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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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