## Compactness results for Schrödinger equations with asymptotically linear terms.(English)Zbl 1387.35246

Summary: We study the nonlinear problem $-\Delta u+V(x)=f(x,u),\quad x\in\mathbb R^N,$
$\lim_{|x|\to\infty}u(x)=0,$ where the Schrödinger operator $$-\Delta+V$$ is semibounded and the nonlinearity $$f$$ is linearly bounded. We establish compactness of Palais-Smale sequences and Cerami sequences for the associated energy functional under general spectral-theoretic assumptions. Applying these results, we obtain existence of three nontrivial solutions if the energy functional has a mountain-pass geometry.

### MSC:

 35J60 Nonlinear elliptic 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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### References:

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