On the Existence and Multiplicity of Solutions for Double Phase Equations with Robin Boundary Conditions

Abstract

This paper studies the following new class of nonlinear double-phase equations with a defined Robin boundary condition (the nonlinear disturbances that drive them to obey a suitable condition at the origin and on the boundary)\begin{equation*}\left\{\begin{array}{lcr}-\mbox{div} \left( \mathcal{A} \left( \nabla u \right) \right) -\mbox{div} \left( \mathcal{B} \left( \nabla u \right) \right) = \mu f \left( x, u \right) & \mbox{ in} \ \Omega,\\ \left( \mathcal{A} \left( \nabla u \right) + \mathcal{B} \left( \nabla u \right) \right).\eta + b \left( x \right) \vert u \vert^{p \left( x \right)-2} u + d \left( x \right) \vert u \vert^{q \left( x \right)-2} u = g \left( x, u \right) & \mbox{on} \ \partial \Omega,\end{array}\right.\end{equation*}where $\mu>0$, $\Omega \subset I\!\!R^{N} (N\geqslant2)$ is a bounded open domain with smooth boundary $\partial\Omega$, $\eta$ is the outer unit normal vector on $\partial\Omega$, $b \left( . \right) $ and $d \left( . \right)$ are positive and continuous functions on $\partial\Omega$. At first, via a generalized mountain-pass approach with Critical point theory, we prove that this problem with superlinear nonlinearity has a solution and infinitely many solutions. Furthermore, we rigorously verify the existence of infinitely many solutions by imposing sufficient constraints on the functions $f$ and $g$. Our results comprehensively expand and generalize specific recent contributions within the existing literature.