Topological Methods in Nonlinear Analysis

An indefinite concave-convex equation under a Neumann boundary condition II

Abstract

We proceed with the investigation of the problem $$-\Delta u = \lambda b(x)|u|^{q-2}u +a(x)|u|^{p-2}u \quad \mbox{in } \Omega, \qquad \frac{\partial u}{\partial n} = 0\quad \mbox{on } \partial \Omega, \tag{{\rm P}_\lambda}$$ 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.

Article information

Source
Topol. Methods Nonlinear Anal., Volume 49, Number 2 (2017), 739-756.

Dates
First available in Project Euclid: 28 May 2017

https://projecteuclid.org/euclid.tmna/1495936816

Digital Object Identifier
doi:10.12775/TMNA.2017.007

Mathematical Reviews number (MathSciNet)
MR3670484

Zentralblatt MATH identifier
1376.35033

Citation

Ramos Quoirin, Humberto; Umezu, Kenichiro. An indefinite concave-convex equation under a Neumann boundary condition II. Topol. Methods Nonlinear Anal. 49 (2017), no. 2, 739--756. doi:10.12775/TMNA.2017.007. https://projecteuclid.org/euclid.tmna/1495936816