Taiwanese Journal of Mathematics

Weak Solutions for Nonlinear Neumann Boundary Value Problems with $p(x)$-Laplacian Operators

Lingju Kong

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We study the nonlinear Neumann boundary value problem with a $p(x)$-Laplacian operator \[ \begin{cases} \Delta_{p(x)}u + a(x)|u|^{p(x)-2}u = f(x,u) &\textrm{in $\Omega$}, \\ |\nabla u|^{p(x)-2} \dfrac{\partial u}{\partial\nu} = |u|^{q(x)-2}u + \lambda |u|^{w(x)-2}u &\textrm{on $\partial \Omega$}, \end{cases} \] where $\Omega \subset \mathbb{R}^N$, with $N \geq 2$, is a bounded domain with smooth boundary and $q(x)$ is critical in the context of variable exponent $p_*(x) = (N-1)p(x)/(N-p(x))$. Using the variational method and a version of the concentration-compactness principle for the Sobolev trace immersion with variable exponents, we establish the existence and multiplicity of weak solutions for the above problem.

Article information

Taiwanese J. Math., Volume 21, Number 6 (2017), 1355-1379.

Received: 16 August 2016
Revised: 25 January 2017
Accepted: 9 February 2017
First available in Project Euclid: 17 August 2017

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Zentralblatt MATH identifier

Primary: 35J20: Variational methods for second-order elliptic equations 35J25: Boundary value problems for second-order elliptic equations 35J92: Quasilinear elliptic equations with p-Laplacian

nonlinear boundary conditions weak solutions concentration-compactness principle variable exponent spaces critical growth mountain pass lemma dual fountain theorem


Kong, Lingju. Weak Solutions for Nonlinear Neumann Boundary Value Problems with $p(x)$-Laplacian Operators. Taiwanese J. Math. 21 (2017), no. 6, 1355--1379. doi:10.11650/tjm/7995. https://projecteuclid.org/euclid.twjm/1502935252

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