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Zeitschrift für Analysis und ihre Anwendungen


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Volume 37, Issue 1, 2018, pp. 25–38
DOI: 10.4171/ZAA/1600

Published online: 2018-01-08

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

Khaled Kefi[1]

(1) Northern Border University, Kingdom of Saudi Arabia

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.

Keywords: Robin problem, Ekeland's variational principle, generalized Sobolev spaces, weak solution

Kefi Khaled: On the Robin Problem with Indefinite Weight in Sobolev Spaces with Variable Exponents. Z. Anal. Anwend. 37 (2018), 25-38. doi: 10.4171/ZAA/1600