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arXiv · 1905.04542

Bound states for the Schr\"{o}dinger equation with mixed-type nonlinearites

Bartosz Bieganowski·Jarosław Mederski
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Abstract

We prove the existence results for the Schr\"odinger equation of the form $$ -\Delta u + V(x) u = g(x,u), \quad x \in \mathbb{R}^N, $$ where $g$ is superlinear and subcritical in some periodic set $K$ and linear in $\mathbb{R}^N \setminus K$ for sufficiently large $|u|$. The periodic potential $V$ is such that $0$ lies in a spectral gap of $-\Delta+V$. We find a solution with the energy bounded by a certain min-max level, and infinitely many geometrically distinct solutions provided that $g$ is odd in $u$.

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Bartosz Bieganowski, Jarosław Mederski. 2019-05-11. Bound states for the Schr\"{o}dinger equation with mixed-type nonlinearites. https://doi.org/10.1512/iumj.2022.71.8662

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