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Weak linking theorems and Schrödinger equations with critical Sobolev exponent

Martin SchechterWenming Zou — 2003

ESAIM: Control, Optimisation and Calculus of Variations

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.

Weak Linking Theorems and Schrödinger Equations with Critical Sobolev Exponent

Martin SchechterWenming Zou — 2010

ESAIM: Control, Optimisation and Calculus of Variations

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 ≥ 4; are periodic in x for 1 ≤ ≤ and 0 is in a gap of the spectrum of -Δ + ; . If 0 < g ( x , u ) u c | u | 2 * for an appropriate constant , we show that this equation has a nontrivial solution.

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