Weak linking theorems and Schrödinger equations with critical Sobolev exponent

Martin Schechter; Wenming Zou

ESAIM: Control, Optimisation and Calculus of Variations (2003)

  • Volume: 9, page 601-619
  • ISSN: 1292-8119

Abstract

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In this paper we establish a variant and generalized weak linking theorem, which contains more delicate result and insures the existence of bounded Palais–Smale sequences of a strongly indefinite functional. The abstract result will be used to study the semilinear Schrödinger equation - Δ u + V ( x ) u = K ( x ) | u | 2 * - 2 u + g ( x , u ) , u ∈ W 1 , 2 ( 𝐑 N ) , where N ≥ 4 ; V , K , g are periodic in x j for 1 ≤ j ≤ N and 0 is in a gap of the spectrum of - Δ + V ; K > 0 . If 0 < g ( x , u ) u ≤ c | u | 2 * for an appropriate constant c , we show that this equation has a nontrivial solution.

How to cite

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Schechter, Martin, and Zou, Wenming. "Weak linking theorems and Schrödinger equations with critical Sobolev exponent." ESAIM: Control, Optimisation and Calculus of Variations 9 (2003): 601-619. <http://eudml.org/doc/244774>.

@article{Schechter2003,
abstract = {In this paper we establish a variant and generalized weak linking theorem, which contains more delicate result and insures the existence of bounded Palais–Smale sequences of a strongly indefinite functional. The abstract result will be used to study the semilinear Schrödinger equation $-\Delta u+V(x)u=K(x)|u|^\{2^\ast -2\}u+g(x, u), u\in W^\{1,2\}(\{\bf R\}^N)$, where $N\ge 4; V, K, g $ are periodic in $ x_j $ for $1\le j\le N$ and 0 is in a gap of the spectrum of $-\Delta +V$; $K&gt;0$. If $0&lt;g(x, u)u\le c|u|^\{2^\ast \}$ for an appropriate constant $c$, we show that this equation has a nontrivial solution.},
author = {Schechter, Martin, Zou, Wenming},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {linking; Schrödinger equations; critical Sobolev exponent},
language = {eng},
pages = {601-619},
publisher = {EDP-Sciences},
title = {Weak linking theorems and Schrödinger equations with critical Sobolev exponent},
url = {http://eudml.org/doc/244774},
volume = {9},
year = {2003},
}

TY - JOUR
AU - Schechter, Martin
AU - Zou, Wenming
TI - Weak linking theorems and Schrödinger equations with critical Sobolev exponent
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2003
PB - EDP-Sciences
VL - 9
SP - 601
EP - 619
AB - In this paper we establish a variant and generalized weak linking theorem, which contains more delicate result and insures the existence of bounded Palais–Smale sequences of a strongly indefinite functional. The abstract result will be used to study the semilinear Schrödinger equation $-\Delta u+V(x)u=K(x)|u|^{2^\ast -2}u+g(x, u), u\in W^{1,2}({\bf R}^N)$, where $N\ge 4; V, K, g $ are periodic in $ x_j $ for $1\le j\le N$ and 0 is in a gap of the spectrum of $-\Delta +V$; $K&gt;0$. If $0&lt;g(x, u)u\le c|u|^{2^\ast }$ for an appropriate constant $c$, we show that this equation has a nontrivial solution.
LA - eng
KW - linking; Schrödinger equations; critical Sobolev exponent
UR - http://eudml.org/doc/244774
ER -

References

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