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2017 An indefinite concave-convex equation under a Neumann boundary condition II
Humberto Ramos Quoirin, Kenichiro Umezu
Topol. Methods Nonlinear Anal. 49(2): 739-756 (2017). DOI: 10.12775/TMNA.2017.007

Abstract

We proceed with the investigation of the problem \begin{equation} -\Delta u = \lambda b(x)|u|^{q-2}u +a(x)|u|^{p-2}u \quad \mbox{in } \Omega, \qquad \frac{\partial u}{\partial n} = 0\quad \mbox{on } \partial \Omega, \tag{${\rm P}_\lambda$} \end{equation} where $\Omega$ is a bounded smooth domain in $\mathbb R^N$ ($N \geq 2$), $a,b \in C^\alpha(\overline{\Omega})$ with $0<\alpha<1$. Dealing now with the case $b \geq 0$, $b \not \equiv 0$, we show the existence (and several properties) of an unbounded subcontinuum of nontrivial nonnegative solutions of $({\rm P}_\lambda)$. Our approach is based on a priori bounds, a regularisation procedure, and Whyburn's topological method.

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Humberto Ramos Quoirin. Kenichiro Umezu. "An indefinite concave-convex equation under a Neumann boundary condition II." Topol. Methods Nonlinear Anal. 49 (2) 739 - 756, 2017. https://doi.org/10.12775/TMNA.2017.007

Information

Published: 2017
First available in Project Euclid: 28 May 2017

zbMATH: 1376.35033
MathSciNet: MR3670484
Digital Object Identifier: 10.12775/TMNA.2017.007

Rights: Copyright © 2017 Juliusz P. Schauder Centre for Nonlinear Studies

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Vol.49 • No. 2 • 2017
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