Uniqueness and asymptotical behavior of solutions to a Choquard equation with singularity

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Abstract

In this paper, we consider a nonautonomous Choquard equation with singularity Δu+V(x)u+λ(Iα|u|p)|u|p2u=f(x)uγ,xR3,u>0,xR3,where Iα is the Riesz potential of order α(0,3) and 1+α3p<3+α, 0<γ<1. Under certain assumptions on V and f, we show the existence and uniqueness of positive solution for λ>0 by using variational method. We also study the asymptotic behavior of solutions as λ0.

Introduction

In this paper, we are interested in the nonautonomous Choquard equation Δu+V(x)u+λ(Iα|u|p)|u|p2u=f(x)uγ,xR3,u>0,xR3,where 1+α3p<3+α, 0<γ<1, λ>0 and Iα with α(0,3) is the Riesz potential defined by Iα=Γ(3α2)Γ(α2)2απ32|x|3α,xR3{0}. Here, Γ denotes the Gamma function. Throughout the paper, we suppose V and f satisfy:

(V1) VC(R3) satisfies infxR3V(x)>V0>0, where V0 is a constant.

(V2) meas{xR3:<V(x)h}<+ for all hR.

(f1) fL6p6p(1γ)(3+α)(R3) is a nonnegative function.

Recently, many scholars pay attention to the following more general Choquard equation Δu+V(x)u+λ(Iα|u|p)|u|p2u=h(x,u),xRN,where NN and α(0,N). Problem (1.1) with N=3, V(x)=1, λ=1, p=α=2 and h(x,u)=0 was proposed by Pekar [1] to describe the quantum theory of a polaron at rest and as an approximation to Hartree–Fock theory of one component plasma by Choquard (see [2]). Many scholars investigated problem (1.1) with λ=1: when V(x)=1, 2pN+αN2 and h(x,u)=0, Ruiz and Van Schaftingen [3] proved that least energy nodal solutions for problem (1.1) have an odd symmetry with respect to a hyperplane when α0 or αN. Based on [3], Seok [4] further studied limit profiles of ground states as α0 or αN. When N3, V(x)=1 and p>1, Seok [5] considered problem (1.1) with a critical local term and showed the existence of radially symmetric nontrivial solution and concentration results as α0. When N3 and V(x)=1+μg(x) is a potential well, Lü [6] obtained the existence of ground state solutions and concentration results as μ+ for problem (1.1) with subcritical exponents and h(x,u)=0. Li et al. [7] extended the results of Lü [6] to critical case and obtained the existence of ground state solutions and concentration results as α0. As for λ=1, Mercuri et al. [8] obtained the existence and regularity of ground state solutions and radial solutions for problem (1.1) with V(x)=0, p>1 and h(x,u)=|u|q2u,q>1. When N3, p1+αN,NN2, Wu [9] investigated the existence, multiplicity and asymptotic behavior of positive solution for problem (1.1) with V(x) and h(x,u) satisfying some suitable conditions. For more related topics, we refer to the survey paper [10] and the references therein.

When p=α=2, problem (Pλ) reduces to a Schrödinger–Poisson system with singularity (0<γ<1). On the bounded domain ΩR3, when V(x)=0 and f(x)=μ is a positive parameter, Zhang [11] investigated problem (Pλ) and obtained the existence and uniqueness results for λ=1 and multiplicity of solutions for λ=1. When λ=1 and f(x)=μ|x|β, Lei and Liao [12] generalized a part of results in Zhang [11] to the critical problem and obtained two positive solutions for problem (Pλ) with critical exponent. Mukherjee and Sreenadh [13] investigated a nonlinear Choquard equation with upper critical exponent and singularity. On the unbounded domain RN, Sun and Li [14] established some existence, symmetry and uniqueness results for positive ground state solutions to a singular semilinear elliptic equation. Lei, Suo and Chu [15] studied a Schrödinger–Newton system with singularity and critical growth terms.

To the best of our knowledge, so far few results are known to Choquard equations with singularity on unbounded domains. Moreover, although many works considered concentration results for Choquard equations [3], [4], [5], [6], [7], [9], there are few papers investigated the relationships between Choquard equations involving and without convolution term. By the variational method, we obtain the following results.

Theorem 1.1

Assume that 0<γ<1,0<α<3,1+α3p<3+α and(V1),(V2),(f1) hold, then problem(Pλ) admits a unique positive solution for all λ>0.

Theorem 1.2

Assume that 0<γ<1,0<α<3,1+α3p<3+α and(V1),(V2),(f1) hold, for any sequence {λn}>0 with λn0 asn,uλn are the corresponding solutions of problem (Pλ) obtained in Theorem 1.1 with λ=λn, then uλnv0 inE, wherev0 is the unique positive solution to problem Δu+V(x)u=f(x)uγ,xR3.

Section snippets

Preliminaries and proofs of the main results

Throughout the paper, we use the following notations. Ls(R3) is a Lebesgue space with the norm us=(R3|u|sdx)1s. u+=max{u,0} for any function u. Set dα=Γ(3α2)2απ32Γ(3+α2)(Γ(32)Γ(3))α3 and pα=6p3+α. Define the function space E={uL2(R3):uL2(R3),uE<+}, where uE=(R3(|u|2+V(x)u2)dx)12, then E is a Hilbert space with the inner product (u,v)E=R3(uv+V(x)uv)dx. Obviously, for s[2,6], the embedding ELs(R3) is continuous and so there exist constants Ss>0 such that usSsuE,uE.By

Acknowledgments

The authors appreciate unknown referees for many valuable comments.

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This work was supported by the NNSF of China (Nos. 11871152, 11671085) and NSF of Fujian Province (No. 2019J01089).

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