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Session 38. Variational Methods in Nonlinear Analysis
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A logarithmic Schrödinger equation with periodic potential |
Andrzej Szulkin, Department of Mathematics, Stockholm University,
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The talk is based on the joint work with Marco Squassina
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We consider the logarithmic Schrödinger equation
\[
-\Delta u + V(x)u = Q(x)u\log u^2, \quad u\in H^1(\mathbb{R}^N),
\]
where \(V,Q\) are periodic in \(x_1,\ldots,x_N\), \(Q>0\) and \(V+Q>0\). We show that this equation has infinitely many geometrically distinct solutions and that one of these solutions is positive. The main difficulty here is that the functional associated with this problem is lower semicontinuous and takes the value \(+\infty\) for some \(u\in H^1(\mathbb{R}^N)\).
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