## 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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### References:

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