Positive solutions for nonlinear Schrödinger equations with deepening potential well

Abstract

Consider the following nonlinear Schrödinger equation:(*) -Δ+(1+λg(x))u and u > 0 in ℝN, u ε H 1( ℝN), N ≥ 3, where λ ≥ 0 is a parameter, g ε L∞ ( ℝN) vanishes on a bounded domain in ℝN, and the function f is such that lim δ→0f(s)/s = 0 and 1 ≤ α + 1=lim δ→∞ 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. © European Mathematical Society 2009.