Some Remarks on a Class of p(x)-Laplacian Robin Eigenvalue Problems

Some Remarks on a Class of p(x)-Laplacian Robin Eigenvalue Problems We consider the p(x)-Laplacian Robin eigenvalue problem $$\begin{aligned} \left\{ \begin{array}{ll} - \Delta _{p(x)}u = \lambda V(x) |u|^{q(x)-2}u, \quad x\in \Omega ,\\ |\nabla u|^{p(x)-2}\frac{\partial u}{\partial \nu }+\beta (x)|u|^{p(x)-2}u=0,\quad x\in \partial \Omega , \end{array}\right. \end{aligned}$$ - Δ p ( x ) u = λ V ( x ) | u | q ( x ) - 2 u , x ∈ Ω , | ∇ u | p ( x ) - 2 ∂ u ∂ ν + β ( x ) | u | p ( x ) - 2 u = 0 , x ∈ ∂ Ω , where $$\Omega $$ Ω is a bounded domain in $$\mathbb {R}^N$$ R N with smooth boundary $$\partial \Omega $$ ∂ Ω , $$N\ge 2$$ N ≥ 2 , $$\frac{\partial u}{\partial \nu }$$ ∂ u ∂ ν is the outer normal derivative of u with respect to $$\partial \Omega $$ ∂ Ω , $$p,q\in C_+(\overline{\Omega })$$ p , q ∈ C + ( Ω ¯ ) , $$1<p^-:= \inf _{x\in \overline{\Omega }}p(x) \le p^+:=\sup _{x\in \overline{\Omega }}p(x)<N$$ 1 < p - : = inf x ∈ Ω ¯ p ( x ) ≤ p + : = sup x ∈ Ω ¯ p ( x ) < N , $$\beta \in L^\infty (\partial \Omega )$$ β ∈ L ∞ ( ∂ Ω ) , $$\beta ^-:=\inf _{x\in \partial \Omega }\beta (x)>0$$ β - : = inf x ∈ ∂ Ω β ( x ) > 0 , and $$\lambda >0$$ λ > 0 is a parameter. Under some suitable conditions on the functions q and V, we establish the existence of a continuous family of eigenvalues in a neighborhood of the origin using variational methods. The main results of this paper improve and generalize the previous ones introduced in Deng (J Math Anal Appl 360:548–560, 2009), Kefi (Zeitschrift für Analysis und ihre Anwendungen (ZAA) 37:25–38, 2018). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Mediterranean Journal of Mathematics Springer Journals

Some Remarks on a Class of p(x)-Laplacian Robin Eigenvalue Problems

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Publisher
Springer Journals
Copyright
Copyright © 2018 by Springer International Publishing AG, part of Springer Nature
Subject
Mathematics; Mathematics, general
ISSN
1660-5446
eISSN
1660-5454
D.O.I.
10.1007/s00009-018-1196-7
Publisher site
See Article on Publisher Site

Abstract

We consider the p(x)-Laplacian Robin eigenvalue problem $$\begin{aligned} \left\{ \begin{array}{ll} - \Delta _{p(x)}u = \lambda V(x) |u|^{q(x)-2}u, \quad x\in \Omega ,\\ |\nabla u|^{p(x)-2}\frac{\partial u}{\partial \nu }+\beta (x)|u|^{p(x)-2}u=0,\quad x\in \partial \Omega , \end{array}\right. \end{aligned}$$ - Δ p ( x ) u = λ V ( x ) | u | q ( x ) - 2 u , x ∈ Ω , | ∇ u | p ( x ) - 2 ∂ u ∂ ν + β ( x ) | u | p ( x ) - 2 u = 0 , x ∈ ∂ Ω , where $$\Omega $$ Ω is a bounded domain in $$\mathbb {R}^N$$ R N with smooth boundary $$\partial \Omega $$ ∂ Ω , $$N\ge 2$$ N ≥ 2 , $$\frac{\partial u}{\partial \nu }$$ ∂ u ∂ ν is the outer normal derivative of u with respect to $$\partial \Omega $$ ∂ Ω , $$p,q\in C_+(\overline{\Omega })$$ p , q ∈ C + ( Ω ¯ ) , $$1<p^-:= \inf _{x\in \overline{\Omega }}p(x) \le p^+:=\sup _{x\in \overline{\Omega }}p(x)<N$$ 1 < p - : = inf x ∈ Ω ¯ p ( x ) ≤ p + : = sup x ∈ Ω ¯ p ( x ) < N , $$\beta \in L^\infty (\partial \Omega )$$ β ∈ L ∞ ( ∂ Ω ) , $$\beta ^-:=\inf _{x\in \partial \Omega }\beta (x)>0$$ β - : = inf x ∈ ∂ Ω β ( x ) > 0 , and $$\lambda >0$$ λ > 0 is a parameter. Under some suitable conditions on the functions q and V, we establish the existence of a continuous family of eigenvalues in a neighborhood of the origin using variational methods. The main results of this paper improve and generalize the previous ones introduced in Deng (J Math Anal Appl 360:548–560, 2009), Kefi (Zeitschrift für Analysis und ihre Anwendungen (ZAA) 37:25–38, 2018).

Journal

Mediterranean Journal of MathematicsSpringer Journals

Published: Jun 6, 2018

References

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