Browsing by Subject "Steklov problem"

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Access status: Open Access ,
Continuous spectrum of Steklov nonhomogeneous elliptic problem
(2015) Allaoui, Mostafa
By applying two versions of the mountain pass theorem and Ekeland’s variational principle, we prove three different situations of the existence of solutions for the following Steklov problem: $\begin{aligned}\Delta_{p(x)} u&=|u|^{p(x)-2}u \phantom{\lambda} \quad\text{in}\;\Omega, \\ |\nabla u|^{p(x)-2}\frac{\partial u}{\partial \nu}&= \lambda|u|^{q(x)-2}u \quad\text{on}\;\partial\Omega,\end{aligned}$, where $\Omega \subset \mathbb{R}^N$ is a bounded smooth domain and $p,q: \overline{\Omega}\rightarrow(1,+\infty)$ are continuous functions.
Access status: Open Access ,
On the Steklov problem involving the p(x)-Laplacian with indefinite weight
(Wydawnictwa AGH, 2017) Ali, Khaled Ben; Ghanmi, Abdeljabbar; Kefi, Khaled
Under suitable assumptions, we study the existence of a weak nontrivial solution for the following Steklov problem involving the $p(x)$-Laplacian $\begin{cases}\Delta_{p(x)}u=a(x)|u|^{p(x)-2}u \quad \text{in }\Omega, \\ |\nabla u|^{p(x)-2}\frac{\partial u}{\partial \nu}=\lambda V(x)|u|^{q(x)-2}u \quad \text{on }\partial \Omega.\end{cases}$ Our approach is based on min-max method and Ekeland's variational principle.