Intermediate reduction method and infinitely many positive solutions of nonlinear Schrödinger equations with non-symmetric potentials
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2015Metadata
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Pino Manresa, Manuel del
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Intermediate reduction method and infinitely many positive solutions of nonlinear Schrödinger equations with non-symmetric potentials
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Abstract
We consider the standing-wave problem for a nonlinear Schrödinger equation,
corresponding to the semilinear elliptic problem
− u + V(x)u = |u|p−1u, u ∈ H1(R2),
where V(x) is a uniformly positive potential and p > 1. Assuming that
V(x) = V∞ + a
|x|m
+ O
1
|x|m+σ
, as |x| → +∞,
for instance if p > 2, m > 2 andσ > 1 we prove the existence of infinitely many positive
solutions. If V(x) is radially symmetric, this result was proved in [43]. The proof without
symmetries is much more difficult, and for that we develop a new intermediate Lyapunov–
Schmidt reductionmethod,which is a compromise between the finite and infinite dimensional
versions of it.
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Artículo de publicación ISI
Patrocinador
Fondecyt Grant 110181, Fondo Basal CMM and Fondecyt Grant 3130543
Identifier
URI: https://repositorio.uchile.cl/handle/2250/132671
DOI: DOI: 10.1007/s00526-014-0756-3
ISSN: 0944-2669
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Calc. Var. (2015) 53:473–523
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