Standing waves for supercritical nonlinear Schrodinger equations
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Date
2007
Journal Title
Journal ISSN
Volume Title
Publisher
ACADEMIC PRESS INC ELSEVIER SCIENCE
Abstract
Let V (x) be a non-negative, bounded potential in R-N, N >= 3 and p supercritical, p > N+2/N-2. We look for positive solutions of the standing-wave nonlinear Schrodinger equation Delta u - V(x)u + u(P) = 0 in R-N, with u(x) -> 0 as vertical bar x vertical bar -> +infinity. We prove that if V(x) = 0(vertical bar x vertical bar(-2)) as vertical bar x vertical bar -> +infinity, then for N >= 4 and p > N+1/N-3 this problem admits a continuum of solutions. If in addition we have, for instance, V (x) = 0 (vertical bar x vertical bar-mu) with mu > N, then this result still holds provided that N >= 3 and p > N+2/N-2. Other conditions for solvability, involving behavior of V at infinity, are also provided. (C) 2007 Elsevier Inc. All rights reserved.
Description
Keywords
BOUND-STATES, SEMICLASSICAL STATES, ELLIPTIC PROBLEMS, GROUND-STATES, R-N, EXISTENCE, POTENTIALS, STABILITY, V(INFINITY)=0, INFINITY