Abstract
In this paper, we use variant fountain theorems to study the existence of infinitely many solutions for the fractional p-Laplacian equation
where \(\lambda,\mu \) are two positive parameters, \(N,p\ge 2\), \(q\in (1,p)\), \(\alpha \in (0,1)\), \((-\Delta )_{p}^{\alpha }\) is the fractional p-Laplacian, and \(V,g,u:\mathbb{R}^{N}\to \mathbb{R}\), \(f:\mathbb{R}^{N}\times \mathbb{R}\to \mathbb{R}\).
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1 Introduction
In this paper we investigate the existence of infinitely many solutions for the fractional p-Laplacian equation
where \(\lambda,\mu \) are two positive parameters, \(N,p\ge 2\), \(\alpha \in (0,1)\), \((-\Delta )_{p}^{\alpha }\) is the fractional p-Laplacian, and the potential function \(V:\mathbb{R}^{N}\to \mathbb{R}\) satisfies the following conditions:
-
(V1)
\(V\in C(\mathbb{R}^{N}, \mathbb{R})\) and \(\inf_{x\in \mathbb{R} ^{N}} {V}(x)\ge V_{0}>0\), where \(V_{0}\) is a positive constant.
-
(V2)
There exists \(b>0\) such that \(\text{meas}\{x\in \mathbb{R}^{N}: {V}(x)\le b\}\) is finite, where meas denotes the Lebesgue measures.
The functions \(f:\mathbb{R}^{N}\times \mathbb{R}\to \mathbb{R}\), \(g:\mathbb{R}^{N}\to \mathbb{R}\) satisfy the conditions:
-
(f1)
\({f}\in C(\mathbb{R}^{N}\times \mathbb{R},\mathbb{R})\) and \(\lim_{|u|\to 0} \frac{f(x,u)}{|u|^{p-2}u}=0\) uniformly in \(x\in \mathbb{R}^{N}\).
-
(f2)
\({F}(x,u)=\int _{0}^{u} {f}(x,s)\,\mathrm{d}s\ge 0\) and \(\mathscr{{F}}(x,u)= \frac{1}{p}{f}(x,u)u-{F}(x,u)\ge 0\) for all \((x,u)\in \mathbb{R}^{N} \times \mathbb{R}\).
-
(f3)
\(\lim_{|u|\to \infty }\frac{{f}(x,u)u}{|u|^{p}}=+\infty \) uniformly in \(x\in \mathbb{R}^{N}\).
-
(f4)
There exist \(d_{1},r_{0}>0\) and \(\tau > \frac{p_{\alpha }^{*}}{p _{\alpha }^{*}-p} \) with \(p_{\alpha }^{*}= \bigl\{ \scriptsize{ \begin{array}{l@{\quad}l} \frac{Np}{N-\alpha p} & \text{if }\alpha p< N, \\ \infty &\text{if }\alpha p\ge N, \end{array}} \) such that
$$ \bigl\vert {f}(x,u) \bigr\vert ^{\tau }\le d_{1} \mathscr{{F}}(x,u) \vert u \vert ^{(p-1)\tau }\quad\text{for all } x\in \mathbb{R}^{N} \text{ and } \vert u \vert \ge r_{0}. $$ -
(f5)
\(f(x,-u)=-f(x,u)\) for all \((x,u)\in \mathbb{R}^{N}\times \mathbb{R}\).
-
(g)
\(g\in L^{q'}(\mathbb{R}^{N})\) and \(g(x)\ge 0\) \((\not \equiv 0)\) for a.e. \(x\in \mathbb{R}^{N}\), where \(q'\in (\frac{p_{\alpha }^{*}}{p _{\alpha }^{*}-q}, \frac{p}{p-q} ], q\in (1,p)\).
Fractional systems arise for example in phase transitions, chaos, diffusion, finance, flame propagation, and wave propagation. In [1], the authors introduced a fractional order modified Duffing system
where \(\frac{\mathrm{d}^{q_{1}}x}{\mathrm{d}t^{q_{1}}},\frac{\mathrm{d}^{q_{2}}y}{\mathrm{d}t ^{q_{2}}}\) are fractional derivatives, and via phase portraits and bifurcation diagrams, they studied chaotic behaviors for this system; we also refer the reader to the books [2,3,4] and the papers [5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23]. Variational methods and critical point theory were used to study fractional Schrödinger equations in the literature [24,25,26,27,28,29,30,31,32,33,34,35,36,37]; for results on Schrödinger equations, we refer the reader to [38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66]. In [24, 25], Ambrosio and Torres used the mountain pass theorem and a variant of the fountain theorem to obtain the existence of nontrivial solutions for (1) with \(\lambda =1,\mu =0\), where f is p-superlinear at infinity. In [27], Tang et al. obtained infinitely many solutions for the following fractional p-Laplacian equations of Schrödinger–Kirchhoff type:
where they used the condition:
(Tang) There exist \(c_{0}>0,r_{0}>0\), and \(\kappa >\{1, \frac{N}{p \alpha }\}\) such that
to ensure that the energy functional satisfies the Palais–Smale condition, i.e., (PS) sequence has a convergent subsequence; this condition can also be found in [26, 28, 40,41,42]. There are only a few papers on (1) with a sublinear perturbation. For example, in [29] the authors used the famous Ambrosetti–Rabinowitz condition:
(AR) There exists \(\mu >p^{2} \) such that
to obtain nontrivial solutions for (2) with a perturbation g (\(g\in L^{\frac{p}{p-1}}(\mathbb{R}^{N})\)). In [30,31,32, 38, 39] similar methods were used to study various Schrödinger equations with perturbations.
Motivated by the above papers, in this paper we use variant fountain theorems to study the existence of nontrivial solutions for the fractional p-Laplacian equation (1). The novelty is two-fold: (i) the condition (Tang) is adopted to ensure that bounded sequences have convergent subsequences, (ii) we consider the influence of parameters and perturbation terms on the existence of solutions.
Now, we state our main result.
Theorem 1.1
Suppose that (V1)–(V2), (f1)–(f5), and (g) hold. Then, for sufficiently small \(\mu >0\), there exists \(\varLambda >0\) such that system (1) possesses infinitely many solutions when \(\lambda \ge \varLambda \).
Remark 1.2
Note that (f1), (f2), and (f4) imply that f has subcritical growth. From (f2), (f4), for all \(x\in \mathbb{R}^{N}, |u| \ge r_{0}\), we find
This shows that
Let \(\frac{(p-1)\tau +1}{\tau -1}=s-1\). Then \(s= \frac{p\tau }{\tau -1}\in (p,p_{\alpha }^{*})\). On the other hand, from (f1) for all \(\varepsilon >0\), we have
and hence, there exists \(c_{\varepsilon }=\sqrt[\tau -1]{ \frac{d_{1}}{p}}>0\) such that
and from \({F}(x,u)=\int _{0}^{u} {f}(x,s)\,\mathrm{d}s\) we have
Remark 1.3
Consider the Ambrosetti–Rabinowitz condition (see [29,30,31,32, 38, 39]):
(AR) There exists \(\theta >p\) such that
Let \(F(x,u)=|\sin x||u|^{p}\ln (1+|u|), \forall x\in \mathbb{R}^{N}, u\in \mathbb{R}\). Then \(f(x,u)=|\sin x| (p|u|^{p-2}u\ln (1+|u|) + \frac{|u|^{p-1}u}{1+|u|} )\). Consequently, for all \(x\in \mathbb{R}^{N}\), we have
and this is impossible for large \(|u|\). However, this function satisfies conditions (f1)–(f5).
2 Preliminaries
We first discuss the space \(W^{\alpha,p}(\mathbb{R}^{N})\) (for more details, we refer the reader to [67]). When \(u:\mathbb{R}^{N} \to \mathbb{R}\) is a measurable function, we define the Gagliardo seminorm as follows:
Now, the fractional Sobolev space is given by
with the norm
where \(\|u\|_{p}\) is the norm for the usual Lebesgue space \(L^{p}( \mathbb{R}^{N})\), denoted by
For the potential function V, we consider the following fractional Sobolev space:
with the norm
Note that the parameter λ can be chosen large enough, so this norm can be replaced by
In summary, throughout our paper we use the space \((E,\|\cdot \|)\).
Lemma 2.1
(see [67, Theorem 6.5] and [25, Lemma 2.1])
The embedding \(E\hookrightarrow L^{t}( \mathbb{R}^{N})\) is continuous if \(t\in [p,p_{\alpha }^{*}]\) and compact if \(t\in [p,p_{\alpha }^{*})\).
Hence, there exists \(C_{t}>0 \) such that
Let X be a reflexive and separable Banach space and \(X^{*}\) be its dual space. Then there are (see [68, Sect. 17]) \(\{\phi _{n}\} _{n\in \mathbb{N}}\subset X\) and \(\{\phi _{n}^{*}\}_{n\in \mathbb{N}} \subset X^{*}\) such that \(X=\overline{\text{span}\{\phi _{n}:n\in \mathbb{N}\}}\), \(X^{*}=\overline{\text{span}\{\phi _{n}^{*}:n\in \mathbb{N}\}}\), and \(\langle \phi _{n},\phi _{m}\rangle = \bigl\{ \scriptsize{ \begin{array}{l@{\quad}l} 1,& n=m, \\ 0,& n\neq m. \end{array}} \) For \(k=1,2,\ldots\) , let \(Y_{k}=\text{span}\{\phi _{1},\ldots,\phi _{k}\}\) and \(Z_{k}=\overline{\text{span}\{\phi _{k},\phi _{k+1},\ldots\}}\).
Lemma 2.2
(see [69])
Let X be a Banach space, and \(X=\overline{\bigoplus_{j\in \mathbb{N}}X_{j}}\) with \(\dim X_{j}<\infty \) for any \(j\in \mathbb{N}\). Set \(Y_{k}=\bigoplus_{j=0}^{k} X_{j}, Z_{k}=\overline{ \bigoplus_{j=k+1}^{\infty }X_{j}}\). Consider the following \(C^{1}\) functional \(\varPhi _{\lambda }: X\to \mathbb{R}\) defined by
Suppose that
-
(Z1)
\(\varPhi _{\lambda }\) maps bounded sets to bounded sets uniformly for \(\lambda \in [1,2]\). Furthermore, \(\varPhi _{\lambda }(-u)=\varPhi _{\lambda }(u)\) for \((\lambda,u)\in [1,2]\times X\);
-
(Z2)
\(B(u)\ge 0\); \(B(u)\to \infty \) as \(\|u\|\to \infty \) on any finite dimensional subspace of X;
-
(Z3)
There exist \(\rho _{k}>r_{k}>0\) such that \(a_{k}(\lambda )= \inf_{u\in Z_{k},\|u\|=\rho _{k}}\varPhi _{\lambda }(u)\ge 0>b_{k}(\lambda )=\max_{u\in Y_{k},\|u\|=r_{k}}\varPhi _{\lambda }(u)\) for \(\lambda \in [1,2]\), \(d_{k}(\lambda )=\inf_{u\in Z_{k},\|u\|\le \rho _{k}} \varPhi _{\lambda }(u)\to 0\) as \(k\to \infty \), uniformly for \(\lambda \in [1,2]\).
Then there exist \(\lambda _{n}\to 1\), \(u(\lambda _{n})\in Y_{n}\) such that \(\varPhi '_{\lambda _{n}}|_{Y_{n}}(u(\lambda _{n}))=0\), \(\varPhi _{\lambda _{n}}(u( \lambda _{n}))\to c_{k}\in [d_{k}(2),b_{k}(1)]\) as \(n\to \infty \). In particular, if \(\{u(\lambda _{n})\}\) has a convergent subsequence for every k, then \(\varPhi _{1} \) has infinitely many nontrivial critical points \(\{u_{k}\}\subset X\backslash \{0\}\) satisfying \(\varPhi _{1}(u _{k})\to 0^{-}\) as \(k\to \infty \).
3 Main results
Now, we can define the energy functional J on E as follows:
From (4), (V1)–(V2), and (g) we have that J is well defined and of class \(C^{1}\). Moreover,
From the definition of \(J'\), we see that the critical points of J are weak solutions for (1). From [30], we know that the space E can be decomposed as X in Lemma 2.2, so we can consider the family of functionals \(J_{\nu }: E \to \mathbb{R}\) defined by
Then \(B(u)\ge 0\) for \(u\in E\), and \(J_{\nu }(-u)=J_{\nu }(u)\) for \((\nu,u)\in [1,2]\times E\). Also, it is easy to see that \(J_{\nu }\) maps bounded sets to bounded sets uniformly on \(\nu \in [1,2]\).
Lemma 3.1
Suppose that the assumptions of Theorem 1.1 hold. Then \(B(u)\to \infty \) as \(\|u\|\to \infty \) on any finite dimensional subspace of E.
Proof
For any finite dimensional subspace \(\widetilde{E} \subset E\), there exists \(\varepsilon _{1}>0\) such that
If (8) is not true, then for all \(n\in \mathbb{N}\), there exists \(u_{n}\in \widetilde{E}\backslash \{0\}\) such that
Define \(v_{n}(x)=\frac{u_{n}(x)}{\|u_{n}\|}\in \widetilde{E}\backslash \{0\}\), then for all \(n\in \mathbb{N}\), \(\|v_{n}\|=1\), and we obtain
Since \(\dim \widetilde{E}<\infty \), passing to a subsequence if necessary, we may assume that \(v_{n}\to v_{0}\) in Ẽ. Moreover, \(\|v_{0}\|=1\). From the equivalence of all norms on the finite dimensional space Ẽ, we have
Thus, there exist \(\xi _{1},\xi _{2}>0\) such that
If not, for all \(n\in \mathbb{N}\), we obtain
This implies that
Hence, \(v_{0}=0\), contradicting \(\|v_{0}\|=1\), and then (11) holds.
Now let
For n large enough (for example, taking n such that \(\xi _{2}- \frac{1}{n}\ge \frac{1}{2}\xi _{2},\frac{1}{2^{p-1}} \xi _{1} - \frac{1}{n}\ge \frac{1}{2^{p}} \xi _{1}\)), using the inequality \(|v_{n}|^{p}=|v_{n}-v_{0}+v_{0}|^{p} \le 2^{p-1} |v_{n}-v_{0}|^{p} +2^{p-1} |v_{0}|^{p} \), for \(p\ge 2\), we have
This contradicts (10). As a result, (8) holds. For \(\varepsilon _{1}\) in (8), let
Then we have \(\text{meas}(\varOmega _{u})\ge \varepsilon _{1}\). On the other hand, from L’Hospital rule and (f3) we have
Hence, there exists sufficiently large \(d_{2}>0\) such that
From (4) with \(s\in (p,p_{\alpha }^{*})\), we have
As a result, there exists \(d_{3}\in (0,d_{2})\) such that
This, together with (8), implies that
Thus \(B(u)\to \infty \) as \(\|u\|\to \infty \) on any finite dimensional subspace of E. This completes the proof. □
Lemma 3.2
Suppose that the assumptions of Theorem 1.1 hold. Then there exists a sequence \(\rho _{k}\to 0^{+}\) as \(k\to \infty \) such that
and
where \(Z_{k}=\overline{\bigoplus_{j=k}^{\infty }X_{j}}\) for all \(k\in \mathbb{N}\).
Proof
Let \(\beta _{s}(k)=\sup_{u\in Z_{k},\|u\|=1}\|u\|_{s}\) with \(s\in (p,p_{\alpha }^{*})\). Then from Lemma 3.8 of [70] and Lemma 2.1, we have \(\beta _{s}(k)\to 0, k\to \infty \). Now, for \(u\in Z_{k}\), from (4), (5), we obtain
Let \(\|u\|=\rho _{k}=\beta _{s}(k), u\in Z_{k}\), note that \(\beta _{s}(k)\) can be chosen arbitrarily small when k is large, and if \(\varepsilon =\frac{p}{2C_{p}^{p}} [\frac{1}{p}-\frac{3c_{\varepsilon }}{s} \beta _{s}^{s} (k) ]\), we have
On the other hand, for any \(u\in Z_{k}\) with \(\|u\|\le \rho _{k}\), we have
Hence,
Since, \(\rho _{k}\to 0\) as \(k\to \infty \), we have
This completes the proof. □
Lemma 3.3
Suppose that all the assumptions of Theorem 1.1 hold (and μ is sufficiently small). For the sequence \(\{\rho _{k}\} _{k\in \mathbb{N}}\) in Lemma 3.2, there exists \(r_{k}\in (0,\rho _{k})\) for \(k\in \mathbb{N}\) such that
where \(Y_{k}=\overline{\bigoplus_{j=1}^{k} X_{j}}\) for \(k\in \mathbb{N}\).
Proof
For \(u\in Y_{k}\), from (13) and (5) we have
Note that we can take sufficiently large \(d_{2}\) (and μ sufficiently small) such that
This completes the proof. □
From Lemmas 3.1–3.3, we see (Z1)–(Z3) of Lemma 2.2 hold. Therefore, there exist \(\nu _{n}\to 1\), \(u(\nu _{n})\in Y_{n}\) such that
For convenience, we denote \(u_{n}=u(\nu _{n})\) for all \(n\in \mathbb{N}\).
Lemma 3.4
Suppose that all the assumptions of Theorem 1.1 hold. Then the sequence \(\{u_{n}\}\) is bounded in E.
Proof
Note that \(J_{\nu _{n}}(u(\nu _{n}))\) is bounded, and we have
We will argue by contradiction. If \(\|u_{n}\|\) is unbounded in E, we assume that \(\|u_{n}\|\to \infty \). Put \(v_{n}= \frac{u_{n}}{\|u_{n}\|}\), and then \(\|v_{n}\|=1\). Passing to a subsequence, there exists \(v\in E\) such that \(v_{n}\rightharpoonup v\) weakly in E, \(v_{n}\to v\) strongly in \(L^{r}(\mathbb{R}^{N})\) with \(r\in [p,p_{\alpha }^{*})\), \(v_{n}(x)\to v(x)\) for a.e. \(x\in \mathbb{R}^{N}\). For \(0\le a< b\), let \(\varOmega _{n}(a,b)=\{x\in \mathbb{R}^{N}:a\le |u_{n}(x)|< b\}\). Next we consider two cases.
Case 1: Suppose \(v=0\).
Then \(v_{n}\to 0\) \(\text{ in }L^{r}(\mathbb{R}^{N})\text{ with } r \in [p,p_{\alpha }^{*})\), and \(v_{n}(x)\to 0\text{ for a.e. }x\in \mathbb{R}^{N}\). Let \(r_{0}\) be as in (f4), and from (3) we have
From (f4), we know \(\tau >\frac{p_{\alpha }^{*}}{p_{\alpha }^{*}-p}\). Thus, if we set \(\tau '=\tau /(\tau -1)\), then \(p\tau '\in (p,p_{ \alpha }^{*})\). From the Hölder inequality and (18), we obtain
Combining (19) and (20), we have
On the other hand, note that \(\nu _{n}\to 1\), from (5) and (g) we have
which contradicts (21).
Case 2: Suppose \(v\neq0\).
Set \(A=\{x\in \mathbb{R}^{N}: v(x)\neq0\}\) and \(\text{meas}(A)>0\). For \(x\in A\), we have \(\lim_{n\to \infty }|u_{n}(x)|=\infty \). Hence \(A\subset \varOmega _{n}(r_{0},\infty )\) for large n. From (3) and (f3), note the nonnegativity of \({f}(x,u)u\), Fatou’s lemma enables us to obtain
This is also a contradiction.
Thus \(\{u_{n}\}_{n\in \mathbb{N}}\) is bounded in E. This completes the proof. □
Lemma 3.5
Suppose that all the assumptions of Theorem 1.1 hold. For some \(\varLambda >0\), the sequence \(\{u_{n}\}\) possesses a strong convergent subsequence in E.
Proof
From Lemma 3.4, the sequence \(\{u_{n}\}_{n\in \mathbb{N}}\) is bounded in E. Then there exists \(u\in E\) such that \(u_{n}\rightharpoonup u\) weakly in E, \(u_{n}\rightarrow u \) strongly in \(L^{r}(\mathbb{R}^{N})\) for \(r\in [p,p_{\alpha }^{*})\) and \(u_{n}(x)\rightarrow u(x)\) for a.e. \(x \in \mathbb{R}^{N} \) after passing to a subsequence if necessary. Next, we prove two claims.
Claim 1. \(\langle J_{\nu _{n}}'(u_{n}-u),u_{n}-u\rangle =o(1)\) as \(n\to \infty \).
Let \(w_{n}=u_{n}-u\). Then \(w_{n}\rightharpoonup 0\) weakly in E, \(w_{n}\rightarrow 0 \) strongly in \(L^{r}(\mathbb{R}^{N})\) for \(r\in [p,p_{\alpha }^{*})\), and \(w_{n}(x)\rightarrow 0\) for a.e. \(x \in \mathbb{R}^{N} \) after passing to a subsequence. Recall that \(u_{n}\rightharpoonup u\) weakly in E, we have \(\|w_{n}\|=\|u_{n}\|- \|u\|+o(1)\), and from (7) we only need to show
In fact, from (3) we have
and
Claim 2. There is \(M>0\) such that
From Lemma A.1 of [70], there exists \(\sigma (x)\in L^{r}( \mathbb{R}^{N}) \) with \(r\in [p,p_{\alpha }^{*})\) such that
Note that \(w_{n}=u_{n}-u\), by (3), (4), and (22) we have
where \({M}>0\), \(\sigma _{1}\in L^{p}(\mathbb{R}^{N}), \sigma _{2}\in L ^{s}(\mathbb{R}^{N})\) with \(s\in (p,p_{\alpha }^{*})\).
Now, we prove that the sequence \(\{u_{n}\}_{n\in \mathbb{N}}\) has a convergent subsequence. Note \({V}(x)< b\) on a set of finite measure and \(w_{n}\rightarrow 0 \) strongly in \(L^{r}(\mathbb{R}^{N})\), \(r\in [p,p_{\alpha }^{*})\), and we have
Combining this and the Hölder inequality, for \(s=\frac{p\tau }{ \tau -1}\in [p,p_{\alpha }^{*}) \), fixed \(\nu \in (s,p_{\alpha }^{*})\), and we have
From (f1), for any \(\varepsilon >0\), there exists \(\delta =\delta ( \varepsilon )>0\) such that \(|{f}(x,u)|\le \varepsilon |u|^{p-1}\) for \(x\in \mathbb{R}^{N}\) and \(|u|\le \delta \). Moreover, (f4) is also satisfied for some suitable δ. Therefore, we have
and
Consequently, we have
Thus there exists \(\varLambda >0\) such that \(w_{n}\to 0\) in E when \(\lambda >\varLambda \). This implies that \(u_{n}\to u\) in E. This completes the proof. □
Proof of Theorem 1.1
From the last assertion of Lemma 2.2, we know that \(J=J_{1}\) has infinitely many nontrivial critical points. Therefore, (1) possesses infinitely many small negative-energy solutions. This completes the proof. □
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Research supported by the National Natural Science Foundation of China (Grant No. 11601048), Natural Science Foundation of Chongqing (Grant No. cstc2016jcyjA0181), Natural Science Foundation of Chongqing Normal University (Grant No. 16XYY24), and Natural Science Foundation of Shandong Province (Grant No. ZR2018MA009 and ZR2015AM014).
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Zhang, K., O’Regan, D., Xu, J. et al. Infinitely many solutions via critical points for a fractional p-Laplacian equation with perturbations. Adv Differ Equ 2019, 166 (2019). https://doi.org/10.1186/s13662-019-2113-5
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DOI: https://doi.org/10.1186/s13662-019-2113-5