Least energy solutions of the generalized Hénon equation in reflectionally symmetric or point symmetric domains.(English)Zbl 1248.35062

The paper under review deals with the generalized Hénon equation in reflectionally symmetric or point symmetric domains. The main result establishes that a least energy solution is neither reflectionally symmetric nor even. Moreover, the existence of a positive solution with prescribed exact symmetry is argued. In the first part of this paper a function is constructed that has a lower energy than the symmetric least energy, provided that a symmetric least energy solution satisfies a certain inequality. A central auxiliary result is an a priori $$L^\infty$$ estimate of a symmetric least energy solution. The proofs of the main results are done in the final part of this paper.

MSC:

 35J20 Variational methods for second-order elliptic equations 35J25 Boundary value problems for second-order elliptic equations
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References:

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