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Weak linking theorems and Schrödinger equations with critical Sobolev exponent. (English) Zbl 1173.35482
Summary: 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 \[ -\Delta u+V(x)u=K(x)| u|^{2^\ast-2}u+g(x, u), \qquad u\in W^{1,2}(\mathbb R^N), \] where \(N\geq 4\), \(2^\ast:=2N/(N-2)\) is the critical Sobolev exponent, \(V, K, g \) are periodic in \(x_j\) for \(1\leq j\leq N\) and 0 is in a gap of the spectrum of \(-\Delta+V\), \(K>0\). If \(0<g(x, u)u\leq c\,| u|^{2^\ast}\) for an appropriate constant \(c\), we show that this equation has a nontrivial solution.

MSC:
35J60 Nonlinear elliptic equations
35B33 Critical exponents in context of PDEs
35Q55 NLS equations (nonlinear Schrödinger equations)
47J30 Variational methods involving nonlinear operators
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
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