On nontrivial solutions of nonlinear Schrödinger equations with sign-changing potential

In this paper, we consider the superlinear Schrödinger equation with bounded potential well. The potential here is allowed to be sign-changing. Without assuming the Ambrosetti–Rabinowitz-type condition, we prove the existence of a nontrivial solution and multiplicity results.

Existence and concentration of positive solutions for Schrödinger-Poisson systems with steep well potential

This paper is devoted to study the following Schrödinger-Poisson system where λ is a positive parameter, a ∈ C(R3,R+) has a bounded potential well Ω = a−1(0), b ∈ C(R3, R) is allowed to be sign-changing, K ∈ C(R3, R+) and f ∈ C(R, R). Without the monotonicity of f(t)=/|t|3 and the Ambrosetti-Rabinowitz type condition, we establish the existence and exponential decay of positive multi-bump solutions of the above system for , and obtain the concentration of a family of solutions as λ →+∞, where is determined by terms of a, b, K and f. Our results improve and generalize the ones obtained by C. O. Alves, M. B. Yang [3] and X. Zhang, S. W. Ma [38].