| Abstract: |
| In this talk we consider a Schr\"{o}dinger equation of the type
\begin{equation*}
-\Delta u+V(x)u=f(x,u)+\lambda u\quad \text{for}~~x\in\mathbb R^N, N\geq 2
\end{equation*}
where $V$ is a periodic and non constant potential and $f$ is a nonlinear term with critical power growth if $N\geq 3$ or critical exponential growth if $N=2$. The spectrum of the self-adjoint operator $S=-\Delta +V$ is purely continuous and may contain gaps. Assuming that $0$ lies on the boundary of a spectral gap, we prove the existence of a solution in $H^2_{loc}(\mathbb R^N)$. |
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