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Nonradial positive solutions of the p ‐Laplace Emden–Fowler equation with sign‐changing weight
Author(s) -
Kajikiya Ryuji
Publication year - 2016
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201500103
Subject(s) - mathematics , laplace's equation , mathematical analysis , dirichlet boundary condition , unit sphere , sign (mathematics) , laplace transform , ball (mathematics) , dirichlet problem , boundary value problem , weight function
In this paper we study the p ‐Laplace Emden–Fowler equation with a radial and sign‐changing weight in the unit ball under the Dirichlet boundary condition. We show that if the weight function is negative in the unit ball except for a small neighborhood of the boundary and positive at somewhere in this neighborhood, then no least energy solution is radially symmetric. Therefore the equation has both a positive radial solution and a positive nonradial solution. Moreover, we prove in the one dimensional case that if the neighborhood is large, then a positive solution is unique.