On qualitative properties of solutions for elliptic problems with the p-Laplacian through domain perturbations

We study the dependence of least nontrivial critical levels of the energy functional corresponding to the zero Dirichlet problem in a bounded domain upon domain perturbations. Assuming that the nonlinearity f is superlinear and subcritical, we establish Hadamard-type formulas for such critical level...

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Bibliographic Details
Published in:Communications in partial differential equations 2020-03, Vol.45 (3), p.230-252
Main Authors: Bobkov, Vladimir, Kolonitskii, Sergey
Format: Article
Language:English
Subjects:
Annuli
Dirichlet problem
domain derivative
Domains
Hadamard formula
least energy solution
Nehari manifold
nodal solution
nonradiality
p-Laplacian
shape optimization
superlinear nonlinearity
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Summary:We study the dependence of least nontrivial critical levels of the energy functional corresponding to the zero Dirichlet problem in a bounded domain upon domain perturbations. Assuming that the nonlinearity f is superlinear and subcritical, we establish Hadamard-type formulas for such critical levels. As an application, we show that among all (generally eccentric) spherical annuli Ω least nontrivial critical levels attain maximum if and only if Ω is concentric. As a consequence of this fact, we prove the nonradiality of least energy nodal solutions whenever Ω is a ball or concentric annulus.
ISSN:0360-5302
1532-4133
DOI:10.1080/03605302.2019.1670674