Existence of multi-bump solutions for a semilinear Schrödinger–Poisson system

Abstract

In this paper, we study the existence of multi-bump solutions for the semilinear Schrödinger-Poisson systemFor any positive integer K, we prove that there exists ε(K) > 0 such that, for 0 < ε < ε(K), the system has a K-bump solution. Then the equation has more and more multi-bump solutions as ε → 0.

Bound state solutions of Schrödinger-Poisson system with critical exponent

“…And the results for (1) with positive non-radial potential and different assumptions on nonlinearities can be found in [3], [26] and [31]. Other existence or multiplicity results can be found in [15,16,20,23,29] with variant assumptions on the potential and nonlinearities. The semiclassical solutions of the system have also been discussed and we refer the readers to [11,12,13,14,19] for details.…”

Existence of ground state solutions for asymptotically linear Schrödinger–Poisson systems

“…More precisely, the first equation of system (SP) looks like 2 u C V.x/u C K.x/ u D a.x/juj p 1 u, and the solutions exhibit concentration phenomena as the parameter goes to zero. See, for example, [26][27][28][29][30][31][32][33][34][35] and the references therein on this subject.…”

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

“…Recently, in [20], the authors proved that (SP ) 1 has positive radially symmetric solutions for λ < 0 and g(x, u) ≤ 0. Also, the paper [37] proved the existence of multi-bump solutions for (1.1) with subcritical growth nonlinearity.…”

“…If ε = 1, the papers [1,2,10,22,37] proved the existence and multiplicity results for (1.1) when g(x, u) = c(x)h(u), a and c are positive constants or radially symmetric functions. In [19], the author studied the existence and orbital stability of standing wave solutions for system (1.1), where h is given by…”