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Steklov Eigenvalue Problem with a -Harmonic Solutions and Variable Exponents by Belhadj Karim; Abdellah Zerouali; Omar Chakrone is a Mathematics article available to read on EtoBox.

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## Abstract Using the Ljusternik–Schnirelmann principle and a new variational technique, we prove that the following Steklov eigenvalue problem has infinitely many positive eigenvalue sequences: { div ⁡ ( a ⁢ ( x , ∇ ⁡ u ) ) = 0 in ⁢ Ω , a ⁢ ( x , ∇ ⁡ u ) ⋅ ν = λ ⁢ m ⁢ ( x ) ⁢ | u | p ⁢ ( x ) - 2 ⁢ u on ⁢ ∂ ⁡ Ω , \left\{\begin{aligned} &\displaystyle\operatorname{div}(a(x,\nabla u))=0&&% \displaystyle\phantom{}\text{in }\Omega,\\ &\displaystyle a(x,\nabla u)\cdot\nu=\lambda m(x)|u|^{p(x)-2}u&&\displaystyle% \phantom{}\text{on }\partial\Omega,\end{aligned}\right. where Ω ⊂ R N {\Omega\subset\mathbb{R}^{N}} ( N ≥ 2 ) {(N\geq 2)} is a bounded domain of smooth boundary ∂ ⁡ Ω {\partial\Omega} and ν is the outward unit normal vector on ∂ ⁡ Ω {\partial\Omega} . The functions m ∈ L ∞ ⁢ ( ∂ ⁡ Ω ) {m\in L^{\infty}(\partial\Omega)} , p : Ω ̄ ↦ R {p\colon\overline{\Omega}\mapsto\mathbb{R}} and a : Ω ̄ × R N ↦ R N {a\colon\overline{\Omega}\times\mathbb{R}^{N}\mapsto\mathbb{R}^{N}} satisfy appropriate conditions.

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Author

Belhadj Karim; Abdellah Zerouali; Omar Chakrone

Publisher

Walter de Gruyter GmbH

Published

2020

Language

EN

Field

Mathematics (Physical Sciences)