On the Sobolev embedding theorem for variable exponent spaces in the critical range (Q439115)

Let \(\Omega\) be an open subset of the Euclidean space \(\mathbb{R}^N\), where \(N\) is a positive integer, and let \(p(x)\), \(q(x)\), \(x\in\Omega\), be variable Lebesgue space exponents with conjugate exponents \(p'(x)\), \(q'(x)\), defined by \(p'(x)= p(x)/(p(x)-1)\), and critical exponents defined by \(p^*(x)= Np(x)/(N- p(x))\) if \(p(x)< N\), \(p^*(x)=\infty\) if \(p(x)\geq N\).NEWLINENEWLINE The variable exponent Lebesgue space \(L^{p(.)}(\Omega)\) with norm \(\|.\|_{p(.)}\) and the variable exponent Sobolev space \(W^{1,p(.)}(\Omega)\) with norm \(\|.\|_{Wp(.)}\) are defined by NEWLINE\[NEWLINEL^{p(.)}(\Omega)= \Biggl\{u\text{ locally integrable}: \int_\Omega|u(x)|^{p(x)} dx< \infty\Biggr\},NEWLINE\]NEWLINE NEWLINE\[NEWLINE\| u\|_{p(.)}= \text{inf}\Biggl\{\lambda> 0: \int_\Omega |u(x)/\lambda|^{p(x)} dx\leq 1\Biggr\},NEWLINE\]NEWLINE NEWLINE\[NEWLINEW^{1,p(.)}(\Omega)= \{u\text{ locally integrable}: u\in L^{p(.)}(\Omega),\;|\nabla(u)|\in L^{p(.)}(\Omega)\},NEWLINE\]NEWLINE \(\| u\|_{Wp(.)}=\| u\|_{p(.)}+ \|\nabla(u)\|_{p(.)}\), where \(\nabla\) is the gradient operator.NEWLINENEWLINE The Raleigh quotient \(Q_{p,q,\Omega}(v)\), the Sobolev immersion constant \(S(p(.),q(.),\Omega)\), the localized Sobolev constant \(\overline S_x\), \(x\in\Omega\), the critical constant \(\overline S\), and the Sobolev constant for constant exponents \(K^{-1}_r\), are defined by NEWLINE\[NEWLINEQ_{p(), q(),\Omega}(v)= \| v\|^{-1}_{q(.)}\|\nabla(v)\|_{p(.)};NEWLINE\]NEWLINE NEWLINE\[NEWLINES(p(.),q(.),\Omega)= \text{inf}\{Q_{p(),q(),\Omega}(v),\, v\in w^{1,p(.)}_0(\Omega)\};NEWLINE\]NEWLINE NEWLINE\[NEWLINES_x= \lim_{\varepsilon\to 0} S(p(),q(), B_\varepsilon(x));NEWLINE\]NEWLINE NEWLINE\[NEWLINE\overline S= \underset{x\in{\mathcal A}}{}{\text{inf}}\overline S_x,\;{\mathcal A}= \{x\in \Omega: q(x)= p^*(x), \;p(x)< N\};NEWLINE\]NEWLINE NEWLINE\[NEWLINEK^{-1}_r= \text{inf}\{Q_{p,q,\mathbb{R}^n}(v):v\in C^\infty(\mathbb{R}^n)\}.NEWLINE\]NEWLINE The main theorems of this paper include the statements that: if \(p(x)\geq 1\), \(q(x)\geq 1\), \(x\in\Omega\), with moduli of continuity which satisfy \(\rho(t)\log 1/t\to 0\) as \(t\to 0\), then NEWLINENEWLINENEWLINE\[NEWLINES(p(),q(),\Omega)<\overline S<\text{inf}\{K^{-1}_r:\{p_{\mathcal {A}^+} \leq r \leq p_{\mathcal {A}^-}\},NEWLINE\]NEWLINE NEWLINENEWLINEwhere \(p_{{\mathcal A}^-}= \text{inf}_{x\in{\mathcal A}} p(x)\), \(p_{{\mathcal A}^+}= \sup_{x\in{\mathcal A}} p(x)\).