Solvability of Nonlinear Elliptic Type Equation With Two Unrelated Non standard Growths

TL;DR

This paper investigates the conditions under which nonlinear elliptic equations with complex growth conditions have solutions, using advanced mathematical techniques to establish existence results.

Contribution

It introduces new sufficient conditions for the existence of solutions to nonlinear elliptic equations with two unrelated non-standard growths.

Findings

Established existence of weak solutions under specified conditions

Applied monotonicity and compactness methods effectively

Extended solvability results to more general growth scenarios

Abstract

In this paper, we study the solvability of the nonlinear Dirichlet problem with sum of the operators of independent non standard growths in a bounded domain $\Omega \subset \mathbb{R}^{n}$. We obtain sufficient conditions and show the existence of weak solutions of the considered problem by using monotonicity and compactness methods together.

Equations

$$-div\left({\left|\nabla \ddot{o}u\right|}^{p_1\left(x\right)-2}\nabla \ddot{o}u\right)-\sum _{i=1}^n{D}_i\left(\left|\ddot{o}u\right|{\ddot{o}}^{p_0\left(x\right)-2}{D}_{i}u\right)+c\left(\ddot{o}x,u\right)\ddot{o}=h\left(\ddot{o}x\right),\text{o¨ }x\in \ddot{o}\Omega$$

$$\left\{\begin{array}[]{l}-div\left(\left|\nabla\"{o}u\right|^{p_{1}\left(x\right)-2}\nabla\"{o}u\right)-\sum\limits_{i=1}^{n}D_{i}\left(\left|\"{o}u\right|\"{o}^{p_{0}\left(x\right)-2}D_{i}u\right)+c\left(\"{o}x,u\right)\"{o}=h\left(\"{o}x\right)\"{o}\\ u\mid\"{o}_{\partial\"{o}\Omega\"{o}}=0\end{array}\right.\"{o}\"{o}$$

$$\left\{\begin{array}[]{l}-div\left(\left|\nabla\"{o}u\right|^{p_{1}\left(x\right)-2}\nabla\"{o}u\right)-\sum\limits_{i=1}^{n}D_{i}\left(\left|\"{o}u\right|\"{o}^{p_{0}\left(x\right)-2}D_{i}u\right)+c\left(\"{o}x,u\right)\"{o}=h\left(\"{o}x\right),\hskip 8.5359ptx\in\"{o}\Omega\"{o}\\ u\mid\"{o}_{\partial\"{o}\Omega\"{o}}=0\end{array}\right.$$

$$\left|\ddot{o}c\left(\ddot{o}x,\tau \ddot{o}\right)\ddot{o}\right|\ddot{o}\le \ddot{o}{c}_0\left(\ddot{o}x\right)\left|\ddot{o}\tau \ddot{o}\right|{\ddot{o}}^{\alpha \ddot{o}\left(\ddot{o}x\right)\ddot{o}-1}+c_1\left(\ddot{o}x\right)$$

$$c\left(\ddot{o}x,\tau \ddot{o}\right)\ddot{o}\tau \ddot{o}\ge \ddot{o}{c}_2\left(\ddot{o}x\right)\ddot{o}\left|\ddot{o}\tau \right|{\ddot{o}}^{\alpha \ddot{o}\left(\ddot{o}x\right)\ddot{o}},$$

$$M_0\left(\ddot{o}\Omega \ddot{o}\right)\ddot{o}:=\left\{\ddot{o}p\in \ddot{o}M\left(\ddot{o}\Omega \ddot{o}\right):1\le \ddot{o}{p}^{-}\le \ddot{o}p\left(\ddot{o}x\right)\ddot{o}\le \ddot{o}{p}^{+}<\infty \ddot{o},\text{o¨a.e. }x\in \ddot{o}\Omega \ddot{o}\right\}\ddot{o}.$$

$$\sigma {\ddot{o}}_p\left(\ddot{o}u\right)\ddot{o}\equiv \ddot{o}\underset{\Omega \ddot{o}\backslash \ddot{o}{\Omega }_{\infty \ddot{o}}}{\int }\left|\ddot{o}u\right|{\ddot{o}}^{p\left(\ddot{o}x\right)\ddot{o}}dx+\underset{{\Omega }_{\infty \ddot{o}}}{\mathrm{ess}}\sup \ddot{o}\left|\ddot{o}u\left(\ddot{o}x\right)\ddot{o}\right|$$

$$\Vert \ddot{o}u\Vert {\ddot{o}}_{L^{p\left(\ddot{o}x\right)\ddot{o}}\left(\ddot{o}\Omega \ddot{o}\right)\ddot{o}}\equiv \inf \ddot{o}\left\{\ddot{o}\lambda \ddot{o}>0:\text{o¨}\sigma {\ddot{o}}_p\left(\ddot{o}\frac{u}{\lambda \ddot{o}}\right)\le \ddot{o}1\right\}\ddot{o}.$$

$$L^{p\left(\ddot{o}x\right)\ddot{o}}\left(\ddot{o}\Omega \ddot{o}\right)\ddot{o}:=\ddot{o}\left\{\ddot{o}u:\ddot{o}u\text{o¨}\text{iso¨ao¨measurableo¨real-valuedo¨functiono¨sucho¨that}\text{o¨}\sigma {\ddot{o}}_p\left(\ddot{o}u\right)\ddot{o}<\infty \ddot{o}\right\}\ddot{o}.$$

$$\underset{\Omega \ddot{o}}{\int }\left|\ddot{o}uv\right|\ddot{o}dx\le \ddot{o}2\Vert u\Vert {\ddot{o}}_{L^{p\left(\ddot{o}x\right)\ddot{o}}\left(\ddot{o}\Omega \ddot{o}\right)\ddot{o}}\Vert v\Vert {\ddot{o}}_{L^{q\left(\ddot{o}x\right)\ddot{o}}\left(\ddot{o}\Omega \ddot{o}\right),\ddot{o}}\ddot{o}\text{ (generalizedo¨Ho¨ldero¨inequality)}\ddot{o}\ddot{o}$$

$$\min \ddot{o}\{\Vert \ddot{o}u\Vert {\ddot{o}}_{L^{p\left(\ddot{o}x\right)\ddot{o}}\left(\ddot{o}\Omega \ddot{o}\right)}^{p^{-}},\ddot{o}\Vert \ddot{o}u\Vert {\ddot{o}}_{L^{p\left(\ddot{o}x\right)\ddot{o}}\left(\ddot{o}\Omega \right)\ddot{o}}^{p^{+}}\}\le \ddot{o}\sigma {\ddot{o}}_p\left(\ddot{o}u\right)\ddot{o}\le \ddot{o}\max \ddot{o}\{\Vert u\Vert {\ddot{o}}_{L^{p\left(\ddot{o}x\right)\ddot{o}}\left(\ddot{o}\Omega \ddot{o}\right)\ddot{o}}^{p^{-}},\Vert \ddot{o}u\Vert {\ddot{o}}_{L^{p\left(\ddot{o}x\right)\ddot{o}}\left(\ddot{o}\Omega \ddot{o}\right)}^{p^{+}}\}\ddot{o}\ddot{o}$$

$$W^{1,\text{o¨}p\left(\ddot{o}x\right)\ddot{o}}\left(\ddot{o}\Omega \ddot{o}\right)\ddot{o}:=\left\{\ddot{o}u\in L^{p\left(\ddot{o}x\right)\ddot{o}}\left(\ddot{o}\Omega \ddot{o}\right)\ddot{o}:\text{o¨}\left|\ddot{o}\nabla u\right|\ddot{o}\in \ddot{o}{L}^{p\left(\ddot{o}x\right)\ddot{o}}\left(\ddot{o}\Omega \ddot{o}\right)\ddot{o}\right\}$$

$$\Vert \ddot{o}u\Vert {\ddot{o}}_{W^{1,\text{o¨}p\left(\ddot{o}x\right)\ddot{o}}\left(\ddot{o}\Omega \right)\ddot{o}}\equiv \ddot{o}\Vert \ddot{o}u\Vert {\ddot{o}}_{L^{p\left(\ddot{o}x\right)\ddot{o}}\left(\ddot{o}\Omega \right)\ddot{o}}+\Vert \ddot{o}\nabla \ddot{o}u\Vert {\ddot{o}}_{L^{p\left(\ddot{o}x\right)\ddot{o}}\left(\Omega \ddot{o}\right)\ddot{o}}$$

$$\Vert \ddot{o}u\Vert {\ddot{o}}_{W_0^{1,\text{o¨}p\left(\ddot{o}x\right)}\left(\ddot{o}\Omega \ddot{o}\right)\ddot{o}}\equiv \ddot{o}{\Vert \ddot{o}\nabla \ddot{o}u\Vert }_{L^{p\left(\ddot{o}x\right)\ddot{o}}\left(\ddot{o}\Omega \ddot{o}\right)}\equiv \sum _{i=1}^n{\Vert \ddot{o}{D}_{i}u\Vert }_{L^{p\left(\ddot{o}x\right)}\left(\ddot{o}\Omega \ddot{o}\right)}.$$

$$-\left|\ddot{o}p\left(\ddot{o}x\right)\ddot{o}-p\left(\ddot{o}y\right)\ddot{o}\right|\ddot{o}\log \left|\ddot{o}x-y\right|\ddot{o}\le \ddot{o}L,\text{o¨ }\forall x,y\in \ddot{o}\Omega$$

$$S_{1,\gamma \ddot{o}\left(\ddot{o}x\right)\ddot{o},\beta \ddot{o}\left(\ddot{o}x\right)\ddot{o}}\left(\ddot{o}\Omega \ddot{o}\right):=\left\{\ddot{o}u\in \ddot{o}{L}^1\left(\ddot{o}\Omega \ddot{o}\right)\ddot{o}:\underset{\Omega \ddot{o}}{\int }\left|u\right|{\ddot{o}}^{\gamma \ddot{o}\left(\ddot{o}x\right)\ddot{o}+\beta \ddot{o}\left(\ddot{o}x\right)}dx+\sum _{i=1}^n\underset{\Omega \ddot{o}}{\int }\left|\ddot{o}u\right|{\ddot{o}}^{\gamma \left(\ddot{o}x\right)\ddot{o}}\left|\ddot{o}{D}_{i}u\right|{\ddot{o}}^{\beta \ddot{o}\left(\ddot{o}x\right)}dx<\infty \ddot{o}\right\}\ddot{o},$$

$$[\ddot{o}u{]}_{S_{\gamma \ddot{o},\beta \ddot{o}}}:=\inf \ddot{o}\left\{\ddot{o}\lambda \ddot{o}>0:\underset{\Omega }{\int }\left|\ddot{o}\frac{u}{\lambda \ddot{o}}\right|{\ddot{o}}^{\gamma \ddot{o}\left(\ddot{o}x\right)\ddot{o}+\beta \left(\ddot{o}x\right)\ddot{o}}dx+\sum _{i=1}^n\left(\ddot{o}\underset{\Omega \ddot{o}}{\int }\left|\frac{\left|\ddot{o}u\right|{\ddot{o}}^{\frac{\gamma \ddot{o}\left(\ddot{o}x\right)\ddot{o}}{\beta \ddot{o}\left(x\right)\ddot{o}}}{D}_{i}u}{\lambda {\ddot{o}}^{\frac{\gamma \ddot{o}\left(\ddot{o}x\right)\ddot{o}}{\beta \ddot{o}\left(x\right)\ddot{o}}+1}}\right|{\ddot{o}}^{\beta \ddot{o}\left(\ddot{o}x\right)\ddot{o}}\right)\ddot{o}dx\le \ddot{o}1\right\}\ddot{o}.$$

$$S_{1,\gamma \ddot{o}\left(\ddot{o}x\right)\ddot{o},\beta \ddot{o}\left(\ddot{o}x\right)\ddot{o},\theta \ddot{o}\left(\ddot{o}x\right)}\left(\ddot{o}\Omega \ddot{o}\right)\ddot{o}:=S_{1,\gamma \ddot{o}\left(\ddot{o}x\right)\ddot{o},\beta \ddot{o}\left(\ddot{o}x\right)}\left(\ddot{o}\Omega \ddot{o}\right)\ddot{o}\cap \ddot{o}{L}^{\theta \ddot{o}\left(\ddot{o}x\right)\ddot{o}}\left(\ddot{o}\Omega \ddot{o}\right),\ddot{o}\ddot{o}$$

$$[\ddot{o}u{]}_{S_{\gamma \ddot{o},\beta \ddot{o},\theta }\ddot{o}}:=[u{]}_{S_{\gamma \ddot{o},\beta \ddot{o}}}+\Vert u\Vert {\ddot{o}}_{L^{\theta \ddot{o}\left(\ddot{o}x\right)\ddot{o}}\left(\ddot{o}\Omega \ddot{o}\right)\ddot{o}},\text{o¨ }\forall \ddot{o}u\in \ddot{o}{S}_{1,\gamma \ddot{o}\left(\ddot{o}x\right)\ddot{o},\beta \ddot{o}\left(\ddot{o}x\right)\ddot{o},\theta \left(\ddot{o}x\right)\ddot{o}}\left(\ddot{o}\Omega \ddot{o}\right)\ddot{o}.$$

$$S_{1,\gamma \ddot{o}\left(\ddot{o}x\right)\ddot{o},\beta \ddot{o}\left(\ddot{o}x\right)\ddot{o},\theta \ddot{o}\left(\ddot{o}x\right)}\left(\ddot{o}\Omega \ddot{o}\right)\ddot{o}\equiv \ddot{o}\left\{\ddot{o}u\in \ddot{o}{L}^1\left(\ddot{o}\Omega \ddot{o}\right)\ddot{o}:{\Re }^{\gamma \ddot{o},\beta \ddot{o},\theta \ddot{o}}\left(\ddot{o}u\right)\ddot{o}:=\underset{\Omega \ddot{o}}{\int }\left|u\right|{\ddot{o}}^{\theta \ddot{o}\left(\ddot{o}x\right)\ddot{o}}dx+\sum _{i=1}^n\underset{\Omega }{\int }\left|\ddot{o}u\right|{\ddot{o}}^{\gamma \ddot{o}\left(\ddot{o}x\right)\ddot{o}}\left|D_{i}u\right|{\ddot{o}}^{\beta \ddot{o}\left(\ddot{o}x\right)\ddot{o}}dx<\infty \ddot{o}\right\}\ddot{o},$$

$$[\ddot{o}u{]}_{S_{\gamma \ddot{o},\beta \ddot{o},\theta \ddot{o}}}\equiv \ddot{o}\inf \ddot{o}\left\{\ddot{o}\lambda >0:\underset{\Omega \ddot{o}}{\int }\left|\ddot{o}\frac{u}{\lambda \ddot{o}}\right|{\ddot{o}}^{\theta \left(\ddot{o}x\right)\ddot{o}}dx+\sum _{i=1}^n\left(\ddot{o}\underset{\Omega \ddot{o}}{\int }\left|\frac{\left|\ddot{o}u\right|{\ddot{o}}^{\frac{\gamma \ddot{o}\left(\ddot{o}x\right)\ddot{o}}{\beta \ddot{o}\left(x\right)\ddot{o}}}{D}_{i}u}{\lambda {\ddot{o}}^{\frac{\gamma \ddot{o}\left(\ddot{o}x\right)\ddot{o}}{\beta \ddot{o}\left(x\right)\ddot{o}}+1}}\right|{\ddot{o}}^{\beta \ddot{o}\left(\ddot{o}x\right)\ddot{o}}\right)\ddot{o}dx\le \ddot{o}1\right\}\ddot{o}.$$

$$Q\left(\ddot{o}\Omega \ddot{o}\right)\ddot{o}:={\mathring{S}}_{1,q_0\left(\ddot{o}x\right)\ddot{o}\left(p_0\left(\ddot{o}x\right)\ddot{o}-2\right)\ddot{o},q_0\left(\ddot{o}x\right)\ddot{o},\alpha \ddot{o}\left(\ddot{o}x\right)}\left(\ddot{o}\Omega \ddot{o}\right)\ddot{o}\cap \ddot{o}{W}_0^{1,\text{o¨}{p}_1\left(\ddot{o}x\right)\ddot{o}}\left(\Omega \ddot{o}\right)$$

$$\underset{\Omega \ddot{o}}{\int }\left({\left|\nabla u\right|}^{p_1\left(\ddot{o}x\right)-2}\nabla \ddot{o}u\right)\cdot \nabla vdx+\sum _{i=1}^n\underset{\Omega \ddot{o}}{\int }\left(\ddot{o}\left|u\right|{\ddot{o}}^{p_0\left(\ddot{o}x\right)\ddot{o}-2}{D}_{i}u\right)D_{i}vdx+$$

$$+\underset{\Omega \ddot{o}}{\int }c\left(\ddot{o}x,u\right)\ddot{o}vdx=\underset{\Omega }{\int }hvdx\ddot{o}\ddot{o}$$

$$\text{liminf}⟨\ddot{o}A\left(\ddot{o}{u}_m\right)\ddot{o},\text{o¨}{u}_m-v⟩\ge \ddot{o}⟨\ddot{o}A\left(\ddot{o}u\right)\ddot{o},\text{o¨}u-v⟩\ddot{o},\text{o¨ }\forall \ddot{o}v\in \ddot{o}X.$$

$$⟨\ddot{o}B\left(\ddot{o}u\right)\ddot{o},\text{o¨}u⟩\ddot{o}=\phi \ddot{o}\left(\Vert \ddot{o}{B}_0\left(\ddot{o}u\right)\ddot{o}\Vert {\ddot{o}}_{Y_2}\right)\ddot{o},\text{o¨ }\forall \ddot{o}u\in \ddot{o}{S}_{g{Y}_0}.$$

$$⟨\ddot{o}T\left(\ddot{o}u\right)\ddot{o},\text{o¨}u⟩\ddot{o}=⟨\ddot{o}A\left(u\right)\ddot{o}+B\left(\ddot{o}u\right)\ddot{o},\text{o¨}u⟩\ddot{o}\ge \ddot{o}\lambda {\ddot{o}}_0\left(\Vert \ddot{o}u\Vert {\ddot{o}}_X\right)\ddot{o}\Vert \ddot{o}u\Vert {\ddot{o}}_X+{\lambda }_1\left(\ddot{o}[u{]}_{S_{g{Y}_0}}\right)\ddot{o}[u{]}_{S_{g{Y}_0}}$$

$$T\left(\ddot{o}u\right)\ddot{o}=A\left(\ddot{o}u\right)\ddot{o}+B\left(\ddot{o}u\right)\ddot{o}=y,\text{o¨ }y\in X^{\ast \ddot{o}}+Y,$$

$$\sup \ddot{o}\left\{\ddot{o}\frac{⟨\ddot{o}y,\text{o¨}u⟩\ddot{o}}{\Vert u\Vert {\ddot{o}}_X+[u{]}_{S_{g{Y}_0}}}:\text{o¨}u\in \ddot{o}{X}_0\right\}\ddot{o}<\infty \ddot{o}.$$

$$\underset{\Omega \ddot{o}}{\int }\left|\ddot{o}u\right|{\ddot{o}}^{\xi \ddot{o}\left(\ddot{o}x\right)\ddot{o}}dx\le \underset{\Omega \ddot{o}}{\int }\left|\ddot{o}u\right|{\ddot{o}}^{\zeta \ddot{o}\left(\ddot{o}x\right)}dx+\left|\ddot{o}\Omega \ddot{o}\right|\ddot{o},\text{o¨ }\forall \ddot{o}u\in \ddot{o}{L}^{\zeta \ddot{o}\left(x\right)\ddot{o}}\left(\ddot{o}\Omega \ddot{o}\right)\ddot{o}\ddot{o}$$

$$\underset{\Omega \ddot{o}}{\int }\left|\ddot{o}u\right|{\ddot{o}}^{\zeta \ddot{o}\left(x\right)\ddot{o}}{\left|\ddot{o}\ln \ddot{o}\left|\ddot{o}u\right|\ddot{o}\right|}^{\eta \ddot{o}}dx\le \ddot{o}{N}_1\underset{\Omega \ddot{o}}{\int }{\left|\ddot{o}u\right|}^{\zeta \ddot{o}\left(\ddot{o}x\right)\ddot{o}+ϵ\ddot{o}}dx+N_2\ddot{o}\ddot{o}$$

Full text

öSolvabilityöoföNonlinearöEllipticöTypeöEquation WithöTwoöUnrelatedöNonöstandardöGrowths

UğuröSertöandöKamalöSoltanovö††footnotetext: U.öSert(✉):öFacultyöoföScience,öDepartmentöoföMathematics, HacettepeöUniversity,ö06800,öBeytepe,öAnkara,öTurkey.öe-mail: usert@hacettepe.edu.tr K.öSoltanov:öFacultyöoföScience,öDepartmentöoföMathematics,öHacettepe University,ö06800,öBeytepe,öAnkara,öTurkey.öe-mail:ösoltanov@hacettepe.edu.tr

öAbstract.öInöthisöpaper,öweöstudyöthe solvabilityöofötheönonlinearöDirichletöproblemöwithösumöoföthe operatorsöoföindependentönonöstandardögrowths

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inöaöboundedödomainö$\Omega\"{o}\subset\"{o}\mathbb{R}^{n}$.öHere,öoneöof theöoperatorsöinötheösumöisömonotoneöandötheöotheröisöweakly compact.öWeöobtainösufficientöconditionsöandöshowötheöexistenceöof weakösolutionsöofötheöconsideredöproblemöbyöusingömonotonicityöand compactnessömethodsötogether.

öKeywords:öEllipticöPDEs,önonöstandard nonlinearity,övariableöexponent,ösolvabilityötheorem,öembedding theorems.

**AMSöSubjectöClassification:**ö35J60,ö35J66.

1 Introduction

Inöthisöwork,öweöinvestigateötheöDirichletöproblemöforötheönonlinear ellipticöequationöwithövariableönonlinearity

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whereö$x\in\Omega\"{o}\subset\"{o}\mathbb{R}^{n}\left(\"{o}n\geq\"{o}3\right)\"{o}$öis aöboundedödomainöwhichöhasösufficientlyösmoothöboundaryö(atöleast Lipschitzöboundary),ö$D_{i}\equiv\partial/\partial\"{o}x_{i}$, $p_{0},$ö$p_{1\text{\"{o}}}$areönonnegativeömeasurableöfunctions definedöonö$\Omega\"{o},$ö$h$öisöaögeneralizedöfunctionöandö$c:\Omega\times\mathbb{R}\rightarrow\mathbb{R}$,ö$c\left(\"{o}x,\tau\"{o}\right)\"{o}$öisöaöfunctionöwithövariableönonlinearityöinö$\tau\"{o}$ö(foröexample,ö$c\left(\"{o}x,u\right)\"{o}=c_{0}\left(\"{o}x\right)\"{o}\left|\"{o}u\right|\"{o}^{\alpha\left(\"{o}x\right)\"{o}-2}u+c_{1}\left(\"{o}x\right)\"{o}$,öseeöSectionö2).

Weödenoteötheöoperatorsö$A$öand$\ B$öwithö

ö$A\left(u\right)\"{o}:=-div\left(\left|\nabla\"{o}u\right|^{p_{1}\left(x\right)-2}\nabla\"{o}u\right),$ö$B\left(\"{o}u\right):=-\sum\limits_{i=1}^{n}D_{i}\left(\"{o}\left|\"{o}u\right|^{p_{0}\left(\"{o}x\right)\"{o}-2}D_{i}u\right)\"{o}+c\left(\"{o}x,u\right).\"{o}$

Thereöhasöbeenörecentlyöaöconsiderableöinterestöinötheöstudyöof equationsöandövariationalöproblemsöwithövariableöexponentsöof nonlinearitiesödueötoötheiröapplications.öNonlinearöequations includingötheöoperatorö$A\left(\"{o}u\right)\"{o}$öisöaöratheröcommon nonlinearöproblemöwithövariableöexponentöwhichöisöknownöas $p_{1}\left(\"{o}.\right)\"{o}$-Laplacianöequation.öThisökindöoföproblems haveöbeenöstudiedöinövariousöcontextsöbyömanyöauthorsö[1,ö3,ö4,ö6, 27]öandöhaveöwideörangeöoföapplicationöareasöinötheömathematical modelingöofönon-Newtonianöfluidsö[7,ö18,ö19],ötheoryöoföelasticity andöhydrodynamicsö[28],öthermistoröproblemö[26]öandöinöimage restorationö[9]öetc.ö(Indeed,öallöapplicationöareasömentioned aboveöareöalsoövalidöforöproblemö(1.1)).öOnötheöotheröhand,öthe equationsöofötheötypeö$B\left(\"{o}u\right)\"{o}=h$öareörarelyöresearched. Foröinstanceöinö[5],öaösimilarötypeöoföproblemöforö$B\left(u\right)\"{o}=h$öwasöstudiedöbyöAntontsevöandöShmarevöwhoöinvestigated theöregularizedöproblemötoöshowötheöexistenceöoföweakösolution. Suchöequationsömayöappearöinötheömathematicalömodelingöoföthe processöofönonstableöfiltrationöoföanöidealöbarotropicögasöinöa nonhomogeneousöporousömediumö(forösampleöseeö[3]).öWeöalsoörefer [14,ö17,ö19]öforötheöseveralöofötheömostöimportantöapplicationsöof (1.1)öandönonlinearöpartialödifferentialöequationsöwithövariable exponentöariseöfromömathematicalömodelingöofösuitableöprocessesöin mechanics,ömathematicalöphysics,öimageöprocessingöetc.

Toötheöbestöoföouröknowledge,öbyönowöthereöhasönotöbeenöany studiesöonötheöexistenceöofösolutionsöforötheöellipticöequations ofötheötypeö(1.1)öwithövariableöexponentsöofönonlinearity. However,öweönoteöthatöaösimilaröproblemötoö(1.1)öwithöconstant exponentsöwasöinvestigatedöinö[24].öMoreöexactly,öinö[24]öthe questionöofötheösolvabilityöoföanöoperatoröequationöwhenötheöcase theöoperatoröisöinötheöformöofösumöoföaöweaklyöcompactöand pseudo-monotoneöoperatorsöwasöanswered.öInötheöpresentöpaper,öwe studyötheösimilarötypeöoföoperatoröequationöinöaömodelöproblem whenötheöoperatorsöinötheösumöwithövariableönonlinearityöand obtainötheösufficientöconditionsöforösolvability.öTheöourögoalöof studyingötheömodelöproblemöisötoöprovideöaömoreöunderstandableöand explicitöwayöforötheöestablishedöresultsöinöthisöarticle.

Theömainöfeatureöofötheöequationö$A\left(\"{o}u\right)\"{o}+B\left(u\right)\"{o}=h$öisöthat,ötheöexponentsö$p_{0}\left(\"{o}x\right)\"{o}$öand $p_{1}\left(\"{o}x\right)\"{o}$öareöindependentöoföeachöother.öThus, neitherö$A$önorö$B\"{o}$öisötheömainöpartöoföthisöequation.öItöisöneed toönoteöthatöifö$A$öisötheömainöpartöofötheöequation,öi.eöthe exponentsöareödependentöeachöother,ötheöresultsöforötheötheoryöof pseudo-monotoneöoperatorsöcanöbeöusedötoöinvestigateötheöproblem. However,öinötheöcaseöthatöweöconsider,öanyömethodsöwhichöisömerely relatedötoömonotonicityöcanönotöbeöused.

Weöuseötheöbasicögeneralösolvabilityötheoremö[24],ö(Theoremö2.5)ötoöprove theöexistenceöoföweakösolutionöofötheöproblemö(1.1).öInöorderötoöapplyöthis theoremötoöexistenceötheoremö(Theoremö2.4)öforöproblemö(1.1),öweöobtain sufficientöconditionsöandöproveötheömonotonicityöofötheöoperatorö$A$öand weaköcompactnessöofö$B$öonöproperöspacesöunderötheseöconditions,öandöthenöwe getötheösolvabilityöoföposedöproblemöbyösimultaneouslyöusingömonotonicity andöcompactness.

Thisöpaperöisöorganizedöasöfollows:öInötheönextösection,öwe presentötheöassumptions,ödefinitionöofötheöweakösolutionöand descriptionöofötheömainöresult.öForöthisöpurpose,öweöalsoödefine someöfunctionöclassesöwhichöareörequiredötoöstudyötheöposed problem.öInöSectionö3,öfirstly,öweöestablishösomeöintegral inequalitiesötoöinvestigateötheöfunctionöclassesö(pseudo-norm spaces)ödefinedöinöpreviousösectionöandöafterwardsöverifyösome necessaryölemmasöandötheoremsöwhichöindicateötheörelationöoföthese spacesöwithötheöLebesgueöandöSobolevöspacesöwithövariableöexponent andötheöcontinuousöandöcompactöembeddingsöofötheseöfunctionöspaces etc.öInöSectionö4,öweögiveötheöprooföofötheömainötheoremö(Theorem 2.4)öoföthisöpaperöbyötheöhelpöofötheöembeddingöresultsöobtained inöSectionö3.

2 StatementöoföTheöProblemöandöTheöMainöResult

Letö$\Omega\"{o}\subset\"{o}\mathbb{R}\"{o}^{n}\left(\"{o}n\geq\"{o}3\right)\"{o}$öbeöaöbounded domainöwithösufficientlyösmoothöboundaryö$\partial\Omega.$öWeöstudyöthe problemö(1.1)

[TABLE]

underötheöfollowingöconditions:

1. (U1)

ö$2\leq\"{o}p_{0}^{-}\leq\"{o}p_{0}\left(\"{o}x\right)\"{o}\leq p_{0}^{+}<\infty\"{o},$ö$1<\"{o}p_{1}^{-}\leq\"{o}p_{1}\left(\"{o}x\right)\"{o}\leq p_{1}^{+}<\infty\"{o}$öandö$p_{0}\in\"{o}C^{1}\left(\bar{\Omega}\right)\"{o},$ö$p_{1}\in\"{o}C^{0}\left(\"{o}\bar{\Omega}\right).$ 2. (U2)

öThereöexistsöaömeasurableöfunctionö$\alpha\"{o}:\Omega\"{o}\longrightarrow\"{o}\left[\"{o}1,\infty\"{o}\right)\"{o},$ö$1\leq\"{o}\alpha^{-}\leq\"{o}\alpha\"{o}\left(\"{o}x\right)\"{o}\leq\"{o}\alpha\"{o}^{+}<\infty\"{o}$ösuchöthat theöfollowingöinequalitiesöhold

[TABLE]

and

[TABLE]

a.e.ö$\left(\"{o}x,\tau\"{o}\right)\"{o}\in\"{o}\Omega\"{o}\times\mathbb{R}.$

Hereö$c\left(\"{o}x,\tau\"{o}\right)\"{o}$öisöaöCaratheodoryöfunctionöandö$c_{i}\"{o}$,ö$i=0,1,2$öareönonnegativeömeasurableöfunctionsödefinedöonö$\Omega\"{o}$,öbesidesöforö$\varepsilon\"{o}>0,$ö$\alpha\"{o}\left(\"{o}x\right)\geq\"{o}p_{0}\left(\"{o}x\right)\"{o}+\varepsilon\"{o}$öandö$c_{2}\left(\"{o}x\right)\geq\"{o}\tilde{C}>0$,ö$x\in\"{o}\Omega\"{o}$öisösatisfied.

Inöorderötoögiveötheödefinitionöoföweakösolutionöofötheöproblem (1.1),öweöintroduceötheörequiredöspaces.öForöthis,öfirstöweöremind someöbasicöfactsöaboutögeneralizedöLebesgueöandöSobolevöspacesö[2, 8,ö10,ö12,ö15].

Letö$\Omega\"{o}$öbeöaöLebesgueömeasurableösubsetöofö$\mathbb{R}^{n}$ösuchöthatö$\left|\"{o}\Omega\"{o}\right|\"{o}>0$ö(Throughoutötheöpaper,öwe denoteöbyö$\left|\"{o}\Omega\"{o}\right|\"{o}$ötheöLebesgueömeasureöofö$\Omega\"{o}$).öByö$M\left(\"{o}\Omega\"{o}\right)\"{o}$ödenoteötheöfamilyöoföall measurable functionsö$p:\Omega\"{o}\longrightarrow\"{o}\left[\"{o}1,\infty\"{o}\right]\"{o}$öandöbyö$M_{0}\left(\"{o}\Omega\"{o}\right)\"{o}$,

[TABLE]

whereö$p^{-}:=\underset{\Omega\"{o}}{ess}\inf\"{o}\left|\"{o}p\left(\"{o}x\right)\"{o}\right|,\text{\"{o}}\"{o}p^{+}:=\underset{\Omega\"{o}}{ess}\sup\"{o}\left|\"{o}p\left(\"{o}x\right)\right|$.ö

öForö$p\in\"{o}$ö$M\left(\"{o}\Omega\"{o}\right)\"{o},$ö$\Omega_{\infty\"{o}}^{p}\equiv\"{o}\Omega\"{o}_{\infty\"{o}}\equiv\"{o}\left\{\"{o}x\in\"{o}\Omega|\text{\"{o}}p\left(\"{o}x\right)\"{o}=\infty\"{o}\right\}\"{o}$öthenöonötheösetöof allöfunctionsöonö$\Omega\"{o}$ödefineöthe functionalö$\sigma\"{o}_{p}$öandö$\left\|\"{o}.\right\|\"{o}_{p}$öby

[TABLE]

and

[TABLE]

Clearlyöifö$p\in\"{o}L^{\infty\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o}$öthenö$p\in M_{0}\left(\"{o}\Omega\"{o}\right)\"{o}$,ö$\sigma\"{o}_{p}\left(\"{o}u\right)\"{o}\equiv\int\limits_{\Omega\"{o}}\left|\"{o}u\right|\"{o}^{p\left(\"{o}x\right)}dx$öandötheögeneralized Lebesgueöspaceöisödefinedöasöfollows:

[TABLE]

Ifö$p^{-}>1,$öthenötheöspaceö$L^{p\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\right)\"{o}$öbecomesöaöreflexiveöandöseparableöBanachöspaceöunderöthe normö$\left\|\"{o}.\right\|\"{o}_{L^{p\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\right)\"{o}}$whichöisöso-calledöLuxemburgönorm.ö

öIfö$0<\left|\Omega\"{o}\right|\"{o}<\infty\"{o},$öand $p_{1},$ö$p_{2}\in\"{o}M\left(\"{o}\Omega\"{o}\right)\"{o}$öthenötheöcontinuousöembeddingöö$L^{p_{1}\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o}\subset\"{o}L^{p_{2}\left(x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o}$öexists$\iff\"{o}$ö$p_{2}\left(\"{o}x\right)\"{o}\leq p_{1}\left(\"{o}x\right)\"{o}$öforöa.eö$x\in\"{o}\Omega\"{o}.$

Forö$u\in\"{o}L^{p\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o}$öandö$v\in L^{q\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o}$öwhereö$p,$ö$q\in\"{o}$ $M_{0}\left(\"{o}\Omega\"{o}\right)\"{o}$öandö$\frac{1}{p\left(\"{o}x\right)}+\frac{1}{q\left(\"{o}x\right)\"{o}}=1$ötheöfollowingöinequalitiesöholds

[TABLE]

and

[TABLE]

Letö$\Omega\"{o}\subset\mathbb{R}^{n}$öbeöaöboundedödomainöandö$p\in\"{o}L^{\infty\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o}$öthen generalizedöSobolevöspaceöisödefinedöasöfollows:

[TABLE]

andöthisöspaceöisöaöseparableöBanachöspaceöunderötheönorm:

[TABLE]

$W_{0}^{1,\text{\"{o}}p\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o}$ödefinesöasöthe closureöofö$C_{0}^{\infty\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o}$öinö$W^{1,\text{\"{o}}p\left(x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o}.$öIfö$p^{-}>1$öthenö$W^{1,\text{\"{o}}p\left(x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o}$öandö$W_{0}^{1,\text{\"{o}}p\left(\"{o}x\right)}\left(\"{o}\Omega\"{o}\right)\"{o}$öareöreflexiveöandöseparableöBanachöspaces.öIfö$\partial\"{o}\Omega\"{o}$öisöLipschitzöboundary  andö$p\in\"{o}C^{0}\left(\"{o}\bar{\Omega}\right)\"{o},$öthenöequivalentönormöinö$W_{0}^{1,\text{\"{o}}p\left(\"{o}x\right)}\left(\"{o}\Omega\"{o}\right)\"{o}$öisögivenöby;

[TABLE]

Letö$p$,ö$q\in\"{o}C\left(\"{o}\bar{\Omega}\right)\"{o}\cap\"{o}M_{0}\left(\"{o}\Omega\right)\"{o}$ andö$p\left(\"{o}x\right)\"{o}<n,$ö$q\left(\"{o}x\right)\"{o}<\frac{np\left(\"{o}x\right)\"{o}}{n-p\left(\"{o}x\right)\"{o}}\equiv\"{o}p^{\ast\"{o}}\left(\"{o}x\right)\"{o}$öisöholdöforöallö$x\in\Omega\"{o},$öthenöthereöisöaöcontinuousöandöcompactöembeddingö$W^{1,\text{\"{o}}p\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o}\hookrightarrow\"{o}L^{q\left(\"{o}x\right)}\left(\"{o}\Omega\"{o}\right)\"{o}.$

Aöfunctionö$p\in\"{o}M_{0}\left(\"{o}\Omega\"{o}\right)\"{o}$öisöcalled log-Hölderöcontinuousöiföthereöisöaöconstantö$L$ösuchöthatöthe inequality

[TABLE]

holds.öIfö$p$öisölog-Hölderöcontinuousöandö$q\in\"{o}M_{0}\left(\Omega\"{o}\right)\"{o}$öthenöweöhaveötheöcontinuousöembedding $W^{1,\text{\"{o}}p\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o}\subset L^{q\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o}$öforöallö$q\leq p^{\ast\"{o}}.$ö

öForömoreödetailsöandöembeddingöresultsöfor theseöspacesöseeö[2,ö8,ö10-12,ö15].ö

Weönowödefineösomeöfunctionöclassesöwhichöareörequiredötoöstudyötheöproblem (1.1).öTheseöclassesöareönonlinearöspacesöwhichöareötheögeneralizationöof theönonlinearöspacesöwithöconstantöexponentöstudiedöinö[21-24](seeöalso referencesöoföthem).öWeöalsoönoteöthatötheönecessaryöpropertiesöoföthese spacesöareöpresentedöinöSectionö3.

Definition 2.1

Letö$\Omega\"{o}\subset\mathbb{R}^{n}\left(\"{o}n\geq\"{o}2\right)\"{o}$öbeöaöboundedödomainöwithöLipschitz boundaryöandö$\gamma\"{o},$ö$\beta\"{o}$ö$\in\"{o}P_{0}\left(\"{o}\Omega\"{o}\right).$öWeöintroduceö$S_{1,\gamma\"{o}\left(\"{o}x\right)\"{o},\beta\"{o}\left(x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o},$ötheöclassöoföfunctions $u:\Omega\"{o}\rightarrow\mathbb{R}$öandötheöfunctionalö$[.]_{S_{\gamma\"{o},\beta\"{o}}}:S_{1,\gamma\"{o}\left(\"{o}x\right),\beta\"{o}\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o}\longrightarrow\mathbb{R}_{+}$öasöfollows:

[TABLE]

[TABLE]

$[.]_{S_{\gamma\"{o},\beta\"{o}}}$ödefinesöaöpseudo-normöonö$S_{1,\gamma\left(\"{o}x\right)\"{o},\beta\"{o}\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o},$ actuallyöitöcanöbeöclearlyöseenöthatö$[.]_{S_{\gamma\"{o},\beta\"{o}}}$ fulfillsöallöconditionsöof pseudo-normö(pn)öseeö[21]öi.e.ö$[u]_{S_{\gamma\"{o},\beta\"{o}}}\geq\"{o}0,$ö$u=0\Rightarrow\"{o}[\"{o}u]_{S_{\gamma\"{o},\beta\"{o}}}=0,$ö$[u]_{S_{\gamma\"{o},\beta}}\neq\"{o}[\"{o}v]_{S_{\gamma\"{o},\beta\"{o}}}\Rightarrow\"{o}u\neq\"{o}v$öandö$[u]_{S_{\gamma\"{o},\beta\"{o}}}=0\Rightarrow\"{o}u=0.$

Letö$S_{1,\gamma\"{o}\left(\"{o}x\right)\"{o},\beta\"{o}\left(\"{o}x\right)\"{o}}\left(\Omega\"{o}\right)\"{o}$öbeötheöspaceögivenöinötheöDefinitionö2.1öand $\theta\"{o}\left(\"{o}x\right)\in\"{o}M_{0}\left(\"{o}\Omega\"{o}\right)$,öweödenote $S_{1,\gamma\"{o}\left(\"{o}x\right)\"{o},\beta\"{o}\left(\"{o}x\right)\"{o},\theta\"{o}\left(x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o},$ötheöclassöoföfunctions $u:\Omega\"{o}\rightarrow\"{o}\mathbb{R}\"{o}$öbyötheöfollowingöintersection

[TABLE]

withötheöpseudo-norm

[TABLE]

Weönowöstateöaöpropositionöwhichöcanöbeöeasilyöprovedöbyötheöembedding resultsöforötheöLebesgueöspacesöwithövariableöexponent.

Proposition 2.2

Ifö$\gamma\"{o}$,ö$\beta\"{o}$,ö$\theta\"{o}$ö$\in\"{o}M_{0}\left(\"{o}\Omega\"{o}\right)\"{o}$öandö$\theta\"{o}\left(\"{o}x\right)\"{o}\geq\"{o}\gamma\"{o}\left(\"{o}x\right)\"{o}+\beta\"{o}\left(\"{o}x\right)+\varepsilon\"{o}_{0}$öa.e.ö$x\in\"{o}\Omega\"{o}$öforösomeö$\varepsilon\"{o}_{0}>0,$öthen weöhaveötheöfollowingöequivalence;

[TABLE]

withötheöpseudo-norm

[TABLE]

öAlsoödenoteötheödualöspaces,ö$W^{-1,\text{\"{o}}q_{0}\left(\"{o}x\right)}\left(\"{o}\Omega\"{o}\right)\"{o}:=\left(\"{o}W_{0}^{1,\text{\"{o}}p_{0}\left(\"{o}x\right)}\left(\"{o}\Omega\"{o}\right)\"{o}\right)\"{o}^{\ast\"{o}},$ö$W^{-1,\text{\"{o}}q_{1}\left(x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o}:=\left(\"{o}W_{0}^{1,\text{\"{o}}p_{1}\left(x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o}\right)\"{o}^{\ast\"{o}}$öandö$L^{\alpha\"{o}^{\prime}\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o}:=\left(\"{o}L^{\alpha\"{o}\left(\"{o}x\right)}\left(\"{o}\Omega\"{o}\right)\"{o}\right)\"{o}^{\ast\"{o}},$öhereö$q_{0}\left(\"{o}x\right)\"{o}:=\frac{p_{0}\left(\"{o}x\right)\"{o}}{p_{0}\left(\"{o}x\right)\"{o}-1},\"{o}$ö$q_{1}\left(\"{o}x\right)\"{o}:=\frac{p_{1}\left(\"{o}x\right)\"{o}}{p_{1}\left(\"{o}x\right)\"{o}-1}$öandö$\alpha\"{o}^{\prime}\left(\"{o}x\right)\"{o}:=\frac{\alpha\"{o}\left(\"{o}x\right)\"{o}}{\alpha\"{o}\left(\"{o}x\right)\"{o}-1}.\"{o}$

Weöinvestigateötheöproblemö(1.1)öforöfunctionsö$h\in\"{o}W^{-1,\text{\"{o}}q_{0}\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o}+L^{\alpha\"{o}^{\prime\"{o}}\left(x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o}+W^{-1,\text{\"{o}}q_{1}\left(\"{o}x\right)\"{o}}\left(\Omega\"{o}\right)\"{o}.$öLetöusödenoteö$Q\left(\"{o}\Omega\"{o}\right)$öby

[TABLE]

whereö$\mathring{S}_{1,q_{0}\left(\"{o}x\right)\"{o}\left(\"{o}p_{0}\left(x\right)\"{o}-2\right)\"{o},q_{0}\left(\"{o}x\right)\"{o},\alpha\"{o}\left(\"{o}x\right)}\left(\"{o}\Omega\"{o}\right):=\left\{\"{o}u\in\"{o}S_{1,q_{0}\left(\"{o}x\right)\left(\"{o}p_{0}\left(\"{o}x\right)\"{o}-2\right)\"{o},q_{0}\left(\"{o}x\right),\alpha\"{o}\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o}:\"{o}u\mid\"{o}_{\partial\Omega\"{o}}=0\right\}\"{o}.$öNoticedöthatötheöconditionöonö$\alpha\"{o}\left(x\right)$öinö*(U2)*öindicatesöthatöthatöPropositionö2.2öis validöforö$\mathring{S}_{1,q_{0}\left(\"{o}x\right)\"{o}\left(\"{o}p_{0}\left(x\right)\"{o}-2\right)\"{o},q_{0}\left(\"{o}x\right)\"{o},\alpha\"{o}\left(\"{o}x\right)}\left(\"{o}\Omega\"{o}\right)$.

öWeöareöreadyötoögiveötheödefinitionöofötheöweak solutionöoföproblemö(1.1).ö

Definition 2.3

Ifötheöfunctionö$u\in\"{o}Q\left(\"{o}\Omega\"{o}\right)\"{o}$ösatisfiesötheöfollowing equality

[TABLE]

foröeveryö$v\in\"{o}W_{0}^{1,\text{\"{o}}p_{0}\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\cap\"{o}W_{0}^{1,\text{\"{o}}p_{1}\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o}\cap L^{\alpha\"{o}\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o},$öthenötheöfunctionö$u$öis calledötheöweakösolutionöofötheöproblemö(1.1).

öNoteöthatöitöisöclearöunderötheöconditionsö*(U1)* andö*(U2)*,öallötheöintegralsöinö(2.4)ömakeösense.

öNowöweöstateötheömainötheoremöoföthisöarticleöthatöis theösolvabilityötheoremöforöproblemö(1.1):ö

Theorem 2.4

(ExistenceöTheorem)öLetötheöconditions (U1)-(U2) fulfillöandö$c_{0},\"{o}$ö$c_{2}\in\"{o}L^{\infty\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o},$ö$c_{1}\in\"{o}$ö$L^{\alpha\"{o}^{\prime\"{o}}\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o}.$ Thenöforöeveryö$h\in\"{o}$ö$W^{-1,\text{\"{o}}q_{0}\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\right)\"{o}+L^{\alpha\"{o}^{\prime\"{o}}\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o}+W^{-1,\text{\"{o}}q_{1}\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o},$öproblemö(1.1)öhasöa weakösolutionöinötheöspaceö$Q\left(\"{o}\Omega\"{o}\right)\"{o}.$

Weöwillöuseötheöfollowingösolvabilityötheoremö[24]ötoöproveötheöTheoremö2.4. Letö$X$öandö$Y_{0}$öbeöreflexiveöBanachöspaces,ö$Y$öisöanöarbitraryöBanach spaceöandö$S_{gY_{0}}$öisöpn-space(pseudo-normöspace)ö[21].öLetö$A:X\longrightarrow\"{o}X^{\ast\"{o}}$öandö$B:S_{gY_{0}}\longrightarrow\"{o}Y$öbe nonlinearöoperators.

öAssumeöthatötheöfollowingöconditionsöareösatisfied:

ö$A:X\longrightarrow\"{o}X^{\ast\"{o}}$öisöaöpseudo-monotoneöoperator,öi.e.,

1. (i)

ö$A\"{o}$öisöboundedöoperatoröand 2. (ii)

ötheöconditionsö$u_{m}\overset{X}{\rightharpoonup\"{o}}u_{0}$öand limsup$\left\langle\"{o}A\left(\"{o}u_{m}\right)\"{o},\text{\"{o}}u_{m}-u\right\rangle\"{o}\leq 0\"{o}$öimply

[TABLE] 2. 2)

ö$B:S_{gY_{0}}\longrightarrow\"{o}Y$öisöweaklyöcompact.öFurthermore, thereöexistsöaömappingö$B_{0}:X_{0}\cap\"{o}S_{gY_{0}}\longrightarrow Y_{2}\subseteq\"{o}Y$ösuchöthatö$B_{0}$öisöweaklyöcompactöfromö$X_{0}\cap S_{gY_{0}}$ötoö$Y_{2}$öwhereö$X_{0}$öisöaöseparableötopologicalövectoröspace whichöisödenseöinö$X,$ö$Y^{\ast\"{o}}$öandö$S_{gY_{0}}$öandöthereöexistsöa continuousönondecreasingöfunctionö$\varphi\"{o}:\mathbb{R}_{+}^{1}\longrightarrow\mathbb{R}_{+}^{1},$ösuchöthatö$\varphi\"{o}\in\"{o}C^{0}$öand

[TABLE] 3. 3)

öTheöoperatorö$T:=A+B$öisöcoerciveöinötheögeneralizedösenseöonö$X_{0},$öi.e.öforöeachö$u\in\"{o}X_{0}$öwithö$\left\|\"{o}u\right\|\"{o}_{X}$,ö$[u]_{S_{gY_{0}}}\geq\"{o}M$öweöhave

[TABLE]

whereö$\lambda\"{o}_{0},$ö$\lambda\"{o}_{1}\in\"{o}C^{0}$ösuchöthatöasö$\tau\"{o}\nearrow\infty\"{o},$ö$\lambda\"{o}_{0}\left(\"{o}\tau\"{o}\right)\"{o},$ö$\lambda\"{o}_{1}\left(\"{o}\tau\right)\"{o}\nearrow\"{o}\infty\"{o}$öandö$M>0$öisösomeönumber.

Theorem 2.5

$[24]$öLetöconditionsö1)-3)öbeösatisfied.öThenötheöequation

[TABLE]

isösolvableöinö$X\cap\"{o}S_{gY_{0}}$öforöanyö$y\in\"{o}X^{\ast\"{o}}+Y$ösatisfying

[TABLE]

3 PreliminaryöResults

Inöthisösection,öweöstudyötheöfunctionöclassesöwhichöareödefined inöSectionö2öthatöactuallyöisörequiredötoöinvestigateötheöproblem (1.1).öFirst,öweöestablishösomeöintegralöinequalitiesötoörealize theöstructureöofötheseöspaces.öAfterwards,öweöshowöthatöthese spacesöareöcompleteömetricöspaces.öMoreover,öweöproveösomeölemmas andötheoremsöonöcontinuousöandöcompactöembeddingöetc.öforöthese spacesöandöalsoöindicateötheirörelationöwithötheöLebesgueöand Sobolevöspacesöwithövariableöexponent.

3.1 SomeöIntegralöInequalities

Inöthisösubsection,öweöderiveösomeöinequalitiesöwhichöareögivenöasölemmas. AsötheöproofsöofötheseölemmasöcanöbeöobtainedöeasilyöbyöusingöYoung’s,öHölderöinequalitiesöandöbyöcalculationsö(andöalsoöseeö[20,öLemmaö4.1])öhence, weöskipötheöproofsöforötheösakeöoföbrevity.öThroughoutöthisösectionöwe assumeöthatö$\Omega\"{o}\subset\mathbb{R}^{n}\left(\"{o}n\geq\"{o}2\right)\"{o}$öisöaöboundedödomainöwithöLipschitzöboundary.

Lemma 3.1

Letö$\zeta\"{o}$,ö$\xi$ö$\in\"{o}M_{0}\left(\"{o}\Omega\"{o}\right)$öandö$\zeta\left(x\right)\"{o}\geq\"{o}\xi\"{o}\left(\"{o}x\right)\"{o}$öa.e.ö$x\in\"{o}\Omega\"{o}.$öThenötheöinequality

[TABLE]

holds.

Lemma 3.2

Assumeöthatö$\zeta\"{o}$ö$\in\"{o}M_{0}\left(\"{o}\Omega\"{o}\right)$öandöthe numbersö$\eta$öandö$\epsilon$ösatisfyö$\eta\geq\"{o}1,$ö$\epsilon\"{o}>0.$ Thenöforöeveryö$u\in L^{\zeta\"{o}\left(\"{o}x\right)\"{o}+\epsilon\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o}$

[TABLE]

isösatisfied.öHereö$N_{1}\equiv\"{o}N_{1}\left(\"{o}\epsilon\"{o},\eta\"{o}\right)\"{o}>0$öandö$N_{2}\equiv\"{o}N_{2}\left(\"{o}\epsilon\"{o},\eta\"{o},\left|\"{o}\Omega\right|\"{o}\right)\"{o}>0$öareöconstants.

Corollary 3.3

Letö$\zeta\"{o}$,ö$\eta\"{o}$ö$\in\"{o}M_{0}\left(\"{o}\Omega\"{o}\right)$.öThenöfor $\epsilon>0,$ötheöinequality

[TABLE]

holds.öHereö$N_{3}\equiv\"{o}N_{3}\left(\"{o}\epsilon\"{o},\text{\"{o}}\eta^{+}\right)\"{o}>0$öandö$N_{4}\equiv\"{o}N_{4}\left(\"{o}\epsilon\"{o},\"{o}\eta\"{o}^{+},\left|\"{o}\Omega\"{o}\right|\"{o}\right)\"{o}>0$öareöconstants.

3.2 GeneralizedöNonlinearöSpacesöandöEmbedding

Theorems

Inöthisösection,öweöexamineötheöpropertiesöofötheöspacesö$S_{1,\gamma\"{o}\left(x\right)\"{o},\beta\"{o}\left(\"{o}x\right)\"{o},\theta\"{o}\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\right)\"{o}$öandötheiröconnectionöwithötheöknownöspaces.öInvestigatingömostöof boundaryövalueöproblemsöonöitsöownöspaceöleadsötoöobtainöbetteröresults. Henceforthöconsideredöproblemö(1.1)öisöinvestigatedöonöitsöownöspaceö(i.e.ö$S_{1,\gamma\"{o}\left(\"{o}x\right)\"{o},\beta\"{o}\left(\"{o}x\right)\"{o},\theta\"{o}\left(\"{o}x\right)}\left(\"{o}\Omega\"{o}\right)\"{o}$).öUnlikeölinearöboundaryövalueöproblems,ötheösets generatedöbyönonlinearöproblemsöareösubsetsöofölinearöspaces,öbutönot possessingötheölinearöstructureö[21-24].

Noteöthat,öfromönowöonöunlessöadditionalöconditionsöareöimposed,öallöthe functionsö$\gamma\"{o},$ö$\beta\"{o}$öandö$\theta\"{o}$öwillösatisfyötheöconditions givenöinöPropositionö2.2.

Lemma 3.4

Letö$u\in\"{o}$ö$S_{1,\gamma\"{o}\left(\"{o}x\right)\"{o},\beta\"{o}\left(\"{o}x\right)\"{o},\theta\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o}$öandö$\lambda\"{o}_{u}:=[u]_{S_{\gamma,\beta\"{o},\theta\"{o}}}$öthenötheöfollowingöinequality

[TABLE]

holds.

**Proof.**öForö$u\in\"{o}$ö$S_{1,\gamma\"{o}\left(\"{o}x\right)\"{o},\beta\"{o}\left(\"{o}x\right)\"{o},\theta\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o},$

[TABLE]

[TABLE]

ifö$\lambda\"{o}_{u}\geq\"{o}1,$öweöhave

[TABLE]

[TABLE]

fromötheödefinitionöofö$[.]_{S_{\gamma\"{o},\beta\"{o}},\theta\"{o}}$öandölast inequality,öweöobtain

[TABLE]

Obviously,öifö$0<\lambda\"{o}_{u}<1$öthenö$\Re\"{o}^{\gamma\"{o},\beta\"{o},\theta\"{o}}\left(u\right)\"{o}\geq\"{o}\lambda\"{o}_{u}^{\theta\"{o}^{+}}.$öSimilarly,öweöcanöshowötheöother sideöofötheöinequalityöthus,ötheöprooföisöcomplete.

Theorem 3.5

Assumeöthatö$p\"{o}\in\"{o}M_{0}\left(\"{o}\Omega\"{o}\right)$öandö$p\left(x\right)\"{o}\geq\theta\"{o}\left(\"{o}x\right)\"{o}$öa.e.ö$x\in\"{o}\Omega\"{o}.$öThen,öweöhaveötheöembedding

[TABLE]

**Proof.**öLetö$u\in\"{o}W^{1,\text{\"{o}}p\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o},$ asöaöconsequenceöoföLemmaö3.4ötoöobtainötheöembeddingö(3.4)öitöis sufficientötoöshowöthatö$\Re\"{o}^{\gamma\"{o},\beta\"{o},\theta\"{o}}\left(u\right)\"{o}$öisöfiniteö(i.e.ö$\Re\"{o}^{\gamma\"{o},\beta\"{o},\theta\"{o}}\left(u\right)\"{o}<\infty$)

[TABLE]

byöLemmaö3.1öandöusingöYoung’söinequalityöweöget

[TABLE]

estimatingötheöthirdöintegralöonötheörightösideöofötheölastöinequalityöby usingöLemmaö3.1,öweöobtain

[TABLE]

thus,öweögetötheödesiredöresultöfromötheölastöinequalityöandö(2.2).

Ifö$p\left(\"{o}x\right)\"{o}=\theta\"{o}\left(\"{o}x\right)\"{o}$öa.e.ö$x\in\"{o}\Omega,$öbyöusingötheösameöoperationsöasödoneöaboveöweöcanöobtainö(3.4).

öWeöomitötheöprooföofötheöfollowingölemmaöasöitöisöstraightforward.

Lemma 3.6

Letö$\gamma\"{o},$ö$\beta\"{o}:\Omega\"{o}\longrightarrow\"{o}\left[\"{o}1,\text{\"{o}}\infty\right)\"{o}$öbeöfunctionsösatisfyingö$1\leq\"{o}\gamma\"{o}^{-}\leq\"{o}\gamma\"{o}\left(x\right)\"{o}\leq\"{o}\gamma\"{o}^{+}<\infty\"{o}$,ö$1\leq\"{o}\beta\"{o}^{-}\leq\"{o}\beta\"{o}\left(x\right)\"{o}\leq\"{o}\beta\"{o}^{+}<\infty\"{o}$öa.e.ö$x\in\"{o}\Omega\"{o}$öandö$\gamma\"{o},$ö$\beta\in\"{o}C^{1}\left(\"{o}\bar{\Omega}\right)\"{o}$.öThenötheöfunctionö$\varphi\"{o}:\Omega\times\mathbb{R}\longrightarrow\mathbb{R},$ö$\varphi\"{o}\left(\"{o}x,t\right)\"{o}:=\left|\"{o}t\right|\"{o}^{\frac{\gamma\left(\"{o}x\right)\"{o}}{\beta\"{o}\left(\"{o}x\right)\"{o}}}t$ösatisfiesötheöfollowing:

1. (i)

öForöeveryöfixedö$x_{0}\in\"{o}\Omega\"{o},$ö$\varphi\"{o}\left(x_{0},.\right)\"{o}:\mathbb{R}\longrightarrow\mathbb{R}$öisöcontinuouslyödifferentiable;öhasöanöinverseöandöinverseöfunctionöis alsoöcontinuouslyödifferentiable. 2. (ii)

öForöeveryöfixedö$t_{0}\in\mathbb{R}-\left\{\"{o}0\right\}\"{o},$ö$\varphi\"{o}$öandö$\varphi\"{o}^{-1}$öisöcontinuousöonö$\Omega\"{o},$öandöforö$\forall\"{o}i=\overline{1,n},$öpartialöderivativesö$\varphi_{x_{i}}\left(\"{o}x,t_{0}\right)\"{o},$ö$\varphi\"{o}_{x_{i}}^{-1}\left(\"{o}x,t_{0}\right)$öexistöandöareöcontinuous.

Definition 3.7

Letö$\eta\"{o}$ö$\in\"{o}M_{0}\left(\"{o}\Omega\"{o}\right),$öweöintroduceö$L^{1,\text{\"{o}}\eta\"{o}\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)$,ötheötheöclassöoföfunctionsö$u:\Omega\"{o}\rightarrow\"{o}\mathbb{R}\"{o}$

[TABLE]

Theorem 3.8

Letötheöfunctionsö$\gamma\"{o},$ö$\beta\"{o}$öandö$\varphi\"{o}$ösatisfyöthe conditionsöoföLemmaö3.6öandö$L^{1,\text{\"{o}}\beta\"{o}\left(\"{o}x\right)}\left(\"{o}\Omega\"{o}\right)\"{o}$öbeötheöspaceögivenöinöDefinitionö3.7. Then,ö$\varphi\"{o}$öisöaöbijectiveömappingöbetweenö$S_{1,\gamma\left(\"{o}x\right)\"{o},\beta\"{o}\left(x\right)\"{o},\theta\"{o}\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o}$öandö$L^{1,\text{\"{o}}\beta\"{o}\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o}\cap\"{o}L^{\psi\"{o}\left(\"{o}x\right)}\left(\"{o}\Omega\"{o}\right)\"{o}$öwhereö$\psi\"{o}\left(\"{o}x\right)\"{o}:=\frac{\theta\"{o}\left(x\right)\"{o}\beta\"{o}\left(\"{o}x\right)\"{o}}{\gamma\"{o}\left(\"{o}x\right)\"{o}+\beta\"{o}\left(x\right)\"{o}}.$

**Proof.**öFirstöletöusöverifyöthatöforöeveryö$u\in\"{o}S_{1,\gamma\"{o}\left(\"{o}x\right)\"{o},\beta\left(\"{o}x\right)\"{o},\theta\"{o}\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o},$

[TABLE]

toöshowöthis,öfromötheödefinitionöofötheöspacesö$L^{1,\text{ }\beta\"{o}\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o}$öandö$L^{\psi\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o},$öitöisösufficientöto proveöthatö$\forall\"{o}i=\overline{1,n},$ö$\sigma\"{o}_{\beta\"{o}}\left(D_{i}v\right)\"{o}$öandö$\sigma\"{o}_{\psi\"{o}}\left(\"{o}v\right)\"{o}$öareöfinite. As

[TABLE]

theöaboveöequationöensuresöthatö$\sigma\"{o}_{\psi\"{o}}\left(\"{o}v\right)\"{o}$öisöfinite.

öNowöforö$\forall\"{o}i=\overline{1,n},$öletöusöshowöthat $\sigma\"{o}_{\beta\"{o}}\left(\"{o}D_{i}v\right)\"{o}$öisöfinite,

[TABLE]

estimatingötheöcorrespondingöcoefficientsöandörightöhandösideöoföabove equationöbyöusingöCorollaryö3.3,öweöobtain

[TABLE]

hereö$C_{1}=C_{1}\left(\"{o}\beta\"{o}^{\pm\"{o}},\gamma\"{o}^{+}\right)\"{o},$ö$C_{2}=C_{2}\left(\"{o}\beta\"{o}^{\pm\"{o}},\gamma\"{o}^{+},\text{\"{o}}\left\|\"{o}\gamma\right\|\"{o}_{C^{1}\left(\"{o}\bar{\Omega}\right)\"{o}},\left\|\"{o}\beta\"{o}\right\|_{C^{1}\left(\"{o}\bar{\Omega}\right)\"{o}},\varepsilon\"{o}_{0}\right)\"{o}$öandö$C_{3}=C_{3}\left(\"{o}\beta\"{o}^{+},\left|\"{o}\Omega\"{o}\right|\"{o},\varepsilon_{0}\right)\"{o}>0$öareöconstants.ö($\varepsilon\"{o}_{0}>0,$öcomesöfromöthe Propositionö2.2öwhichösatisfyö$\theta\"{o}\left(\"{o}x\right)\"{o}\geq\"{o}\gamma\"{o}\left(x\right)\"{o}+\beta\"{o}\left(\"{o}x\right)\"{o}+\varepsilon\"{o}_{0}$).

Forö$C_{4}:=\max\"{o}\left\{\"{o}C_{1},C_{2}\right\}\"{o},$öweöhave

[TABLE]

sinceö$u\in\"{o}S_{1,\gamma\"{o}\left(\"{o}x\right)\"{o},\beta\"{o}\left(\"{o}x\right),\theta\"{o}\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o},$ösoöweögetöthe desiredöresultöbyö(3.5).

öNow,öconverselyöweöneedötoöshowöthatöforö$\forall\"{o}v\in\"{o}L^{1,\text{ }\beta\"{o}\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o}\cap\"{o}L^{\psi\"{o}\left(\"{o}x\right)}\left(\"{o}\Omega\"{o}\right)\"{o}$

[TABLE]

Fromötheödefinitionöofötheöspaceö$S_{1,\gamma\"{o}\left(\"{o}x\right),\beta\"{o}\left(\"{o}x\right)\"{o},\theta\"{o}\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\right)\"{o},$öitöisösufficientötoöproveö$\Re\"{o}^{\gamma\"{o},\beta\"{o},\theta}\left(\"{o}w\right)\"{o}$öisöfinite.öByöusingösimilaröprocessöandöresults asömentionedöabove,öweöobtain

[TABLE]

hereö$C_{5}=C_{5}\left(\"{o}\beta\"{o}^{+}\right)\"{o}>0,$ö$C_{6}=C_{6}\left(\"{o}\beta^{+},\varepsilon\"{o}_{1},\left\|\"{o}\gamma\"{o}\right\|\"{o}_{C^{1}\left(\"{o}\bar{\Omega}\right)\"{o}},\left\|\"{o}\beta\"{o}\right\|\"{o}_{C^{1}\left(\bar{\Omega}\right)\"{o}}\right)\"{o}>0$öandö$C_{7}=C_{7}\left(\"{o}\beta^{+},\varepsilon\"{o}_{1},\left|\"{o}\Omega\"{o}\right|\"{o}\right)\"{o}>0$ areöconstants.ö

öSinceö$v\in\"{o}L^{1,\text{\"{o}}\beta\"{o}\left(x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o}\cap\"{o}L^{\psi\"{o}\left(\"{o}x\right)}\left(\"{o}\Omega\"{o}\right)\"{o},$öthusö$w\in\"{o}S_{1,\gamma\"{o}\left(\"{o}x\right),\beta\"{o}\left(\"{o}x\right)\"{o},\theta\"{o}\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\right)$öbyö(3.6).

Toöendötheöproof,öitönowöremainsötoöverifyöthatö$\varphi$öis bijective,öasöweöhaveöshownöinöLemmaö3.6öthatöforöfixedö$x_{0}\in\Omega\"{o}$,ö$\varphi\"{o}\left(\"{o}t\right)\"{o}:=\varphi\"{o}\left(x_{0},t\right)$öandö$\varphi\"{o}^{-1}\left(\tau\"{o}\right)\"{o}:=\varphi\"{o}^{-1}\left(\"{o}x_{0},\tau\"{o}\right)\"{o}$öareöstrictlyömonotoneöthenöforöeveryö$v\in\"{o}L^{1,\text{\"{o}}\beta\"{o}\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o}\cap\"{o}L^{\psi\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o},$öthereöexistsöanöunique $u\in\"{o}S_{1,\gamma\"{o}\left(\"{o}x\right)\"{o},\beta\"{o}\left(\"{o}x\right)\"{o},\theta\left(\"{o}x\right)\"{o}}\left(\Omega\"{o}\right)\"{o}$ösuchöthatö$u=\varphi\"{o}^{-1}\left(\"{o}v\right)\"{o},$öandöforöeveryö$u\in\"{o}S_{1,\gamma\"{o}\left(\"{o}x\right)\"{o},\beta\"{o}\left(\"{o}x\right)\"{o},\theta\"{o}\left(x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o},$öthereöexistsöanöuniqueö$v\in\"{o}L^{1,\text{\"{o}}\beta\"{o}\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o}\cap\"{o}L^{\psi\"{o}\left(\"{o}x\right)}\left(\"{o}\Omega\"{o}\right)\"{o}$ösuchöthatö$v=\varphi\"{o}\left(\"{o}u\right)\"{o}$ösoöthat showsötheöbijectivityöofö$\varphi$.

öNow,öweögiveötwoöimportantöresultsöofötheöTheoremö3.8öwhichöhelpöus toöunderstandötheötopologyöofötheöspaceö$S_{1,\gamma\"{o}\left(\"{o}x\right)\"{o},\beta\left(\"{o}x\right)\"{o},\theta\"{o}\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o}.$

Corollary 3.9

*Letö$\beta\"{o},$ö$\gamma\"{o}$öandö$\psi\"{o}$ösatisfyötheöconditionsöof Theoremö3.8,öthenö

ö$S_{1,\gamma\"{o}\left(\"{o}x\right)\"{o},\beta\left(\"{o}x\right)\"{o},\theta\"{o}\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o}$ isöaöcompleteömetricöspaceöwithötheömetricöwhichöisödefinedöbelow: $\forall\"{o}u,$ö$v\in\"{o}S_{1,\gamma\"{o}\left(\"{o}x\right)\"{o},\beta\"{o}\left(x\right),\theta\"{o}\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o}$*

[TABLE]

hereö$\varphi\"{o}\left(\"{o}x,t\right)\"{o}=\left|\"{o}t\right|\"{o}^{\frac{\gamma\left(\"{o}x\right)\"{o}}{\beta\"{o}\left(\"{o}x\right)\"{o}}}t$öandöforöeveryöfixedö$x\in\Omega$,ö$\varphi\"{o}_{t}^{\prime\"{o}}\left(\"{o}t\right)\"{o}=\left(\"{o}\frac{\gamma\"{o}\left(x\right)\"{o}}{\beta\"{o}\left(\"{o}x\right)\"{o}}+1\right)\"{o}\left|\"{o}t\right|\"{o}^{\frac{\gamma\"{o}\left(\"{o}x\right)\"{o}}{\beta\"{o}\left(\"{o}x\right)\"{o}}}.$

Corollary 3.10

UnderötheöconditionsöoföCorollaryö3.9,ö$\varphi\"{o}$öisöa homeomorphismöbetweenötheöspacesö$S_{1,\gamma\"{o}\left(\"{o}x\right),\beta\"{o}\left(x\right)\"{o},\theta\"{o}\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o}$öandö$L^{1,\text{\"{o}}\beta\"{o}\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o}\cap\"{o}L^{\psi\"{o}\left(\"{o}x\right)}\left(\"{o}\Omega\"{o}\right)\"{o}.$

**Proof.**ö(Sketchöofötheöproof)öSinceöweöhaveöshowedöthatö$\varphi\"{o}$öisöa bijectionöbetweenö$S_{1,\gamma\"{o}\left(\"{o}x\right)\"{o},\beta\"{o}\left(\"{o}x\right),\theta\"{o}\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o}$öandö$L^{1,\text{\"{o}}\beta\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o}\cap\"{o}L^{\psi\"{o}\left(\"{o}x\right)\"{o}}\left(\Omega\"{o}\right)\"{o},$öitöisösufficientötoöproveötheöcontinuityöofö$\varphi\"{o}$öas wellöasö$\varphi\"{o}^{-1}$öinötheösenseöofötopologyöinducedöbyötheömetricö$d_{S_{1}}\left(\"{o}.,.\right)\"{o}.$öForöthis,öweöneedötoöshowöthatö

**(a)**ö$d_{S_{1}}\left(\"{o}u_{m},u_{0}\right)\"{o}\underset{m\nearrow\"{o}\infty\"{o}}{\longrightarrow\"{o}}0\Rightarrow\"{o}\varphi\"{o}\left(\"{o}u_{m}\right)\"{o}{\underset{m\nearrow\"{o}\infty\"{o}}{\overset{L^{1,\text{\"{o}}\beta\"{o}\left(\"{o}x\right)\"{o}}\left(\Omega\"{o}\right)\"{o}\cap\"{o}L^{\psi\"{o}\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o}}{\longrightarrow\"{o}}}\varphi\"{o}\left(\"{o}u_{0}\right)\"{o}}$öforöeveryö

$\left\{\"{o}u_{m}\right\}\"{o}_{m=1}^{\infty\"{o}}\in\"{o}S_{1,\gamma\"{o}\left(\"{o}x\right),\beta\"{o}\left(\"{o}x\right)\"{o},\theta\"{o}\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o}$ whichöconvergesötoö$u_{0}$öandöö

**(b)**ö$v_{m}{\underset{m\nearrow\"{o}\infty\"{o}}{\overset{L^{1,\text{\"{o}}\beta\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o}\cap\"{o}L^{\psi\"{o}\left(\"{o}x\right)\"{o}}\left(\Omega\"{o}\right)\"{o}}{\longrightarrow\"{o}}}}v_{0}\Rightarrow\"{o}d_{S_{1}}\left(\"{o}\varphi^{-1}\left(\"{o}v_{m}\right)\"{o},\varphi\"{o}^{-1}\left(\"{o}v_{0}\right)\"{o}\right)\"{o}\underset{m\nearrow\"{o}\infty\"{o}}{\longrightarrow\"{o}}0$öforöeveryö$\left\{\"{o}v_{m}\right\}_{m=1}^{\infty\"{o}}\in\"{o}L^{1,\text{\"{o}}\beta\"{o}\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\right)\"{o}\cap\"{o}L^{\psi\"{o}\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o}$öwhich convergesötoö$v_{0}.$

öSinceöforöeveryö$v_{m}$öandö$v_{0}$,öthereöexistöuniqueö$u_{m}$ andö$u_{0}\in\"{o}S_{1,\gamma\"{o}\left(\"{o}x\right)\"{o},\beta\"{o}\left(\"{o}x\right)\"{o},\theta\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o}$ösuchöthatö$\varphi\"{o}\left(u_{m}\right)\"{o}=v_{m}$öandö$\varphi\"{o}\left(\"{o}u_{0}\right)\"{o}=v_{0},$öthe implicationö**(b)**öcanöbeöwrittenöequivalently,ö

$\varphi\"{o}\left(\"{o}u_{m}\right)\"{o}{\underset{m\nearrow\"{o}\infty\"{o}}{\overset{L^{1,\text{\"{o}}\beta\"{o}\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o}\cap\"{o}L^{\psi\"{o}\left(x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o}}{\longrightarrow\"{o}}}\varphi\"{o}}\left(u_{0}\right)\"{o}\Rightarrow\"{o}d_{S_{1}}\left(\"{o}u_{m},u_{0}\right)\"{o}\underset{m\nearrow\"{o}\infty\"{o}}{\longrightarrow\"{o}}0$öforöeveryö$\left\{u_{m}\right\}\in\"{o}S_{1,\gamma\"{o}\left(\"{o}x\right)\"{o},\beta\"{o}\left(x\right)\"{o},\theta\"{o}\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o}$öwhich convergesötoö$u_{0}.$öSinceötheöproofsöofö**(a)**öand **(b)**öareösimilar,öweöonlyöprove (b):öLetö$v_{0},$ö$\left\{\"{o}v_{m}\right\}\"{o}_{m=1}^{\infty\"{o}}\in\"{o}L^{1,\text{\"{o}}\beta\"{o}\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o}\cap\"{o}L^{\psi\"{o}\left(x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o}$öandö$v_{m}\overset{L^{1,\text{\"{o}}\beta\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o}\cap\"{o}L^{\psi\"{o}\left(\"{o}x\right)\"{o}}\left(\Omega\"{o}\right)\"{o}}{{\longrightarrow\"{o}}}v_{0}\Leftrightarrow\"{o}\varphi\"{o}\left(u_{m}\right)\"{o}\overset{L^{1,\text{\"{o}}\beta\"{o}\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\right)\"{o}\cap\"{o}L^{\psi\"{o}\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o}}{{\longrightarrow\"{o}}}{\varphi\"{o}}\left(\"{o}u_{0}\right)\"{o}.$ö

öTo verifyö$d_{S_{1}}\left(\"{o}u_{m},u_{0}\right)\"{o}\rightarrow\"{o}0,$öby definitionöofö$d_{S_{1}}$öitöisösufficientötoöestablishöthat

[TABLE]

asö$m\nearrow\"{o}\infty\"{o}.$öTheösecondöconvergenceöaboveöisöobviousöby definitionöofö$d_{S_{1}}$öandötheöfirstöoneöcanöbeöprovedöby applyingöTheoremö3.8öandöVitaliöconvergenceötheoremöbyövirtueöof theöequivalenceö$\left\|\"{o}\varphi\"{o}_{t}^{\prime\"{o}}\left(u_{m}\right)\"{o}D_{i}u_{m}-\varphi\"{o}_{t}^{\prime\"{o}}\left(\"{o}u_{0}\right)D_{i}u_{0}\right\|\"{o}_{L^{\beta\"{o}\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\right)\"{o}}\rightarrow\"{o}0\text{\"{o}}\Leftrightarrow\"{o}\sigma\"{o}_{\beta}\left(\"{o}\varphi\"{o}_{t}^{\prime\"{o}}\left(\"{o}u_{m}\right)D_{i}u_{m}-\varphi\"{o}_{t}^{\prime\"{o}}\left(\"{o}u_{0}\right)D_{i}u_{0}\right)\"{o}\rightarrow\"{o}0\"{o}$.

Theorem 3.11

SupposeöthatöconditionsöoföTheoremö3.8öareösatisfied.öLetö$p\in M_{0}\left(\"{o}\Omega\"{o}\right)\"{o}$öandöadditionallyö$\beta\"{o}$ösatisfies $1\leq\"{o}\beta\"{o}^{-}\leq\"{o}\beta\"{o}\left(\"{o}x\right)\"{o}<n,$ö$x\in\"{o}\Omega\"{o}$. Assumeöthatöforö$\varepsilon\"{o}>0,$ötheöinequality

[TABLE]

holds.öThenöweöhaveötheöcompactöembedding

[TABLE]

**Proof.**öFirst,öweöshowöthatö$S_{1,\gamma\"{o}\left(\"{o}x\right)\"{o},\beta\"{o}\left(x\right)\"{o},\theta\"{o}\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o}\subset L^{p\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o},$öafteröthatöweöprove theöcompactnessöoföthisöembedding.

Foröeveryö$u\in\"{o}S_{1,\gamma\"{o}\left(\"{o}x\right)\"{o},\beta\"{o}\left(\"{o}x\right)\"{o},\theta\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o},$öbyöTheoremö3.8

[TABLE]

Sinceö$L^{1,\text{\"{o}}\beta\"{o}\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\cap\"{o}L^{\psi\"{o}\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o}\subset W^{1,\text{\"{o}}\beta\"{o}\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)$öandöthe embeddingö[8]ö$W^{1,\text{\"{o}}\beta\"{o}\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\right)\"{o}\subset\"{o}L^{\beta\"{o}^{\ast\"{o}}\left(\"{o}x\right)}\left(\"{o}\Omega\"{o}\right)$öexistsöforö$\beta\"{o}^{\ast\"{o}}\left(\"{o}x\right)\"{o}=\frac{n\beta\"{o}\left(\"{o}x\right)\"{o}}{n-\beta\"{o}\left(\"{o}x\right)\"{o}},$ötherefore,öweögetöthatö$v\in\"{o}L^{\beta^{\ast\"{o}}\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\"{o}\right).$öSoöfromöthe definitionöofö$v$öandötheöspaceö$L^{\beta\"{o}^{\ast\"{o}}\left(\"{o}x\right)}\left(\"{o}\Omega\"{o}\right)\"{o}$öwe attain

[TABLE]

As,öbyötheöconditionsöofötheorem

[TABLE]

thus,ö$u\in\"{o}L^{p\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o}.$

öNow letöusöproveöthatöthisöembeddingöisöcompact.ö

öLetö$\left\{u_{m}\right\}\"{o}_{m=1}^{\infty\"{o}}\in\"{o}S_{1,\gamma\"{o}\left(\"{o}x\right),\beta\"{o}\left(\"{o}x\right)\"{o},\theta\"{o}\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\right)\"{o}$öbeöboundedösequenceö(i.e.ö$[u_{m}]_{\gamma\"{o},\beta,\theta\"{o}}<\infty\"{o},$ö$\forall\"{o}m\geq\"{o}1$).

öFromöTheoremö3.8,öweöhave

[TABLE]

sinceöweöhaveötheöcompactöembeddingö[8]

[TABLE]

whereö$q\left(\"{o}x\right)\"{o}<\beta\"{o}^{\ast\"{o}}\left(\"{o}x\right)\"{o}-\tilde{\varepsilon},$ $x\in\"{o}\Omega\"{o}$öandö$\tilde{\varepsilon}\in\"{o}\left(\"{o}0,n^{\prime\"{o}}\right)\"{o}$ö($n^{\prime\"{o}}=\frac{n}{n-1}$).öThus,öthereöexistsöaösubsequenceö$\left\{v_{m_{j}}\right\}\"{o}\subset\"{o}\left\{\"{o}v_{m}\right\}\"{o},$ösuchöthat

[TABLE]

henceöbyö(3.7),öweöobtain

[TABLE]

asöfromöLemmaö3.6,ö$\varphi\"{o}^{-1}(x,\tau\"{o})=\left|\"{o}\tau\"{o}\right|\"{o}^{\frac{\gamma\"{o}\left(\"{o}x\right)\"{o}}{\gamma\"{o}\left(\"{o}x\right)\"{o}+\beta\"{o}\left(\"{o}x\right)}}\tau\"{o}$öisöcontinuousö(withörespectötoö$\tau\"{o}$öandö$x$)ösoöweöhave

[TABLE]

Toöendötheöproof,öweöuseöLemmaö3.12‡‡‡Lemmaö3.12öLetö$\Lambda\"{o}$öbeöaöfamilyöoförealöfunctions definedöonöbounded domainö$\Omega\"{o}.$öIföthereöisöanöincreasingöfunctionö$\Phi\"{o}:\left[\"{o}0,\text{\"{o}}\infty\"{o}\right)\"{o}\rightarrow\"{o}\left[\"{o}0,\text{\"{o}}\infty\"{o}\right)\"{o}$öthat satisfies

$\lim_{t\rightarrow\"{o}+\infty\"{o}}\Phi\"{o}\left(\"{o}t\right)\"{o}=+\infty$

andöthereöisöaöpositiveöconstantö$L$ösuchöthat

$\int\limits_{\Omega\"{o}}\left|\"{o}f_{\alpha\"{o}}\left(\"{o}x\right)\right|\"{o}\Phi\left(\"{o}\left|\"{o}f_{\alpha\"{o}}\left(\"{o}x\right)\"{o}\right|\"{o}\right)\"{o}dx\leq\"{o}L,\text{\"{o}\ \ }\forall\"{o}f_{\alpha\"{o}}\in\"{o}\Lambda\"{o},$

thenöeveryöfunctionöinö$\Lambda\"{o}$öisöLebesgueöintegrable,öandöthe functionsöfamilyö$\Lambda\"{o}$öpossessesöabsolutelyöequicontinuous integralsöonö$\Omega\"{o}.$ö[16,öTheoremö7].

öDenoteö$u_{0}:=\varphi\"{o}^{-1}\left(\"{o}v_{0}\right)\"{o}$öandötheöset

[TABLE]

andötheöfunction

[TABLE]

Clearlyö$\Phi\"{o}:[0,\infty\"{o})\rightarrow\"{o}[\"{o}0,\infty\"{o})$öisöincreasingöandö$\underset{t\rightarrow\"{o}+\infty\"{o}}{\lim\"{o}}\Phi\"{o}\left(\"{o}t\right)\"{o}=+\infty\"{o}$

Furthermoreöforöeveryö$f_{j}\in\"{o}\Lambda\"{o}$öweöhave,

[TABLE]

[TABLE]

EstimatingötheölastöintegralöbyöusingöLemmaö3.1,öweöarriveöat

[TABLE]

usingötheöwellöknownöinequalityöforöabsoluteövalueöabove,öweöget

[TABLE]

Sinceö$u_{0},\left\{\"{o}u_{m_{j}}\right\}\"{o}\subset\"{o}L^{p\left(\"{o}x\right)+\varepsilon\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o}$öisöbounded,öfromö(3.8),öthereöexistsöa numberö$L>0$ösuchöthat

[TABLE]

hereö$L=L\left(\"{o}\left|\"{o}\Omega\"{o}\right|\"{o},p^{+},\varepsilon\"{o},\left\|u_{0}\right\|\"{o}_{L^{p\left(\"{o}x\right)\"{o}+\varepsilon\"{o}}\left(\"{o}\Omega\"{o}\right)},\left\|\"{o}u_{m_{j}}\right\|\"{o}_{L^{p\left(\"{o}x\right)\"{o}+\varepsilon\"{o}}\left(\Omega\"{o}\right)\"{o}}\right)\"{o}.$

Consequentlyöbyö(3.9),öweöobtainöthatötheöfamilyöoföfunctions $\Lambda\"{o}$öpossessesöabsolutelyöequicontinuousöintegralsöon $\Omega\"{o}.$öHenceöusing thisöandö$u_{m_{j}}\overset{a.e.}{\underset{\Omega\"{o}}{\longrightarrow\"{o}}}u_{0},\"{o}$öweöhaveö[16]

[TABLE]

thatöimpliesö$\left\|\"{o}u_{m_{j}}-u_{0}\right\|\"{o}_{L^{p\left(x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o}}{\rightarrow\"{o}}0$,ösoötheöprooföis complete.

4 ProoföoföTheöExistenceöTheorem

öTheöprooföisöbasedöonöTheoremö2.5.öWeöintroduceötheöfollowing spacesöandömappingsöinöorderötoöapplyöTheoremö2.5ötoöproveöTheoremö2.4.

[TABLE]

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

[TABLE]

[TABLE]

WeöshowöthatöallötheöconditionsöoföTheoremö2.5öareösatisfiedöby provingösomeölemmas.öThenöbasedöonötheseölemmas,öweöestablishöthe prooföoföTheoremö2.4.

Lemma 4.1

UnderötheöconditionsöoföTheoremö2.4,ötheöoperatorö$T$ödefinedöbyö(4.3)öis coerciveöinötheögeneralizedösenseöonö$X_{0}.$

**Proof.**öForöeveryö$u\in\"{o}W_{0}^{1,\text{\"{o}}p_{0}\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\cap\"{o}W_{0}^{1,\text{\"{o}}p_{1}\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o}\cap L^{\alpha\"{o}\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o},$öweöhave

[TABLE]

[TABLE]

Iföweötakeöaccountötheöconditionö*(U2)*öintoötheöthird integralöofö(4.4)öandöapplyötheöfollowingösimpleöcalculated inequality

[TABLE]

weöget,

[TABLE]

hereö$C_{8}=C_{8}\left(\"{o}n,\tilde{C}\right)\"{o}>0$öandö$C_{9}=$ $C_{9}\left(\"{o}n,\left|\"{o}\Omega\"{o}\right|\"{o},\tilde{C}\right)\"{o}>0$ areöconstants.

öApplyingöLemmaö3.4ötoöestimateöthe right-handösideöofötheöinequalityö(4.6),öweöobtain

[TABLE]

byötheödefinitionsöofö$\"{o}[.]_{S_{q_{0}\left(\"{o}p_{0}-2\right),q_{0},\alpha\"{o}}}$öandö$\left\|\"{o}.\right\|\"{o}_{W_{0}^{1,\text{\"{o}}p_{1}\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)}.$ö

öSinceö$q_{0}^{-}+1>2$öandö$p_{1}^{-}>1$öthus,ö$\lambda\"{o}_{0}\left(\"{o}\tau\"{o}\right)\"{o}=\tau\"{o}^{q_{0}^{-}}$öandö$\lambda\"{o}_{1}\left(\"{o}\tau\"{o}\right)\"{o}=\tau\"{o}^{p_{1}^{-}-1}$ötendsöto infinityöwhenö$\tau\"{o}\nearrow\"{o}\infty\"{o},$ö(seeöTheoremö2.5)ösoöthat meansöoperatorö$T$öisöcoerciveöinötheögeneralizedösenseöon $X_{0}.$

Lemma 4.2

UnderötheöconditionsöoföTheoremö2.4,ötheöoperatorö$A$öisömonotone andöboundedöfrom $W_{0}^{1,\text{\"{o}}p_{1}\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o}$öintoö$W^{-1,\text{\"{o}}q_{1}\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o}.$

**Proof.**öFirstöweöproveöthatö$A:$ö$W_{0}^{1,\text{\"{o}}p_{1}\left(\"{o}x\right)}\left(\"{o}\Omega\"{o}\right)\"{o}$ö$\rightarrow\"{o}W^{-1,\text{\"{o}}q_{1}\left(x\right)\"{o}}\left(\Omega\"{o}\right)\"{o}$öisöbounded.öForöthis,öitöisösufficientötoöinvestigateötheödualöformö$\langle\"{o}A\left(\"{o}u\right)\"{o},v\rangle\"{o}$öforöeveryö$v\in\"{o}W_{0}^{1,\text{\"{o}}p_{1}\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o},$

[TABLE]

usingötheögeneralizedöHölderöinequalityötoötheörightöhandösideöoföthe aboveöequation,öweöget

[TABLE]

Thusöbyö(4.8)öweödemonstrateötheöboundednessöofö$A$öfrom $W_{0}^{1,\text{\"{o}}p_{1}\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o}$öto $W^{-1,\text{\"{o}}q_{1}\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o}.$

öNowöletöusöshowöthatö$A:$ö$W_{0}^{1,\text{\"{o}}p_{1}\left(\"{o}x\right)}\left(\"{o}\Omega\"{o}\right)\"{o}$ö$\rightarrow\"{o}W^{-1,\text{\"{o}}q_{1}\left(\"{o}x\right)}\left(\"{o}\Omega\"{o}\right)\"{o}$öisöaömonotoneöoperator.

Indeedöforöeveryö$u,$ö$v\in\"{o}W_{0}^{1,\text{\"{o}}p_{1}\left(\"{o}x\right)\"{o}}\left(\Omega\"{o}\right)\"{o}$öweöhave,

[TABLE]

Sinceötheöinequalityö$\"{o}\left(\"{o}\left|\"{o}a\right|^{p-2}a-\left|b\right|\"{o}^{p-2}b\right)\cdot\left(\"{o}a-b\right)\geq\"{o}0\"{o}$öisövalidöforö$1<\"{o}p<\infty,$ö$a,$ö$b\in\"{o}\mathbb{R}^{n}$öfromötheölastöequality,öweöattain

[TABLE]

whichöcompletesötheöproof.ö§§§Here,öweönoteöthatösinceö$A$öisömonotoneöandöhemicontinuousöthen itöisöpseudo-monotoneö[25].

Lemma 4.3

UnderötheöconditionsöoföTheoremö2.4,ö$B$öisöaöboundedöoperatoröfromö$\mathring{S}_{1,q_{0}\left(\"{o}x\right)\"{o}\left(\"{o}p_{0}\left(\"{o}x\right)\"{o}-2\right),q_{0}\left(\"{o}x\right)\"{o},\alpha\"{o}\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o}$öintoö$W^{-1,\text{\"{o}}q_{0}\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o}+L^{\alpha^{\prime\"{o}}\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o}.$

**Proof.**öSinceö$B=B_{1}+B_{2},$öweöshallöshowöthatöbothö$B_{1}$öandö$B_{2}$öare bounded.

Firstöletöusöverifyö$B_{2}:$ö$\mathring{S}_{1,q_{0}\left(\"{o}x\right)\"{o}\left(p_{0}\left(\"{o}x\right)\"{o}-2\right)\"{o},q_{0}\left(\"{o}x\right)\"{o},\alpha\"{o}\left(\"{o}x\right)}\left(\"{o}\Omega\"{o}\right)\"{o}$ö$\rightarrow\"{o}W^{-1,\text{\"{o}}q_{0}\left(\"{o}x\right)}\left(\"{o}\Omega\"{o}\right)\"{o}+L^{\alpha\"{o}^{\prime\"{o}}\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\right)\"{o}$öisöbounded:

Asö$\mathring{S}_{1,q_{0}\left(\"{o}x\right)\"{o}\left(\"{o}p_{0}\left(\"{o}x\right)-2\right)\"{o},q_{0}\left(\"{o}x\right)\"{o},\alpha\"{o}\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\right)\"{o}\subset\"{o}L^{\alpha\"{o}\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o},$öitöis sufficientötoöshowötheöboundednessöofö$B_{2},$öfromö$L^{\alpha\"{o}\left(x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o}$ötoö$L^{\alpha\"{o}^{\prime\"{o}}\left(\"{o}x\right)}\left(\"{o}\Omega\"{o}\right)\"{o}.$

Foröeveryö$u\in\"{o}L^{\alpha\"{o}\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o}$

[TABLE]

hereötakingötheöconditionsöoföTheoremö2.4öintoöaccountöand estimatingötheöaboveöintegral,öweöobtain

[TABLE]

Soöfromö(4.9),öweöarriveöatö$B_{2}$öisöbounded.

Nowöletöusöproveöthatö$B_{1}:$ö$\mathring{S}_{1,q_{0}\left(x\right)\"{o}\left(\"{o}p_{0}\left(\"{o}x\right)\"{o}-2\right)\"{o},q_{0}\left(x\right)\"{o},\alpha\"{o}\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o}$ $\rightarrow\"{o}W^{-1,\text{\"{o}}q_{0}\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\right)\"{o}+L^{\alpha\"{o}^{\prime\"{o}}\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\right)\"{o}$öisöbounded:ö$\forall\"{o}i=\overline{1,n},$ödenote $b_{i}\left(\"{o}x\right)\"{o}:=\left|\"{o}u\right|\"{o}^{p_{0}\left(x\right)\"{o}-2}\left|\"{o}D_{i}u\left(\"{o}x\right)\"{o}\right|\"{o},$öforöeveryö$v\in\"{o}W_{0}^{1,\text{\"{o}}p_{0}\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)$

[TABLE]

applyingötheögeneralizedöHölderöinequalityötoötheörightöhandösideöoföthe aboveöequation,öweöarriveöat

[TABLE]

Sinceö$u\in\"{o}\mathring{S}_{1,q_{0}\left(\"{o}x\right)\"{o}\left(p_{0}\left(\"{o}x\right)\"{o}-2\right)\"{o},q_{0}\left(\"{o}x\right)\"{o},\alpha\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o},$öfromö(2.3)öandöthe definitionöofötheöfunctionsö$b_{i}\left(\"{o}x\right)$,öobviously $\sum\limits_{i=1}^{n}\left\|\"{o}b_{i}\right\|\"{o}_{L^{q_{0}\left(x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o}}<\infty\"{o}.$öThusöweöverifyöthat $B_{1}$öisöbounded.ö

Consequently,öweöproveöthatö$B$öisöaöboundedöoperatoröfromö$\mathring{S}_{1,q_{0}\left(\"{o}x\right)\"{o}\left(\"{o}p_{0}\left(\"{o}x\right)\"{o}-2\right)\"{o},q_{0}\left(x\right)\"{o},\alpha\"{o}\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o}$ötoö$W^{-1,\text{\"{o}}q_{0}\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o}+L^{\alpha\"{o}^{\prime\"{o}}\left(x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o}.$

Lemma 4.4

UnderötheöconditionsöoföTheoremö2.4,ö$B$öisöaöweaklyöcompactöoperatoröfromö$\mathring{S}_{1,q_{0}\left(\"{o}x\right)\"{o}\left(\"{o}p_{0}\left(\"{o}x\right)\"{o}-2\right),q_{0}\left(\"{o}x\right)\"{o},\alpha\"{o}\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o}$öintoö$W^{-1,\text{\"{o}}q_{0}\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o}+L^{\alpha^{\prime\"{o}}\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o}.$

**Proof.**öSinceö$B=B_{1}+B_{2},$öweöshallöshowöthatöbothö$B_{1}$öandö$B_{2}$ areöweaklyöcompact.ö

öFirstöweöshowötheöweaköcompactnessöof $B_{1}:$öLetö$\left\{\"{o}u_{m}\right\}\"{o}_{m=1}^{\infty\"{o}},$ö$u_{0}$ $\in\"{o}\mathring{S}_{1,q_{0}\left(\"{o}x\right)\"{o}\left(\"{o}p_{0}\left(x\right)\"{o}-2\right)\"{o},q_{0}\left(\"{o}x\right)\"{o}}\left(\Omega\"{o}\right)\"{o}\cap\"{o}L^{\alpha\"{o}\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o}$öandö$u_{m}\overset{S_{gY_{0}}}{\rightharpoonup\"{o}}u_{0}.$öByöTheoremö3.8öweöhave

[TABLE]

Asö$q_{0}^{-}>1$öthatöimpliesö$W_{0}^{1,\text{\"{o}}q_{0}\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o}$öisöaöreflexiveöspaceöthus, thereöexistsöaösubsequenceö$\left\{\"{o}w_{m_{j}}\right\}\"{o}_{j=1}^{\infty\"{o}}$öofö$\left\{\"{o}w_{m}\right\}\"{o}$ösuchöthat

[TABLE]

Letöusöverifyö$\xi\"{o}=\left|\"{o}u_{0}\right|\"{o}^{p_{0}\left(x\right)\"{o}-2}u_{0}$.öSinceö$W_{0}^{1,\text{\"{o}}q_{0}\left(\"{o}x\right)}\left(\"{o}\Omega\"{o}\right)\"{o}\hookrightarrow\"{o}L^{q_{0}\left(\"{o}x\right)}\left(\"{o}\Omega\"{o}\right)\"{o}$öthereforeöthereöexistöaösubsequence $\left\{\"{o}w_{m_{j_{k}}}\right\}\"{o}\subset\"{o}\left\{\"{o}w_{m_{j}}\right\}\"{o}$ (denote thisösubsequenceöbyö$w_{m_{j}}$öinöorderötoöavoidönotationöconfusion)ösuchöthat

[TABLE]

hence

[TABLE]

asöfromöLemmaö3.6,ö$\varphi\"{o}^{-1}(x,\tau\"{o})=\left|\"{o}\tau\"{o}\right|\"{o}^{-\frac{p_{0}\left(\"{o}x\right)\"{o}-2}{p_{0}\left(\"{o}x\right)\"{o}-1}}\tau\"{o}$öisöcontinuous (withörespectötoö$\tau\"{o}$öandö$x$)ösoöusingö(4.10)öweöobtain

[TABLE]

henceöbyö(4.11),öweöarriveöatö$\varphi\"{o}^{-1}(\xi\"{o})=u_{0}$,öequivalentlyö$\xi\"{o}=\left|\"{o}u_{0}\right|\"{o}^{p_{0}\left(\"{o}x\right)\"{o}-2}u_{0}.$

Toöverifyötheöweaköcompactnessöof $B_{1}$,öweömustöshowöthatöforöarbitraryö$v\in\"{o}W_{0}^{1,\text{\"{o}}p_{0}\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o}$

[TABLE]

Byötheödefinitionöoföoperatorö$B_{1},$

[TABLE]

UsingöLemmaö3.6öandöchainöruleöweöhaveötheöfollowingöequality

[TABLE]

iföweöinsertötheöequalityö(4.13)öintoö(4.12),öweöobtain

[TABLE]

Letöusödenoteötheöfirstösumöinö(4.14)öbyö$I_{1}$öandötheösecondöoneöbyö$I_{2}\"{o}$öi.e.

[TABLE]

iföweöuseötheösameömanneröinö[20,öLemmaö3.3]öandöpassötoötheölimitöinö$I_{1},\"{o}$öweöobtain

[TABLE]

ConsideringöLemmaö3.2ötogetheröwithöTheoremö3.11öandöcontinuityöof theöfunctionö$\left|\"{o}t\right|\"{o}^{p_{0}\left(\"{o}x\right)-2}t\ln\"{o}\left|\"{o}t\right|\"{o}$öwithörespectötoö$t$öandöpassöto theölimitöinö$I_{2}$,öwe obtain

[TABLE]

Henceöfromö(4.15)öandö(4.16),öweöhave

[TABLE]

Therefore,öweöproveötheöweaköcompactnessöofö$B_{1}$öfromö$\mathring{S}_{1,q_{0}\left(\"{o}x\right)\"{o}\left(\"{o}p_{0}\left(\"{o}x\right)\"{o}-2\right)\"{o},q_{0}\left(x\right)\"{o},\alpha\"{o}\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o}$ötoö$W^{-1,\text{\"{o}}q_{0}\left(\"{o}x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o}+L^{\alpha\"{o}^{\prime\"{o}}\left(x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)\"{o}.$

Nowöweöproveötheöweaköcompactnessöofö$B_{2}.$öAsö$\left(\"{o}\alpha^{\prime\"{o}}\right)\"{o}^{-}>1,$ö$L^{\alpha\"{o}^{\prime\"{o}}\left(\"{o}x\right)}\left(\"{o}\Omega\"{o}\right)\"{o}$öisöaöreflexiveöspaceöandö$\left\{B_{2}\left(\"{o}u_{m}\right)\"{o}\right\}\"{o}_{m=1}^{\infty\"{o}}:=\left\{\"{o}\eta_{m}\right\}\"{o}_{m=1}^{\infty\"{o}}\subset\"{o}L^{\alpha\"{o}^{\prime\"{o}}\left(x\right)\"{o}}\left(\"{o}\Omega\"{o}\right)$öisöboundedö(see,öLemmaö4.3),öthen thereöexistsöaösubsequenceö$\left\{\"{o}\eta\"{o}_{m_{j}}\right\}\"{o}\subset\left\{\"{o}\eta\"{o}_{m}\right\}$ösuchöthat

[TABLE]

ByöTheoremö3.11,ötheöembedding

[TABLE]

isöcompactöforö$s\left(\"{o}.\right)\"{o}$öwhichösatisfiesötheöinequalityö$s\left(\"{o}x\right)\"{o}<\frac{np_{0}\left(\"{o}x\right)\"{o}}{n-q_{0}\left(\"{o}x\right)\"{o}},$ö$x\in\"{o}\Omega\"{o}.$

Thus,öbyö(4.17)öthereöexistsöaösubsequenceö$\left\{\"{o}u_{m_{j_{k}}}\right\}\subset\"{o}\left\{\"{o}u_{m_{j}}\right\}\"{o}$ö(letöusödenoteöthisösubsequenceöbyö$u_{m_{j}}$öinöorderötoöavoidönotationöconfusion.)ösuchöthat

[TABLE]

so

[TABLE]

Sinceötheöfunctionö$c\left(\"{o}x,\tau\"{o}\right)\"{o}$öisöcontinuousöwithörespectöto variableö$\tau\"{o}$ö($c\left(\"{o}x,\tau\"{o}\right)\"{o}$öisöCarathèdoryöfunction), byö(4.18)

[TABLE]

Therefore,öfromö(4.19)öweöobtainöthatö$\psi\"{o}=B_{2}\left(\"{o}u_{0}\right)\"{o}.$

Finally,öweöarriveöat

[TABLE]

whichöimpliesöthat

[TABLE]

So,öfromö(4.20)öweöobtainötheöweaköcompactnessöofö$B_{2}$öwhichöprovides,öas aöresult,ötheöweaköcompactnessöofötheöoperatorö$B.$

öItönowöremainsötoödefineötheöcorrespondingöoperatorsöforö$B$öinöconditionö”2)”öoföTheorem 2.5ötoöapplyöthisötheoremötoötheöproblemö(1.1).

öSinceö$B=B_{1}+B_{2},$öaccordingötoöconditionö”2)”öweödefine correspondingö$B_{01}$öwithöregardötoö$B_{1}$öandöcorresponding $B_{02}$ withöregardötoö$B_{2}$öasöbelow:

[TABLE]

and

[TABLE]

Noteöthatöhereö$c\left(\"{o}x,\tau\right)\tau>0$öbyötheöcondition (U2).

ByötheösameöargumentsöwhichöareöusedöinötheöprooföoföLemmaö4.3öto establishötheöboundednessöofötheöoperatorsö$B_{1}$öandö$B_{2},$öwe canöshowöthatötheöoperatorsö$B_{01}$öandö$B_{02}$öareöbounded betweenötheöspaces whichöareöintroducedöbelow:

[TABLE]

and

[TABLE]

öHere,öweöhaveötoöproveötheöweaköcompactnessöofö$B_{01}$ andö$B_{02}$ötoöshowöthatöconditionö”2)”öinöTheoremö2.5öis satisfied.

Byöusingötheösimilarömanneröwhichöhasöbeenöestablishedöinöthe prooföoföLemmaö4.4öandöbyötheödefinitionöofötheöfunctionals correspondingötoöoperatorsö$B_{01}$öandö$B_{02}$ö(seeö(4.21), (4.22)),öfollowingölemmasöcanöbeöprovedöstraightforwardly,ösoöwe omitötheöproofsöoföthem.

Lemma 4.5

UnderötheöconditionsöoföTheoremö2.4,ö$B_{01}$öisöweaklyöcompactöoperator fromö$X_{0}$öintoö$W^{-1,\text{\"{o}}2}\left(\"{o}\Omega\"{o}\right)\"{o}.$öMoreoveröforöthe functionö$\mu\"{o}\left(\"{o}\tau\"{o}\right)\"{o}=\tau\"{o}^{2}$öandöforöeveryö$u\in\"{o}X_{0},$ theöequality

[TABLE]

holds.

Lemma 4.6

UnderötheöconditionsöoföTheoremö2.4,ö$B_{02}$öisöweaklyöcompactöoperator fromö$X_{0}$öintoö$L^{2}\left(\"{o}\Omega\"{o}\right)\"{o}.$öMoreoveröforöeveryö$u\in X_{0},$ötheöequality

[TABLE]

holds.

öNowöweöcanögiveötheöprooföoföTheoremö2.4.

**Proof.ö(ProoföoföTheoremö2.4)**öInöLemmasö4.1-4.6,öweöshowöthat allötheöconditionsöoföTheoremö2.5öareösatisfiedöforöproblemö(1.1) underötheöconditionsöoföTheoremö2.4.öConsequently,öweöestablish thatöTheoremö2.5öcanöbeöappliedötoötheöproblemö(1.1).öHenceöusing thisötheorem,öweöobtainötheöexistenceöoföaöweakösolutionöof problemö(1.1)öinötheösenseöoföDefinitionö2.3.