Ground state solutions for a class of fractional Schrodinger-Poisson system with critical growth and vanishing potentials

1 Introduction

In this paper, we study the existence of positive and sign-changing least energy solutions for the following fractional Schrödinger-Poisson system

where s, t ∈ (0, 1), 3 < 4s < 3 + 2t, q ∈ (1, 2s∗ /2) are real numbers, 2s∗:=63−2s is the fractional critical Sobolev exponent. (−Δ)^s is the fractional Laplacian operator defined as

where P.V. is a commonly used abbreviation for in the principal value sense, and C_s is a suitable normalization constant.

We notice that, when s = t = 1 and V(x) ≡ 1, Ghergu and Singh [10] considered problem (1.1) with Choquard nonlinearity,

where I_α(x) = |x|^−(3−α), α ∈ (0, 3), is the Ruiz potential. The authors discussed the existence of a ground state solution by using a variational approach. They also used a Pohozaev type identity to derive conditions in terms of p, q, N, α and K for which no solutions exist.

Formally, the system

can be seen as a formal limit of (1.2) as α → 0. The classical nonlinear Schrödinger equation

is used as an approximation to the Hartree-Fock model of a quantum many body-system of electrons under the presence of an external potential V_ext. In such a setting, (1.4) and its stationary counterpart bear the name of Schrödinger-Poisson-Slater, Schrödinger-Poisson-X_α, or Maxwell-Schrödinger-Poisson equation. The convolution term in (1.4) represents the Coulombic repulsion between the electrons. The local term |u|^2p−2 u was introduced by Slater as a local approximation of the exchange potential in the Hartree-Fock model. We refer to [6, 19, 28] for more applied background for (1.4).

When s = t = 1 and q = 2, system (1.1) reduces to the following Schrödinger-Poisson system

which plays a great role in looking for solitary waves of the nonlinear stationary Schrödinger equations interacting the electrostatic field. It also appears in physics such as quantum mechanics models[22], semiconductor theory[5] and so on. The nonlinearity f represents the particles interacting with each other. The term Kϕ u represents the interaction with the electric field. We refer the interesting readers to[2] and the references therein for more details about the mathematical and physical backgrounds. Recently, there are many studies on the existence of solutions to (1.5) by using the methods of nonlinear analysis. In [47], the authors established the existence of a positive solution of (1.5) with a critical nonlinearity K(x)|u|^2*−2 u + μ Q(x)|u|^q−2 u, by using the concentration-compactness principle of P. L. Lions and methods of Brezis and Nirenberg. Later, He and Zou [12] studied the existence and concentration behavior of ground state solutions for (1.5) with a critical nonlinearity and double parameters perturbation, namely

Under some suitable conditions on the nonlinearity f and the potential V, they proved that for ε small, (1.6) has a ground state solution concentrating around global minimum of the potential V in the semi-classical limit. For more results on the multiple nontrivial solutions, infinitely many nontrivial solutions, ground state solutions, positive solutions, semiclassical state solutions and sign-changing solutions for (1.5) under various conditions on V, K and f, we refer to [4, 8, 14, 15, 23, 25, 29, 30, 31, 38, 39, 41, 46] and references therein.

In recent years, there is an increasing interest in the studying for the fractional Schrödinger-Poisson system (1.1) with q = 2, i.e.,

where s, t ∈ (0, 1). For example, in [34], Teng considered (1.7) with f(x, u) = μ|u|^p−1 u + |u|2s∗−2u, 1 < p < 2s∗ − 1. The author used the method of Pohozaev-Nehari manifold and global compactness Lemma to obtain a ground state solutions. In [33], Teng studied the existence of ground states to (1.7) with subcritical growth, i.e., f(x, u) = |u|^p−1 u, by using a monotonicity trick and global compactness principle.

In [21], Murcia and Siciliano considered the following semiclassical problem

and established the multiplicity of solutions for small ε via Ljusternik-Schnirelmann category theory, where g is subcritical at infinity. In [43], the authors obtained the multiplicity of solution for (1.8) when g is critical growth. For more results on the concentration behavior of semiclassical solutions for (1.8), we refer to [18, 42, 44]. For the existence results on ground state solutions, sign-changing solutions for (1.7), we refer to [17, 21, 37, 45] and the references therein.

When q ≠ 2, we only note that in [36], Teng and Agarwal studied the existence and non-existence of ground state states via variational method, for (1.1) with the subcritical Choquard nonlinearity (I_α ∗ |u|^p)|u|^p−2u, that is,

where 3+α3<p<3+α3−2s, 1 < q < p. The assumption on V in [32] was formulated as follows:

1. V(x) ∈ C(ℝ³, ℝ), lim_|x|→∞ V(x) = V_∞ > 0;

2. 0 < V₀ = inf_x∈ℝ³ V(x) ≤ V_∞, ∀ x ∈ ℝ³;

3. V(x) ≥ V_∞, ∀ x ∈ ℝ³, W(x) := V(x) − V_∞ ∈ L32s (ℝ³) and V(x) > V_∞ on a positive measure set.

Under some suitable conditions on p, q and K, the authors proved the existence of a nonnegative ground states to (1.9), by using the hypotheses (V₀), (V₁) and a minimization argument; and the existence of bound states by employing the hypotheses (V₀), (V₂) and a linking theorem.

In light of the above cited works, the main purpose of this paper is to study the existence of (sign-changing) solutions of (1.1) under the simultaneous presence of nonlinearities having critical growth and potentials V, h are permitted to vanishing asymptotically as |x| → ∞, without any symmetry assumptions made on V, h. To the best of our knowledge, there is not any result for the existence of positive and sign-changing least energy solutions for the fractional Schrödinger-Poisson system (1.1) in the existing literature.

In order to state the main results, we introduce the assumptions on the potentials K, V and h. The non-negative function K satisfies following condition:

1. K ∈ L^r(ℝ³) ∩ L^∞(ℝ³) ∖ {0} for some r ∈ [6/(3 + 2t − q(3 − 2s)), ∞).

The functions V, h : ℝ³ → ℝ, are continuous and we say that (V, h) ∈ 𝓗 if the following conditions hold true:

1. V(x), h(x) > 0 for all x ∈ ℝ³ and h(x) ∈ L^∞(ℝ³);

2. If {A_n}_n∈ℕ ⊂ ℝ³ is a sequence of Borel sets such that the Lebesgue measure m(A_n) is less than or equal to R, for all n ∈ ℕ and some R > 0, we have

   limr→+∞∫An∩Brc(0)h(x)dx=0,

   uniformly in n ∈ ℕ, where Brc (0) := ℝ³ ∖ B_r(0).

3. Either h(x)V(x) ∈ L^∞(ℝ³); or

4. there exists p ∈ (2q, 2s∗ ), such that

   h(x)[V(x)](2s∗−p)/(2s∗−2)→0  as  |x|→+∞,

For what concerns the nonlinearity f, we assume that f fulfills the following conditions:

1. lim|t|→0f(t)|t|2q−1 = 0 if (Vh₂) holds; or

2. lim|t|→0f(t)|t|p−1 < + ∞ if (Vh₃) holds, for some p ∈ (2q, 2s∗ );

3. lim|t|→+∞F(t)t2q = +∞, where F(t):=∫0tf(τ)dτ;

4. f(t)|t|2q−1 is strictly increasing function of t ∈ ℝ ∖ {0};

5. there exist μ, ν ∈ (2q, 2s∗ ) and λ > 0, such that F(t) ≥ λ t^μ, ∀ t ≥ 0, and

   lim|t|→+∞f(t)|t|ν−1=0.

The main results of this paper can be stated as follows.

This paper is organized as follows. In Section 2 we present some facts about the involved fractional Sobolev spaces and technical results, which are useful to prove the main results. In Section 3, we show the existence of positive solution for problem (1.1) and complete the proof of Theorem 1.1. In Section 4, we show the existence of sign-changing solutions of problem (1.1), and prove Theorem 1.2.

2 Variational setting and preliminary lemmas

In this section we outline the variational framework for studying problem (1.1) and present some preliminary results, which are useful for the proof of the main results in Sections 3, 4.

We first fix some notations as follows. The homogeneous fractional Sobolev space 𝓓^s,2(ℝ³)(s ∈ (0, 1)) is defined by

which is the completion of C0∞ (ℝ³) under the norm

The embedding 𝓓^s,2(ℝ³) ↪ L2s∗ (ℝ³) is continuous and the best constant 𝓢_s can be defined as

The fractional Sobolev space H^s(ℝ³) is defined by

endowed with the norm

where

We define the work space for (1.1) by

with the norm

We recall the following well-known Hardy-Littlewood-Sobolev inequality as follows:

Now, we are going to reduce system (1.1) to a single equation. To this aim, for any u ∈ H^s(ℝ³), we define the map 𝓛_u : D^t,2(ℝ³) → ℝ as

Using the Hölder’s inequality, (K) and the Sobolev embedding inequality, we get

Here we have used the fact 2 < 6q/(3 + 2t) < 6qr/[r(3 + 2t) − 6] ≤ 2s∗ , and r ∈ [6/(3 + 2t − q(3 − 2s)), ∞). Hence, 𝓛_u is a linear and continuous map on D^t,2(ℝ³). Therefore, by the Lax-Milgram theorem, there exists a unique ϕut ∈ 𝓓^t,2(ℝ³) such that 𝓛_u(v) = 〈 ϕut , v〉 for all v ∈ 𝓓^t,2(ℝ³). In other words, ϕut is the weak solution of

and the representation formula holds

which is called t-Riesz potential, which can be rewritten as

In the sequel, we often omit the constant C_t for convenience. Substituting ϕut in (1.1), we have the following single fractional Schrödinger equation

So, the energy functional I : H → ℝ associated with (2.4),

is well defined on H. It is easy to see that I is of class C¹ and for any critical point of I is a weak solution of (2.4). We shall seek the ground state solution of problem (2.4) on the Nehari manifold

and define the following minimization problem:

We call u is a least energy sign-changing solution to problem (2.4) if u is a solution of problem (2.5) with u^± ≠ 0 and

where u⁺(x) := max{u(x), 0}, u⁻(x) := min{u(x), 0}. Since there exist nonlocal terms (−Δ)^su and ϕut in (2.4), we have

for u^± ≠ 0, a direct computation yields that

So the methods to obtain sign-changing solutions of the local problems and to estimate the energy of the sign-changing solutions seem not suitable for our nonlocal one (2.4). In order to get a sign-changing solution for problem (2.4), we will consider the following minimization problem:

Finally, we introduce the weighted Banach space:

equipped with the norm: