Positive solutions for semipositone (p,N)-Laplacian problems with critical Trudinger–Moser nonlinearities

In this paper, we deal with the existence of positive solutions for semipositone (p,N)-Laplacian problems with critical Trudinger–Moser nonlinearities in a bounded domain: -Δpu-ΔNu=λuN-1eβuN′-μ,inΩ;u>0,inΩ;u=0,on∂Ω.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{array}{clll} -\varDelta _p u-\varDelta _N u=\lambda u^{N-1}e^{\beta u^{N'}} - \mu , &{} \text {in}\,\varOmega ;\\ u>0, &{} \text {in}\,\varOmega ;\\ u=0,&{} \text {on}\,\partial \varOmega . \end{array} \right. \end{aligned}$$\end{document}We obtain the positive solutions by combining variational methods with regularity arguments. And the main novelty here is to obtain a uniform C1,α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {C}^{1,\alpha }$$\end{document} priori estimate of the weak solution. Our arguments can be also adapted to seek positive solutions of more general semipositone problems.