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May/June 2010 Constant-sign and sign-changing solutions for nonlinear elliptic equations with Neumann boundary values
Patrick Winkert
Adv. Differential Equations 15(5/6): 561-599 (May/June 2010).

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.

Citation

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Patrick Winkert. "Constant-sign and sign-changing solutions for nonlinear elliptic equations with Neumann boundary values." Adv. Differential Equations 15 (5/6) 561 - 599, May/June 2010.

Information

Published: May/June 2010
First available in Project Euclid: 18 December 2012

zbMATH: 1208.35065
MathSciNet: MR2643235

Subjects:
Primary: 35B38, 35J20, 47J10

Rights: Copyright © 2010 Khayyam Publishing, Inc.

JOURNAL ARTICLE
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Vol.15 • No. 5/6 • May/June 2010
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