Operator equations and duality mappings in Sobolev spaces with variable exponents (Q379887)

The authors point out that all vectors and functions are real. Let \(\Omega\) be a bounded connected open subset of \(\mathbb R^n\), \(n\geq 2\), whose boundary \(\partial\Omega=\Gamma\) is Lipschitz continuous, with the set \(\Omega\) being locally on the same side of \(\Gamma\). A defined measure, on \(\Gamma\), is denoted by \(d\Gamma\). Let \(\Gamma_0\) be a subset of \(\Gamma\) whose \(d\Gamma\)-measure is strictly positive. Let \(p\in\mathcal C(\bar{\Omega})\), with \(p(x)>1\) for all \(x\in\bar{\Omega}\). Let \(U_{\Gamma_0}\) be the separable reflexive Banach space \[ U_{\Gamma_0}=\{u\in W^{1,p(\cdot)}(\Omega):\,u=0 \text{ on }\Gamma_0\} \] with the norm \[ \|u\|_{U_{\Gamma_0}}=\|u\|_{L^{p(\cdot)}}+\|\nabla u\|_{L^{p(\cdot)}}. \] For a normed space \(X\), let \(X^*\) denote the dual space of \(X\). Definitions and properties of duality mappings \(J_{\varphi}:X\to X^*\) subordinate to a gauge function \(\varphi\) are recalled in Section 3. Properties of the space \(U_{\Gamma_0}\), and of duality mappings on \(U_{\Gamma_0}\) are studied in Section 4. For \(p\in L^{\infty}(\Omega)\), let \(p^-=\underset{x\in\Omega}{\text{ ess}\inf}p(x)\). Let \(p,q \in C(\bar\Omega)\) with \(p^->1\), \(q^->1\) and \(q^{\prime}\) be defined by \(\frac{1}{q(x)}+\frac{1}{q^{\prime}(x)}=1\). Let \(f:\Omega\times\mathbb R\to\mathbb R\) be a Carathéodory function and \(N_f:L^{q(\cdot)}(\Omega)\to L^{q^{\prime}(\cdot)}(\Omega)\) be defined by \((N_fu)(x)=f(x,u(x))\) for almost all \(x\in\Omega\). The main result (Theorem 6.1) states sufficient conditions on \(p,q,f\) and on a gauge function \(\varphi\) for the solution set of the equation \[ J_{\varphi} u=N_f u\tag{1} \] to be a non-empty and compact subset of \(U_{\Gamma_0}\). The existence of suitable solutions to equation (1) is proved by three different methods based, respectively, on reflexivity and smoothness of the space \(U_{\Gamma_0}\), the Schauder fixed point theorem, or the Leray-Schauder degree.