Schrödinger–Poisson system with steep potential well

Abstract

In this paper, we consider the following Schrödinger-Poisson systemwhere λ, μ are positive parameters, p ∈ (1, 5), g(x) ∈ L ∞ (R 3 ) is nonnegative and g(x) ≡ 0 on a bounded domain in R 3 . In this case, μg(x) represents a potential well that steepens as μ getting large. If μ = 0, (P λ ) was well studied in Ruiz (2006) [18]. If μ = 0 and g(x) is not radially symmetric, it is unknown whether (P λ ) has a nontrivial solution for p ∈ (1, 2). By priori estimates and approximation methods we prove that (P λ ) with p ∈ (1, 2) has a ground state if μ large and λ small. In the meantime, we prove also that (P λ ) with p ∈ [3, 5) has a nontrivial solution for any λ > 0 and μ large. Moreover, some behaviors of the solutions of (P λ ) as λ → 0, μ → +∞ and |x| → +∞ are discussed.

Ground state solutions for a class of generalized quasilinear Schrödinger–Poisson systems

“…We refer the reader to [29][30][31][32][33][34] for some related and important results. In view of this, it is also reasonable to consider the generalized quasilinear Schrödinger-Poisson system.…”

Multiplicity and concentration results for fractional Schrödinger‐Poisson systems involving a Bessel operator

“…Taking s = t = 1 into account, then can reduce to be the following system: introduced by Benci‐Fortunato to represent solitary waves for nonlinear Schrödinger‐type equations and look for the existence of standing waves interacting with an unknown electrostatic field. The aforementioned system has been studied extensively by many scholars in the last several decades (see other works and their references therein for some related and important results).…”

Ground states for asymptotically periodic Schrödinger-Poisson systems with critical growth

“…The nonlinearity | | −2 ( > 3) is C 1 smooth. As regards other relevant papers for smooth nonlinearities about , we mention here [7,8,17,28]. Later, the differentiability of the nonlinearity was weaken by Alves et al in [1] and they dealt (SP) with K ( ) = 1 and ( ) = ( ) continuous and discussed the existence of ground states when V is periodic and asymptotically periodic in the meaning that there exists a periodic function V such that lim | |→∞ |V ( ) − V ( )| = 0.…”