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Morse index and symmetry for elliptic problems with nonlinear mixed boundary conditions

Part of: Elliptic equations and systems Spectral theory and eigenvalue problems Partial differential equations

Published online by Cambridge University Press:  27 December 2018

Lucio Damascelli  and
Filomena Pacella
Lucio Damascelli
Affiliation:
Dipartimento di Matematica, Università di Roma ‘Tor Vergata’ Via della Ricerca Scientifica 1-00173, Roma, Italy (damascel@mat.uniroma2.it)
Filomena Pacella
Affiliation:
Dipartimento di Matematica, Università di Roma ‘La Sapienza’ P. le A. Moro 2-00185, Roma, Italy (pacella@mat.uniroma1.it)
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Abstract

We consider an elliptic problem of the type

$$\left\{ {\matrix{ {-\Delta u = f(x,u)\quad } \hfill & {{\rm in}\,\Omega } \hfill \cr {u = 0} \hfill & {{\rm on}\,\Gamma _1} \hfill \cr {\displaystyle{{\partial u} \over {\partial \nu }} = g(x,u)} \hfill & {{\rm on}\,\Gamma _2} \hfill \cr } } \right.$$
where Ω is a bounded Lipschitz domain in ℝN with a cylindrical symmetry, ν stands for the outer normal and $\partial \Omega = \overline {\Gamma _1} \cup \overline {\Gamma _2} $.

Under a Morse index condition, we prove cylindrical symmetry results for solutions of the above problem.

As an intermediate step, we relate the Morse index of a solution of the nonlinear problem to the eigenvalues of the following linear eigenvalue problem

$$\left\{ {\matrix{ {-\Delta w_j + c(x)w_j = \lambda _jw_j} \hfill & {{\rm in }\Omega } \hfill \cr {w_j = 0} \hfill & {{\rm on }\Gamma _1} \hfill \cr {\displaystyle{{\partial w_j} \over {\partial \nu }} + d(x)w_j = \lambda _jw_j} \hfill & {{\rm on }\Gamma _2} \hfill \cr } } \right.$$
For this one, we construct sequences of eigenvalues and provide variational characterization of them, following the usual approach for the Dirichlet case, but working in the product Hilbert space L2 (Ω) × L2(Γ2).

Keywords

mixed elliptic problemsnonlinear boundary conditionssymmetrymaximum principleMorse index

MSC classification

Primary: 35J66: Nonlinear boundary value problems for nonlinear elliptic equations 35B06: Symmetries, invariants, etc.
Secondary: 35B07: Axially symmetric solutions 35B50: Maximum principles 35P05: General topics in linear spectral theory
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 149 , Issue 2 , April 2019 , pp. 305 - 324
Copyright
Copyright © Royal Society of Edinburgh 2018 

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