Abstract
In this paper, we study the following nonlinear Schrödinger-Poisson system \begin{document}$\left\{\begin{array}{ll} -\Delta u+u+\epsilon K(x)\Phi(x)u = f(u),& x\in \mathbb{R}^{3} , \\ -\Delta \Phi = K(x)u^{2},\,\,& x\in \mathbb{R}^{3}, \\\end{array}\right.$ \end{document} where $K(x)$ is a positive and continuous potential and $f(u)$ is a nonlinearity satisfying some decay condition and some non-degeneracy condition, respectively. Under some suitable conditions, which are given in section 1, we prove that there exists some $\epsilon_{0}>0$ such that for $0<\epsilon<\epsilon_{0}$, the above problem has infinitely many positive solutions by applying localized energy method. Our main result can be viewed as an extension to a recent result Theorem 1.1 of Ao and Wei in [3] and a result of Li, Peng and Wang in [26].
Highlights
Introduction and the main resultIn this paper, we are mainly motivated by the following problem∂ψ i = − ∆ψ + (V (x) + E)ψ + K(x)Φ(x)ψ − f (x, ψ),∂t 2m (x, t) ∈ R3 × R+,−∆Φ = K(x)|ψ|2, x ∈ R3, where is the Planck constant, i is the imaginary unit, m is a positive constant, E is a real number, > 0, ψ : R3 × [0, T ] → C
This paper was partially supported by NSFC (No 11671162, No 11601194) and CCNU18CXTD04. ∗ Corresponding author: Chunhua Wang
(1.2) with f (x, u) = Q(x)|u|p−1u, 3 < p < 5. They have proved the existence of positive solutions to (1.2) when Q and K were nonsymmetric and nonnegative functions satisfying lim Q(x) = Q∞ and lim K(x) = 0
Summary
In [5], the existence of ground state solutions for problem (1.2) has been proved in several situations, including the positive constant potential case. In [14], when = 1 and V (x) ≡ 1, Cerami and Vaira studied (1.2) with f (x, u) = Q(x)|u|p−1u, 3 < p < 5 They have proved the existence of positive solutions to (1.2) when Q and K were nonsymmetric and nonnegative functions satisfying lim Q(x) = Q∞ and lim K(x) = 0. F (x, u) = Q(x)|u|p−1u, 1 < p < 5, in [25], Li, Peng and Yan proved that (1.2) has infinitely many non-radial positive solutions under the assumptions that K(x), Q(x) were positive bounded radial functions in R3 satisfying some decaying conditions.
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