subseteq W^{1,A}_0(\Omega )\) is such that \(\Vert u\Vert _{W^{1,A}_0(\Omega )}\le M_0\) for some \(M_0>0\), then \(\Vert u\Vert _{L^{A_n}(\Omega )}\le M_1\) for all \(u\in {{{\mathcal {C}}}}\). From (2.55) and Lemma 2.7 of [10], there exist \(c_1, L>0\) such that
Then
and
Let now \(\{u_k\}\) be a sequence in \({W_0^{1,A}(\Omega )}\) converging to \(u\in {W_0^{1,A}(\Omega )}\). Then, we can find a subsequence, say still \(\{u_k\}\), and two functions \(g_1\in L^{A_n}(\Omega )\), \(g_2\in L^A(\Omega )\) such that \(|u_k(x)|\le g_1(x)\), \(|u(x)|\le g_1(x)\), \(|\nabla u_k(x)|\le g_2(x)\) and \(|\nabla u(x)|\le g_2(x)\) for a.a. \(x\in \Omega \) and for all \(k\in N\). Let \(\lambda >0\). Using (2.36) and the condition \({{\widetilde{A}}}\in \Delta _2\) near infinity we can find \(c_1>0\) such that
The function \(v_\lambda \in L^1(\Omega )\), inasmuch both \(A,{{\widetilde{A}}}\in \Delta _2 \) near inifnity and \(F\ll A_n\) near infinity. The continuity of \({\mathcal {A}}(x,\cdot , \cdot )\) guarantees that \(\lim _{k\rightarrow +\infty }{\mathcal {A}}(x,u_k(x),\nabla u_k(x))={\mathcal {A}}(x,u(x),\nabla u(x))\) a.e. in \(\Omega \). We can thus apply the Lebesgue theorem to obtain
The arguments above apply to any subsequence of \(\{u_k\}\). This means that given any subsequence, we can extract a subsubsequence for which (2.58) holds. Thus (2.58) holds for the whole sequence. Note that
Thus
This proves the continuity af \({{{\mathcal {S}}}}\).
Let us now demonstrate the \((S)_+\) property. Let \(\{u_k\}\) be a sequence in \({W_0^{1,A}(\Omega )}\), \(u_k\rightharpoonup u\) and
We divide the proof in four steps.
Step 1 \(\nabla u_k\rightarrow \nabla u\) a.e.
Let F be as in (2.36). From Proposition 2.6 there exists a Young function \(F_1\) such that \(F\ll F_1\ll A_n\) near infinity. Thus \(\{u_k\}\) strongly converges to u in \(L^{F_1}(\Omega )\), and we can find a function \(g\in L^{F_1}(\Omega )\) and a subsequence of \(\{u_k\}\), say still \(\{u_k\}\), such that \(|u_k(x)|,|u(x)|\le g(x)\) a.e. in \(\Omega \). Thus
The function \(v_\lambda \in L^1(\Omega )\) because of (2.14) and standard arguments used several times in this proof. Thus, arguing as for (2.58), \({\mathcal {A}}(x,u_k,\nabla u)-{\mathcal {A}}(x,u,\nabla u)\rightarrow 0\) in \(L^{{{\widetilde{A}}}}(\Omega )\) and
By definition of weak convergence
From (2.37)
Passing to the limit in (2.64) and using (2.60), (2.63) and (2.62)
Thus the sequence \(({\mathcal {A}}(x,u_k,\nabla u_k)-{\mathcal {A}}(x,u_k,\nabla u))(\nabla u_k-\nabla u)\rightarrow 0\) in \(L^1(\Omega )\), and we can find a subsequence, say still \(\{u_k\}\), and a set \(\Omega _0\subset \Omega \), such that \(|\Omega _0|=0\) and (recall that \(u_k\rightarrow u\) in \(L^{F_1}(\Omega )\))
We prove that for every \(x\in \Omega \setminus \Omega _0\) there exists \(M>0\) such that \(|\nabla u_k(x)|\le M\) for all \(k\in N\). We argue by contradiction. Assume that there exists \(x\in \Omega \setminus \Omega _0\) such that for every \(h>|\nabla u(x)|+1\) there exists \(k_h\in N\) such that \(|\nabla u_{k_h}(x)|>h\). In particular
The sequence \(\left\{ \frac{\nabla u_{k_h}(x)-\nabla u(x)}{|\nabla u_{k_h}(x)-\nabla u(x)|}\right\} \) converges to \(\xi \in {{{\mathbb {R}}}^n}\), up to a subsequence. We keep the same notation as above for the subsequence and use (2.37) and (2.67)
and this leads to \(\left( {\mathcal {A}}(x,u(x),\nabla u(x)+\xi )-{\mathcal {A}}(x,u(x),\nabla u(x))\right) \xi =0\), namely \(\xi =0\). This contradicts \(|\xi |=1\), thus \(|\nabla u_{k_h}(x)|\le M\), for some \(M>0\) and we can find a subsequence, say still \(\{\nabla u_{k_h}\}\), converging to \(\eta \in {{{\mathbb {R}}}^n}\). Then from (2.66) and the convergence of \(\{\nabla u_{k_h}\}\) to \(\eta \in {{{\mathbb {R}}}^n}\)
From (2.37) we deduce \(\eta =\nabla u(x)\). We have so proved that every subsequence of \(\{\nabla u_k\}\) has a subsequence converging to \(\nabla u(x)\). Thus the whole sequence converges to \(\nabla u(x)\) in \(\Omega \setminus \Omega _0\).
Step 2 \(|a_i(x,u_k,\nabla u_k)-a_i(x,u,\nabla u)|\rightharpoonup 0\) in \(L^{{\widetilde{A}}}(\Omega )\).
Due to step 1 and to the continuity of \({\mathcal {A}}(x,\cdot ,\cdot )\), we have \({\mathcal {A}}(x,u_k,\nabla u_k)\rightarrow {\mathcal {A}}(x,u,\nabla u)\) a.e. in \(\Omega \). \(\{u_k\}_{k\in N}\) is bounded in \({W_0^{1,A}(\Omega )}\) and, from (2.56), \(\{|{\mathcal {A}}(x,u_k,\nabla u_k)|\}_{k\in N}\) is bounded in \(L^{{\widetilde{A}}}(\Omega )\). Since \({\widetilde{A}}\in \nabla _2\) near infinity we can apply Lemma 2.10 to obtain
Step 3. \({\mathcal {A}}(x,u_k,\nabla u_k)\cdot \nabla u_k\rightarrow {\mathcal {A}}(x,u,\nabla u)\cdot \nabla u\) in \(L^{1}(\Omega )\).
It holds
Put \(\sigma _k(x)={\mathcal {A}}(x,u,\nabla u)\cdot \nabla u-{\mathcal {A}}(x,u_k,\nabla u_k)\cdot \nabla u_k\). One has \(\sigma _k^+\le {\mathcal {A}}(x,u,\nabla u)\cdot \nabla u\) and \(\sigma _k\rightarrow 0\) a.e in \(\Omega \). Then
From (2.71)
So, using (2.72), (2.60) and Step 2
Step 4 \(\nabla u_k\rightarrow \nabla u\) in \(L^A(\Omega )\).
From (2.39), \(cA(|\nabla u_k(x)|)\le {\mathcal {A}}(x,u_k,\nabla u_k)\cdot \nabla u_k+dG(d|u_k|)+r(x)\) for all \(x\in \Omega \), all \(k\in N\). From Step 3 and the assumption \(G\ll A_n\), we know that there exists a subsequence of \(\{u_k\}\), say still \(\{u_k\}\) and a function \(g\in L^1(\Omega )\), such that \({\mathcal {A}}(x,u_k,\nabla u_k)\cdot \nabla u_k+dG(d|u_k|)\le g(x)\) a.e. in \(\Omega \). Taking into account that \(A(|\nabla u_k(x)|)\rightarrow A(|\nabla u(x)|)\) a.e. in \(\Omega \), we deduce that
A standard and repeatedly used argument shows that it holds for the whole sequence. Equation (2.74) and the \(\Delta _2\) condition on A, guarantees that \(\nabla u_k\rightarrow \nabla u\) in \(L^A(\Omega )\). \(\square \)
We now construct the truncation of \({{{\mathcal {S}}}}\) that we use in the proof of our results. For every \(r\in {\mathbb {R}}\), we set \(r^+=\max \{r,0\}\), \(r^-=\max \{-r,0\}\).
Let \({\underline{u}},\,{\overline{u}}\in W^{1,A}(\Omega )\) be such that \(({\overline{u}})^-,\,{\underline{u}}^+\in {W_0^{1,A}(\Omega )}\), and \({\underline{u}}\le {\overline{u}}\) a.e. in \(\Omega \). The truncation operator \(T:{W_0^{1,A}(\Omega )}\rightarrow {W_0^{1,A}(\Omega )}\) is defined by
The properties of \({\underline{u}},\,{\overline{u}}\) guarantee that \(T(u),\, (u-{\overline{u}})^+, \,(u-{\underline{u}})^-\in {W_0^{1,A}(\Omega )}\). In particular T is well defined.
It is known (see [34], p.20) that
Given the functions \({\underline{u}},\,{\overline{u}}\in W^{1,A}(\Omega )\) as above, and such that \({\underline{u}},\,{\overline{u}}\in L^{A_n}(\Omega )\), let us define the operator \({{{\mathcal {S}}}}_T: W_0^{1,A}(\Omega )\rightarrow (W_0^{1,A}(\Omega ))^*\), as
Corollary 2.18
Assume that \(A:[0,+\infty [\rightarrow 0,+\infty [\) is a Young function, \(A\in \Delta _2\cap \nabla _2\) near infinity, and that \({\mathcal {A}}\) satisfies (2.36), (2.37) and (2.38). Then the operator \({{{\mathcal {S}}}}_T\), introduced in (2.77) is well defined, bounded, continuous and has the \((S_+)\) property.
Proof
The inequality \(|T(u(x))|\le |u(x)|\) for all \(x\in \Omega \) guarantees that the operator \({{{\mathcal {S}}}}_T\) is well defined and bounded. Thanks to Lemma 4.1 in [10], the arguments used in Proposition 2.17 work also for \({{{\mathcal {S}}}}_{T}\). Thus \({{{\mathcal {S}}}}_{T}\) is continuous and has the \((S)_+\) property. \(\square \)
3 Main results
In this Section we state two of the main results of the paper (Theorems 3.4 and 3.9).
First we give the fundamental definitions of weak solution, subsolution and supersolution to (1.1).
Definition 3.1
A function \(u\in {W_0^{1,A}(\Omega )}\) is a weak solution to problem (1.1) if
and \(\int _{\Omega }{\mathcal {A}}(x,u,\nabla u)\cdot \nabla v dx\in {\mathbb {R}}\) for all \(v\in {W_0^{1,A}(\Omega )}\).
Definition 3.2
We say that \({{\overline{u}}}\in W^{1,A}(\Omega )\) is a supersolution to (1.1) if \(({\overline{u}})^{-} \in {W_0^{1,A}(\Omega )}\),
for all \(v\in {W_0^{1,A}(\Omega )}\), \(v\ge 0\) a.e. in \(\Omega \).
Definition 3.3
We say that \({\underline{u}}\in W^{1,A}(\Omega )\) is a subsolution to (1.1) if \({\underline{u}}^+\in {W_0^{1,A}(\Omega )}\) and
for all \(v\in {W_0^{1,A}(\Omega )}\), \(v\ge 0\) a.e. in \(\Omega \).
Theorem 3.4
Let \(\Omega \) be an open set in \({{{\mathbb {R}}}^n}\), with \(n\ge 2\), such that \(|\Omega |<\infty \). Let \(A\in C^1([0,+\infty ))\) be a Young function, \(A\in \Delta _2\cap \nabla _2\) near infinity. Assume also that A satisfies (2.24) and (2.27), or (2.28). Let \({\underline{u}}\) and \({\overline{u}}\) be a subsolution and a supersolution of problem (1.1), respectively, with \({\underline{u}}\le {\overline{u}}\) a.e. in \(\Omega \), and \({\underline{u}},{\overline{u}}\in L^{A_n}(\Omega )\). Assume that the function \({\mathcal {A}}\) satisfies (2.36), (2.37), (2.39). Let \(f:\Omega \times {\mathbb {R}}\times {\mathbb {R}}^n\rightarrow {\mathbb {R}}\) be a Carathéodory function fulfilling
where \(\sigma \in L^{{{\widetilde{A}}}_n}(\Omega )\), \({{\overline{\gamma }}}>0\) and \(E:[0,+\infty [\rightarrow [0,+\infty [\) is a Young function, \(E\ll A_n\) near infinity.
Then problem (P) has a solution \(u\in {W_0^{1,A}(\Omega )}\) such that \({\underline{u}}\le u\le {\overline{u}}\) a.e. in \(\Omega \).
Remark 3.5
In Remark 5.3 of [42], the authors prove that the strongly non linear equation
has a solution. Here \(\Omega \) is a bounded open set in \(R^n\) and the \(a_\alpha \) are Carathéodory functions satisfying
- \((A_2)\):
-
there exist \(c_1,\,c_2>0\), \(k_\alpha \in E^{{{\widetilde{A}}}}(\Omega )\) if \(|\alpha |=m\), \(k_\alpha \in L^{{{\widetilde{A}}}}(\Omega )\) if \(|\alpha |<m\), and a function \(P\ll A\), such that for a.e. \(x\in \Omega \), all \(\xi \in {\mathbb {R}}^{N_0}\)
$$\begin{aligned} |a_\alpha (x,\xi )|&\le k_\alpha (x)+c_1\sum _{|\beta |=m}{{\widetilde{A}}}^{-1}(A(c_2\xi _\beta ))+c_1\sum _{|\beta |<m}{{\widetilde{P}}}^{-1}(A(c_2\xi _\beta ))\ {}&\ \hbox {if}\ |\alpha |\nonumber \\&=m\\ |a_\alpha (x,\xi )|&\le k_\alpha (x)+c_1\sum _{|\beta |=m}{{\widetilde{A}}}^{-1}(P(c_2\xi _\beta ))+c_1\sum _{|\beta |<m}{{\widetilde{A}}}^{-1}(A(c_2\xi _\beta ))\ {}&\ \hbox {if}\ |\alpha |\nonumber \\&<m\,. \end{aligned}$$(3.4) - \((A_3)\):
-
\(\sum _{|\alpha |=m}\left( a_\alpha (x,\eta ,\zeta )-a_\alpha (x,\eta ,\zeta ')\right) \cdot (\zeta _\alpha -\zeta _\alpha ')>0\) for a.e. \(x\in \Omega \), all \(\eta \in {\mathbb {R}}^{N_1}\), all \(\zeta ,\,\zeta '\in {\mathbb {R}}^{N_2},\ \xi \ne \xi '\). The vector \(\zeta \) is the top order part of \(\xi \).
- \((A_4)\):
-
There exist functions \(b_\alpha (x) \in E^{{{\widetilde{A}}}}(\Omega )\) for \(|\alpha |=m\), \(b\in L^1(\Omega )\), constants \(d_1,d_2>0\) and some fixed element \(\varphi \in W_0^mE^{{{\widetilde{A}}}}(\Omega )\) such that for a.e. \(x\in \Omega \), all \(\xi \in {\mathbb {R}}^{N_0}\)
$$\begin{aligned} \sum _{|\alpha |=m}a_\alpha (x,\xi )(\xi _\alpha -D^\alpha \varphi (x)) \ge d_1 \sum _{|\alpha |=m}A(d_2\xi _\alpha )-\sum _{|\alpha |=m}b_\alpha (x)\xi _\alpha -b(x). \end{aligned}$$(3.5) - \((A_5)\):
-
\(f\in E^{{{\widetilde{A}}}}(\Omega )\).
The authors do not need the \(\Delta _2\) condition on the function A, so \(E^A(\Omega )\subseteq L^A(\Omega )\).
The result in [42] involves a very general operator, but it is possible to (partially) compare our result in Theorem 3.4, with that of [42], for \(m=1\) and for a function \(A\in \Delta _2\) near infinity.
In this situation the operator \({{\mathcal {A}}}\) in (1.1) coincides with the principal part of the operator in (3.2). The growth condition in (2.36) is more general than (3.3). The inequality (3.5), does not involve the s variable, but has some additional terms that do not appear in our (2.38). As regards the other terms, namely the lower part of the operator, and the function f in (3.2), and our function f in (1.1), we stress that our assumptions involve sub and super solutions, that are far from the framework of [42]. So we can simply note that if we assume the existence of a sub and a super solution for the problem than the lower part and the function f in (3.2) satisfy (3.1).
To prove Theorem 3.4, we perturb problem (1.1) and formulate an auxiliary one. Let \(\Pi :W_0^{1,A}(\Omega )\rightarrow ( W_0^{1,A}(\Omega ))^*\), given by
where
Let \({{{\mathcal {N}}}}_f\circ T:{W_0^{1,A}(\Omega )}\rightarrow ({W_0^{1,A}(\Omega )})^*\) be the operator defined as
Given \(\mu >0\), we consider the following problem
The result below guarantees that problem (3.7) has a solution, provided the parameter \(\mu >0\) is sufficiently large.
Theorem 3.6
Under the same assumtions of Theorem 3.4, there exists \(\mu _0>0\) such that (3.7) admits a solution whenever \(\mu \ge \mu _0\).
Proof
For all \(\mu >0\) consider the operator \({{{\mathcal {A}}}}_\mu :W_0^{1,A}(\Omega )\rightarrow (W_0^{1,A}(\Omega ))^*\), defined by
We prove that \({{{\mathcal {A}}}}_\mu \) is well defined, bounded, pseudomonotone and there is \(\mu _0>0\) such that \({{{\mathcal {A}}}}_\mu \) is coercive for all \(\mu >\mu _0\).
Due to Corollary 2.18, Propositions 4.3 and 4.5 of [10], \({{{\mathcal {A}}}}_\mu \) is well defined, bounded and continuous. To prove that it is pseudomonotone, we take \(u\in {W_0^{1,A}(\Omega )}\), and a sequence \(\{u_k\}\subset {W_0^{1,A}(\Omega )}\) such that
Equations (4.6) and (4.17) of [10] allow to write
Thus \(u_k\rightarrow u\) in \({W_0^{1,A}(\Omega )}\) (see Corollary 2.18), and consequently
so \(\langle {{{\mathcal {A}}}}_\mu (u_k),u_k\rangle \rightarrow \langle {{{\mathcal {A}}}}_\mu (u),u\rangle \), \(\langle {{{\mathcal {A}}}}_\mu (u_k),v\rangle \rightarrow \langle {{{\mathcal {A}}}}_\mu (u),v\rangle \) for all \(v\in {W_0^{1,A}(\Omega )}\) and \({{{\mathcal {A}}}}_\mu \) is a pseudomonotone operator.
It remains to prove that \({{{\mathcal {A}}}}_\mu \) is coercive for some \(\mu >0\). Arguing like for Eq. (5.3) of [10] we can find a constant \(c_1>0\) such that
Here c is that of (2.39). From (3.1), (2.12), (2.26), and (3.8)
Let \(v(x)=\max \{|{\underline{u}}(x)|, |{{\overline{u}}}(x)|\}\) Then \(|T(u)(x)|\le v(x)\) for all \(x\in \Omega \), and \(\int _\Omega G(dv(x))dx<+\infty \). From (2.39)
So, choosing \(\mu >a\), \(u\in {W_0^{1,A}(\Omega )}\) with \(\Vert u\Vert _{{W_0^{1,A}(\Omega )}} \gg 1\), and using (3.9), (3.10), Lemma 4.6 of [10], and (2.16)
Thus
Theorem 2.14 guarantees that there exists \(u\in W^{1,A}_0(\Omega )\) such that \({{{\mathcal {A}}}}_\mu (u)\equiv 0\). Thus
for all \(v\in W_0^{1,A}(\Omega )\). \(\square \)
Remark 3.7
The proof above works also if we weaken (2.36)\(\ldots \)(2.38), requiring them to hold for \(s\in [{\underline{u}}(x), {{\overline{u}}}(x)]\) rather than for all \(s\in {\mathbb {R}}\). This is because in the proof we consider only the truncated function \({\mathcal {A}}(x,Tu,\nabla u)\).
Proof of Theorem 3.4
By Theorem 3.6 there exists a solution \(u\in {W_0^{1,A}(\Omega )}\) of the truncated auxiliary problem (3.7) provided \(\mu >0\) is sufficiently large. Let us fix such a \(\mu \) and u.
Via the same comparison arguments of the proof of Theorem 3.6 of [10] we can prove that the solution of (3.7) has the enclosure property \(u\in [{\underline{u}},{\overline{u}}]\). Thus, it follows from (2.75) and (3.6) that \(Tu=u\) and \(\Pi (u)=0\). Consequently, u is a solution of(1.1). \(\square \)
The proof of the Corollary below follows the same lines as that of Corollary 5.2 of [10].
Corollary 3.8
Let \(\Omega \) be an open set in \({{{\mathbb {R}}}^n}\), with \(n\ge 2\), such that \(|\Omega |<\infty \). Let \(A\in C^1([0,+\infty ))\) be a Young function, \(A\in \Delta _2\cap \nabla _2\) near infinity. Assume also that A satisfies (2.24) and (2.27), or (2.28). Let \({\underline{u}}\) and \({\overline{u}}\) be a subsolution and a supersolution of problem (1.1), respectively, with \({\underline{u}}\le {\overline{u}}\) a.e. in \(\Omega \), \({\underline{u}},{\overline{u}}\in {W_0^{1,A}(\Omega )}\), and such that the Carathéodory function \(f:\Omega \times {\mathbb {R}}\times {\mathbb {R}}^n\rightarrow {\mathbb {R}}\) fulfills
where \(\rho \in L^{{{\widetilde{A}}}_n}(\Omega )\), \({\overline{\gamma }},\,E,\) are as in Theorem 3.4, and \(g:[0,+\infty [\rightarrow [0,+\infty [\) is a nondecreasing function such that \(g(0)=0\) and there exist \(s_0,\,h>0\) such that \(g(|s|)|s|\le A_n(h|s|)\) for all \(|s|\ge s_0\).
Then problem (P) possesses a nontrivial solution \(u\in {W_0^{1,A}(\Omega )}\).
We consider now a special instance of (1.1), in which \({\mathcal {A}}\) does not depend on s and has a potential with respect to \(\xi \). So, let \(\Omega \subset {{{\mathbb {R}}}^n}\) be a set of finite measure and let A, B be two Young functions such that \(A\in \Delta _2\cap \nabla _2\) near infinity, \(B\in \nabla _2\) near zero, and \(A\circ B^{-1}\) is a Young function too. We assume that \({\mathcal {A}}:\Omega \times {{{\mathbb {R}}}^n}\rightarrow {{{\mathbb {R}}}^n}\), \({\mathcal {A}}=(a_1,\ldots a_n)\), is such that each \(a_i(x,\xi )\) is a Carathéodory function, and
Furthermore, we assume that there exists a measurable function \(\Phi :\Omega \times {{{\mathbb {R}}}^n}\rightarrow {\mathbb {R}}\), even with respect to \(\xi \in {{{\mathbb {R}}}^n}\) and such that
Since \(A\circ B^{-1}\) is a Young function, it follows that A dominates B near infinity and B dominates A near zero. Thus
Also, if \(u\in L^A(\Omega )\) then \(B(k|u|)\in L^{A\circ B^{-1}}(\Omega )\) for all \(k>0\).
Condition (3.14) ensures that \(\Phi (x,\cdot )\) is convex for every \(x\in \Omega \). From (3.13) and (3.15) there exist \(k_4,k_5>0\) such that
Now, problem (1.1) reads as
For functions f satisfying suitable growth conditions we can construct a sub or a supersolution for problem (3.19), via variational methods. The \(\Delta _2\) and \(\nabla _2\) conditions play a crucial role here.
Theorem 3.9
Let \(\Omega \) be an open set in \({{{\mathbb {R}}}^n}\), with \(n\ge 2\), such that \(|\Omega |<\infty \). Let \(A\in C^1([0,+\infty ))\) be a Young function, \(A\in \Delta _2\cap \nabla _2\) at infinity. Assume also that A satisfies (2.24) and (2.27), or (2.28). Let \({\mathcal {A}}: \Omega \times {{{\mathbb {R}}}^n}\rightarrow {{{\mathbb {R}}}^n}\) and \(\Phi : \Omega \times {{{\mathbb {R}}}^n}\rightarrow {\mathbb {R}}\) be two Carathéodory functions satisfying (3.13)\(\ldots \)(3.16). Let \(f:\Omega \times {\mathbb {R}}\times {\mathbb {R}}^n\rightarrow {\mathbb {R}}\) be a Carathéodory function fulfilling
or
where \({{\overline{\gamma }}}>0\), E is a Young function, \(E\ll A_n\) near infinity, \(\rho _1,\rho _2:\Omega \rightarrow [0,+\infty [\) are two measurable functions, \(\rho _i\in L^{{{\widetilde{A}}}_n}(\Omega ),\ i=1,2\), \(g_1,g_2:[0,+\infty [\rightarrow [0,+\infty [\) are two non decreasing functions such that \(g_1(0)=g_2(0)=0\) and there exist \(s_0>0,\,h_0\in \left]0,\tau \omega _n^{\frac{1}{n}}|\Omega |^{-\frac{1}{n}}\right[\), \(h_1>0\) such that
Here \(\omega _n\) is the measure of the unit ball in \({{{\mathbb {R}}}^n}\), \(\tau =\min \{1,k_4^2\}\) where \(k_4\) is that of (3.18).
Then problem (3.19) possesses a nontrivial constant sign solution \(u\in {W_0^{1,A}(\Omega )}\).
Proof
Suppose that (3.20) is in force. We construct a subsolution \({\underline{u}}\le 0\) a.e., \({\underline{u}}\not \equiv 0\), and show that \({{\overline{u}}}\equiv 0\) is a supersolution but not a solution to (3.19). Then, we show that f satisfies (3.12).
Put \(G_1(t)=\int _0^t g_1(\tau )d\tau ,\ t\ge 0\) and consider the functional \(J:{W_0^{1,A}(\Omega )}\rightarrow {\mathbb {R}}\), defined as
We prove that J is well defined, weakly lower semicontinuous, coercive and
for all \(u,v\in {W_0^{1,A}(\Omega )}\). We examine separately the three integrals.
Due to (3.18), the fact that \(A\in \Delta _2\) at infinity, and the convexity of \(\Phi (x,\cdot )\), for all \(x\in \Omega \), the functional \(u\mapsto \int _\Omega \Phi (x,\nabla u)dx\) is well defined in \({W_0^{1,A}(\Omega )}\), convex. We briefly sketch the proof of its regularity, because it makes use of standard arguments like the Lebesgue Theorem, and the properties of Young’s functions. Let \(u,v\in {W_0^{1,A}(\Omega )}\). For all \(x\in \Omega \), all \(t>0\), \(t<<1\), there exists \(\mu _{t,x}\in (0,1)\) such that
We used (3.13), the monotonicity of \({{\widetilde{A}}}^{-1}\circ A\) and \({{\widetilde{B}}}^{-1}\circ B\), and (2.2). Now, the condition \(A\in \Delta _2\) near infinity, (3.17) and the Lebesgue Theorem, allow to prove that the functional \(u\mapsto \int _\Omega \Phi (x,\nabla u)dx\) is \(C^1\).
Thus the weak lower semicontinuity of J and Eq. (3.23) follow.
To prove the coercivity of J we need the following inequality, that can be found in [8, Proposition 3.2],
Let \(\varepsilon >0\) be such that \(h_0\omega _n^{-\frac{1}{n}}|\Omega |^{\frac{1}{n}}<\tau -\varepsilon \). Using (3.22) and the inequality above
Take now \(u\in {W_0^{1,A}(\Omega )}\), \(\Vert u\Vert _{{W_0^{1,A}(\Omega )}}>1\) and use (3.18), (2.12), (3.25) and (2.16)
This proves that J is coercive. Thus it has a global minimum. Let \({\underline{u}}\) be a global minimum point for J. We prove that \({\underline{u}}\not \equiv 0\). To this end consider a function \(v\in C_0^1(\Omega )\), such that \(b|\nabla v(x)|\le {{\overline{t}}}\) and \(k_5{{\overline{k}}}|\nabla v(x)|\le {{\overline{t}}}\) for all \(x\in \Omega \). Also, \(v\le 0\) and \(\rho _1(x)v(x)\not \equiv 0\) in \(\Omega \). The inequality \(\frac{B(t_1)}{B(t_0)}>\left( \frac{t_1}{t_0}\right) ^{k_B}\) holds for \(0<t_0<t_1<{{\overline{t}}}\), and some \(k_B>1\), by virtue of the \(\nabla _2\) condition near zero. Then, choosing once \(t_1=b|\nabla v|\), \(t_0=bt|\nabla v|\), and secondly \(t_1=k_5{{\overline{k}}}|\nabla v|\), \(t_0=tt_1\), with \(t<1\), and taking into account (3.18)
and this proves that \(J({\underline{u}})<0\). Using \(J(-|{\underline{u}}|)\ge J({\underline{u}})\) and the fact that \(\Phi (x,\cdot )\) is even, we obtain \({\underline{u}}\le 0\) a.e. in \(\Omega \).
Now we prove that \({\underline{u}}\) is a subsolution and \(u\equiv 0\) is a supersolution but not a solution to (1.1). Note that
Acting with any \(v\in {W_0^{1,A}(\Omega )}\), \(v\ge 0\), in (3.26) and using (3.20)
that is \({\underline{u}}\) is a subsolution to (3.19). Using (3.20) and choosing \(v\in {W_0^{1,A}(\Omega )}\), \(v\ge 0\)
thus \(u\equiv 0\) is a supersolution to (1.1) and the assumption on f(x, 0, 0) guarantees that it is not a solution.
We put \(\rho (x)=\max \{\rho _i(x),\ i=1,2\}\), \(g(|s|)=\max \{g_i(|s|),\ i=1,2\}\) and use (3.20)
Then f satisfies (3.12) and from Corollary 3.8 problem (3.19) has a nontrivial solution \(u\in {W_0^{1,A}(\Omega )}\) and \(u\in [{\underline{u}},0]\).
When (3.21) is in force we consider \(f_1(x,s,\xi )=-f(x,-s,-\xi )\). Then, by virtue of the proof above, problem
has a nontrivial solution \(v\in {W_0^{1,A}(\Omega )}\), \(v\le 0\) a.e. in \(\Omega \). Then the function \(u=-v\) is a nontrivial solution to (1.1) and \(u\ge 0\) a.e. in \(\Omega \). \(\square \)
Remark 3.10
From [9, Theorem 3.1], when \(\rho _1,\ \rho _2\in L^{M,\infty }(\Omega )\), for a suitable Young function M (see Eq. (3.9) in [9]), then the solution u is essentially bounded.
4 Regularity results
In this section we give some existence and regularity results, Theorems 4.4 and 4.5. We strenghten the hypotheses on \(\Omega \) and on A (see (4.1)), in order to apply regularity theory (see Proposition 4.1).
The proof of the existence is based on sub and supersolution methods, while the main tool for the regularity is Theorem 1.7 of [39] (see also the remark after that result and [38]), that we recall below.
Proposition 4.1
(see [39, Theorem 1.7]) Let \(\Omega \) be a bounded domain in \(R^n\) with a \(C^{1,\alpha }\) boundary, for some \(0<\alpha \le 1\). Let A be a Young function satisfying
Let \({\mathcal {A}}:\Omega \times {\mathbb {R}}\times {{{\mathbb {R}}}^n}\rightarrow {{{\mathbb {R}}}^n}\) be a vector valued function, with Carathéodory components, \(a_i\), \(i=1,\ldots ,n\). Consider the problem
Suppose \({\mathcal {A}}\) and f satisfy the structure conditions (here \(a_{ij}(x,s,\eta )=\frac{\partial a_i}{\partial \eta _j}\))
for some positive constants \(\Lambda \), \(\Lambda _1\), \(M_0\), for all x and \(y\in \Omega \), for all \(s,w\in [-M_0,M_0]\), and for all \(\xi \in {\mathbb {R}}^n\). Then, any solution \(u\in W^{1,A}(\Omega )\), with \(|u|\le M_0\) in \(\Omega \), is \(C^{1,\beta }({\overline{\Omega }})\) for some positive \(\beta \).
We point out that (4.1) guarantees that \(A'(0)=0\) and \(A\in \nabla _2\cap \Delta _2\) globally.
Lemma 4.2
Let A be a Young function satisfying (4.1). If \({\mathcal {A}}:\Omega \times {\mathbb {R}}\times {{{\mathbb {R}}}^n}\rightarrow {{{\mathbb {R}}}^n}\) has Carathéodory components, \(a_i(x,s,0)\equiv 0\), for a.e. \(x\in \Omega \), all \(s\in {\mathbb {R}}\), all \(i=1,\ldots ,n\), and satisfies (4.2) and (4.3) for all \(s\in {\mathbb {R}}\), then (2.36), (2.37) and (2.38) hold with \(q(x)\equiv r(x)\equiv 0\), \(F\equiv 0\), and \(d=0\). bluespecificare se le (4.2) e (4.3) valgono per qche \(M>0\) allora anche le altre valgono nell’intervallo
Proof
Let \(H(t)=a_i(x,s,t\xi ),\ t>0\). Using (4.3), \(a_i(x,s,0)\equiv 0\), and (4.1)
Then, thanks to Eq. (6.23) in [8]
and (2.36) is proved.
For any \(\xi ,\xi '\in {{{\mathbb {R}}}^n}\), write \(\xi =\xi '+\eta \), \(\eta \ne 0_{{{{\mathbb {R}}}^n}}\). Define \(H(t)=\sum _{i=1}^n\left( a_{i}(x,s,\xi '+t\eta )-a_{i}(x,s,\xi ')\right) \eta _i\), for \(t\in [0,1]\). From (4.2), \(H'(t)\ge \frac{A'(|\xi '+t\eta |)}{|\xi '+t\eta |}\cdot |\eta |^2\) and this guarantees that \(H(1)>H(0)=0\), namely (2.37).
Choose now \(\xi '=0\) in the function H just used. For \(\eta \ne 0_{{{{\mathbb {R}}}^n}}\), (4.2) and (4.1) give
If \(\eta =0_{{{{\mathbb {R}}}^n}}\) then \(H(1)=0\), and (2.38) is proved. \(\square \)
Following the spirit of Example 2.16, we present a function \({\mathcal {A}}\) satisfying (4.2), (4.3) and (4.4). This general function will be used in Example 5.3, but we prefere to introduce this function here, to emphasize its general structure.
Example 4.3
Let \(p>1\), \(q\in {\mathbb {R}}\) and \(p+q-1>0\). Consider the Young function \(A:[0,+\infty [\rightarrow [0,+\infty [\) complying with
Then A satisfies (4.1) because
Let us define
Here \(\Omega \) is a bounded domain with a \(C^{1,\alpha }\) boundary, \(\beta ,\delta \ge \alpha \) and p, q are like above. We show that \({\mathcal {A}}\) satisfies (4.2), (4.3) and (4.4). It holds
Let now \(\mu =\min \{1,p+q-1,p-1\}\). Then
This guarantees that
and (4.2) holds. A simple calculation shows that (4.3) holds too. For what concerns (4.4), let \(M>0\) and take \(x,y \in \Omega ,\ s,w \in [-M,M],\ \xi \in {{{\mathbb {R}}}^n}\). Then
For the last inequality in (4.12) we used (4.15) and (4.16) further down. They are obtained via the inequalities below.
If \(\rho \le 1\) then there exists \(c>0\) such that
If \(\rho > 1\) then there exists \(c>0\) such that
Take \(s,w\in [-M,M]\), for \(M>0\). If \(0<\delta \le 1\), from \((|s|^{1-\delta }+|w|^{1-\delta })^{\frac{1}{1-\delta }}\ge |s|+|w|\) and (4.134.14)
A similar argument, when \( 1\le \delta \le \alpha \), via (4.134.14), leads to
The same holds for \(|\Vert x\Vert ^\gamma -\Vert y\Vert ^\gamma |,\ x,y\in \Omega \). Thus (2.39) remains true.
For the first existence and regularity Theorem we assume that problem (1.1) admits a subsolution and a supersolution \({\underline{u}}\), \({\overline{u}}\in W^{1,\infty }(\Omega )\).
Theorem 4.4
Let \(\Omega \) be a bounded domain in \({{{\mathbb {R}}}^n}\) with a \(C^{1,\alpha }\) boundary. Let the functions A and \({\mathcal {A}}\) be as in Proposition 4.1. Assume further that (4.2), (4.3) and (4.4) hold for all \(s\in {\mathbb {R}}\). Let \({\underline{u}}\), \({\overline{u}}\in W^{1,\infty }(\Omega )\) be a subsolution and a supersolution for problem (1.1), with \({\underline{u}}(x)<{\overline{u}}(x)\) a.e. \(x\in \Omega \). Let \(f:\Omega \times {\mathbb {R}}\times {{{\mathbb {R}}}^n}\rightarrow {\mathbb {R}}\) be a Carathéodory function satisfying
where \(\sigma \in L^{\infty }(\Omega )\) and \({{\overline{\gamma }}}:[0,+\infty [\rightarrow [0,+\infty [\) is locally essentially bounded.
Then problem (1.1) admits at least a solution \(u\in C_0^{1,\beta } ({{\overline{\Omega }}})\). Moreover \({\underline{u}}(x)\le u(x)\le {{\overline{u}}}(x)\) a.e in \(\Omega \).
Proof
Let \(M=\max \{\Vert {{\overline{u}}}\Vert _\infty ,\Vert {\underline{u}}\Vert _\infty \}+1>0\), \(R>\max \{\Vert \nabla {{\overline{u}}}\Vert _\infty ,\Vert \nabla {\underline{u}}\Vert _\infty \}\). Let \(I=[-M,M]\) and \({{\overline{\gamma }}}=\Vert {{\overline{\gamma }}}\Vert _{L^\infty (I)}\). Then \({{\overline{\gamma }}}<+\infty \) and \({{\overline{\gamma }}}(s)\le {{\overline{\gamma }}}\) for a.e. \(s\in I\). Let \(I_0\subset I\) be a set of null measure, such that \({{\overline{\gamma }}}(s)>{{\overline{\gamma }}}\) for all \(s\in I_0\). For \(s\in I_0\) it holds
Let the truncated functions \(a_i^M\) and \(f_R\) be defined as
and
Consider the problem
In view of the choice of R, \({\underline{u}}\) and \({{\overline{u}}}\) are a subsolution and a supersolution to (4.18) respectively. The function f satisfies satisfies the hypotheses of Proposition 4.1. In particular, (4.5) holds ith \(\Lambda _1=\max \{\Vert \sigma \Vert _\infty , {{\overline{\gamma }}}\}\). Since \(|f_R|\le |f|\) the same holds for \(f_R\) whatever R is. Also \({\mathcal {A}}^M\) satisfies (4.2), (4.3) and (4.4). Due to Proposition 4.1 there exist two positive constants \(0<\beta \le 1\) and C, independent from R, such that any solution to (4.18) belongs in \(C_0^{1,\beta }({{\overline{\Omega }}})\) and \(\Vert u\Vert _{C_0^{1,\beta }({{\overline{\Omega }}})}\le C\). Choosing \(R>C\) we deduce that u is a solution to (1.1). \(\square \)
The next Theorem is related to Theorem 3.9, because it deals with problem (3.19). Now, we are concerned with the existence of a regular solution to (3.19).
Theorem 4.5
Let \(\Omega \) be a bounded domain in \({{{\mathbb {R}}}^n}\) with \(C^{1,\alpha }\) boundary. Let A be a Young function satisfying (4.1), and let \({\mathcal {A}}:\Omega \times {{{\mathbb {R}}}^n}\rightarrow {{{\mathbb {R}}}^n}\) be a vector valued function satisfying (4.2) and (4.3) for a.e. \(x\in \Omega \), all \(\xi \in {{{\mathbb {R}}}^n}\), and such that \(a_i(x,0)\equiv 0\) for a.e. \(x\in \Omega \), all \(i=1,\ldots ,n\). We further assume that (3.16) holds. Let \(f:\Omega \times {\mathbb {R}}\times {\mathbb {R}}^n\rightarrow {\mathbb {R}}\) be a Carathéodory function fulfilling
or
Here \(\rho _1,\rho _2:\Omega \rightarrow [0,+\infty [\) are two measurable functions, \(\rho _1,\,\rho _2\in L^\infty (\Omega )\); \(g_1\) is like in Theorem 3.9, \(g_2:[0,+\infty [\rightarrow [0,+\infty [\) is a non-decreasing function such that \(g_2(0)=0\) and \({{\overline{\gamma }}} (s)\) is a locally essentially bounded function.
Then problem (P) has a nontrivial, solution \(u\in C_0^{1,\beta }({{\overline{\Omega }}})\).
If (4.194.20) jolds, then \(u\ge 0\) in \(\Omega \). In the other case \(u\le 0\) in \(\Omega \).
Proof
From the proof of Theorem 3.9 we know that, when (4.194.20) is in force, there exists a nontrivial solution \({\overline{u}}\ge 0\), to problem (3.19).
Remark 3.10 guarantees that \({\overline{u}}\) is bounded. From Proposition 4.1, we have that \({\overline{u}}\in C_0^{1,\beta }({\overline{\Omega }})\). The inequalities in (4.194.20) show that \({\overline{u}}\) is a supersolution to problem (1.1) and \({\underline{u}}=0\) is a subsolution to problem (1.1). The assumptions on \(\rho _2\) guarantee that \({\underline{u}}=0\) is not a solution. If we put \(M=\Vert {{\overline{u}}}\Vert _\infty \), \({{\overline{\gamma }}}=\Vert {{\overline{\gamma }}}\Vert _{L^\infty (0,M)}\), \(\sigma (x)=\max \{\rho _1(x)+g_1(M),\; \rho _2(x)+g_2(M)\}\) for a.e. \(x\in \Omega \), then (4.194.20) leads to
So, from Theorem 4.4, problem (3.19) admits at least a nontrivial solution \(u\in C_0^{1,\beta }({\overline{\Omega }})\) such that \(0\le u\le {\overline{u}}\).
For the other case, it is enough to put \(f_1(x,s,\xi )=-f(x,-s,-\xi )\) and to use the first part of the proof. \(\square \)
5 Examples
In this Section we present some applications of Theorem 3.4 (Sect. 5.1) where the structure of f and the condition \({\mathcal {A}}(x,s,0)\equiv 0_{{{{\mathbb {R}}}^n}}\) guarantee the existence of constant sub and supersolutions for (1.1). This situation is really interesting and meaningful because hilights how in this setting the growth of \({\mathcal {A}}\) with respect to s can be whatever we want. Then we present an application of Theorem 3.9 (Sect. 5.2) and finally some applications of Theorem 4.5 (Sect. 5.3). In the latter case the growth of \({\mathcal {A}}\) with respect to s comes once again into play (see Eq. (4.4)).
In all the examples \(\Omega \) is an open subset in \({{{\mathbb {R}}}^n}\) with finite measure.
5.1 Applications of Theorem 3.4
In this example we do not impose growth conditions with respect to s for \({\mathcal {A}}\), but only with respect to x and \(\xi \). The structure of the convection term guarantees that (1.1) has a pair of constant sub and supersolutions.
Let \(0<\delta <p-1\), \(p+q>1\), \(a:\Omega \rightarrow [0,+\infty [\) be a measurable function, \(a\in L^{\frac{p}{\delta }}(\Omega )\), and let \(b:[0,+\infty [\rightarrow [0,+\infty [\) be a continuous function.
Consider the problem
The Young function A governing the differential operator \({\mathcal {A}}\) obeys (2.33).
The convection term \(f:\Omega \times {\mathbb {R}}\times {{{\mathbb {R}}}^n}\rightarrow {\mathbb {R}}\) is defined as
Here \(h:\Omega \rightarrow [0,+\infty [\) is a measurable function, \(h\in L^{{{\widetilde{A}}}_n}(\Omega )\), \(k:[0,+\infty ]\rightarrow {\mathbb {R}}\) is a continuous function, \(k(0)>0\), and k has the following behavior near infinity
The function \(g:{\mathbb {R}}\rightarrow {\mathbb {R}}\) is continuous, \(g(s)>0\) for \(s\in [0,{{\overline{s}}})\) and \(g({{\overline{s}}})=0\). First of all we note that \(u_1=0\) and \(u_2={{\overline{s}}}\) are a subsolution and a supersolution to (5.1) and \(u\equiv 0\) is not a solution. The growth of the functions h and k quarantee that f satisfies (3.1).
Due to the continuity of b, and taking into account Example 2.16 (with \(\beta =\beta _1=0\)), we see that conditions (2.36), (2.37) and (2.38) hold, for a.e. \(x\in \Omega \), all \(s\in [0,{{\overline{s}}}]\), all \(\xi \in {{{\mathbb {R}}}^n}\). Thus, by Theorem 3.4, problem (5.1) has a nontrivial solution \(u\in [u_1,u_2]\).
The same arguments work for different choices of h and g. In particular we see that Theorem 3.4 works well with all nonlinearities having two zeros \(s_1\) and \(s_2\), with \(s_1<0<s_2\), (\(f(x,s_1,0)=f(x,s_2,0)=0\) for all \(x\in \Omega \)) and f(x, s, 0) has constant sign for \(s\in ]s_1,s_2[\), or for which f(x, 0, 0) has constant sign and there exists \(s\in {\mathbb {R}}\) such that \(f(x,s,0)\equiv 0\) and \(s\cdot f(x,0,0)>0\).
5.2 Applications of Theorem 3.9
Let \(p,q,r\in {\mathbb {R}}\) be such that \(1<r<p<n\), \(1<q<p\), and let \(m<0\).
Let \(a,\rho :\Omega \rightarrow [0,+\infty [\) be two measurable functions, \(a\in L^{\frac{p}{p-r}}(\Omega )\) and \(\rho >0\) on a subset of \(\Omega \) having positive measure.
We show that problem
has a nontrivial solution \(u\in W_0^{1,p}(\Omega )\), \(u\ge 0\).
The functions \(\Phi (x,\xi )=a(x)\frac{|\xi |^{r}}{r} +\frac{|\xi |^{p}}{p}\), and \({\mathcal {A}}(x,\xi )=\left( a(x)|\xi |^{r-2} +|\xi |^{p-2}\right) \xi \) satisfy (3.13), (3.14), (3.15) and (3.16). Condition (3.21) holds too, with \(\rho _1(x)\equiv \rho (x)\), \(g_1(|s|)=|s|^{q-1}\), \(\rho _2(x)=-M^{\frac{p}{{p^*}'}}|\lg (|M|)|^{\frac{m}{p}^*}\), for a suitable constant \(M>0\), \(g_2(|s|)\equiv 0\) and \(E(t)\approx t^{p^*}\lg ^m(t)\), for \(t\gg 1\).
Thus from Theorem 3.9 problem (5.4) has a nontrivial solution \(u\in W_0^{1,p}(\Omega )\), \(u\ge 0\).
5.3 Applications of Theorem 4.4
Let \(\Omega \) be a bounded domain with a \(C^{1,\alpha }\) boundary. Let \(\gamma \ge \alpha \), \(p>1\) and \(q\in {\mathbb {R}}\), satisfying \(p+q-1>0\). Let \(h:\Omega \rightarrow [0,+\infty [\) be a measurable function, \(h\in L^{\infty }(\Omega )\), \(h\ne 0\) on a subset of \(\Omega \) having finite measure, and let \(g:{\mathbb {R}}\rightarrow {\mathbb {R}}\) be a continuous function such that \(g(s)>0\) for all \(s\in [0,{{\overline{s}}}[\) and \(g({{\overline{s}}})\equiv 0\). We consider the problem
and prove that it has a nontrivial solution \(u\in W_0^{1,A}(\Omega )\), \(u\ge 0\). The Young function A is defined via (4.8).
Using the Mac Laurin expansion of \(k(t)=e^t\) we see that \(|e^{|s|}-e^{|w|}|\le \frac{e^M(e^M-1)}{M^\alpha }|s-w|^\alpha \) for all \(s,w\in [-M,M]\). Thus, following the arguments used in the Example 4.3, we can prove that the operator \({\mathcal {A}}\) and the function f, defined respectively as \({\mathcal {A}}(x,s,\xi )=\Vert x\Vert ^\gamma e^{|s|}|\xi |^{p-2} \lg ^q(1+|\xi |)\xi \) and \(f(x,s,\xi )=\left( h(x)+|\xi |^p\lg ^{q}(1+|\xi |)\right) g(s)\) satisfy the hypotheses of Theorem 4.4. Furthermore, \({\underline{u}}\equiv 0\) and \({{\overline{u}}}={{\overline{s}}}\) are a subsolution and a supersolution to (5.5), and \({\underline{u}}\equiv 0\) is not a solution. Thus, from Theorem 4.4, the problem which we are dealing with, has a regular solution \(u:\Omega \rightarrow [0,{{\overline{s}}}]\).
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Acknowledgements
The author thanks the referee for his/her careful reading of the paper, for useful suggestions about bibliographical references, and for requesting a detailed comparison with other results on the topic.
Funding
Open access funding provided by Università degli Studi Mediterranea di Reggio Calabria within the CRUI-CARE Agreement. This research is partially supported by the Ministry of Education, University and Research of Italy, Prin 2017 Nonlinear Differential Problems via Variational, Topological and Set-valued Methods (Project no. 2017AYM8XW).
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Giuseppina Barletta wrote and reviewed the manuscript.
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Barletta, G. Existence and regularity results for nonlinear elliptic equations in Orlicz spaces. Nonlinear Differ. Equ. Appl. 31, 29 (2024). https://doi.org/10.1007/s00030-024-00922-x
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DOI: https://doi.org/10.1007/s00030-024-00922-x
Keywords
- Nonlinear elliptic equations
- Orlicz–Sobolev spaces
- Gradient dependence
- Leray–Lions operators
- Subsolution and supersolution