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Symmetry and concentration behavior of ground state in axially symmetric domains. (English) Zbl 1129.35409

Summary: Let \(\Omega(r)\) be axially symmetric bounded domains which satisfy some suitable conditions. Then the ground-state solutions of a semilinear elliptic equation in \(\Omega(r)\) are, in general, nonaxially symmetric and concentrated on one side. We prove a necessary and sufficient condition for the symmetry of ground-state solutions.

MSC:

35J60 Nonlinear elliptic equations
35A30 Geometric theory, characteristics, transformations in context of PDEs
35J20 Variational methods for second-order elliptic equations
47J30 Variational methods involving nonlinear operators
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