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