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Multiple solutions of Schrödinger equations with indefinite linear part and super or asymptotically linear terms. (English) Zbl 1090.35077
Based on new information concerning strongly indefinite functionals without Palais-Smale conditions, we study existence and multiplicity of solutions of the Schrödinger equation $$-\Delta u+V(x)u=g(x,u)\quad\text{for }x\in\Bbb R^N,\quad u(x)\to 0\quad \text{as }|x|\to\infty,$$ where $V$ and $g$ are periodic with respect to $x$ and 0 lies in a gap of $\sigma(-\Delta+V)$. Supposing $g$ is asymptotically linear as $|u|\to \infty$ and symmetric in $u$, we obtain infinitely many geometrically distinct solutions. We also consider the situation where $g$ is superlinear with mild assumptions different from those studied previously, and establish the existence and multiplicity.

35J60Nonlinear elliptic equations
35Q55NLS-like (nonlinear Schrödinger) equations
Full Text: DOI
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