Uniqueness and asymptotical behavior of solutions to a Choquard equation with singularity☆
Introduction
In this paper, we are interested in the nonautonomous Choquard equation where , , and with is the Riesz potential defined by . Here, denotes the Gamma function. Throughout the paper, we suppose and satisfy:
satisfies , where is a constant.
for all .
is a nonnegative function.
Recently, many scholars pay attention to the following more general Choquard equation where and . Problem (1.1) with , , , and 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 : when , and , 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 or . Based on [3], Seok [4] further studied limit profiles of ground states as or . When , and , Seok [5] considered problem (1.1) with a critical local term and showed the existence of radially symmetric nontrivial solution and concentration results as . When and is a potential well, Lü [6] obtained the existence of ground state solutions and concentration results as for problem (1.1) with subcritical exponents and . 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 . As for , Mercuri et al. [8] obtained the existence and regularity of ground state solutions and radial solutions for problem (1.1) with , and . When , , Wu [9] investigated the existence, multiplicity and asymptotic behavior of positive solution for problem (1.1) with and satisfying some suitable conditions. For more related topics, we refer to the survey paper [10] and the references therein.
When , problem () reduces to a Schrödinger–Poisson system with singularity . On the bounded domain , when and is a positive parameter, Zhang [11] investigated problem () and obtained the existence and uniqueness results for and multiplicity of solutions for . When and , Lei and Liao [12] generalized a part of results in Zhang [11] to the critical problem and obtained two positive solutions for problem () with critical exponent. Mukherjee and Sreenadh [13] investigated a nonlinear Choquard equation with upper critical exponent and singularity. On the unbounded domain , 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 ,, and,, hold, then problem() admits a unique positive solution for all .
Theorem 1.2 Assume that ,, and,, hold, for any sequence with as, are the corresponding solutions of problem () obtained in Theorem 1.1 with , then in, where is the unique positive solution to problem
Section snippets
Preliminaries and proofs of the main results
Throughout the paper, we use the following notations. is a Lebesgue space with the norm . for any function . Set and . Define the function space , where , then is a Hilbert space with the inner product . Obviously, for , the embedding is continuous and so there exist constants such that 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).