Let \(\Omega \subseteq \mathbb{R}^N\) (\(N \geq 2\)) be a bounded domain with a Lipschitz continuous boundary \(\Gamma := \partial \Omega\). Let \(\Gamma\) be divided into three mutually disjoint parts \(\Gamma_i\) (\(i=1,2,3\)) with \(\Gamma_1\) having positive Lebesgue measure. The authors consider the following problem \[ \begin{cases} -\Delta_{p,q}^{\mu}u+|u|^{p-2}u+\mu(x)|u|^{q-2}u \in U_1(x,u)+f(x,u,\nabla u) \mbox{ in } \Omega,\\ \\ u\big|_{\Gamma_1}=0, \, \dfrac{\partial u}{\partial \nu_a}\in U_2(x,u) \mbox{ on } \Gamma_2, \, -\dfrac{\partial u}{\partial \nu_a}\in \partial_c \phi(x,u) \mbox{ on } \Gamma_3, \, L(u)\leq J(u), \end{cases}\] where \(\Delta_{p,q}^{\mu} u= \operatorname{div}(|\nabla u|^{p-2}\nabla u+\mu(x)|\nabla u|^{q-2}\nabla u)\) for all \(u \in W^{1,\mathcal{H}}(\Omega)\) (i.e., the Musielak-Orlicz-Sobolev space) is the double phase operator, \(\mu : \overline{\Omega}\to [0,+\infty)\), \(1<p<q<N\), \(f: \Omega \times \mathbb{R}\times \mathbb{R}^N \to \mathbb{R}\), \(U_1:\Omega \times \mathbb{R} \to 2^{\mathbb{R}}\), \(U_2:\Gamma_2 \times \mathbb{R} \to 2^{\mathbb{R}}\), \(\dfrac{\partial u}{\partial \nu_a}=(|\nabla u|^{p-2}\nabla u+\mu(x)|\nabla u|^{q-2}\nabla u)\cdot \nu \) with \(\nu\) being the unit normal vector on \(\Gamma\), \(\phi: \Gamma_3 \times \mathbb{R}\to \mathbb{R}\) with \(\partial_c \phi(x,u)\) being the convex subdifferential of \(s \to \phi(x,s)\), and \(L,J: W^{1,\mathcal{H}}(\Omega) \to \mathbb{R}\) are suitable functions.
The study is concerned with the combined effects of a nonstandard operator with unbalanced growth \(-\Delta_{p,q}^{\mu}\), a convection nonlinearity \(f\), certain multivalued terms, and an implicit obstacle constraint. Using the Kakutani-Ky Fan fixed point theorem for multivalued operators, the theory of nonsmooth analysis, and variational methods for pseudomonotone operators, the authors establish the existence of a (weak) solution to the above problem.