Abstract
We consider a nonlinear Dirichlet elliptic problem driven by the sum of a p-Laplacian and a Laplacian [a (p, 2)-equation] and with a reaction term, which is superlinear in the positive direction (without satisfying the Ambrosetti–Rabinowitz condition) and sublinear resonant in the negative direction. Resonance can also occur asymptotically at zero. So, we have a double resonance situation. Using variational methods based on the critical point theory and Morse theory (critical groups), we establish the existence of at least three nontrivial smooth solutions.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
Let \(\Omega \subset \mathbb {R}^N\) be a bounded domain with a \(C^2\)-boundary \(\partial \Omega \). In this paper, we study the following nonlinear Dirichlet problem
In this problem \(2<p\) and \(\Delta _p\) denotes the p-Laplace differential operator defined by
The reaction term \(f(z,\zeta )\) is a measurable function on \(\Omega \times \mathbb {R}\) and for almost all \(z\in \Omega \), \(f(z,\cdot )\in C^1(\mathbb {R})\). The interesting feature of our work here is that \(f(z,\cdot )\) exhibits asymmetric behaviour as \(\zeta \rightarrow \pm \infty \). More precisely, \(f(z,\cdot )\) is \((p-1)\)-superlinear as \(\zeta \rightarrow +\infty \) but need not satisfy the usual for superlinear problems Ambrosetti–Rabinowitz condition. Instead, we use a weaker condition, which incorporates in the framework of our work also problems in which the forcing term is \((p-1)\)-superlinear but with “slower” growth near \(+\infty \). Such a function fails to satisfy the Ambrosetti–Rabinowitz condition. Near \(-\infty \) the reaction term \(f(z,\cdot )\) is \((p-1)\)-sublinear and resonance can occur with respect to the principal eigenvalue of \((-\Delta _p,W^{1,p}_0(\Omega ))\). Resonance can occur also at zero. Thus, our problem exhibits double resonance.
Problems with asymmetric reaction term of the form described above, were studied by Arcoya and Villegas [4], Cuesta et al. [11], D’Aguì et al. [12], de Figueiredo and Ruf [14], Motreanu et al. [24, 25], de Paiva and Presoto [29], Papageorgiou and Rǎdulescu [31], Recôva and Rumbos [35].
In problem (1.1), the differential operator \(u\longmapsto -\Delta _p u-\Delta u\) is nonhomogeneous and this is a source of difficulties in the analysis on (1.1). We mention that (p, 2)-Laplace equations (that is, elliptic problems driven by the sum of a p-Laplacian and a Laplacian), arise naturally in problems of mathematical physics. We mention the works of Benci et al. [6] (quantum physics) and Cherfils and Il\('\)yasov [8] (plasma physics). Recently there have been some existence and multiplicity results for such equations. In this direction, we mention the works of Aizicovici et al. [2], Barile and Figueiredo [5], Cingolani and Degiovanni [9], Gasiński and Papageorgiou [17, 20], Gasiński et al. [21], Mugnai and Papageorgiou [28], Papageorgiou and Rǎdulescu [31, 32], Papageorgiou and Smyrlis [33], Sun [37] and Sun et al. [38]. Of the aforementioned works, only Papageorgiou and Rǎdulescu [31] and Gasiński and Papageorgiou [20] deal with problems having an asymmetric reaction term. However, the conditions are different, since in [31] it is assumed that \(f(z,\cdot )\) is \((p-1)\)-sublinear in both directions (crossing nonlinearity) and no resonance is allowed asymptotically at \(-\infty \) or near zero (nonresonant problem; see Theorem 12 in [31]), while in [20], \(f(z,\cdot )\) is \((p-1)\)-sublinear as \(\zeta \rightarrow +\infty \) and \((p-1)\)-superlinear as \(\zeta \rightarrow -\infty \). A more general problem with a (p, q)-Laplacian operator was studied in Gasiński and Papageorgiou [18, 19].
Our approach combines variational methods based on the critical point theory, together with Morse theory (critical groups theory) and the use of suitable truncation and comparison techniques. In the next section, for the convenience of the reader, we review the main mathematical tools which we will use in the sequel.
2 Mathematical background
Let X be a Banach space and let \(X^*\) be its topological dual. By \(\langle \cdot ,\cdot \rangle \) we denote the duality brackets for the pair \((X^*,X)\). Given \(\varphi \in C^1(X)\), we say that \(\varphi \) satisfies the Cerami condition, if the following is true:
Every sequence \(\{u_n\}_{n\geqslant 1}\subset X\), such that \(\big \{\varphi (u_n)\big \}_{n\geqslant 1}\subset \mathbb {R}\) is bounded and \((1+\Vert u_n\Vert )\varphi '(u_n)\longrightarrow 0 \quad \text {in} X^*\), admits a strongly convergent subsequence.
Evidently this is a compactness type condition on the functional \(\varphi \) which compensates for the fact that the ambient space which in applications is infinite dimensional, is not locally compact. Using this condition, one can prove a deformation theorem from which the minimax theory of the critical values of \(\varphi \) follows. One of the most important results in this theory is the so called mountain pass theorem due to Ambrosetti and Rabinowitz [3]. Here we state it in a slightly stronger form (see Gasiński and Papageorgiou [15]).
Theorem 2.1
If X is a Banach space, \(\varphi \in C^1(X;\mathbb {R})\) satisfies the Cerami condition, \(u_0,u_1\in X\), \(\Vert u_1-u_0\Vert>\varrho >0\),
and \(\displaystyle c=\inf _{\gamma \in \Gamma }\max _{0\leqslant t\leqslant 1} \varphi \left( \gamma (t)\right) \), where
then \(c\geqslant m_{\varrho }\) and c is a critical value of \(\varphi \), that is there exists \(\widehat{u}\in X\) such that
In the analysis of problem (1.1), in addition to the Sobolev spaces \(W^{1,p}_0(\Omega )\) and \(H^1_0(\Omega )\), we will also use the Banach space
This is an ordered Banach space with order cone
This cone has a nonempty interior, given by
Here \(\frac{\partial u}{\partial n}=(\nabla u,n)_{{}_{\mathbb {R}^N}}\) with \(n(\cdot )\) being the outward unit normal on \(\partial \Omega \) (the normal derivative of u). The space \(C^1_0(\overline{\Omega })\) is dense in \(W^{1,p}_0(\Omega )\) and in \(H^1_0(\Omega )\).
We will also use some elementary facts on the spectrum of \((-\Delta _p,W^{1,p}_0(\Omega ))\). So, we consider the following nonlinear eigenvalue problem
where \(1<p<+\infty \). We say that \(\widehat{\lambda }\in \mathbb {R}\) is an eigenvalue of \((-\Delta _p,W^{1,p}_0(\Omega ))\), provided (2.1) admits a nontrivial solution \(\widehat{u}\in W^{1,p}_0(\Omega )\), which is known as an eigenfunction corresponding to \(\widehat{\lambda }\). There exists a smallest eigenvalue \(\widehat{\lambda }_1(p)>0\) which has the following properties
-
we have
$$\begin{aligned} \widehat{\lambda }_1(p) = \inf \bigg \{\frac{\Vert \nabla u\Vert _p^p}{\Vert u\Vert _p^p}: u\in W^{1,p}_0(\Omega ), u\ne 0\bigg \}; \end{aligned}$$(2.2) -
\(\widehat{\lambda }_1(p)\) is isolated (that is, we can find \(\varepsilon >0\) such that \((\widehat{\lambda }_1(p),\widehat{\lambda }_1(p)+\varepsilon )\) contains no eigenvalues of \((-\Delta _p,W^{1,p}_0(\Omega ))\));
-
\(\widehat{\lambda }_1(p)\) is simple (that is, if \(\widehat{u},\widehat{v}\in W^{1,p}_0(\Omega )\) are two eigenfunctions corresponding to \(\widehat{\lambda }_1(p)\), we have \(\widehat{u}=\xi \widehat{v}\) for some \(\xi \in \mathbb {R}{\setminus } \{0\}\)).
In (2.2) the infimum is realized on the corresponding one dimensional eigenspace. It is clear from (2.2) that the elements of this eigenspace do not change sign. In what follows by \(\widehat{u}_1(p)\) we denote the \(L^p\)-normalized (that is, \(\Vert \widehat{u}_1(p)\Vert _p^p=1\)) positive eigenfunction corresponding to \(\widehat{\lambda }_1(p)\). The nonlinear regularity theory and the nonlinear maximum principle (see, for example Gasiński and Papageorgiou [15, pp. 737, 738]) imply that \(\widehat{u}_1(p)\in \mathrm {int}\, C_+\).
The Ljusternik–Schnirelmann minimax scheme, gives in addition to \(\widehat{\lambda }_1(p)\) a whole strictly increasing sequence \(\{\widehat{\lambda }_k(p)\}_{k\geqslant 1}\) of eigenvalues such that \(\widehat{\lambda }_k(p)\longrightarrow +\infty \) as \(k\rightarrow +\infty \). It is not known if this sequence exhausts the spectrum of \((-\Delta _p,W^{1,p}_0(\Omega ))\). This is the case if \(p=2\) (linear eigenvalue problem) or if \(N=1\) (ordinary differential equation). For the linear eigenvalue problem (\(p=2\)), every eigenvalue \(\widehat{\lambda }_k(2)\), \(k\geqslant 1\), has an eigenspace, denoted by \(E(\widehat{\lambda }_k(2))\), which is a finite dimensional linear subspace of \(H^1_0(\Omega )\). We have that
Also, for every \(k\geqslant 1\), we set
Then
All the eigenvalues \(\widehat{\lambda }_k(2)\), \(k\geqslant 1\), admit variational characterizations
Both the infimum and supremum are realized on \(E(\widehat{\lambda }_k(2))\). Each eigenspace exhibits the unique continuation property, which says that, if \(u\in E(\widehat{\lambda }_i(2))\) vanishes on a set of positive measure, then \(u\equiv 0\). Standard regularity theory implies that \(E(\widehat{\lambda }_i (2))\subset C^1_0(\overline{\Omega })\).
The next lemma can be found in Motreanu et al. [26, p. 305]. It is an easy consequence of the properties of the eigenvalue \(\widehat{\lambda }_1(p)>0\) mentioned above.
Lemma 2.2
If \(\vartheta \in L^{\infty }(\Omega )_+\), \(\vartheta (z)\leqslant \widehat{\lambda }_1(p)\) for almost all \(z\in \Omega \) and the inequality is strict on a set of positive measure, then there exists \(c_0>0\) such that
Let \(A_p:W^{1,p}_0(\Omega )\longrightarrow W^{-1,p'}(\Omega )\) (\(\frac{1}{p}+\frac{1}{p'}=1\)) be the nonlinear map defined by
This map has the following properties (see Gasiński and Papageorgiou [15, p. 746]).
Proposition 2.3
The map \(A_p:W^{1,p}_0(\Omega )\longrightarrow W^{-1,p'}(\Omega ) (1<p<+\infty )\) is bounded (that is, maps bounded sets to bounded sets), continuous, strictly monotone (hence maximal monotone too) and of type \((S)_+\), that is,
if \(u_n\longrightarrow u\) weakly in \(W^{1,p}_0(\Omega )\) and \(\limsup \limits _{n\rightarrow +\infty }\langle A(u_n),u_n-u\rangle \leqslant 0\),
then \(u_n\longrightarrow u\) in \(W^{1,p}_0(\Omega )\).
When \(p=2\), we write \(A_2=A\) and we have \(A\in \mathcal {L}(H^1_0(\Omega ),H^{-1}(\Omega ))\). Let \(f_0:\Omega \times \mathbb {R}\longrightarrow \mathbb {R}\) be a Carathéodory function with subcritical growth, that is,
with \(a_0\in L^{\infty }(\Omega )_+\) and \(1<r<p^*\), where
(the critical Sobolev exponent). We set \(F_0(z,\zeta )=\int _0^{\zeta }f_0(z,s)\,ds\) and consider the \(C^1\)-functional \(\varphi _0:W^{1,p}_0(\Omega )\longrightarrow \mathbb {R}\) defined by
The next proposition is a special case of a more general result of Gasiński and Papageorgiou [16]. Its proof is an outgrowth of the nonlinear regularity theory (see Lieberman [23]).
Proposition 2.4
If \(u_0\in W^{1,p}_0(\Omega )\) is a local \(C^1_0(\overline{\Omega })\)-minimizer of \(\varphi _0\), that is, there exists \(\varrho _0>0\) such that
then \(u_0\in C^{1,\alpha }_0(\overline{\Omega })\) for some \(\alpha \in (0,1)\) and it is a local \(W^{1,p}_0(\Omega )\)-minimizer of \(\varphi _0\), that is, there exists \(\varrho _1>0\) such that
Hereafter, by \(\Vert \cdot \Vert \) we denote the norm of the Sobolev space \(W^{1,p}_0(\Omega )\). Because of the Poincaré inequality, we can have
Also, by \(|\cdot |_N\) we denote that Lebesgue measure on \(\mathbb {R}^N\).
For \(\zeta \in \mathbb {R}\), we set \(\zeta ^{\pm }=\max \{\pm \zeta ,0\}\). Then given \(u\in W^{1,p}_0(\Omega )\) we define \(u^{\pm }(\cdot )=u(\cdot )^{\pm }\). We know that
Given a measurable function \(h:\Omega \times \mathbb {R}\longrightarrow \mathbb {R}\) (for example, a Carathéodory function), we set
the Nemytskii (or superposition) map corresponding to the function \(h(z,\zeta )\).
Finally, we recall some basic facts about critical groups (Morse theory). For details we refer to the book of Motreanu et al. [26].
So, let X be a Banach space, \(\varphi \in C^1(X;\mathbb {R})\) and \(c\in \mathbb {R}\). We introduce the following sets
Let \((Y_1,Y_2)\) be a topological pair such that \(Y_2\subseteq Y_1\subseteq X\) and \(k\geqslant 0\). By \(H_k(Y_1,Y_2)\) we denote the k-th relative singular homology group for the pair \((Y_1,Y_2)\) with integer coefficients. Given \(u\in K_{\varphi }\) isolated with \(\varphi (u)=c\) (that is \(u\in K_{\varphi }^c\)), the critical groups of \(\varphi \) at u are defined by
where U is a neighbourhood of u such that \(K_{\varphi }\cap \varphi ^c\cap U=\{u\}\). The excision property of singular homology, implies that the above definition of critical groups is independent of the particular choice of the neighbourhood U.
Suppose that \(\varphi \) satisfies the Cerami condition and \(c<\inf \varphi (K_{\varphi })\). The critical groups of \(\varphi \) at infinity are defined by
The second deformation theorem (see Gasiński and Papageorgiou [15, p. 628]), implies that this definition is independent of the choice of the level \(c<\inf \varphi (K_{\varphi })\).
Let \(\varphi \in C^1(X;\mathbb {R})\) and assume that \(\varphi \) satisfies the Cerami condition and that \(K_{\varphi }\) is finite. We define
Then the Morse relation says that
where \(Q(t)=\sum \limits _{k\geqslant 0}\beta _k t^k\) is a formal series in \(t\in \mathbb {R}\) with nonnegative integer coefficients.
3 Multiplicity theorem
In this section we prove a multiplicity theorem for problem (1.1) producing three nontrivial smooth solutions.
To obtain the first two solutions, we will not need the continuous differentiability of \(f(z,\cdot )\). So, our hypothesis on the reaction term \(f(z,\zeta )\) are the following:
H(f): \(f:\Omega \times \mathbb {R}\longrightarrow \mathbb {R}\) is a Carathéodory function such that \(f(z,0)=0\) for almost all \(z\in \Omega \) and
-
(i)
\(|f(z,\zeta )|\leqslant a(z)(1+|\zeta |^{r-1})\) for almost all \(z\in \Omega \), all \(\zeta \in \mathbb {R}\), with \(a\in L^{\infty }(\Omega )_+\), \(p<r<p^*\);
-
(ii)
if \(F(z,\zeta )=\int _0^{\zeta }f(z,s)\,ds\), then
$$\begin{aligned} \lim _{\zeta \rightarrow +\infty }\frac{F(z,\zeta )}{\zeta ^p}\ =\ +\infty \end{aligned}$$uniformly for almost all \(z\in \Omega \) and there exist \(q\in ((r-p)\max \{\frac{N}{p},1\},p^*)\) and \(\xi _0>0\) such that
$$\begin{aligned} 0\ <\ \xi _0\ \leqslant \ \liminf _{\zeta \rightarrow +\infty }\frac{f(z,\zeta )\zeta -p F(z,\zeta )}{\zeta ^q} \end{aligned}$$uniformly for almost all \(z\in \Omega \);
-
(iii)
there exist \(\xi _1>0\) and \(c_1>0\) such that
$$\begin{aligned} -\xi _1 \ \leqslant \ \liminf _{\zeta \rightarrow -\infty }\frac{f(z,\zeta )}{|\zeta |^{p-2}\zeta } \ \leqslant \ \limsup _{\zeta \rightarrow -\infty }\frac{f(z,\zeta )}{|\zeta |^{p-2}\zeta } \ \leqslant \ \widehat{\lambda }_1(p) \end{aligned}$$uniformly for almost all \(z\in \Omega \) and
$$\begin{aligned} -c_1\ \leqslant \ f(z,\zeta )\zeta -p F(z,\zeta ) \quad \text {for almost all}\ z\in \Omega ,\ \text {all}\ \zeta \leqslant 0; \end{aligned}$$ -
(iv)
there exist integer \(m\geqslant 2\) and \(\delta >0\) such that
$$\begin{aligned} \widehat{\lambda }_m(2)\zeta ^2\ \leqslant \ f(z,\zeta )\zeta \ \leqslant \ \widehat{\lambda }_{m+1}(2)\zeta ^2 \quad \text {for almost all}\ z\in \Omega ,\ \text {all}\ |\zeta |\leqslant \delta . \end{aligned}$$
Remark 3.1
Hypotheses \(H(f)(\textit{ii})\) and (iii) imply that the reaction term \(f(z,\cdot )\) exhibits an asymmetric behaviour as \(\zeta \rightarrow \pm \infty \). So, \(f(z,\cdot )\) is \((p-1)\)-superlinear as \(\zeta \rightarrow +\infty \) [see hypothesis \(H(f)(\textit{ii})\)] and \(f(z,\cdot )\) is \((p-1)\)-sublinear as \(\zeta \rightarrow -\infty \) [see hypothesis \(H(f)(\textit{iii})\)]. Note that the \((p-1)\)-superlinearity in the positive direction, is not expressed using the common is such cases (unilateral) Ambrosetti–Rabinowitz condition. We recall that the Ambrosetti–Rabinowitz condition (unilateral version that is, valid only in the positive semiaxis), says that there exist \(\tau >p\) and \(M>0\) such that
and
(see Ambrosetti and Rabinowitz [3] and Mugnai [27]). Integrating (3.1) and using (3.2), we obtain
for some \(c_2>0\). From (3.3) and (3.1), we see that \(f(z,\cdot )\) has at least \((\tau -1)\)-polynomial growth near \(+\infty \) and so
uniformly for almost all \(z\in \Omega \). Hypothesis \(H(f)(\textit{ii})\) is weaker than the unilateral Ambrosetti–Rabinowitz condition [see (3.1) and (3.2)]. Indeed, we may take \(\tau >(r-p)\max \{\frac{N}{p},1\}\) and then using (3.1) we have
[see (3.1) and (3.3)]. So, assuming the unilateral Ambrosetti–Rabinowitz condition, we have just seen that hypothesis \(H(f)(\textit{ii})\) holds. Our hypothesis allows the consideration of \((p-1)\)-superlinear at \(+\infty \) nonlinearities with slower growth, which fail to satisfy the Ambrosetti–Rabinowitz condition (see the examples below). Hypothesis \(H(f)(\textit{iii})\) implies that in the negative direction \(f(z,\cdot )\) is \((p-1)\)-sublinear and asymptotically at \(-\infty \) we can have resonance with respect to the principal eigenvalue \(\widehat{\lambda }_1(p)>0\) of \((-\Delta _p, W^{1,p}_0(\Omega ))\). Hypothesis \(H(f)(\textit{iv})\) says that at zero we can have resonance with respect to any nonprincipal eigenvalue of \((-\Delta , H^1_0(\Omega ))\).
Example 3.2
The following functions satisfy hypotheses H(f). For the sake of simplicity, we drop the z-dependence.
-
(a)
For \(2<p<\tau <p^*\), \(m\geqslant 2\), \(\widetilde{c}_1=\widehat{\lambda }_1(p)-\widehat{\lambda }_m(2)\), \(\widetilde{c}_2=\widehat{\lambda }_m(2)-1\), we consider
$$\begin{aligned} f_1(\zeta ) = \left\{ \begin{array}{ll} \widehat{\lambda }_1(p)|\zeta |^{p-2}\zeta +\widetilde{c}_1 &{} \quad \text {if} \; \zeta<-1,\\ \widehat{\lambda }_m(2)\zeta &{} \quad \text {if} \; -1\leqslant \zeta \leqslant 1,\\ \zeta ^{\tau -1}+\widetilde{c}_2 &{} \quad \text {if} \; 1<\zeta . \end{array} \right. \end{aligned}$$This function satisfies the unilateral Ambrosetti–Rabinowitz condition.
-
(b)
For \(2<p\), \(\widehat{c}_1=\widehat{\lambda }_1(p)-\widehat{\lambda }_m(p)\), \(\widehat{c}_2=\widehat{\lambda }_m(2)-\frac{1}{p}\), we consider
$$\begin{aligned} f_2(\zeta ) = \left\{ \begin{array}{ll} \widehat{\lambda }_1(p)|\zeta |^{p-2}\zeta +\widehat{c}_1 &{} \quad \text {if} \; \zeta<-1,\\ \widehat{\lambda }_m(2)\zeta &{} \quad \text {if} \; -1\leqslant \zeta \leqslant 1,\\ \zeta ^{p-1}\left( \ln \zeta +\frac{1}{p}\right) +\widehat{c}_2 &{} \quad \text {if} \; 1<\zeta . \end{array} \right. \end{aligned}$$This function fails to satisfy the unilateral Ambrosetti–Rabinowitz condition.
Let \(\varphi :W^{1,p}_0(\Omega )\longrightarrow \mathbb {R}\) be the \(C^1\)-functional defined by
Proposition 3.3
If hypotheses H(f) hold, then the functional \(\varphi \) satisfies the Cerami condition.
Proof
Let \(\{u_n\}_{n\geqslant 1}\subset W^{1,p}_0(\Omega )\) be a sequence such that
for some \(M_1>0\) and
From (3.5), we have
with \(\varepsilon _n\searrow 0\). We will show that the sequence \(\{u_n\}_{n\geqslant 1}\subset W^{1,p}_0(\Omega )\) is bounded. To this end note that
[see (3.4) and use the fact that \(p>2\)]. In (3.6) we choose \(h=-u_n^-\in W^{1,p}_0(\Omega )\) and obtain
for some \(M_2>0\). Note that
It follows that
for some \(M_3>0\) [see hypothesis \(H(f)(\textit{iii})\)]. In (3.6) we choose \(h=u_n^+\in W^{1,p}_0(\Omega )\). Then
On the other hand from (3.9), we have
We add (3.10) and (3.11) and recalling that \(p>2\), we infer that
for some \(M_4>0\). Hypotheses H(f)(i) and \((\textit{ii})\) imply that we can find \(\xi _2\in (0,\xi _0)\) and \(c_4>0\) such that
Using (3.13) in (3.12), we infer that
First we assume that \(p\ne N\). From hypothesis \(H(f)(\textit{ii})\), it is clear that without any loss of generality, we may assume that \(q<r<p^*\). Let \(t\in (0,1)\) be such that
The interpolation inequality (see, for example Gasiński and Papageorgiou [15, p. 905]) implies that
so
for some \(M_5>0\) [see (3.14) and use the Sobolev embedding theorem]. In (3.6) we choose \(h\in u_n^+\in W^{1,p}_0(\Omega )\). Then
for some \(c_5>0\) [see hypothesis H(f)(i) and (3.16)]. Using (3.15) and hypothesis \(H(f)(\textit{ii})\), we see that \(tr<p\), so
[see (3.17)].
Now assume that \(N=p\). In this case \(p^*=+\infty \), but the Sobolev embedding theorem says that \(W^{1,p}_0(\Omega )\hookrightarrow L^{\tau }(\Omega )\) for all \(\tau \in [1,+\infty )\). Let \(\tau>r>q\) and choose \(t\in (0,1)\) such that
so
Note that
Since, by hypothesis \(H(f)(\textit{ii})\) we have \(r-q<p\) (recall \(N=p\)), the previous argument remains valid if we replace \(p^*\) be \(\tau >r\) big such that \(tr<p\) [see (3.19) and (3.20)]. Then again we conclude that (3.18) holds.
Next we show that the sequence \(\{u_n^-\}_{n\geqslant 1}\subset W^{1,p}_0(\Omega )\) is bounded. Arguing by contradiction, suppose that at least for a subsequence, we have \(\Vert u_n^-\Vert \longrightarrow +\infty \). Let \(y_n=\frac{u_n^-}{\Vert u_n^-\Vert }\) for all \(n\geqslant 1\). Then
So, we may assume that
From (3.6) with \(h=-u_n^-\in W^{1,p}_0(\Omega )\), we have
so
Hypotheses H(f)(i) and \((\textit{iii})\) imply that
with \(c_6>0\), so the sequence \(\left\{ \frac{N_f(-u_n^-)}{\Vert u_n^-\Vert ^{p-1}}\right\} _{n\geqslant 1}\subset L^{p'}(\Omega )\) is bounded.
Therefore, by passing to a subsequence if necessary and using hypothesis \(H(f)(\textit{iii})\), we obtain
with \(\eta \in L^{\infty }(\Omega )\), \(\eta (z)\leqslant \widehat{\lambda }_1(p)\) for almost all \(z\in \Omega \) (see Aizicovici et al. [1, proof of Proposition 16]). Therefore, if in (3.22) we pass to the limit as \(n\rightarrow +\infty \) and use (3.21) and (3.23), then
(recall that \(p>2\)). If \(\eta \not \equiv \widehat{\lambda }_1(p)\), then from (3.24) and Lemma 2.2, we have
so \(y=0\). Then from (3.22), it follows that \(\Vert \nabla y_n\Vert _p \longrightarrow 0\), so
a contradiction to the fact that \(\Vert y_n\Vert =1\) for \(n\geqslant 1\).
If \(\eta (z)=\widehat{\lambda }_1(p)\) for almost all \(z\in \Omega \), then from (3.24) and (2.2), we have
with \(\vartheta \geqslant 0\). If \(\vartheta =0\), then \(y=0\) and so as above we reach a contradiction to the fact that \(\Vert y_n\Vert =1\) for \(n\geqslant 1\). So, suppose that \(\vartheta >0\). Then \(y\in \mathrm {int}\, C_+\) and so
From (3.4) and (3.18), we have
for some \(M_6>0\), so
For almost all \(z\in \Omega \) and all \(\zeta \leqslant 0\), we have
[see hypothesis \(H(f)(\textit{iii})\)], so for almost all \(z\in \Omega \) and all \(\zeta<y<0\), we have
Hypothesis \(H(f)(\textit{iii})\) implies that
So, if in (3.26) we pass to the limit as \(\zeta \rightarrow -\infty \) and use (3.27), then
so
We return to (3.25) and use (3.28). Then
for some \(M_7>0\), so
[see (2.3)]. But recall that \(u_n^-(z)\longrightarrow +\infty \) for almost all \(z\in \Omega \). Then using Fatou’s lemma we contradict (3.29). This proves that the sequence \(\{u_n^-\}_{n\geqslant 1}\subset W^{1,p}_0(\Omega )\) is bounded, and thus the sequence \(\{u_n\}_{n\geqslant 1}\subset W^{1,p}_0(\Omega )\) is bounded [see (3.18)].
By passing to a suitable subsequence if necessary, we may assume that
In (3.6) we choose \(h=u_n-u\in W^{1,p}_0(\Omega )\) and pass to the limit as \(n\rightarrow +\infty \). Using (3.30) we obtain
so
(recall that A is monotone), thus
[see (3.30)] and hence
(see Proposition 2.3).
This proves that functional \(\varphi \) satisfies the Cerami condition. \(\square \)
We introduce the \(C^1\)-functional \(\varphi _-:W^{1,p}_0(\Omega )\longrightarrow \mathbb {R}\) defined by
Proposition 3.4
If hypotheses H(f) hold, then the functional \(\varphi _-\) is coercive.
Proof
We argue indirectly. So, suppose that \(\varphi _-\) is not coercive. Then we can find a sequence \(\{u_n\}_{n\geqslant 1}\subset W^{1,p}_0(\Omega )\) and \(M_8>0\) such that
So, we have
Let \(y_n=\frac{u_n}{\Vert u_n\Vert }\) for \(n\geqslant 1\). Then \(\Vert y_n\Vert =1\) for all \(n\geqslant 1\) and so we may assume that
From (3.32), we have
Hypotheses H(f)(i) and \((\textit{iii})\) imply that
for some \(c_7>0\), so the sequence \(\left\{ \frac{N_F(-u_n^-)}{\Vert u_n\Vert ^p}\right\} _{n\geqslant 1}\subset L^1(\Omega )\) is uniformly integrable. Hence, by the Dunford–Pettis theorem, we may assume that
Using (3.27), we have
with \(-\xi _1\leqslant \gamma (z)\leqslant \widehat{\lambda }_1(p)\) for almost all \(z\in \Omega \) (see Aizicovici et al. [1, proof of Proposition 16]). If in (3.34) we pass to the limit as \(n\rightarrow +\infty \) and use (3.33), (3.35) and (3.36), then
(recall that \(p>2\)), so
If \(\gamma \not \equiv \widehat{\lambda }_1(p)\), then from (3.38) and Lemma 2.2, we have \(c_0\Vert y^-\Vert ^p\leqslant 0\) so \(y\geqslant 0\). Using this in (3.37), we obtain
so \(y=0\). Then from (3.34) it follows that
so \(y_n\longrightarrow 0\) in \(W^{1,p}_0(\Omega )\), a contradiction to the fact that \(\Vert y_n\Vert =1\) for all \(n\geqslant 1\).
If \(\gamma (z)=\widehat{\lambda }_1(p)\) for almost all \(z\in \Omega \), then from (3.38), we have
so \(y^-=\widetilde{\xi }\widehat{u}_1(p)\), \(\widetilde{\xi }\geqslant 0\).
If \(\widetilde{\xi }=0\), then \(y^-=0\) and from (3.37) we also have \(y^+=0\), hence \(y=0\). From this as above, we reach a contradiction to the fact that \(\Vert y_n\Vert =1\) for all \(n\geqslant 1\).
If \(\widetilde{\xi }>0\), then \(y^-\in \mathrm {int}\, C_+\) and so
From (3.32), (2.2) and (2.3), we have
so
[see (3.28)]. From (3.39) and Fatou’s lemma, we have
From (3.40) and (3.41) we reach a contradiction. This proves the coercivity of \(\varphi _-\). \(\square \)
Using Proposition 3.4 and the direct method of the calculus of variations, we can produce a negative smooth solution.
Proposition 3.5
If hypotheses H(f) hold, then problem (1.1) admits a negative solution \(u_0\in -\mathrm {int}\, C_+\) which is a local minimizer of \(\varphi \).
Proof
From Proposition 3.4 we know that the functional \(\varphi _-\) is coercive. Also, using the Sobolev embedding theorem, we see that \(\varphi _-\) is sequentially weakly lower semicontinuous. So, by the Weierstrass theorem, we can find \(u_0\in W^{1,p}_0(\Omega )\) such that
Since \(\widehat{u}_1(2)\in \mathrm {int}\, C_+\), we can find \(t\in (0,1)\) small such that
with \(\delta >0\) as in hypothesis \(H(f)(\textit{iv})\). From that hypothesis, we have
Therefore
[see (3.37) and recall that \(\Vert \widehat{u}_1(2)\Vert _2=1\)]. Since \(m\geqslant 2\) and \(p>2\), choosing \(t\in (0,1)\) even smaller, we have
so
[see (3.42)], hence \(u_0\ne 0\). From (3.42), we have
so
On (3.44) we act with \(u_0^+\in W^{1,p}_0(\Omega )\). Then
so \(u_0\leqslant 0\) and \(u_0\ne 0\).
So, equation (3.44) becomes
thus
From Ladyzhenskaya and Uraltseva [22, p. 289], we have that \(u_0\in L^{\infty }\). Then Theorem 1 of Lieberman [23] implies that \(u_0\in (-C_+){\setminus } \{0\}\).
Let \(a(y)=|y|^{p-2}y+y\) for all \(y\in \mathbb {R}^N\). Then \(a\in C^1(\mathbb {R}^N;\mathbb {R}^N)\) and
(recall that \(p>2\)). We have
so
Then we can use the tangency principle of Pucci and Serrin [34, p. 35] on (3.38) and infer that
Using the boundary point theorem of Pucci and Serrin [34, p. 120], we have
Note that
So, \(u_0\in -\mathrm {int}\, C_+\) is a local \(C_0^1(\overline{\Omega })\)-minimizer of \(\varphi \). Then Proposition 2.4 implies that \(u_0\) is a local \(W^{1,p}_0(\Omega )\)-minimizer of \(\varphi \). \(\square \)
Corollary 3.6
If hypotheses H(f) hold and \(u_0\in -\mathrm {int}\, C_+\) is the negative solution from Proposition 3.5, then
Using \(u_0\in -\mathrm {int}\, C_+\) from Proposition 3.5 and the mountain pass theorem (see Theorem 2.1), we can produce a second nontrivial solution for problem (1.1). Of course we assume that \(K_{\varphi }\) is finite or otherwise we already have infinitely many of smooth solutions.
First we compute the critical groups of \(\varphi \) at zero.
Proposition 3.7
If hypotheses H(f) hold, then
with \(d_m=\dim \bigoplus \limits _{i=1}^m E(\widehat{\lambda }_i(2))\).
Proof
Recall that
and
Then every \(u\in H^1_0(\Omega )\) can be written in a unique way as
Let \(\widetilde{\psi }:H^1_0(\Omega )\longrightarrow \mathbb {R}\) be the \(C^2\)-functional defined by
Evidently, we have
Therefore from Proposition 2.3 of Su [36], we have
with \(d_m=\dim \bigoplus \limits _{i=1}^m E(\widehat{\lambda }_i(2))\).
Next, let \(\lambda \in (\widehat{\lambda }_m(2),\widehat{\lambda }_{m+1}(2))\) and let \(\widetilde{\varphi }:H^1_0(\Omega )\longrightarrow \mathbb {R}\) be the \(C^2\)-functional defined by
Evidently \(K_{\widetilde{\varphi }}=\{0\}\) and \(\widetilde{\varphi }\) satisfies the Cerami condition. Let \(\widehat{\varphi }=\widetilde{\varphi }|_{W^{1,p}_0(\Omega )}\). The density of the embedding of \(W^{1,p}_0(\Omega )\) into \(H^1_0(\Omega )\) implies that
(see Chang [7, p. 14] and Palais [30]). Note that hypotheses H(f)(i) and \((\textit{iv})\) imply that
with \(c_8>0\). Then we have
for some \(c_9,c_{10}>0\) [see (3.48)]. It follows that
for some \(c_{11}>0\). Also, for all \(h\in W^{1,p}_0(\Omega )\), we have
for some \(c_{12},c_{13}>0\) (use the Sobolev embedding theorem). Again we have
for some \(c_{14}>0\), so
From (3.49), (3.50) and the continuity of the critical groups with respect to the \(C^1\)-topology (see Corvellec and Hantoute [10, Theorem 5.1]), we have
so
[see (3.47)]. Consider the homotopy
By \(\langle \cdot ,\cdot \rangle _0\) we denote the duality brackets for the pair \((H^{-1}(\Omega ),H^1_0(\Omega ))\). Then for \(u\in C^1_0(\overline{\Omega })\) with \(\Vert u\Vert _{C^1_0(\overline{\Omega })}\leqslant \delta \) (here \(\delta >0\) is as in hypothesis \(H(f)(\textit{iv})\)) we have
[see hypothesis \(H(f)(\textit{iv})\) and (2.4)]. Also for all \(u\in H^1_0(\Omega )\), we have
Using (3.52) and (3.53) we see that for \(u\in C^1_0(\overline{\Omega })\) with \(\Vert u\Vert _{C^1_0(\overline{\Omega })}\leqslant \delta \), we have
So, if \(t>0\), then \(h_u'(t,u)\ne 0\) for all \(u\in C^1_0(\overline{\Omega })\), \(u\ne 0\), \(\Vert u\Vert _{C^1_0(\overline{\Omega })}\leqslant \delta \).
For \(t=0\), we have \(h(0,u)=\widetilde{\varphi }(u)\) for all \(u\in H^1_0(\Omega )\) and \(K_{\widetilde{\varphi }}=\{0\}\). So, we can use the homotopy invariance property of critical groups (see Corvellec and Hantoute [10, Theorem 5.2]) and have that
so
(see Chang [7] and Palais [30]), thus
[see (3.51) and (3.46)]. \(\square \)
Now we are ready to produce the second nontrivial smooth solution.
Proposition 3.8
If hypotheses H(f) hold, then problem (1.1) admits a second nontrivial solution \(\widehat{u}\in C^1_0(\overline{\Omega })\).
Proof
From Proposition 3.5 we have a solution \(u_0\in -\mathrm {int}\, C_+\) which is a local minimizer of the functional \(\varphi \). So, we can find \(\varrho \in (0,1)\) small such that
(see Aizicovici et al. [1, proof of Proposition 29]). Because of hypothesis \(H(f)(\textit{ii})\) we have
Moreover, from Proposition 3.3, we know that
Because of (3.54), (3.55) and (3.56), we can use the mountain pass theorem (see Theorem 2.1) and find \(\widehat{u}\in W^{1,p}_0(\Omega )\) such that
so \(\widehat{u}\) is a solution of (1.1), hence \(\widehat{u}\in C^1_0(\overline{\Omega })\) (nonlinear regularity; see Lieberman [23]) and \(\widehat{u}\ne u_0\) [see (3.54)]. Since \(\widehat{u}\in K_{\varphi }\) is of mountain pass type, we have
(see Motreanu et al. [26, p. 176]).
On the other hand from Proposition 3.7, we have
with \(d_m=\dim \bigoplus \limits _{i=1}^m E(\widehat{\lambda }_i(2))\geqslant 2\). Comparing (3.57) and (3.58) we conclude that \(\widehat{u}\ne 0\). \(\square \)
We can produce a third nontrivial smooth solution provided we strengthen the regularity of f. To this end, first we compute the critical groups of \(\varphi \) at infinity. For this we do not need additional assumptions on f.
Proposition 3.9
If hypotheses H(f) hold, then
Proof
Consider the set
and the deformation \(h:[0,1]\times \partial B_1^+\longrightarrow \partial B_1^+\) defined by
We have
so \(\partial B_1^+\) is contractible in itself.
Hypothesis \(H(f)(\textit{ii})\) implies that given \(\xi >0\), we can find \(M_9=M_9(\xi )>0\) such that
Hypothesis \(H(f)(\textit{iii})\) implies that we can find \(c_{15}>0\) and \(M_{10}>0\) such that
Finally hypothesis H(f)(i) implies that we can find \(c_{16}>0\) such that
Let \(t\geqslant 1\) and \(u\in \partial B_1^+\) and consider the sets
We have
for some \(c_{17}>0\) [see (3.59), (3.60) and (3.61) and recall that \(\Vert u\Vert =1\)]. Since \(u^+\ne 0\) (recall that \(u\in \partial B_1^+\)), we can find \(t_0>0\) and \(\widehat{\xi }_0>0\) such that
Using (3.63) in (3.62), we obtain
Choose \(\xi >\frac{c_{17}}{\widehat{\xi }_0}\). Then from (3.64) and since \(p>2\), we infer that
From (3.13) and hypothesis \(H(f)(\textit{iii})\), we see that there exists \(c_{18}>0\) such that
Using the chain rule, (3.66) and since \(p>2\), we have
Because of (3.65), for \(t\geqslant 1\) big, we will have
Let \(\eta <\min \{-\frac{c_{18}|\Omega |_N}{p},\inf \limits _{\overline{B}_1^+}\varphi \}\). The implicit function theorem implies that there is a unique \(\sigma \in C(\partial B_1^+)\), \(\sigma \geqslant 1\) such that
From (3.67) and the choice of \(\eta \), we have
Let \(E_+=\{tu:\ u\in \partial B_1^+,\ t\geqslant 1\}\). We have \(\varphi ^{\eta }\subseteq E_+\). Let \(\widehat{h}:[0,1]\times E_+\longrightarrow E_+\) be the deformation defined by
We have
[see (3.67)], so \(\varphi ^{\eta }\) is a strong deformation retract of \(E_+\), and thus
Using the radial retraction and Theorem 6.5 of Dugundji [13, p. 325], we see that
so
Recall that \(\partial B_1^+\) is contractible in itself. Hence
(see Motreanu et al. [26, p.147]), thus
(choosing \(\eta <0\) even more negative if necessary). \(\square \)
Now we introduce the stronger regularity conditions on f.
H(f)’: \(f:\Omega \times \mathbb {R}\longrightarrow \mathbb {R}\) is a measurable function such that \(f(z,0)=0\) for almost all \(z\in \Omega \), \(f(z,\cdot )\in C^1(\mathbb {R})\) and (i) \(|f_{\zeta }'(z,\zeta )|\leqslant a(z)(1+|\zeta |^{r-2})\) for almost all \(z\in \Omega \), all \(\zeta \in \mathbb {R}\), with \(a\in L^{\infty }(\Omega )_+\), \(p<r<p^*\);
(ii)–(iv) are the same as the corresponding hypotheses \(H(f)(\textit{ii})\)–\((\textit{iv})\).
Then we can have the full multiplicity theorem for problem (1.1).
Theorem 3.10
If hypotheses \(H(f)'\) hold, then problem (1.1) admits at least three nontrivial solutions
Proof
From Proposition 3.8 we already have two nontrivial smooth solutions
Hypotheses \(H(f)'\) imply that \(\varphi \in C^2(W^{1,p}_0(\Omega ))\). Also, recall that
[see (3.57)]. So, from Papageorgiou and Smyrlis [33] (see also Papageorgiou and Rǎdulescu [32]), we have
Also, from Corollary 3.6, we have
Finally, from Propositions 3.7 and 3.9, we have
Suppose that \(K_{\varphi }=\{0,u_0,\widehat{u}\}\). Then from (3.70), (3.71), (3.72), (3.73) and the Morse relation with \(t=-1\) [see (2.5)], we have
so \((-1)^{d_m}=0\), a contradiction.
So, there exists \(\widehat{y}\in K_{\varphi }\), \(\widehat{y}\not \in \{0,u_0,\widehat{u}\}\). This means that \(\widehat{y}\) is the third nontrivial solution of problem (1.1) and using Theorem 1 of Lieberman [23], we conclude that \(\widehat{y}\in C^1_0(\overline{\Omega })\). \(\square \)
References
Aizicovici, S., Papageorgiou, N.S., Staicu, V.: Degree theory for operators of monotone type and nonlinear elliptic equations with inequality constraints. Mem. Am. Math. Soc. 196 (2008)
Aizicovici, S., Papageorgiou, N.S., Staicu, V.: Nodal solutions for \((p,2)\)-equations. Trans. Am. Math. Soc. 367, 7343–7372 (2015)
Ambrosetti, A., Rabinowitz, P.H.: Dual variational methods in critical point theory and applications. J. Funct. Anal. 14, 349–381 (1973)
Arcoya, D., Villegas, S.: Nontrivial solutions for a Neumann problem with a nonlinear term asymptotically linear at \(-\infty \) and superlinear at \(+\infty \). Math. Z. 219, 499–513 (1995)
Barile, S., Figueiredo, G.M.: Existence of least energy positive, negative and nodal solutions for a class of \(p\) and \(q\)-problems with potentials vanishing at infinity. J. Math. Anal. Appl. 427, 1205–1233 (2015)
Benci, V., D’Avenia, P., Fortunato, D., Pisani, L.: Solitons in several space dimensions: Derrick’s problem and infinitely many solutions. Arch. Ration. Mech. Anal. 154, 297–324 (2000)
Chang, K.C.: Infinite Dimensional Morse Theory and Multiple Solutions Problems. Birkhäuser, Boston (1993)
Cherfils, L., Il’yasov, Y.: On the stationary solutions of generalized reaction diffusion equations with \(p\) and \(q\)-Laplacian. Appl. Anal. 4, 9–22 (2005)
Cingolani, S., Degiovanni, M.: Nontrivial solutions for \(p\)-Laplace equations with right-hand side having \(p\)-linear growth at infinity. Commun. Partial Differ. Equ. 30, 1191–1203 (2005)
Corvellec, J.-N., Hantoute, A.: Homotopical stability of isolated critical points of continuous functionals. Set-Valued Anal. 10, 143–164 (2002)
Cuesta, M., de Figueiredo, D.G., Srikanth, P.N.: On a resonant-superlinear elliptic problem. Calc. Var. Partial Differ. Equ. 17, 221–233 (2003)
D’Aguì, G., Marano, S.A., Papageorgiou, N.S.: Multiple solutions to a Robin problem with indefinite weight and asymmetric reaction. J. Math. Anal. Appl. 433, 1821–1845 (2016)
Dugundji, J.: Topology. Allyn and Bacon Inc., Boston (1966)
de Figueiredo, D., Ruf, B.: On a superlinear Sturm–Liouville equation and a related bouncing problem. J. Reine Angew. Math. 421, 1–22 (1991)
Gasiński, L., Papageorgiou, N.S.: Nonlinear Analysis. CRC Press, Boca Raton, FL (2006)
Gasiński, L., Papageorgiou, N.S.: Multiple solutions for nonlinear coercive problems with a nonhomogeneous differential operator and a nonsmooth potential. Set-Valued Var. Anal. 20, 417–443 (2012)
Gasiński, L., Papageorgiou, N.S.: Multiplicity of positive solutions for eigenvalue problems of \((p,2)\)-equations. Bound. Value Probl. 152, 1–17 (2012)
Gasiński, L., Papageorgiou, N.S.: A pair of positive solutions for \((p, q)\)-equations with combined nonlinearities. Commun. Pure Appl. Anal. 13, 203–215 (2014)
Gasiński, L., Papageorgiou, N.S.: Dirichlet \((p, q)\)-equations at resonance. Discrete Contin. Dyn. Syst. 34, 2037–2060 (2014)
Gasiński, L., Papageorgiou, N.S.: Nonlinear elliptic equations with a jumping reaction. J. Math. Anal. Appl. 443, 1033–1070 (2016)
Gasiński, L., O’Regan, D., Papageorgiou, N.S.: A variational approach to nonlinear logistic equations. Commun. Contemp. Math. 17, 1450021-1–37 (2015)
Ladyzhenskaya, O.A., Uraltseva, N.: Linear and Quasilinear Elliptic Equations, Vol. 46 of Mathematics in Science and Engineering. Academic Press, New York (1968)
Lieberman, G.M.: Boundary regularity for solutions of degenerate elliptic equations. Nonlinear Anal. 12, 1203–1219 (1988)
Motreanu, D., Motreanu, V.V., Papageorgiou, N.S.: Multiple solutions for Dirichlet problems which are superlinear at \(+\infty \) and sublinear at \(-\infty \). Comm. Appl. Nonlinear Anal. 13, 341–358 (2009)
Motreanu, D., Motreanu, V.V., Papageorgiou, N.S.: On \(p\)-Laplace equations with concave terms and asymmetric perturbations. Proc. Roy. Soc. Edinb. Sect. A 141, 171–192 (2011)
Motreanu, D., Motreanu, V.V., Papageorgiou, N.S.: Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems. Springer, New York (2014)
Mugnai, D.: Addendum to: multiplicity of critical points in presence of a linking: application to a superlinear boundary value problem, NoDEA. Nonlinear Differ. Equ. Appl. 11(3), 379–391 (2004). NoDEA Nonlinear Differ. Equ. Appl. 19, 299–301 (2012)
Mugnai, D., Papageorgiou, N.S.: Wang’s multiplicity result for superlinear \((p, q)\)-equations without the Ambrosetti–Rabinowitz condition. Trans. Am. Math. Soc. 366, 4919–4937 (2014)
de Paiva, F.O., Presoto, A.E.: Semilinear elliptic problems with asymmetric nonlinearities. J. Math. Anal. Appl. 409, 254–262 (2014)
Palais, R.S.: Homotopy theory of infinite dimensional manifolds. Topology 5, 1–16 (1966)
Papageorgiou, N.S., Rǎdulescu, V.D.: Resonant \((p,2)\)-equations with asymmetric reaction. Anal. Appl. 13, 481–506 (2015)
Papageorgiou, N.S., Rǎdulescu, V.D.: Qualitative phenomena for some classes of quasilinear elliptic equations with multiple resonance. Appl. Math. Optim. 69, 393–430 (2014)
Papageorgiou, N.S., Smyrlis, G.: On nonlinear nonhomogeneous resonant Dirichlet equations. Pac. J. Math. 264, 421–453 (2013)
Pucci, P., Serrin, J.: The Maximum Principle. Birkhäuser Verlag, Basel (2007)
Recôva, L., Rumbos, A.: An asymmetric superlinear elliptic problem at resonance. Nonlinear Anal. 112, 181–198 (2015)
Su, J.: Semilinear elliptic boundary value problems with double resonance between two consecutive eigenvalues. Nonlinear Anal. 48, 881–895 (2002)
Sun, M.: Multiplicity of solutions for a class of the quasilinear elliptic equations at resonance. J. Math. Anal. Appl. 386, 661–668 (2012)
Sun, M., Zhang, M., Su, J.: Critical groups at zero and multiple solutions for a quasilinear elliptic equation. J. Math. Anal. Appl. 428, 696–712 (2015)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by A. Neves.
The research was supported by the National Science Center of Poland under Projects No. 2015/19/B/ST1/01169 and 2012/06/A/ST1/00262.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.