Uniqueness of renormalized solution to nonlinear Neumann problems with variable exponent
Volume 44 / 2017
Applicationes Mathematicae 44 (2017), 1-14
MSC: Primary 35J60; Secondary 35A02, 35J66, 35J70.
DOI: 10.4064/am2287-6-2016
Published online: 21 October 2016
Abstract
We study the uniqueness of renormalized solutions to nonlinear Neumann problems with variable exponents \begin{equation*} \begin{cases} |u|^{p(x)-2}u- \varDelta_{p(x)}(u) =f &\text{in $\varOmega$,}\\ |\nabla u|^{{p(x)}-2}\dfrac{\partial u}{\partial \eta} + \gamma(u)=g &\text{on $\partial\varOmega$,} \end{cases} \end{equation*} where $\varOmega$ is a connected open bounded set in $\mathbb{R}^N$, $p(\cdot)$ is a continuous function defined on $\overline{\varOmega} $ with $p(x) \gt 1$ for all $x \in \overline{\varOmega}, $ $\gamma $ is a nondecreasing continuous function on $\mathbb{R}$ such that $\gamma(0)=0$ and $f,g\in L^1$.
