High perturbations of quasilinear problems with double criticality

This paper is concerned with the qualitative analysis of solutions to the following class of quasilinear problems P $$\begin{aligned} \left\{ \begin{array}{ll} -\Delta _{\Phi }u=f(x,u) &{}\quad \text {in } \Omega ,\\ u=0 &{}\quad \text {on }\partial \Omega , \end{array} \right. \end{aligned}$$ where $$\Delta _{\Phi }u=\mathrm{div}\,(\varphi (x,|\nabla u|)\nabla u)$$ and $$\Phi (x,t)=\int _{0}^{|t|}\varphi (x,s)s\,ds$$ is a generalized N-function. We assume that $$\Omega \subset {\mathbb {R}}^N$$ is a smooth bounded domain that contains two open regions $$\Omega _N,\Omega _p$$ with $${\overline{\Omega }}_N \cap {\overline{\Omega }}_p=\emptyset $$ . The features of this paper are that $$-\Delta _{\Phi }u$$ behaves like $$-\Delta _N u $$ on $$\Omega _N$$ and $$-\Delta _p u $$ on $$\Omega _p$$ , and that the growth of $$f:\Omega \times {\mathbb {R}} \rightarrow {\mathbb {R}}$$ is like that of $$e^{\alpha |t|^{\frac{N}{N-1}}}$$ on $$\Omega _N$$ and as $$|t|^{p^{*}-2}t$$ on $$\Omega _p$$ when |t| is large enough. The main result establishes the existence of solutions in a suitable Musielak–Sobolev space in the case of high perturbations with respect to the values of a positive parameter.