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On a semilinear Schrödinger equation with critical Sobolev exponent. (English) Zbl 0984.35150

Summary: We consider the semilinear Schrödinger equation \[ -\Delta u+V(x)u = K(x)|u|^{2^{*}-2}u+g(x,u), \quad u\in W^{1,2}(\mathbb{R}^{N}), \] where \(N\geq 4\), \(V,K,g\) are periodic in \(x_{j}\) for \(1\leq j\leq N\), \(g\) is of subcritical growth, and 0 is in a gap of the spectrum of \(-\Delta+V\). We show that under suitable hypotheses this equation has a solution \(u\neq 0\). In particular, such a solution exists if \(K\equiv 1\) and \(g\equiv 0\).

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35B33 Critical exponents in context of PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations
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