On critical double phase Kirchhoff problems with singular nonlinearity
Let $\Omega\subset\mathbb{R}^N$ be a bounded domain with Lipschitz boundary $\partial\Omega$. This work combines the effects of a nonlocal Kirchhoff coefficient $m:[0,+\infty)\to [0,+\infty)$ given as \[ m(t)=a_0+b_0t^{\theta-1}\text{ for all }t\geq0,\text{ with }a_0\geq 0, b_0>0\text{ and }\theta \in [1,p^\ast/q) \] ($p^\ast=Np/(N-p)$ is the critical Sobolev exponent to $p$, $1<p<N$ and $p<q<p^\ast$), and a double phase operator \[ \operatorname{div}(|\nabla u|^{p-2}\nabla u+a(x)|\nabla u|^{q-2}\nabla u)\text{ for all }u\in W_0^{1,\mathcal{H}}(\Omega) \] ($W_0^{1,\mathcal{H}}(\Omega)$ is the homogeneous Musielak-Orlicz Sobolev space) with a parametric singular term $\lambda u^{-\gamma}$ ($\lambda>0$ and $0<\gamma<1$) and a critical term $u^{p^\ast-1}$. The proof of main result is based on a minimization argument on the Nehari manifold. Under appropriate hypotheses on the data, the authors obtain the existence of at least a weak solution $u_\lambda \in W_0^{1,\mathcal{H}}(\Omega)$ to the Dirichlet problem for appropriate values of $\lambda$.
scientific article; zbMATH DE number 7606144
scientific article