Steklov problems involving the \(p(x)\)-Laplacian (Q2877620)

In this article, the authors study the problem NEWLINE\[NEWLINE\Delta_{p(x)}u=a(x)|u|^{p(x)-2}u\text{, }x\in\OmegaNEWLINE\]NEWLINE subject to the boundary condition NEWLINE\[NEWLINE|\nabla u|^{p(x)-2}\frac{\partial u}{\partial\nu}=\lambda f(x,u)\text{, }x\in\partial\Omega.NEWLINE\]NEWLINE Here the set \(\Omega\subseteq\mathbb{R}^{N}\) is bounded with a smooth boundary, \(\lambda>0\) is a parameter, and \(\Delta_{p(x)}\) denotes the so-called \(p(x)\)-Laplacian operator -- namely, NEWLINE\[NEWLINE\Delta_{p(x)}u=\nabla\cdot\left(|\nabla u|^{p(x)-2}\nabla u\right).NEWLINE\]NEWLINE It should be noted that the map \(p\) is a continuous function on \(\overline{\Omega}\). Thus, the authors allow for a variable exponent in the Laplacian term. All in all, the paper studies the existence of one of more nontrivial solutions to the above mentioned problem. The techniques involved are variational in nature. It is worth noting that \(\lambda\) appears difficult to calculate, and, unfortunately, no examples are given either to explicate or to clarify the use and application of the existence theorem.