Journal of the European Mathematical Society


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Volume 11, Issue 3, 2009, pp. 545–573
DOI: 10.4171/JEMS/160

Published online: 2009-06-30

Positive solutions for nonlinear Schrödinger equations with deepening potential well

Zhengping Wang[1] and Huan-Song Zhou[2]

(1) Chinese Academy of Sciences, Wuhan, China
(2) Chinese Academy of Sciences, Wuhan, China

Consider the following nonlinear Schrödinger equation:

(*) -Δu + (1 + λg(x))u = f(u) and u> 0 in ℝN, uH1.(ℝN), N ≥ 3,

where λ ≥ 0 is a parameter, gL(ℝN) vanishes on a bounded domain in ℝN, 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.

Keywords: Nonlinear Schrödinger equation, mountain pass theorem, potential well, asymptotically linear

Wang Zhengping, Zhou Huan-Song: Positive solutions for nonlinear Schrödinger equations with deepening potential well. J. Eur. Math. Soc. 11 (2009), 545-573. doi: 10.4171/JEMS/160