Existence of multiple positive solutions for Schrödinger–Poisson systems with critical growth

Abstract

In this paper, we are concerned with the existence, multiplicity and concentration of positive ground state solutions for the semilinear Schrödinger-Poisson systemwhere ε > 0 is a small parameter, f is a continuous, superlinear and subcritical nonlinearity, and λ = 0 is a real parameter. Suppose that a(x) has at least one global minimum and b(x) has at least one global maximum. We prove that there are two families of positive solutions for sufficiently small ε > 0, of which one is concentrating on the set of minimal points of a and the other on the sets of maximal points of b. Moreover, we obtain some sufficient conditions for the nonexistence of positive ground state solutions.Mathematics Subject Classification. 35J61 · 35J50 · 35Q55 · 49J40.

Existence and concentration of positive ground state solutions for nonlinear fractional Schrödinger‐Poisson system with critical growth

“…When V ( x ), K ( x ), and Q ( x ) are all bounded and positive functions, Zhao and Zhao considered the critical Schrödinger‐Poisson system with ϵ = 1 and , and the existence of positive ground solutions was obtained. Wang et al proved the existence and concentration of positive solutions for system with , where f is some superlinear‐4 growth nonlinearity. In Liu and Guo, the Kirchhoff‐type problem with competing potential was considered and the existence and concentration behavior of positive solution were established.…”

“…For this reason, (1.2) is referred to as a nonlinear Schrödinger-Poisson system. In recent years, there has been increasing attention to systems like (1.2) on the existence of positive solutions, ground state solutions, multiple solutions, and semiclassical states; see, for example, previous studies [4][5][6][7][8][9][10][11][12][13] and the references therein.…”

Standing waves for Schrödinger-Poisson system with general nonlinearity

“…Indeed, when the parameter ε is small, the limit problem for the latter is the classical nonlinear Schrödinger equation while the former is still the Schrödinger-Poisson system. Recently, problem (1.1) was also considered in [15,16,17,39,40,43]. More specifically, under the subcritical and Ambrosetti-Rabinowitz type growth condition, He [16] studied the existence of positive solutions by using minimax theorems, and proved that these solutions concentrate around the global minimum of the potential V in the semiclassical limit.…”

“…And moreover, some new concentration phenomenons of semi-classical solutions on the minimum points of a(x) and the maximum points of b(x) are also investigated. For the critical case, we refer readers to [40] for details. It is worth pointing out that the global condition (1.3) used in [15,16,39,40] plays a crucial role in proving the existence of positive solutions.…”

“…For the critical case, we refer readers to [40] for details. It is worth pointing out that the global condition (1.3) used in [15,16,39,40] plays a crucial role in proving the existence of positive solutions. Indeed, the key point is that the property of the potential V can help us to restore the necessary compactness by comparing energy levels of original problem and limit problem.…”