In this paper, (i) we present unified approaches to studying the existence of ground state solutions and mountain-pass type solutions for the following quasilinear equation: -Delta Nu+V(x)divided by u divided by N-2u=f(u)inRN,N >= 2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-{\Delta_{N}}u+V(x){|u|<^>{N-2}}u=f(u)\quad {\rm in} \ {{\mathbb R}<^>{N}},\quad N \geqslant 2$$\end{document} in three different cases allowing the potential V is an element of C(RN,R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V \in {\cal C}({{\mathbb R}<^>{N}}, {\mathbb R})$$\end{document} to be periodic, radially symmetric, or asymptotically constant, where Delta Nu:=div(divided by del u divided by N-2 del u)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Delta_{N}}u :={\rm div}({|\nabla u|}<^>{N-2} \nabla u)$$\end{document} and f has critical exponential growth; (ii) two new compactness lemmas in W1,N(& Ropf;N) for general nonlinear functionals are established, which generalize the ones obtained in the radially symmetric space Wrad1,N(RN)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W_{\rm rad}<^>{1,N}({\mathbb R}<^>{N})$$\end{document}; (iii) based on some key observations, we construct a special path allowing us to control the mountain-pass minimax level by a fine threshold under which the compactness can be restored for the critical case. In particular, some delicate analyses are developed to overcome non-standard difficulties due to both the quasilinear characteristic of the equation and the lack of compactness aroused by the critical exponential growth of f. Our results extend and improve the ones of Alves et al. (2012), Ibrahim et al. (2015) (N = 2), and Masmoudi and Sani (2015) (N >= 3) for the constant potential case, Alves and Figueiredo (2009) for the periodic potential case, Lam and Lu (2012) and Yang (2012) for the coercive potential case, and Chen et al. (Sci China Math, 2021) for the degenerate potential case, which are totally new even for the simpler semilinear case of N = 2. We believe that our approaches and strategies may be adapted and modified to attack more variational problems with critical exponential growth.
