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Existence of nodal solutions for quasilinear elliptic problems in ℝN

Part of: Elliptic equations and systems

Published online by Cambridge University Press:  24 August 2015

Ann Derlet  and
François Genoud
Ann Derlet
Affiliation:
Haute École Robert Schuman, Rue de Mageroux, 27 6760 Virton, Belgique (aderlet@hers.be)
François Genoud
Affiliation:
Delft Institute of Applied Mathematics, Delft University of Technology, Mekelweg 4, 2628 CD Delft, Netherlands (S.F.Genoud@tudelft.nl) and Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria
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Abstract

We prove the existence of one positive, one negative and one sign-changing solution of a p-Laplacian equation on ℝN with a p-superlinear subcritical term. Sign-changing solutions of quasilinear elliptic equations set on the whole of ℝN have scarcely been investigated in the literature. Our assumptions here are similar to those previously used by some authors in bounded domains, and our proof uses fairly elementary critical point theory, based on constraint minimization on the nodal Nehari set. The lack of compactness due to the unbounded domain is overcome by working in a suitable weighted Sobolev space.

Keywords

quasilinear elliptic equationsunbounded domainsign-changing solutionsNehari manifoldweighted Sobolev spaces

MSC classification

Primary: 35J20: Variational methods for second-order elliptic equations
Secondary: 35J92: Quasilinear elliptic equations with $p$-Laplacian
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 145 , Issue 5 , October 2015 , pp. 937 - 957
Copyright
Copyright © Royal Society of Edinburgh 2015 

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