## Advances in Differential Equations

- Adv. Differential Equations
- Volume 15, Number 5/6 (2010), 561-599.

### Constant-sign and sign-changing solutions for nonlinear elliptic equations with Neumann boundary values

#### Abstract

In this paper we study the existence of multiple solutions to the equation \begin{align*} -{\Delta_p} u = f(x,u)-|u|^{p-2}u \end{align*} with the nonlinear boundary condition \begin{align*} |\nabla u|^{p-2} \frac{\partial u}{ \partial \nu} = \lambda |u|^{p-2}u+g(x,u). \end{align*} We establish the existence of a smallest positive solution, a greatest negative solution, and a nontrivial sign-changing solution when the parameter $\lambda$ is greater than the second eigenvalue of the Steklov eigenvalue problem. Our approach is based on truncation techniques and comparison principles for nonlinear elliptic differential inequalities. In particular, we make use of variational and topological tools, such as critical point theory, the mountain-pass theorem, the second deformation lemma and variational characterizations of the second eigenvalue of the Steklov eigenvalue problem.

#### Article information

**Source**

Adv. Differential Equations, Volume 15, Number 5/6 (2010), 561-599.

**Dates**

First available in Project Euclid: 18 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.ade/1355854681

**Mathematical Reviews number (MathSciNet)**

MR2643235

**Zentralblatt MATH identifier**

1208.35065

**Subjects**

Primary: 35B38: Critical points 35J20: Variational methods for second-order elliptic equations 47J10: Nonlinear spectral theory, nonlinear eigenvalue problems [See also 49R05]

#### Citation

Winkert, Patrick. Constant-sign and sign-changing solutions for nonlinear elliptic equations with Neumann boundary values. Adv. Differential Equations 15 (2010), no. 5/6, 561--599. https://projecteuclid.org/euclid.ade/1355854681