Abstract
In this paper, we study the existence of weak solutions for fractional p-Laplacian equations with sublinear growth and oscillatory behavior as the following
where \(\mathcal{L}^{p}_{K} \) is a nonlocal operator with singular kernel, Ω is an open bounded smooth domain of \(\mathbb{R}^{N}\). Our purpose is to generalize the known results for fractional Laplacian equations to fractional p-Laplacian equations.
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1 Introduction
Recently, a great attention has been devoted to the research of problems involving fractional and nonlocal operators. This type of operators finds many applications in a lot of fields, such as continuum mechanics, phase transition phenomena, population dynamics and game theory, as they are the typical outcome of stochastic stabilization of Lévy processes; see, for instance, [1–4] and the references therein. There are many works on nonlocal fractional operators and their applications which are very interesting; we refer the interested reader to [5–14] and the references therein. Here we want to generalize the multiplicity existence results for fractional Laplacian equations in [15] and at the same time fix some bugs there.
In this paper we deal with the following fractional problem in a bounded smooth domain \(\Omega\subset\mathbb{R}^{N}\):
where \(\lambda>0\), \(1< p<+\infty\). \(\mathcal{L}^{p}_{K} \) is a nonlocal fractional operator defined as follows:
provided that the limit exists and K is a measurable function having the following property:
Throughout this paper we assume \(N>ps\) with \(s\in(0,1)\). A typical example for K is the singular kernel \(\vert x \vert ^{-(N+ps)}\), in which case problem (1.1) becomes
where \((-\Delta)^{s}_{p}\) is the fractional p-Laplacian operator defined as
for all \(x\in\mathbb{R}^{N}\). In the case \(p=2\), problem (1.3) becomes the fractional Laplacian problem
Following [16, 17], we present the main structural assumptions on the nonlinear term f.
- \((f_{1})\) :
-
\(f\in C(\mathbb{R^{N}}\times\mathbb{R})\), \(f(x,0)=0\), \(\liminf_{t\rightarrow 0}\frac{f(x,t)}{ \vert t \vert ^{p-2}t}>-\infty\).
- \((f_{2})\) :
-
\(\lim_{ \vert t \vert \rightarrow\infty}\frac{f(x,t)}{ \vert t \vert ^{p-1}}=0\) uniformly in x.
- \((f_{3})\) :
-
There exist \(t^{-}<0\) and \(t^{+}>0\) such that \(\min\{f(x,t): (x,t)\in \overline{\Omega}\times\{t^{-}\}\}>0\) and \(\max\{f(x,t): (x,t)\in \overline{\Omega}\times\{t^{+}\}\}<0\).
- \((f_{4})\) :
-
There exist \(\hat{t}^{-}\) and \(\hat{t}^{+}\), with \(\hat{t}^{-}< t^{-}\) and \(\hat{t}^{+}>t^{+}\), such that
$$\min\bigl\{ F(x,t): (x,t)\in\overline{\Omega}\times\bigl\{ \hat{t}^{-},\hat{t}^{+} \bigr\} \bigr\} >\max\bigl\{ f(x,t): (x,t)\in\overline{\Omega}\times\bigl[t^{-},t^{+} \bigr]\bigr\} , $$where \(F(x,t)=\int_{0}^{t}f(x,\xi)\,d\xi\).
A typical example of f satisfying \((f_{1})\)-\((f_{4})\) is
where ε is small.
The natural solution space of problem (1.1) is
where \(X(\mathbb{R}^{N})\) is the fractional Sobolev space given by
and endowed with the norm
For the basic properties of fractional Sobolev spaces, we refer to [18]. We seek the solutions of (1.1) as the critical points of the functional given by
Set
where \(t^{-}\) and \(t^{+}\) are the numbers given in \((f_{3})\).
Theorem 1.1
Assume that \((f_{1})\)-\((f_{4})\) hold. Then there exists \(\Lambda>0\) such that, for every \(\lambda\ge\Lambda\), problem (1.1) admits two positive, two negative and two sign-changing solutions in \(Y\setminus[t^{-},t^{+}]\), where \(Y:=X_{0}(\Omega)\cap C^{\alpha/2}(\overline{\Omega})\).
Here, for two real numbers \(t< s\), the symbol \([t,s]\) denotes either an order interval in Y or a usual interval in \(\mathbb{R}\). In this paper there will be no ambiguous meaning concerning this symbol, for example, in Theorem 1.1 \(Y\setminus[t^{-},t^{+}]\) denotes the difference of Y and \([t^{-},t^{+}]\), where \([t^{-},t^{+}]\) is an order interval in Y.
Theorem 1.2
Assume that \((f_{1})\)-\((f_{4})\) hold and \(f(x,t)\) is odd with respect to t. Then, for every \(k\in\mathbb{N}\), there exists positive \(\Lambda _{k}\) such that, for every \(\lambda\ge\Lambda_{k}\), problem (1.1) admits a sign-changing solution \(u_{i}\) with \(I_{\lambda}(u_{i})<\alpha_{\lambda}\) and a sign-changing solution \(v_{i}\) with \(I_{\lambda}(v_{i})\ge\alpha _{\lambda}\), where \(i=1,\ldots,n\).
Next consider the case that \(f(x,t)\) is not odd with respect to t, but oscillates around 0 in the following manner.
- \((f_{5})\) :
-
There exist \(t_{i}^{-}\) and \(t_{i}^{+}\), \(i=1,\ldots,n\), with \(t_{n}^{-}<\cdots <t_{1}^{-}<0<t_{1}^{+}<\cdots<t_{n}^{+}\), such that \(\min\{f(x,t): (x,t)\in\overline{\Omega}\times\{t_{i}^{-}\}\}>0>\max\{ f(x,t): (x,t)\in\overline{\Omega}\times\{t_{i}^{+}\}\}\) for all \(i=1,\ldots,n\).
- \((f_{6})\) :
-
\(\min\{F(x,t): (x,t)\in\overline{\Omega}\times\{t_{i}^{-},t_{i}^{+}\}\}>\max\{ F(x,t): (x,t)\in\overline{\Omega}\times[t_{i-1}^{-},t_{i-1}^{+}]\}\) for \(i=2,\ldots,n\).
Theorem 1.3
Assume that \((f_{1})\)-\((f_{2})\) and \((f_{5})\)-\((f_{6})\) hold. Then there exists \(\Lambda>0\) such that, for every \(\lambda\ge\Lambda\), problem (1.1) admits positive solutions \(u_{i1}\), \(u_{i2}\), negative solutions \(v_{i1}\), \(v_{i2}\), and sign-changing solutions \(w_{i1}\), \(w_{i2}\) in \([t_{i+1}^{-},t_{i+1}^{+}]\setminus[t_{i}^{-},t_{i}^{+}]\), \(i=1,\ldots,n-1\).
At last, we consider the case of f having infinitely many oscillations in both \((-\infty,0)\) and \((0,+\infty)\).
- \((f_{7})\) :
-
There exist a decreasing (an increasing) sequence \((t_{i}^{-})_{i}\) and an increasing (a decreasing) sequence \((t_{i}^{+})_{i}\) such that \(t_{i}^{-}<0<t_{i}^{+}\), and \(f(x,t_{i}^{+})<0<f(x,t_{i}^{-})\) and \(\max\{F(x,t): (x,t)\in\overline{\Omega}\times[t_{i-1}^{-},t_{i-1}^{+}]\} <\min\{F(x,t): (x,t)\in\overline{\Omega}\times\{t_{i}^{-},t_{i}^{+}\}\}\) for every \(i\in\mathbb{N}\).
Theorem 1.4
Assume that \((f_{1})\)-\((f_{2})\) and \((f_{7})\) hold. Then, for every arbitrarily chosen \(k\in\mathbb{N}\), there exists \(\Lambda_{k}>0\) such that, for every \(\lambda\ge\Lambda_{k}\), problem (1.1) admits at least 2k positive, 2k negative, and 2k sign-changing solutions.
Theorem 1.4 is immediate in view of Theorem 1.3.
Remark 1.1
Bartsch and Liu obtained Theorems 1.1-1.4 for p-Laplacian Dirichlet problems in [16]. Fu and Pucci obtained Theorems 1.1-1.4 for fractional Laplacian Dirichlet problems in [15]. Theorems 1.1-1.4 above are generalizations of the corresponding results in [15].
This article is organized as follows. In Section 2, we introduce the fractional Sobolev spaces and some preliminary results. In Section 3, in a suitably chosen framework, we verify that the conditions in the abstract critical point theorems in [16] are satisfied, then we generalize the existence of multiplicity solutions for fractional Laplacian problems to the one for fractional p-Laplacian problems.
2 Preliminaries
Let us now recall some inequalities.
Lemma 2.1
(See [19])
There exist positive constants \(C_{1}\)-\(C_{4}\) such that, for all \(\xi ,\eta\in\mathbb{R}^{N}\),
The Hölder regularity up to the boundary, strong maximum principles and the Hopf lemma are important in the proof of our results.
Lemma 2.2
(See [20])
Let \(u\in X_{0}\) satisfy \(\vert \mathcal{L}^{p}_{K}u \vert \leq K\) weakly in Ω for some \(K>0\). Then
for some \(C_{\Omega}=C(N,p,s,\Omega)\), \(\delta(x)=\operatorname{dist}(x,\Omega^{C})\). Furthermore, there exists \(\alpha\in(0,s]\) such that, for all weak solutions \(u\in X_{0}\) of problem (1.1), \(u\in C^{\alpha}(\overline{\Omega})\) and
Lemma 2.3
(See [8])
If \(u\in X_{0}\) is such that \(u(x)\geq0\) a.e. in Ω and
for each \(\varphi\in X_{0},\varphi(x)\geq0\) a.e. in Ω, then \(u(x)>0\) a.e. in Ω.
Let \(s\in(0,1)\), \(p\in(1,\infty)\). Consider the problem
Lemma 2.4
(see [21])
Let Ω be bounded and satisfy the interior ball condition on ∂Ω, \(0\ge c\in C(\overline{\Omega})\), and \(u\in X_{0}\) be a weak solution of (2.5), then either \(u=0\) a.e. in \(\mathbb{R}^{N}\) or
where \(\delta_{R}(x)=\operatorname{dist}(x,B_{R}^{C})\).
In the sequel, for any fixed parameter \(\lambda>0\), the space \(X_{0}(\Omega)\) is endowed with the norm
and, for brevity, we put \(X_{\lambda}=(X_{0}(\Omega), \Vert \cdot \Vert _{\lambda})\). The number \(m>0\) will be determined in Section 3. \(Y=X_{\lambda}\cap C^{\alpha/2}(\overline{\Omega})\) is endowed with the \(C^{\alpha /2}\)-norm and \(Z=X_{\lambda}\cap C^{\alpha}(\overline{\Omega})\) with the \(C^{\alpha}\)-norm, where \(\alpha>0\) is from Lemma 2.2.
3 Multiplicity of weak solutions
We say that \(u\in X_{\lambda}\) is a (weak) solution of problem (1.1) if
holds for any \(\varphi\in X_{\lambda}\), where
Fix \(\lambda>0\) and \(m>0\). Define \(\mathcal{L}(v)={\mathcal {L}^{p}_{K}}v+\lambda m \vert v \vert ^{p-2}v\) and
where v is the solution of the linear problem
which is uniquely determined.
In the following we give a series of lemmas to show that the conditions in the abstract critical point theorems in [16] are satisfied. First we study the properties of the operator \(A_{\lambda}\).
Lemma 3.1
\(A_{\lambda}\in C(X_{\lambda},X_{\lambda})\) is well defined and \(A_{\lambda }(Y)\subset Z\).
Proof
First, from
we know that
i.e., the operator \(\mathcal{L}\) is coercive. Second, from
whenever \(v_{1}\neq v_{2}\), we get that \(\mathcal{L}\) is monotone. It is easy to show that \(\mathcal{L}\) is weakly continuous. Thus, by the monotone operator theory, see [22, 23], we conclude that problem (3.1) has a unique (weak) solution in \(X_{\lambda}\). Therefore, \(A_{\lambda}\) is well defined.
Next we prove that \(A_{\lambda}\in C(X_{\lambda},X_{\lambda})\). By \((f_{1})\) and \((f_{2})\), we have \(\vert f(x,u) \vert \le C( \vert u \vert ^{p-1}+1)\) for some constant C depending only on f. Suppose that \(u_{n}\rightarrow u\) in \(X_{\lambda}\). Then \((u_{n})_{n}\) is bounded in \(X_{\lambda}\). By (3.1), Hölder’s and Young’s inequalities, we have
Taking \(\varepsilon={\lambda m}/{2C}\), we conclude that \((v_{n})_{n}\) is bounded in \(X_{\lambda}\). Hence, \((v_{n})_{n}\) admits a weakly convergent subsequence, still denoted by \((v_{n})_{n}\). Suppose that \(v_{n}\rightharpoonup v\) weakly in \(X_{\lambda}\). By Lemma 2.5 in [7], we have \(v_{n}\rightarrow v\) in \(L^{p}(\Omega)\) since Ω is bounded. By [24], we get \(f(x,u_{n})\rightarrow f(x,u)\) in \(L^{p}(\Omega)\) by going to a further subsequence if necessary. By (3.1) and Lemma 2.1, using an argument similar to (3.2), we also have
Letting \(n\rightarrow\infty\), we obtain
This shows that \(A_{\lambda}\in C(X_{\lambda},X_{\lambda})\), as claimed.
If \(u\in Y\), then by \((f_{1})\) and \((f_{2})\) the solution v of (3.1) is in \(L^{\infty}(\Omega)\) thanks to Lemma 2.4. Therefore, \(\Omega\ni x\mapsto g(x)=\lambda (f(x,u(x))+ m \vert u(x) \vert ^{p-2}u(x) )\) is in \(L^{\infty}(\Omega)\) by \((f_{1})\), \((f_{2})\) and the fact that \(u\in L^{\infty}(\Omega)\). Hence, \(v\in C^{\alpha}(\overline{\Omega})\) by Lemma 2.2, since \(v\in X_{\lambda}\), and so \(v\in Z\) by the definition of Z. □
Second, we show that conditions \((J_{1})\) and \((J_{2})\) in [16] are satisfied.
Lemma 3.2
-
(i)
If \(1< p\leq2\), then the functional \(I_{\lambda}\) satisfies
$$\begin{aligned} &\bigl\langle I_{\lambda}^{\prime}(u),u-v\bigr\rangle _{X_{\lambda}^{\ast}, X_{\lambda}} \ge C_{1} \Vert u-v \Vert _{X_{\lambda}}^{2}\bigl( \Vert u \Vert _{X_{\lambda}}+ \Vert v \Vert _{X_{\lambda}} \bigr)^{p-2}, \\ &\bigl\Vert I_{\lambda}^{\prime}(u) \bigr\Vert _{X_{\lambda}^{\ast}}\le C_{2} \Vert u-v \Vert _{X_{\lambda}}^{p-1}. \end{aligned}$$ -
(ii)
If \(p\geq2\), then the functional \(I_{\lambda}\) satisfies
$$\begin{aligned} &\bigl\langle I_{\lambda}^{\prime}(u),u-v\bigr\rangle _{X_{\lambda}^{\ast},X_{\lambda}} \ge C_{3} \Vert u-v \Vert _{X_{\lambda}}^{p}, \\ &\bigl\Vert I_{\lambda}^{\prime}(u) \bigr\Vert _{X_{\lambda}^{\ast}}\le C_{4} \Vert u-v \Vert _{X_{\lambda}}\bigl( \Vert u \Vert _{X_{\lambda}}+ \Vert v \Vert _{X_{\lambda}}\bigr)^{p-2}. \end{aligned}$$
Proof
Let \(u\in X_{\lambda}\). Thus \(v=A_{\lambda}(u)\) implies that
If \(p\geq2\), it follows by Lemma 2.1 that
If \(1< p\leq2\), then from Lemma 2.1 we have
By Hölder’s inequality
and therefore
For \(w\in X_{\lambda}\), we have
If \(1< p\leq2\), then by Lemma 2.1
If \(p\geq2\), then by Lemma 2.1 and Hölder’s inequality, we have
Now we conclude the result. □
Third, we establish the regularity of the critical points of \(I_{\lambda}\).
Lemma 3.3
Let \(K=\{u\in X_{0}(\Omega): I_{\lambda}^{\prime}(u)=0\}\). Then \(K\subset Y\).
Proof
Let u be fixed in K. Then \(u\in L^{\infty}(\Omega)\) by \((f_{1})\), \((f_{2})\) and Lemma 2.2. Hence, \(\Omega\ni x\mapsto\lambda f(x,u(x))\) is in \(L^{\infty}(\Omega)\) again by \((f_{1})\), \((f_{2})\) and the fact that u is in \(L^{\infty}(\Omega )\). Therefore, Lemma 2.2 gives \(u\in C^{\alpha}(\overline{\Omega})\) since \(u\in X_{\lambda}\), and so \(u\in Y\). □
Forth, we get a comparison principle for the operator \(\mathcal{L}\).
Lemma 3.4
If \(u,v\in X_{\lambda}\), with \(v\ge u\) in \(\mathbb{R}^{N}\setminus\Omega\), are such that \(\mathcal{L}v\ge\mathcal {L}u\) in Ω, i.e., \(\langle\mathcal{L}v-\mathcal{L}u,w\rangle _{X_{\lambda}^{\ast},X_{\lambda}}\ge0\) for all \(w\in X_{\lambda}\), with \(w\ge0\) in Ω, then \(v\ge u\) in Ω.
Proof
Let \(u,v\in X_{\lambda}\), with \(v\ge u\) in \(\mathbb {R}^{N}\setminus\Omega\), be such that \(\mathcal{L}v\ge\mathcal{L}u\) in Ω. Take \(\varphi =u-v=[u-v]^{+}-[u-v]^{-}\), \(w=[u-v]^{+}=[\varphi]^{+}\). Clearly, \(w\in X_{\lambda}\) and \(w\ge0\) in Ω. As
we have
where
We see that \(Q(x,y)\geq0\) and \(Q(x,y)=0\) only if \(v(y)=v(x)\) and \(u(y)=u(x)\), and \(R(x)\geq0\). Thus \(\varphi^{+}=0\) in Ω, i.e., \(v\geq u\) in Ω. □
For \(u,v\in Y\), define \(u\ll v\) provided \(u(x)< v(x)\) for any \(x\in \Omega\) and
for any \(x_{0}\in\partial\Omega\), where ν is the unit outward normal vector of ∂Ω at \(x_{0}\). If \(u\le v\) in Ω, then \([u,v]\) denotes the order interval \(\{w\in Y:u\le w\le v\}\).
By \((f_{1})\)-\((f_{3})\), we can choose suitable positive m so that
Fix such an m from now on.
Fifth, we give the following three technical lemmas which are necessary to verify that the conditions in the abstract critical point theorems in [16] (especially \((P_{1})\)-\((P_{3})\) there) are satisfied.
Lemma 3.5
Assume that \((f_{3})\) and \((f_{4})\) also hold. Then there exist ϕ, \(\psi\in Y\) having the following properties.
-
(1)
\(t^{-}\le\phi\ll0\ll\psi\le t^{+}\).
-
(2)
For every \(u\in X_{\lambda}\cap L^{\infty}(\Omega)\) with \(\phi\le u\), \(\phi\ll A_{\lambda}(u)\).
-
(3)
For every \(u\in X_{\lambda}\cap L^{\infty}(\Omega)\) with \(u\le\psi\), \(A_{\lambda}(u)\ll\psi\).
-
(4)
For every \(u\in X_{\lambda}\cap L^{\infty}(\Omega)\) with \(t^{-}\le u\), \(\phi\ll A_{\lambda}(u)\).
-
(5)
For every \(u\in X_{\lambda}\cap L^{\infty}(\Omega)\) with \(u\le t^{+}\), \(A_{\lambda}(u)\ll\psi\).
-
(6)
\(\max\{F(x,t): (x,t)\in\overline{\Omega}\times [- \Vert \phi \Vert _{L^{\infty}(\Omega)}, \Vert \psi \Vert _{L^{\infty}(\Omega)}]\}<\min\{F(x,t): (x,t)\in\overline{\Omega}\times\{s^{-},s^{+}\}\}\).
Proof
Let ψ be the solution of
Thus \(0\ll\psi\le t^{+}\) again by Lemma 3.4 and the strong maximum principle Lemma 2.3 since \(\mathcal{L}^{p}_{K}(\psi)\ge0\). By Lemmas 3.4 and 2.4, we also have \(\liminf_{B_{R}\ni x\rightarrow x_{0}}\frac{\psi(x)}{{\delta_{R}(x)}^{s}}>0\). Let ϕ be the solution of
Therefore \(t^{-}\le\phi\ll0\) by Lemma 3.4 and by the strong maximum principle Lemma 2.3 since \(\mathcal{L}^{p}_{K}(-\phi)\ge0\). By Lemmas 3.4 and 2.4, we also have \(\liminf_{B_{R}\ni x\rightarrow x_{0}}\frac{-\varphi(x)}{{\delta_{R}(x)}^{s}}>0\).
Next we prove the results on \(A_{\lambda}\). By (3.3), we can choose \(\delta\in(0,1)\) small enough such that
Let \(u\in X_{\lambda}\cap L^{\infty}(\Omega)\) such that \(t^{-}\le u\). Denote \(v=A_{\lambda}(u)\). Then
Hence, \(v\ge\delta^{\frac{1}{p-1}}\phi\gg\phi\) by Lemma 3.4.
Similarly, if \(u\in X_{\lambda}\cap L^{\infty}(\Omega)\) with \(u\le t^{+}\), then \(v\le\delta^{\frac{1}{p-1}}\psi\ll\psi\). This completes the proof of (1)-(5).
Conclusion (6) follows directly from \((f_{2})\) and properties (1)-(3). □
Lemma 3.6
Suppose that \((f_{3})\) and \((f_{4})\) also hold. Then there exists \(\Lambda>0\) such that, for every \(\lambda\ge\Lambda \), there exists \(h\in C([0,1],{\tilde{Y}})\) satisfying \(h(1)\le0\le h(0)\) and
where \({\tilde{Y}}=X\cap C^{1}(\overline{\Omega})\), ϕ and ψ are the ones in Lemma 3.5. Furthermore, if \(f(x,t)\) is odd with respect to t, then, for every \(k\in\mathbb{N}\), there exists \(\Lambda_{k}>0\) such that, for every \(\lambda\ge\Lambda_{k}\), there exists an odd map \(h_{k}\in C(S^{k},{\tilde{Y}})\) satisfying
The proof is a minor modification of the corresponding argument given in order to prove Lemma 3.2 of [16].
Lemma 3.7
For any bounded set \(B\subset Y\) and any order interval \([\phi,\psi]\), there exist \(\phi_{1}\), \(\psi_{1}\in Y\) such that
and
Proof
By \((f_{2})\) there exist ε, \(C>0\), which depend on λ, such that
where \(\lambda_{1}\) is the first eigenvalue in Ω of \(\mathcal {L}^{p}_{K}\) with zero Dirichlet boundary condition. By the monotone operator theory, see [22, 23], there exists a unique solution \(\psi_{1}\) to
where
Let \(\phi_{1}=-\psi_{1}\). Then we get \(\phi_{1}\ll0\ll\psi_{1}\) by Lemma 3.4 and the strong maximum principle lemma. Provided that \(C>0\) is large enough, (3.6) follows.
Next, fix \(u\in X_{\lambda}\) which satisfies \(\phi_{1}\le u\le\psi_{1}\). Denote \(v=A_{\lambda}(u)\). We have
Hence, \(v\le\mu^{\frac{1}{p-1}}\psi_{1}\ll\psi_{1}\) by Lemma 3.4. Similarly, \(v\gg\phi_{1}\). This completes the proof of (3.7). □
At last, we show that \(I_{\lambda}\) satisfies the Palais-Smale condition which is crucial to guaranteeing the existence of critical points.
Lemma 3.8
Let \((u_{n})_{n}\subset X_{\lambda}\) be a Palais-Smale sequence of \(I_{\lambda}\), i.e., \((I_{\lambda}(u_{n}))_{n}\) is bounded and \(I_{\lambda}^{\prime}(u_{n})\rightarrow0\) as \(n\rightarrow\infty\). Then \((u_{n})_{n}\) admits a strongly convergent subsequence in \(X_{\lambda}\).
Proof
By \((f_{1})\)-\((f_{2})\), for \(\varepsilon>0\) small enough,
By Lemma 2.3 in [7],
since Ω is bounded.
Let \((u_{n})_{n}\) be a Palais-Smale sequence of \(I_{\lambda}\) in \(X_{\lambda}\). Then there exists \(C>0\) such that, for all n,
where \(\varepsilon=1/{2pC_{1}^{2}\lambda}\) in (3.8). Thus \((u_{n})_{n}\) is bounded in \(X_{\lambda}\) by (3.9). So, up to a subsequence, still denoted by \((u_{n})_{n}\), we have \(u_{n}\rightharpoonup u\) weakly in \(X_{\lambda}\). Then \(\langle I'(u_{n}),u_{n}-u\rangle\rightarrow0\), and further we obtain
as \(n\rightarrow\infty\). Moreover, by Lemma 2.5 in [7], up to a subsequence, \(u_{n}\rightarrow u\) strongly in \(L^{p}(\Omega)\) and a.e. in Ω. Thus, \(f(x,u_{n})(u_{n}-u)\rightarrow0\) a.e. in Ω as \(n\rightarrow\infty\). It is easy to show that the sequence \((f(x,u_{n})(u_{n}-u))_{n}\) is uniformly bounded and equi-integrable in \(L^{1}(\Omega)\). Hence, by the Vitali convergence theorem (see Rudin [25]), we get
Therefore, by (3.10), we have
as \(n\rightarrow\infty\). Thus, by the weak convergence of \((u_{n})_{n}\) in \(X_{\lambda}\), we get
as \(n\rightarrow\infty\). By Lemma 2.1, we obtain, for \(p>2\),
as \(n\rightarrow\infty\). For \(1< p<2\), we have
as \(n\rightarrow\infty\). Combining (3.11) and (3.12), we get that \(u_{n}\rightarrow u\) strongly in \(X_{\lambda}\) as \(n\rightarrow\infty\). Therefore, \(I_{\lambda}\) satisfies the (PS) condition. □
Taking inspiration from [16], we apply Lemmas 3.5-3.8 in order to prove Theorems 1.1-1.3. Let \(\mathrm{int}_{D_{1}}(D_{2})\) refer to the Y-topology on \(D_{1}\).
Proof of Theorem 1.1
Let ϕ, ψ and Λ be the ones in Lemma 3.5 and Lemma 3.6. Fix \(\lambda\ge \Lambda\). Let h be the one in Lemma 3.6. Choose \(B=h([0,1])\) so that \(B\subset\tilde{Y}\subset Y\) by Lemma 3.6. Let the corresponding \(\phi_{1}\) and \(\psi_{1}\) be the ones stated in Lemma 3.7.
Define \(D_{1}^{+}=[0,\psi_{1}]\) and \(D_{2}^{+}=[0,\psi]\). Then
where \(\mathrm{int}_{D_{1}^{+}}(D_{2}^{+})\) refers to the Y-topology on \(D_{1}^{+}\). By Lemmas 3.5 and 3.7 and Lemma 2.4, conditions \((P_{1})\)-\((P_{3})\) in Section 2 are satisfied for \(D_{2}^{+}\subset D_{1}^{+}\) and \(A=A_{\lambda}\). By Lemmas 3.6-3.8 and Theorem 2.1 in [17], there are two positive and two negative solutions of problem (1.1).
Choose \(D_{1}=[\phi_{1},\psi_{1}]\), \(D_{2}=[\phi,\psi_{1}]\), \(D_{3}=[\phi_{1},\psi]\), and \(D_{4}=[\phi,\psi]\). By Lemmas 3.5-3.8, \(D_{i}\), \(i=1,\ldots,4\), and \(A=A_{\lambda}\) satisfy all the assumptions of Theorem 2.2 in [17]. So there are two sign-changing solutions of problem (1.1).
At last, in view of (3) of Lemma 3.5, we complete the proof. □
Proof of Theorem 1.2
Let \(D_{i}\), \(i=1,\ldots,4\), be the ones in the proof of Theorem 1.1. First, in view of Lemmas 3.5-3.8, all the assumptions of Theorem 2.3 in [17] are satisfied. Second, by (2) and (3) of Lemma 3.5,
and
Hence, (3.13)-(3.14) and Proposition 2.4 of [16] yield the result. □
Proof of Theorem 1.3
Define \(f_{i}(x,t)\), \(i=1,\ldots,n-1\), as in [16]. Consider
By Theorem 1.1, problem (3.15) admits two positive, two negative and two sign-changing solutions. Replacing \(t^{-}\), \(t^{+}\) by \(t^{-}_{i}\), \(t^{+}_{i}\), respectively, the six solutions are outside of the order interval \([t^{-}_{i},t^{+}_{i}]\). According to the choice of m, we have
Thus, by the definition of \(f_{i}\), we get
Furthermore, by Lemma 3.4 the six solutions of problem (3.15) are inside the order interval \([t^{-}_{i+1},t^{+}_{i+1}]\) and are obviously solutions of problem (1.1). In this way, we manage to get \(2(n-1)\) positive, \(2(n-1)\) negative and \(2(n-1)\) sign-changing solutions of problem (1.1). □
4 Conclusions
The purpose of this paper is to study the existence of multiplicity solutions for fractional p-Laplacian equations with sublinear growth and oscillatory behavior. The key point is the choice of the framework to study the existence of weak solutions. In the suitably chosen framework, we are able to fulfil our strategy and generalize the corresponding results for fractional Laplacian equations.
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Fu, Y. A note on the existence and multiplicity of solutions for sublinear fractional problems. Bound Value Probl 2017, 171 (2017). https://doi.org/10.1186/s13661-017-0903-9
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DOI: https://doi.org/10.1186/s13661-017-0903-9