A new class of double phase variable exponent problems: existence and uniqueness (Q2124514)

Let \(\Omega \subseteq \mathbb{R}^N\) (\(N \geq 2\)) be a bounded domain with a Lipschitz boundary \(\partial \Omega\). The authors consider a double phase operator of the form \[\operatorname{div}(|\nabla u|^{p(x)-2}\nabla u+\mu(x)|\nabla u|^{q(x)-2}\nabla u), \quad p,q \in C(\overline{\Omega}),\] where \(1<p(x)<N\), \(p(x)<q(x)\) for all \(x \in \overline{\Omega}\) and \(0\leq \mu(\cdot)\in L^1(\Omega)\). The authors first study the function space related to the above double phase operator, then they prove the properties of this operator. Namely, they establish that the operator is continuous, bounded, strictly monotone and satisfies the \((S_+)\)-property. In the last part of the paper, the authors obtain the existence and uniqueness of corresponding double phase elliptic equations with convection term, under suitable hypotheses on the data.