On the Multi-Bump Solutions to (N, q)-Laplacian Equations with Exponential Critical Growth and Logarithmic Nonlinearity in RN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^N$$\end{document}On the multi-bump solutions to (N, q)-Laplacian equations...Y. Tong, S. Liang

In this paper, we are concerned with the existence and multiplicity of multi-bump solutions for the following (N, q)-Laplacian equation -ΔNu-Δqu+(μV(x)+Z(x))(|u|N-2u+|u|q-2u)=h(u)+|u|q-2ulog|u|qinRN,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} & -\Delta _N u-\Delta _q u+(\mu \mathcal {V}(x)+\mathcal {Z}(x))(|u|^{N-2}u+|u|^{q-2}u)\\ & \quad =h(u)+|u|^{q-2}u\log |u|^q\quad \text {in}\ \mathbb {R}^N, \end{aligned}$$\end{document}where 2≤N<q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2\le N<q$$\end{document}, μ∈[1,+∞)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu \in [1,+\infty )$$\end{document}, Δsu=div(|∇u|s-2∇u)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta _\mathfrak {s}u=\text {div}(|\nabla u|^{\mathfrak {s}-2}\nabla u)$$\end{document} is the s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak {s}$$\end{document}-Laplace operator with s∈{N,q}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak {s}\in \{N,q\}$$\end{document}, h is a continuous function with exponential critical growth, V\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {V}$$\end{document} is a nonnegative continuous function with the potential well Ω:=int(V-1(0))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega :=\text {int}(\mathcal {V}^{-1}(0))$$\end{document} consisting of k components, and the nonnegative continuous function Z\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {Z}$$\end{document} verifies some assumptions. With the aid of variational methods, we obtain the existence and multiplicity of multi-bump solutions as μ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document} is large enough. As far as we know, it is the first time that the existence and multiplicity of multi-bump solutions to the (N, q)-Laplacian equation with exponential critical growth and logarithmic nonlinearity are studied. The most obvious and important feature is that we establish some new technique results to prove our results.