The manuscript deals with the existence and multiplicity of weak solutions to the following double phase problem \[ \begin{cases}- \Delta_{p(z)}^1 u(z) - \Delta_{q(z)}^{\mu(z)} u(z) =f(z,u(z)) \quad \mbox{in } \Omega, &\\ u=0 \quad \mbox{on }\partial \Omega, &\end{cases} \] where \(\Omega \subseteq \mathbb{R}^N\) is a bounded domain with a Lipschitz boundary \(\partial \Omega\), and the authors impose a Dirichlet boundary condition. Given \(r \in C(\overline{\Omega})\) with \(1<r(z)\) for all \(z \in \overline{\Omega}\), by \(\Delta_{r(z)}^{\mu(z)}\) we denote the following weighted differential operator \[ \Delta_{r(z)}^{\mu(z)} u=\mathrm{div } [\mu(z)|\nabla u|^{r(z)-2}\nabla u] \quad \mbox{for all } u \in W^{1,r(z)}(\Omega), \] with \(0 \leq \mu(\cdot) \in L^\infty(\Omega)\). When \(\mu \equiv 1\) we write \(\Delta_{r(z)}^1\). In the above problem, the differential operator is the sum of two such operators (double phase problem). The authors work in the variable exponents setting, hence pose the problem in an appropriate Musielak-Orlicz Sobolev space. The exponents \(p,q\) are properly linked each other and also fulfil suitable conditions. The reaction (right hand side) has superlinear type growth at zero and infinity, and satisfies certain technical conditions.
The authors first obtain a priori bounds for weak solutions to a class of suitable general problems, then prove the existence of a positive and a negative weak solution to the above double phase problem via the mountain pass theorem, involving auxiliary functionals truncated at zero. Further, they obtain the existence of a sign-changing solution to the above double phase problem, by solving a minimization problem on a modified version of the Nehari manifold.