On a nonlinear general eigenvalue problem in Musielak–Orlicz spaces

Abstract

Abstract In this paper, we concern the existence result of the following general eigenvalue problem: <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>{</m:mo> <m:mtable columnspacing="0pt" displaystyle="true" rowspacing="0pt"> <m:mtr> <m:mtd columnalign="right"> <m:mrow> <m:mi mathvariant="script">𝒜</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>u</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mtd> <m:mtd columnalign="left"> <m:mrow> <m:mi/> <m:mo>=</m:mo> <m:mrow> <m:mi>λ</m:mi> <m:mo>⁢</m:mo> <m:mi mathvariant="script">ℬ</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>u</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:mtd> <m:mtd/> <m:mtd columnalign="right"> <m:mrow> <m:mrow> <m:mtext>in </m:mtext> <m:mo>⁢</m:mo> <m:mi mathvariant="normal">Ω</m:mi> </m:mrow> <m:mo>,</m:mo> </m:mrow> </m:mtd> </m:mtr> <m:mtr> <m:mtd columnalign="right"> <m:mrow> <m:msup> <m:mi>D</m:mi> <m:mi>α</m:mi> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>u</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mtd> <m:mtd columnalign="left"> <m:mrow> <m:mi/> <m:mo>=</m:mo> <m:mn>0</m:mn> </m:mrow> </m:mtd> <m:mtd/> <m:mtd columnalign="right"> <m:mrow> <m:mrow> <m:mtext>on </m:mtext> <m:mo>⁢</m:mo> <m:mrow> <m:mo>∂</m:mo> <m:mo>⁡</m:mo> <m:mi mathvariant="normal">Ω</m:mi> </m:mrow> </m:mrow> <m:mo>,</m:mo> </m:mrow> </m:mtd> </m:mtr> </m:mtable> </m:mrow> </m:math> \left\{\begin{aligned} \displaystyle{}\mathcal{A}(u)&amp;\displaystyle={\lambda}% \mathcal{B}(u)&amp;&amp;\displaystyle\phantom{}\text{in }{\Omega},\\ \displaystyle D^{\alpha}(u)&amp;\displaystyle=0&amp;&amp;\displaystyle\phantom{}\text{on }% {\partial\Omega},\end{aligned}\right. in an arbitrary Musielak–Orlicz spaces, where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="script">𝒜</m:mi> </m:math> {\mathcal{A}} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="script">ℬ</m:mi> </m:math> {\mathcal{B}} are quasilinear operators in divergence form of order <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mn>2</m:mn> <m:mo>⁢</m:mo> <m:mi>n</m:mi> </m:mrow> </m:math> {2n} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mn>2</m:mn> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>n</m:mi> <m:mo>-</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {2(n-1)} , respectively. The main assumptions in this case are that <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="script">𝒜</m:mi> </m:math> {\mathcal{A}} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="script">ℬ</m:mi> </m:math> {\mathcal{B}} are potential operators with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi math