Positive solutions for nonlinear Schrödinger equations with deepening potential well

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.