On the Robin Problem with Indefinite Weight in Sobolev Spaces with Variable Exponents

The present paper is concerned with a Robin problem involving an indefinite weight in Sobolev spaces with variable exponents \begin{equation*} \left\{\begin{alignedat}{2}-\text{ div}(|\nabla u|^{p(x)-2}\nabla u)&=\lambda V(x)|u|^{q(x)-2}u,& \quad x&\in\Omega\\ |\nabla u|^{p(x)-2} \frac{\partial u}{\partial n}+a(x)|u|^{p(x)-2}u&=0.&\quad x&\in\partial\Omega \end{alignedat}\right. \end{equation*} By means of the variational approach and Ekeland's principle, we establish that the above problem admits a non-trivial weak solution under appropriate conditions.