Abstract
In this paper, we investigate the existence of solutions for the fractional p-Laplace equation
where \(N>sp\), \(0< s<1<p\), \(1< q< p< r< p_{s}^{*}:=\frac{Np}{N-sp}\), and the potential function \(V(x)>0\) and \(h_{1}(x)\), \(h_{2}(x)\) are allowed to change sign in \(\mathbb {R}^{N}\). By using variant fountain theorem, we prove that the above equation admits infinitely many small and high energy solutions.
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1 Introduction and main result
In this paper, we consider the existence and multiplicity of solutions for the following elliptic problem:
where \((-\Delta)_{p}^{s}\) is the fractional p-Laplacian operator with \(0< s<1<p\) and \(sp< N\), \(1< q< p< r< p_{s}^{*}:=\frac{Np}{N-sp}\) and potential function \(V(x)>0\), \(h_{1}\) and \(h_{2}\) are sign-changing weight functions. The exact assumptions will be given below.
The fractional p-Laplacian operator \((-\Delta)_{p}^{s}\) is defined along a function \(u\in C_{0}^{\infty}(\mathbb{R}^{N})\) as follows:
for \(x\in\mathbb{R}^{N}\), where \(B_{\varepsilon}(x)=\{y\in\mathbb{R}^{N}:|x-y|<\varepsilon\}\), see [1–3] and the references therein.
In the last years, since the nonlinear equations involving fractional powers of the Laplacian played an increasingly important role in physics, probability, and finance, a great attention has been focused on the study of problems involving the fractional Laplacian. So there has been a lot of interest in the study of the fractional Schrödinger equation
where the nonlinearity f satisfies some general conditions, see for instance [4–19] and the references therein.
More recently, Xiang et al. [3] investigated the fractional p-Laplacian equation
where \(\lambda>0\), \(p< r<\min\{q,p_{s}*\}\), and \(a(x)\) and \(b(x)\) are related by the condition \(a(a/b)^{(r-p)(q-r)}\in L^{N/ps}(\mathbb{R}^{N})\). By using a direct variational method and the mountain pass theorem, the authors proved the existence of two nontrivial weak solutions of (1.4) for \(\lambda>\lambda^{*}\) (\(\lambda^{*}>0\) is a given constant).
There are also a lot of works about problem (1.3) with concave-convex nonlinearities. Goyal and Sreenadh in [20] considered the following p-fractional Laplace equation:
where Ω is a bounded domain in \(\mathbb{R}^{N}\) with Lipschitz boundary, \(p\geq2\), \(n>p\alpha\), \(1< q< p< r< p_{s}^{*}\), \(\lambda>0\), h and b are sign-changing smooth functions. They proved that there exists \(\lambda_{0}>0\) such that problem (1.5) has at least two nonnegative solutions for all \(\lambda\in(0,\lambda_{0})\).
In [21], the authors considered the problem as follows:
where \(M(t)=a+bt^{\theta-1}\), \(\theta>1\), \(a,b\geq0\), \(a+b>0\), \(\lambda >0\), and \(1< q< p<\theta p<r<p_{s}^{*}\). The functions \(h_{1}(x)\), \(h_{2}(x)\), and \(h(x)\) may change sign on \(\mathbb{R}^{N}\). Note that problem (1.6) is reduced to the fractional p-Laplacian equation with \(a=1\) and \(b=0\). Under some suitable conditions, they obtained the existence of two nontrivial entire solutions by applying the mountain pass theorem and Ekeland’s variational principle.
In this paper, we are interested in the multiplicity of solutions to equation (1.1) and find sufficient conditions to guarantee the existence of infinitely many solutions.
The present article is motivated by the papers [6, 7], as well as by the fact that we do not find in the literature any paper dealing with the existence of infinitely many solutions to equation (1.1). The main tools employed in our works are the variant fountain theorems established in [22]. They are effective tools for studying the existence of infinitely many large or small energy solutions. Moreover, the results about the existence of solutions in the above papers are all related to the number λ. So we are also interested in whether the restriction on λ can be taken out.
Throughout this paper, we make the following assumptions:
- \((H_{1})\):
\(V(x)\in C(\mathbb{R}^{N})\) and there exists \(V_{0}>0\) such that \(V(x)\geq V_{0}\) in \(\mathbb{R}^{N}\);
- \((H_{2})\):
\(h_{1}(x)\in L^{\tau}(\mathbb{R}^{N})\), where \(\tau\in[\frac{p_{s}^{*}}{p_{s}^{*}-q},\frac{p}{p-q}]\) is a positive constant;
- \((H_{3})\):
There exists \(\mu\in[\frac{p_{s}^{*}}{p_{s}^{*}-r},\infty)\) such that \(h_{2}(x)\in L^{\mu}(\mathbb{R}^{N})\);
- \((H_{3}')\):
\(h_{2}(x)\in L^{\infty}(\mathbb{R}^{N})\) and \(h_{2}(x)\rightarrow0\) as \(|x|\rightarrow\infty\).
Our main results in this paper are stated as follows.
Theorem 1.1
Assume\((H_{1})\), \((H_{2})\), \((H_{3})\) (or\((H_{3}')\)), and\(h_{1}(x)>0\), then problem (1.1) admits infinitely many small energy solutions\(u_{k}\in E\)satisfying\(I(u_{k})\rightarrow0^{-}\)as\(k\rightarrow\infty\).
Theorem 1.2
Let\((H_{1})\), \((H_{2})\), and\((H_{3})\) (or\((H_{3}')\)) hold. If\(h_{2}(x)>0\)is satisfied, then problem (1.1) possesses infinitely many high energy solutions\(u_{k}\in E\)such that\(I(u_{k})\rightarrow\infty\)as\(k\rightarrow\infty\).
The functional I that appeared in Theorems 1.1–1.2 is the energy functional for problem (1.1), which will be given below.
By combining Theorems 1.1–1.2, the following corollary is immediate.
Corollary 1.3
Let\((H_{1})\), \((H_{2})\), and\((H_{3})\) (or\((H_{3}')\)) hold. If\(h_{1}(x),h_{2}(x)>0\), then problem (1.1) has two sequences\(\{u_{k}\}\)and\(\{\overline{u}_{k}\}\)of nontrivial solutions such that
The rest of the paper is organized as follows. In the forthcoming section, we set up the variational framework for (1.1) and state the variant fountain theorems that will be used later. In Sect. 3, we study problem (1.1) and give the proof of Theorem 1.1. Section 4 is devoted to the proof of Theorem 1.2.
2 Preliminaries
First of all, we give some basic results of fractional Sobolev space that will be used in the next sections. Let \(0< s<1<p\) be real numbers. The fractional Sobolev space \(W^{s,p}(\mathbb{R}^{N})\) is defined as follows:
equipped with the norm
Then \((W^{s,p}(\mathbb{R}^{N}), \|u\|_{W^{s,p}(\mathbb{R}^{N})})\) is a uniformly convex Banach space and the embedding
is continuous for \(t\in[p,p_{s}^{*}]\). Moreover, the embedding is locally compact whenever \(1\leq t< p_{s}^{*}\), see [1] for details.
For our problem (1.1), consider the subspace \(X\subset W^{s,p}(\mathbb{R}^{N})\) given by
Then X is a separable Banach space with the norm
As usual, for \(t\geq1\), we let
By the embedding \(X\hookrightarrow L^{t}(\mathbb{R}^{N})\), we know that there exists a constant \(S_{t}>0\) such that
Definition 2.1
A function \(u\in X\) is said to be a (weak) solution of (1.1) if, for any \(v\in X\), we have
Let \(I(u): X\rightarrow\mathbb{R}\) be the energy functional associated with (1.1) defined by
Using (2.6), it follows from conditions \((H_{1})\), \((H_{2})\), and \((H_{3})\) (or \((H'_{3})\)) that the functional I is well defined and \(I\in C^{1}(X,\mathbb{R})\) with
for any \(v\in X\). It is standard to verify that the weak solutions of (1.1) correspond to the critical points of I.
Finally, we give the variant fountain theorems.
Let X be a Banach space with the norm \(\|\cdot\|\) and \(X=\overline{\bigoplus_{j=1}^{\infty}X_{j}}\) with \(\dim X_{j}<\infty\), \(j\in\mathbb{Z}\). Define
Consider the following \(C^{1}\)-functional \(I_{\lambda}: X\rightarrow\mathbb{R}\) defined by
The following two variant fountain theorems were established in [22].
Theorem 2.2
Assume that the functional\(I_{\lambda}\)defined above satisfies
- \((A_{1})\):
\(I_{\lambda}\)maps bounded sets into bounded sets uniformly for\(\lambda\in[1,2]\), and\(I_{\lambda}(-u)=I_{\lambda}(u)\)for all\((\lambda,u)\in[1,2]\times X\);
- \((A_{2})\):
\(B(u)\geq0\)for all\(u\in X\), and\(B(u)\rightarrow\infty\)as\(\|u\|\rightarrow\infty\)on any finite dimensional subspace ofX;
- \((A_{3})\):
There exist\(\rho_{k}>r_{k}>0\)such that
$$ a_{k}(\lambda):=\inf_{u\in Z_{k}, \|u\|=\rho_{k}}I_{\lambda}(u) \geq0,\qquad b_{k}(\lambda):=\max_{u\in Y_{k}, \|u\|=r_{k}}I_{\lambda}(u)< 0, \quad \forall\lambda\in[1,2] $$and
$$ d_{k}(\lambda):=\inf_{u\in Z_{k}, \|u\|\leq\rho_{k}}I_{\lambda}(u) \rightarrow0,\quad \textit{as } k\rightarrow\infty\textit{ uniformly for } \lambda \in[1,2]. $$
Then there exist\(\lambda_{n}\rightarrow1\), \(u_{n}(\lambda_{n})\in Y_{n}\)such that
where\(c_{k}\in[d_{k}(2),b_{k}(1)]\). In particular, if\(\{u(\lambda_{n})\}\)has a convergent subsequence for everyk, then\(I_{1}\)has infinitely many nontrivial critical points\(\{u_{k}\}\in X\setminus \{0\}\)satisfying\(I_{1}(u_{k})\rightarrow0^{-}\)as\(k\rightarrow\infty\).
Theorem 2.3
Assume that the functional\(I_{\lambda}\)defined above satisfies:
- \((B_{1})\):
\(I_{\lambda}\)maps bounded sets into bounded sets uniformly for\(\lambda\in[1,2]\), and\(I_{\lambda}(-u)=I_{\lambda}(u)\)for all\((\lambda,u)\in[1,2]\times X\);
- \((B_{2})\):
\(B(u)\geq0\), \(A(u)\rightarrow\infty\)or\(B(u)\rightarrow\infty\)as\(\|u\|\rightarrow\infty\) (or\(B(u)\leq0\), \(B(u)\rightarrow-\infty\)as\(\|u\|\rightarrow\infty\));
- \((B_{3})\):
There exist\(\rho_{k}>r_{k}>0\)such that
$$ b_{k}(\lambda)=\inf_{u\in Z_{k}, \|u\|=r_{k}}I_{\lambda}(u)>a_{k}( \lambda)=\max_{u\in Y_{k}, \|u\|=\rho_{k}}I_{\lambda}(u),\quad \forall \lambda \in[1,2]. $$
Then
where\(B_{k}=\{u\in Y_{k}: \|u\|\leq\rho_{k}\}\)and\(\varGamma_{k}=\{\gamma\in C(B_{k},X): \gamma\textit{ is odd}, \gamma|_{\partial B_{k}=\mathrm{id}}\}\) (\(k\geq2\)). Moreover, for almost every\(\lambda\in[1,2]\), there exists a sequence\(u_{n}^{k}(\lambda)\)such that
3 Proof of Theorem 1.1
In this section, we use Theorem 2.2 to complete the proof.
For the notation in Theorem 2.2, we define the functional A, B, and \(I_{\lambda}\) on our working space X by
and
Since X is a separable and reflexive Banach space, then there exist \(\{e_{i}\}_{i=1}^{\infty}\subset X\) and \(\{e_{i}^{*}\}_{i=1}^{\infty}\subset X^{*}\) such that
and
Let \(X_{i}=\mathbb{R}e_{i}\), and \(Y_{k}\) and \(Z_{k}\) be defined as (2.10).
In the proof of our results, we need the following limits.
Lemma 3.1
Assume\((H_{1})\), \((H_{2})\), and\((H_{3})\) (or\((H'_{3})\)), and let
then\(\alpha_{k},\beta_{k}\rightarrow0\)as\(k\rightarrow\infty\).
Proof
It is clear that \(0<\alpha_{k+1}\leq\alpha_{k}\), so \(\alpha_{k}\rightarrow\alpha_{0}\geq0\) as \(k\rightarrow\infty\). For every \(k\in\mathbb{N}^{+}\), taking \(u_{k}\in Z_{k}\) such that \(\|u_{k}\|=1\) and
As X is reflexive, \(\{u_{k}\}\) has a weakly convergent subsequence, without loss of generality, suppose \(u_{k}\rightharpoonup u\) weakly in X. Then, for every \(i\in\mathbb{N}^{+}\), we have
which implies that \(u=0\) and \(u_{k}\rightharpoonup0\) weakly in X.
Let \((H_{2})\) hold, then for any given small \(\varepsilon>0\), we may find \(R>0\) big enough such that
where \(B_{R}=\{x\in\mathbb{R}^{N}: |x|< r\}\), \(B_{R}^{c}=\mathbb{R}^{N}\setminus B_{R}\), and \(S_{q\tau'}\) is the embedding constant in (2.6).
If \(\frac{p_{s}^{*}}{p_{s}^{*}-q}<\tau\leq\frac{p}{p-q}\), since the embedding \(X\hookrightarrow L^{q\tau'}(B_{R})\) is compact, then \(u_{k}\rightarrow0\) in \(L^{q\tau'}(B_{R})\) and hence there exists \(K_{1}>0\) such that
for \(k>K_{1}\).
Using (3.8) and (3.9), for all \(k>K_{1}\), we get
If \(\tau=\frac{p_{s}^{*}}{p_{s}^{*}-q}\), since \(u_{k}\rightharpoonup0\) in E and the compact embedding \(X\hookrightarrow L^{t}(B_{R})\) (\(1\leq t< p_{s}^{*}\)), we can assume that \(u_{k}\rightarrow0\) a.e. in \(B_{R}\). For each measurable subset \(\varOmega\subset B_{R}\), we have
Then \(\{|h_{1}||u_{k}|^{q}\}\) is uniformly integrable, and the Vitali convergence theorem implies
So, for k big enough, (3.10) still holds.
Then, from (3.6) and (3.10), we conclude that \(\alpha_{k}\rightarrow0\) as \(k\rightarrow\infty\).
Assume \((H_{3})\). Since \(\mu\in[\frac{p_{s}^{*}}{p_{s}^{*}-r},\infty)\) implies \(p< r\mu'\leq p_{s}^{*}\), arguing as in the above proof, one has \(\beta_{k}\rightarrow0\) as \(k\rightarrow\infty\).
Similarly, if assumption \((H_{3}')\) holds, it follows
Since \(h_{2}(x)\in L^{\infty}(\mathbb{R}^{N})\) and \(h_{2}(x)\rightarrow0\) as \(|x|\rightarrow\infty\), \(\beta_{k}\rightarrow0\) can be obtained in a similar way, and we complete the proof. □
In order to apply Theorem 2.2, we give the following lemma first.
Lemma 3.2
Let\((H_{1})\), \((H_{2})\), and\((H_{3})\) (or\((H'_{3})\)) hold. Then there exist two sequences\(0< r_{k}<\rho_{k}\rightarrow0\)as\(k\rightarrow\infty\)such that
and
Proof
From Lemma 3.1 we see that, for every \(u\in Z_{k}\), it holds
Thus
Fix \(K_{2}>0\) big enough such that \(\frac{1}{r}\beta_{k}^{r}<\frac{1}{2p}\) for \(k>K_{2}\), then for \(u\in Z_{k}\) and \(\|u\|<1\), we have
If we choose \(\rho_{k}=(8p\alpha_{k}^{q}/q)^{\frac{1}{p-q}}\), then \(\rho_{k}\rightarrow0^{+}\) as \(k\rightarrow\infty\) and for any \(u\in Z_{k}\) with \(\|u\|=\rho_{k}\), we get that
This inequality implies that
In addition, for all \(\lambda\in[1,2]\), \(k>K_{2}\) and \(u\in Z_{k}\) with \(\|u\|\leq\rho_{k}\), we have
Obviously,
So we have \(d_{k}(\lambda)\rightarrow0\) as \(k\rightarrow\infty\) uniformly for \(\lambda\in[1,2]\).
For all \(u\in Y_{k}\), \(\lambda\in[1,2]\), by the equivalence of any norm in a finite dimensional space, we can derive
where \(C_{1}\geq0\), \(C_{2}>0\). Notice \(q< p< r\), so we can choose \(r_{k}>0\) small enough satisfying \(r_{k}<\rho_{k}\) such that
The proof is completed. □
Proof of Theorem 1.1
At first, we confirm conditions \((A_{1})\)–\((A_{2})\) in Theorem 2.2. It follows from \((H_{1})\)–\((H_{3})\) that \(I_{\lambda}\) maps bounded sets into bounded sets uniformly for \(\lambda\in[1,2]\). Evidently, \(I_{\lambda}(-u)=I_{\lambda}(u)\) for all \((\lambda,u)\in[1,2]\times X\). \(B(u)\geq0\) for all \(u\in X\), and by the equivalence of any norm in a finite dimensional space, we know that \(B(u)\rightarrow\infty\) as \(\|u\|\rightarrow\infty\) on any finite dimensional subspace of X. So \((A_{1})\) and \((A_{2})\) hold.
From Lemma 3.2, we see that \((A_{3})\) in Theorem 2.2 are verified. Consequently, for any \(k\in\mathbb{Z}^{+}\), there exist \(\lambda_{n}\rightarrow1\), \(u(\lambda_{n})\in Y_{n}\) such that
Equation (3.23) implies \(I_{\lambda_{n}}'(u(\lambda_{n}))\rightarrow0\) as \(n\rightarrow\infty\). Then, for large n, from \((H_{1})\)–\((H_{3})\) and (2.6) we have
Since \(p>1\), \(p>q>0\), the above inequality implies that \(\{u(\lambda_{n})\}\) is bounded in X.
Then there exist a constant \(M>0\) and \(u\in X\), and a subsequence \(\{u(\lambda_{n})\}\), denoted by \(\{u_{n}\}\), such that \(\|u\|,\|u_{n}\|\leq M\) and \(u_{n}\rightharpoonup u\) weakly in X. Arguing as in the proof of Lemma 3.1, we have
Using the Hölder inequality, we have
and
Then (3.25) implies
Denote
and
Then the fact \(I_{\lambda_{n}}'(u(\lambda_{n}))\rightarrow0\) shows that \(P_{n}\rightarrow0\) as \(n\rightarrow\infty\). Moreover, \(\{u_{n}\}\) is a bounded sequence and \(u_{n}\rightharpoonup u\) in X, which imply \(Q_{n}\rightarrow0\).
Equations (3.29) and (3.30) show that, for large n,
By using the standard inequalities (see [23]) given by
and
where \(C_{p}\) is a positive constant and \(\langle\cdot,\cdot\rangle\) denotes the inner product in \(\mathbb{R}^{N}\), we can easily deduce that \(\|u_{n}-u\|\rightarrow0\) as \(n\rightarrow\infty\). Now, from Theorem 2.2, we see that \(I=I_{1}\) possesses infinitely many nontrivial critical points \(u_{k}\) for \(k\in\mathbb{Z}^{+}\) satisfying \(I(u_{k})\rightarrow0^{-}\) as \(k\rightarrow\infty\). Therefore, problem (1.1) possesses infinitely many nontrivial solutions, the proof of Theorem 1.1 is completed. □
4 Proof of Theorem 1.2
For the notation in Theorem 2.2, we define the functional A, B and \(I_{\lambda}\) on our working space X by
and
Lemma 4.1
Let\((H_{1})\)–\((H_{3})\)hold and\(h_{2}(x)>0\). Then there exist two sequences\(\rho_{k}>r_{k}>0\)such that
Proof
Similar to the beginning of the proof of Lemma 3.2, by (3.14) we have
Fix \(K_{3}>0\) big enough such that \(\frac{2}{q}\alpha_{k}^{q}<\frac{1}{2p}\) for \(k>K_{3}\), then for \(u\in Z_{k}\) and \(\|u\|>1\), we have
If we choose \(r_{k}=(\frac{r}{8p\beta_{k}^{r}})^{\frac{1}{r-p}}\), then \(r_{k}\rightarrow\infty\) as \(k\rightarrow\infty\), and for any \(u\in Z_{k}\) with \(\|u\|=r_{k}\), we get that
This inequality implies that
For all \(u\in Y_{k}\), \(\lambda\in[1,2]\), by the equivalence of any norm in a finite dimensional space, we can derive
where \(C_{1}\geq0\), \(C_{2}>0\). Notice \(q< p< r\), then \(I_{\lambda}(u)\rightarrow-\infty\) as \(\|u\|\rightarrow\infty\). So we can choose \(\rho_{k}>r_{k}\) big enough such that
The proof is completed. □
Proof of Theorem 1.2
We complete the proof by Theorem 2.3. Let us verify the conditions in Theorem 2.3 firstly.
\((H_{1})\)–\((H_{3})\) imply that \(I_{\lambda}\) maps bounded sets into bounded sets uniformly for \(\lambda\in[1,2]\). Moreover, \(I_{\lambda}(-u)=I_{\lambda}(u)\) for all \(u\in X\) and \(\lambda\in[1,2]\). So condition \((B_{1})\) of Theorem 2.3 holds.
Evidently, \(B(u)\geq0\) for all \(u\in X\), and \(A(u)\rightarrow\infty\) as \(\|u\|\rightarrow\infty\). \((B_{2})\) in Theorem 2.3 is verified.
From Lemma 4.1, we see that condition \((B_{3})\) has been verified. Therefore, by Theorem 2.3, for a.e. \(\lambda\in[1,2]\), there exists a sequence \(u_{n}^{k}(\lambda)_{n=1}^{\infty}\) such that
Moreover, by Theorem 2.3 and (4.6) we see that
Since
Then we have
If we choose a sequence \(\lambda_{m}\rightarrow1\), then (4.9) implies that the sequence \(\{u_{n}^{k}(\lambda_{m})\}_{n=1}^{\infty}\) is bounded. Using similar arguments as those in the proof of Theorem 1.1, we can prove that the sequence \(\{u_{n}^{k}(\lambda_{m})\}_{n=1}^{\infty}\) has a strong convergent subsequence as \(n\rightarrow\infty\). Thus we may assume that \(u_{n}^{k}(\lambda_{m})\rightarrow u^{k}(\lambda_{m})\) in X as \(n\rightarrow\infty\) for every \(m\in\mathbb{N}\). Moreover, by (4.9) and (4.12) we have
As in the proof of Theorem 1.1, we can get the boundedness of \(\{u^{k}(\lambda_{m})\}_{m=1}^{\infty}\) and prove it possesses a strong convergent subsequence with the limit \(u^{k}\in X\). Therefore, the limit \(u^{k}\in X\) is a critical point of I with \(I(u^{k})\in[\overline{b}_{k},\overline{c}_{k}]\). Since \(\overline{b}_{k}\rightarrow\infty\) as \(k\rightarrow\infty\), we get infinitely many nontrivial critical points of I. Consequently, problem (1.1) possesses infinitely many nontrivial solutions with high energy. The proof is completed. □
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Chen, Q., Chen, C. & Shi, Y. Multiple solutions for fractional p-Laplace equation with concave-convex nonlinearities. Bound Value Probl 2020, 63 (2020). https://doi.org/10.1186/s13661-020-01355-y
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DOI: https://doi.org/10.1186/s13661-020-01355-y