On the Robin problem with indefinite weight in Sobolev spaces with variable exponents (Q1708543)

Summary: The present paper is concerned with a Robin problem involving an indefinite weight in Sobolev spaces with variable exponents \[ \begin{cases}\quad\quad-\mathrm{div}(|\nabla u|^{p(x)-2}\nabla u)=\lambda V(x)|u|^{q(x)-2}u,&x\in\Omega\\|\nabla u|^{p(x)-2}\frac{\partial u}{\partial n}+a(x)|u|^{p(x)-2}u=0.& x\in\Omega\end{cases} \] 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.