Opuscula Math. 35, no. 6 (), 853-866
http://dx.doi.org/10.7494/OpMath.2015.35.6.853
Opuscula Mathematica

# Continuous spectrum of Steklov nonhomogeneous elliptic problem

Abstract. 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 $$\Omega \subset \mathbb{R}^N$$ $$(N\geq 2)$$ is a bounded smooth domain and $$p,q: \overline{\Omega}\rightarrow(1,+\infty)$$ are continuous functions.
Keywords: $$p(x)$$-Laplacian, Steklov problem, critical point theorem.
Mathematics Subject Classification: 35J48, 35J66.
Cite this article as:
Mostafa Allaoui, Continuous spectrum of Steklov nonhomogeneous elliptic problem, Opuscula Math. 35, no. 6 (2015), 853-866, http://dx.doi.org/10.7494/OpMath.2015.35.6.853

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ISSN 1232−9274, e-ISSN 2300−6919, DOI http://dx.doi.org/10.7494/OpMath
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