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December, 2017 Weak Solutions for Nonlinear Neumann Boundary Value Problems with $p(x)$-Laplacian Operators
Lingju Kong
Taiwanese J. Math. 21(6): 1355-1379 (December, 2017). DOI: 10.11650/tjm/7995


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.


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Lingju Kong. "Weak Solutions for Nonlinear Neumann Boundary Value Problems with $p(x)$-Laplacian Operators." Taiwanese J. Math. 21 (6) 1355 - 1379, December, 2017.


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

zbMATH: 06871373
MathSciNet: MR3732910
Digital Object Identifier: 10.11650/tjm/7995

Primary: 35J20 , 35J25 , 35J92

Keywords: Concentration-compactness principle , Critical growth , dual Fountain theorem , Mountain Pass Lemma , Nonlinear boundary conditions , variable exponent spaces , weak solutions

Rights: Copyright © 2017 The Mathematical Society of the Republic of China


Vol.21 • No. 6 • December, 2017
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