A priori bounds and existence of non-negative solutions to a quasi-linear Schrödinger equation involving p-Laplacian

In this paper, we focus on the a priori bounds and existence of non-negative solutions to the following quasi-linear-Schrödinger equation with p-Laplacian, 0.1-Δpu-1(1+α)p-1(Δp|u|1+α)|u|α-1u=|u|q-1u+h(x,u)x∈Ω,u>0x∈Ω,u=0x∈∂Ω,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta _pu-\frac{1}{(1+\alpha )^{p-1}} (\Delta _p|u|^{1+\alpha })|u|^{\alpha -1}u=|u|^{q-1}u+h(x,u) &{}\quad x\in \Omega , \\ u>0 &{}\quad x\in \Omega ,\\ u=0 &{}\quad x\in \partial \Omega , \end{array} \right. \end{aligned}$$\end{document}where Δpu=div(|∇u|p-2∇u),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta _pu=\text {div}(|\nabla u|^{p-2}\nabla u),$$\end{document}Ω⊆Rn(n≥3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \subseteq {\mathbb {R}}^n(n\ge 3)$$\end{document} is a bounded smooth domain, p∗=npn-p,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p^*=\frac{np}{n-p},$$\end{document}q∈(pα+p-1,p∗α+p∗-1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q\in (p\alpha +p-1,p^*\alpha +p^*-1)$$\end{document} and 0≤α<min{q,q+1-pp-1}.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0\le \alpha <\min \{q,\frac{q+1-p}{p-1}\}.$$\end{document} Under suitable conditions of h,  we prove the existence of non-negative solutions to problem (0.1) via the theory of topological degree whose crucial factor is the a priori bounds of solutions to an auxiliary problem. We employ the well-known blow-up argument to get the a priori bounds of solutions. It is worth mentioning that the main result of the present paper devotes to getting new result on the nonlinear Schrödinger equations with p-Laplacian.