On a PDE involving the variable exponent operator with nonlinear boundary conditions (Q493311)

This paper is concerned with the study of the following boundary value problem \[ \begin{cases} -\mathcal{A}_{p(x)}u=\lambda f(x)|u|^{q(x)-2}u &\text{ in }\Omega,\\ \alpha(x,u){\langle A\nabla u, \nabla u \rangle}^{\frac{p(x)-2}{2}}\langle A\nabla u, \vec{n} \rangle=g(x)|u|^{r(x)-2}u\quad &\text{ on }\partial\Omega, \end{cases}\leqno{(1)} \] where \(\Omega \subset \mathbb R^N\) is a bounded domain with \(N \geq 2.\) The differential operator \({\mathcal{A}}_{p(x)}\) is defined by \[ {\mathcal{A}}_{p(x)}u=\operatorname{div} \left(\alpha(x,u){\langle A\nabla u, \nabla u \rangle}^{\frac{p(x)-2}{2}} A\nabla u \right), \] where \(A: \overline \Omega \rightarrow \mathbb R^{N^2}\) is a symmetric function matrix with \(a_{ij}\in L^\infty(\Omega)\cap C^1(\overline\Omega)\) and satisfies \[ \langle A\xi, \xi \rangle =\sum_{i, j=1}^Na_{ij}(x)\xi_i\xi_j \geq |\xi|^2 \quad \forall x\in \overline\Omega, \quad\xi\in \mathbb R^N. \] Under some conditions imposed on \(p(.)\), \(q(.)\), \(r(.),\) the weight functions \(f\) and \(g,\) and based on the Nehari manifold, the authors establish the existence and multiplicity of positive solutions for problem (1).