Infinitely many solutions for a class of fractional Schrödinger equations with sign-changing weight functions

In this paper, we study the fractional Schrödinger equation {(−Δ)su+u=a(x)|u|p−2u+b(x)|u|q−2u,u∈Hs(RN),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \textstyle\begin{cases} (-\Delta )^{s}u+u=a(x) \vert u \vert ^{p-2}u+b(x) \vert u \vert ^{q-2}u, \\ u\in H^{s}(\mathbb{R}^{N}), \end{cases} $$\end{document} where (−Δ)s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(-\Delta )^{s}$\end{document} denotes the fractional Laplacian of order s∈(0,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$s\in (0,1)$\end{document}, N>2s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$N>2s$\end{document}, 2<p<q<2s∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$2< p< q<2^{*}_{s}$\end{document}, and 2s∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$2^{*}_{s}$\end{document} is the fractional critical Sobolev exponent. The weight potentials a or b is a sign-changing function and satisfies some valid assumptions. We obtain the existence of infinitely many solutions to the problem by the Nehari manifold.


Introduction and the main results
In this paper, we study the fractional Schrödinger equation ⎧ ⎨ ⎩ (-) s u + u = a(x)|u| p-2 u + b(x)|u| q-2 u, u ∈ H s (R N ), (1.1) where the fractional Laplacian (-) s is defined by P.V . stands for the Cauchy principal value, C N,s is a normalizing constant, S(R N ) is the Schwartz space of rapidly decaying functions, s ∈ (0, 1), N > 2s, 2 < p < q < 2 * s , and 2 * s is the fractional critical Sobolev exponent.
In the last few years, the time-dependent fractional Schrödinger equation has been studied extensively in the literature. It appears widely in optimization, finance, phase transi-tions, stratified materials, crystal dislocation, flame propagation, conservation laws, materials science, and water waves (see [5]). The standing waves have a wide range of applications in the real world. Musical instruments generally emit sound due to the standing wave generated by the vibration of a string. In the nonlinear fractional Schrödinger equation, the question of the existence and stability of the standing wave solution is an important research topic. It has been applied in many areas of physics, such as constructive quantum field theory, plasma physics, nonlinear optics, and so on. A basic motivation for the study of Eq. (1.1) arises in looking for the standing wave solutions of the type (x, t) = e -iEt/ε u(x) for the following time-dependent fractional Schrödinger equation: 2) and other differential problems driven by Laplace-type operators; see [1, 2, 7, 10-13, 15, 19-21, 25, 28-32]. When s = 1, (1.1) is the classical semilinear elliptic equation in R N with sign-changing weight functions. Equations of this type have been studied extensively in recent years, mainly on bounded domains. Below we briefly describe some of this work. Berestycki et al. [3] studied the existence and nonexistence of positive solutions to the problem Here m and a may be sign-changing functions, 1 < p < 2 * , Bu = u, and Bu = ∂ ν u (respectively, the Dirichlet and Neumann boundary conditions). Brown and Zhang [4] considered a problem similar to (1.3) by splitting the Nehari manifold into three parts corresponding to local minima, local maxima, and points of inflection of the fibering map and then looked for minimizers of the energy functional on the first two parts. De Paiva [8] studied the problem where 0 < p < 1 < q ≤ 2 * -1, a changes sign, and b ≥ 0. He proved that there exists λ * ∈ (0, +∞) such that (1.4) has at least one nonnegative solution for 0 < λ < λ * and there is no such solution for λ > λ * . For an unbounded domain, we can mention Wu [27], who studied the multiplicity of positive solutions for the concave-convex elliptic equation where 1 < p < 2 < q < 2 * , the parameters λ, μ ≥ 0, f λ = λf + + f -, (f ± := max{0, ±f }) is signchanging, and g μ (x) = a(x) + μb(x). Here the author used the idea of Brown and Zhang [4] mentioned above and has split the Nehari manifold into three parts considered separately. See also [14,24], where related (singular) problems with sign-changing weights in R N are studied by similar techniques. To our best knowledge, there is no similar result for nonlocal problem (1.2) with sign-changing weight functions, so in this paper, we will fill this gap.
Our purpose here is to study the fractional Schrödinger equation (1.1). We are interested in the situation where one of the weight functions a and b is sign-changing, and unlike the papers mentioned above, we are mainly concerned with the existence of infinitely many solutions. In what follows, we assume that a and b satisfy some of the following hypotheses: Our main results of this paper is the following: Under the assumptions above, we prove that the Nehari manifold is closed and of class C 2 and that the energy functional corresponding to problem (1.1) is bounded below. When (H 1 ) or (H 2 ) is satisfied, we show that the energy functional satisfies the Palais-Smale condition on the Nehari manifold, and then using some arguments based on the Krasnoselskii genus, we establish the existence of infinitely many solutions for problem (1.1).
This paper is organized as follows. In Sect. 2, we describe the functional setting to study problem (1.1) and prove some preliminary lemmas. In Sect. 3, we complete the proof of Theorem 1.1.

Variational settings and preliminary results
We denote by | · | p the usual norm of the space L p (R 3 ), 1 ≤ p < ∞, by B r (x) the open ball with center at x and radius r, and by C or C i (i = 1, 2, . . . ) positive constants that may change from line to line. By a n a and a n → a we mean the weak and strong convergence, respectively, as n → ∞.

The functional space setting
Firstly, fractional Sobolev spaces are convenient for our problem, so we will give some sketches on them; a complete introduction can be found in [9]. We recall that for s ∈ (0, 1), the fractional Sobolev space H s (R 3 ) = W s,2 (R 3 ) is defined as follows: where F denotes the Fourier transform. We also define the homogeneous fractional Sobolev space D s,2 (R 3 ) as the completion of C ∞ 0 (R 3 ) with respect to the norm The fractional Laplacian (-) s u of a smooth function u : that is, for functions φ in the Schwartz class. Also, (-) s u can be equivalently represented as (see [9]) Also, by the Plancherel formula in Fourier analysis we have As a consequence, the norms on H s (R 3 ) for every u ∈ H s (R 3 ), where 2 * s = 6 3-2s is the fractional critical exponent. Moreover, the embedding X → L r (R 3 ) is continuous for any r ∈ [2, 2 * s ] and is locally compact whenever r ∈ [2, 2 * s ).

Lemma 2.2 ([22]) If {u n } is bounded in H s (R 3 ) and for some R
then u n → 0 in L r (R 3 ) for all 2 < r < 2 * s .

Properties of the Nehari manifold
Set X := H s (R N ), and let A, B : X → R be defined by It is clear that problem (1.1) is the Euler-Lagrange equation for the functional J : X → R defined by By this remark the action functional J ∈ C 2 (X, R), and its critical points are weak solutions of problem (1.1). Moreover, for all u, v ∈ X, we have Hence in the following, we consider critical points of I using the variational method.
We first introduce the Nehari manifold associated with the functional J: Now we define the fibering map corresponding to u ∈ X\{0} by setting α u (t) = J(tu), t > 0. Then Moreover, tu ∈ N if and only if α u (t) = 0. It remains to show that the equation α u (t) = 0 has at most one solution. Since α u (t u ) > 0, it will then follow that J(u) > 0 for all u ∈ N . Let α u (t 1 ) = α u (t 2 ) = 0 for t 1 , t 2 > 0. We have

Lemma 2.3 Suppose that either (H 1 ) or (H 2 ) is satisfied.
(2.4) From (2.4) we see that either t 1 = t 2 or B(u) < 0. However, in the second case, α t is never 0 under our hypotheses. where r = r r-1 , s = s s-1 , and C 1 , C 2 are positive constants. Since 2 < p < q < 2 * s , inequality (2.5) implies that the Nehari manifold is bounded away from 0. Now we show that it is a closed C 2 -manifold. Define ψ : X → R as By Remark (2.1), ψ ∈ C 2 (X, R)M and by the definition of ψ, N = ψ -1 (0)\{0}. Since N is bounded away from 0, N is closed. If we show that every point of N is regular for ψ, THEN the proof will be complete. Let u ∈ N = ψ -1 (0)\{0}. Then and It follows from (2.6) and (2.7) that Proof Consider u ∈ N and assume that a, b ∈ L ∞ (R N ). Then by the Sobolev inequality we have where C 1 , C 2 are positive constants. Using (2.9), we deduce that N is bounded away from 0. The rest of the proof is the same as that of Lemma 2.4.
A functional I ∈ C 1 (X, R) is said to satisfy the Palais-Smale condition at the level c ∈ R (the (PS) c -condition for short) if every sequence {u n } ⊂ X such that I(u n ) → c and I (u n ) → 0 (2.10) admits a convergent subsequence. A sequence satisfying (2.10) is called a (PS) c -sequence. Proof It is clear that if u = 0 is a critical point of J, then u ∈ N . Let u ∈ N . By (2.8) we know that ψ (u), u < 0, and therefore X = T u N ⊕ Ru. Since J (u)| Ru ≡ 0 by the definition of N , the conclusion follows.

Proof of Theorem 1.1
To prove Theorem 1.1, we need to recall the definition of the Krasnoselskii genus and an abstract multiplicity result in [23]. A set F ⊂ X is said to be symmetric if F = -F. Let := {F ⊂ X : F is closed and symmetric}.
For F = ∅ and F ∈ , the Krasnoselskii genus of F is the least integer n such that there exists an odd function f ∈ C (F, R N \{0}). The genus of F is denoted by γ (F). Set γ (∅) := 0 and γ (F) := ∞ if for all n, there exists no f with the above property. We first need some auxiliary results. Proof We only prove the lemma for A and A ; for B, B , the proof is similar. Let u n ∈ X and u n u. Using the Rellich-Kondrachov theorem, up to a subsequence, we have u n u in X, As in the preceding proof, let w n := |u n | p-2 u n -|u| p-2 u. We see from the Krasnoselskii theorem (see [26]) and (3.6) that

Lemma 3.1 Suppose that assumption (H 1 ) holds. Then the functionals A and B defined by
It follows from (H 2 ) that for any ε > 0, there exists R > 0 such that a + (x) < ε whenever |x| ≥ R. (3.8) Using the Hölder and the Sobolev inequalities and (3.7), we obtain as n → ∞. By inequality (3.3), {w n } is bounded in L p p-1 (R N ), hence inequality (3.8), the Hölder and the Sobolev inequalities imply that there exists a constant C 2 > 0, independent of ε > 0, such that (3.10) Using (3.9) and (3.10), we deduce that we see as in the proof of Lemma 3.2 that A + is weakly continuous. If (H 3 ) holds, then B(u) ≥ 0, and we have, using (3.11) and the boundedness of J(u n ), for all n large enough.
If (H 4 ) holds, then A(u n ) ≤ 0, and using (3.11) again, we have for all n large enough. Hence in both cases, {u n } is a bounded sequence. So there exists u ∈ X such that passing to a subsequence, u n u. Since J(u n ) → 0, it is easy to see that J(u) = 0, and it follows that J (u n ) -J (u), u nu → 0 or, equivalently, (3.14) If (H 1 ) holds, then by Lemma 3.1, A (u n ) → A (u) and B (u n ) → B (u). Thus from (3.14) we get u n → u in X.
Since A + (u n ) → A + (u) and B + (u n ) → B + (u), we see that u n → u in X again.
Proof of Theorem 1.1 By Lemmas 2.3-2.5 and 3.3, N is a closed symmetric C 2 -manifold, J(u) > 0 for all u ∈ N , and J satisfies the (PS) c -condition on N for all c ∈ R. If we show that for any j ≥ 1, there exists a symmetric compact set F j ⊂ N such that γ (F j ) ≥ j, then the conclusion will follow from Lemma 2.6 and Theorem 3.1.
Let j ≥ 1, let X j be a subspace spanned by j linearly independent functions v k ∈ C ∞ 0 (R N ) with supp v k ⊂ {x ∈ R N : b(x) > 0}, and let S j-1 := X j ∩ x ∈ X : u = 1 .
Then we have B(u) > 0 for all u ∈ S j-1 , and it follows from Lemma 2.3 that the equation α u (t) = 0 has exactly one solution t u ∈ (0, ∞). Hence the mapping ϕ : S j-1 → N given by ϕ(u) := t u u is well defined, and it is obviously odd. If it is continuous, then F j := ϕ(S j-1 ) is homeomorphic to S j-1 , and it follows from the properties of genus that γ (F j ) = γ (S j-1 ) = j.
We need to show that u → t u is continuous. An easy computation shows that if the (necessary and sufficient) conditions for the existence of t u given in Lemma 2.3 are satisfied, then α u (t) < 0 for t = t u . Hence the continuity of t u follows from the implicit function theorem.