Advances in Differential Equations

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

Patrick Winkert

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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.


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