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Partially symmetric solutions of the generalized Hénon equation in symmetric domains. (English) Zbl 1388.35068

The paper is concerned with the existence of partially symmetric positive solutions of the Dirichlet problem for the generalized Hénon equation: \[ -\triangle u=f(x)u^p \] on a bounded domain of the Euclidian space. The author first recalls the definition of least energy solutions in the Nehari manifold as well as the definition of \(G,H\)-invariant least energy solutions. The function \(f\) is bounded but satisfies some sign conditions. The author is interested in the existence of \(H\)-invariant solutions that could or could not be \(G\)-invariant solutions. This is the main result of the paper which is illustrated by five examples in some special situations. In the remaining part of the paper, the author develops very sharp variational inequalities that lead to the proof of the main existence result.

MSC:

35J91 Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian
35J25 Boundary value problems for second-order elliptic equations
35J20 Variational methods for second-order elliptic equations
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