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July/August 2013 Global multiplicity results for $p(x)$-Laplacian equation with nonlinear Neumann boundary condition
K. Sreenadh, Sweta Tiwari
Differential Integral Equations 26(7/8): 815-836 (July/August 2013). DOI: 10.57262/die/1369057818


We study the existence and multiplicity results for the following nonlinear Neumann boundary-value problem involving the $p(x)$-Laplacian $$ (P_\lambda)\hspace{1cm} \left\{\begin{array}{rllll}-\operatorname{div} (|\nabla u|^{p(x)-2}\nabla u)+|u|^{p(x)-2}u & = & f(x,u)~~~\mbox{ in }\Omega,\\ u & < & 0~~~~~~~~~~~\mbox{ in }\Omega,\\ |\nabla u|^{p(x)-2}\frac{\partial u}{\partial\nu} & = & \lambda u^{q(x)}~~~~~\mbox{ on }{\partial}\Omega , \end{array} \right. $$ where $\Omega\subset \mathbb R $ is a bounded domain with smooth boundary, $p(x)\in C(\bar\Omega),$ and $q(x)\in C^{0,\beta}({{\partial}\Omega})$ for some $\beta\in(0,1)$. Under appropriate growth conditions on $f(x,u),$ $p(x),$ and $q(x)$ we show that there exists $\Lambda\in(0,\infty)$ such that $(P_\lambda)$ admits two solutions for $\lambda\in(0,\Lambda)$, one solution for $\lambda=\Lambda$, and no solution for $\lambda>\Lambda$.


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K. Sreenadh. Sweta Tiwari. "Global multiplicity results for $p(x)$-Laplacian equation with nonlinear Neumann boundary condition." Differential Integral Equations 26 (7/8) 815 - 836, July/August 2013.


Published: July/August 2013
First available in Project Euclid: 20 May 2013

zbMATH: 1299.35107
MathSciNet: MR3098988
Digital Object Identifier: 10.57262/die/1369057818

Primary: 35J20 , 35J60

Rights: Copyright © 2013 Khayyam Publishing, Inc.

Vol.26 • No. 7/8 • July/August 2013
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