Solutions of perturbed Schrödinger equations with critical nonlinearity

Yanheng Ding, Fanghua Lin

Research output: Contribution to journalArticle

Abstract

We consider the perturbed Schrödinger equation - ε 2 Δu + V(x)u = P(x)|u|{p - 2} u + k(x)|u|{2 - 2} u & text for, x ∈ ℝ N {u(x) 0} & text{as}, {|x| → ∞ where N ≥ 3, 2*N) Px (N-2) is the Sobolev critical exponent, p ∈ (2, 2*), P(x) and K(x) are bounded positive functions. Under proper conditions on V we show that it has at least one positive solution provided that ≤ Em ; for any m ∈ N, it has m pairs of solutions if ≤ Em; and suppose there exists an orthogonal involution τ ℝ N to ℕ N such that V(x), P(x) and K(x) are τ -invariant, then it has at least one pair of solutions which change sign exactly once provided that E ≤ E, where E and Em are sufficiently small positive numbers. Moreover, these solutions u to 0 in H1N as 0.

Original languageEnglish (US)
Pages (from-to)231-249
Number of pages19
JournalCalculus of Variations and Partial Differential Equations
Volume30
Issue number2
DOIs
StatePublished - Oct 1 2007

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ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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