JournalsjemsVol. 11, No. 3pp. 545–573

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
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Abstract

Consider the following nonlinear Schrödinger equation:

(*) -Δ_u_ + (1 + λ_g_(x))u = f(u) and u> 0 in ℝ_N_, uH_1.(ℝ_N), N ≥ 3,

where λ ≥ 0 is a parameter, gL_∞(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