## Topological Methods in Nonlinear Analysis

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

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

Humberto Ramos Quoirin and Kenichiro Umezu

#### 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.

#### Article information

**Source**

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

**Dates**

First available in Project Euclid: 28 May 2017

**Permanent link to this document**

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