# Positive solutions for nonlinear Schrödinger equations with deepening potential well

### Zhengping Wang

Chinese Academy of Sciences, Wuhan, China### Huan-Song Zhou

Chinese Academy of Sciences, Wuhan, China

## Abstract

Consider the following nonlinear Schrödinger equation:

(*) -Δ_u_ + (1 + λ_g_(x))*u* = *f*(*u*) and *u*> 0 in ℝ_N_, *u* ∈ *H_1.(ℝ_N*), *N* ≥ 3,

where λ ≥ 0 is a parameter, *g* ∈ *L_∞(ℝ_N*) vanishes on a bounded domain in ℝ_N_, and the function *f* is such that

lim(_s_→0) *f*(*s*)/*s* = 0 and 1 ≤ α + 1 = lim(_s_→∞) *f*(*s*)/*s* < ∞.

We are interested in whether problem (*) has a solution for any given α, λ ≥ 0. It is shown in [14] and [31] that problem (*) has solutions for some α and λ. In this paper, we establish the existence of solution of (*) for all α and λ by using a variant of the Mountain Pass Theorem. Based on these results, we give a diagram in the (λ,α)-plane showing how the solvability of problem (*) depends on the parameters α and λ.

## Cite this article

Zhengping Wang, Huan-Song Zhou, Positive solutions for nonlinear Schrödinger equations with deepening potential well. J. Eur. Math. Soc. 11 (2009), no. 3, pp. 545–573

DOI 10.4171/JEMS/160