Weak linking theorems and Schrödinger equations with critical Sobolev exponent

Schechter, Martin; Zou, Wenming

doi:10.1051/cocv:2003029

Schechter, Martin ; Zou, Wenming

ESAIM: Control, Optimisation and Calculus of Variations, Tome 9 (2003), pp. 601-619.

Résumé

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 $- Δ u + V (x) u = {K (x) | u |}^{2^{*} - 2} u + g (x, u), u \in W^{1, 2} (𝐑^{N})$ , where $N \geq 4; V, K, g$ are periodic in $x_{j}$ for $1 \leq j \leq N$ and 0 is in a gap of the spectrum of $- Δ + V$ ; $K > 0$ . If $0 < g (x, u) u \leq {c | u |}^{2^{*}}$ for an appropriate constant $c$ , we show that this equation has a nontrivial solution.

DOI : 10.1051/cocv:2003029

Classification : 35B33, 35J65, 35Q55
Mots clés : linking, Schrödinger equations, critical Sobolev exponent

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     author = {Schechter, Martin and Zou, Wenming},
     title = {Weak linking theorems and {Schr\"odinger} equations with critical {Sobolev} exponent},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {601--619},
     publisher = {EDP-Sciences},
     volume = {9},
     year = {2003},
     doi = {10.1051/cocv:2003029},
     mrnumber = {1998717},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv:2003029/}
}

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Schechter, Martin; Zou, Wenming. Weak linking theorems and Schrödinger equations with critical Sobolev exponent. ESAIM: Control, Optimisation and Calculus of Variations, Tome 9 (2003), pp. 601-619. doi : 10.1051/cocv:2003029. http://www.numdam.org/articles/10.1051/cocv:2003029/

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