On multi-bump solutions for a class of (N,q)-Laplacian equation with critical exponential growth in [formula omitted]

Abstract

This article deals with the following (N,q)-Laplacian equation with exponential critical growth in RN of the form:−ΔNu−Δqu+(μV(x)+B(x))(|u|N−2u+|u|q−2u)=h(u)inRN, where 2≤N<q, μ∈(0,∞), h is a continuous function with exponential critical growth and V:RN→R is a continuous function verifying some hypotheses. Assuming that the nonnegative function B has a potential well int(B−1({0})) composed of k disjoint components Ω1,Ω2,⋯,Ωk. With the help of the variational methods and Morse iteration technique, the existence and multiplicity of positive multi-bump solutions are obtained if the parameter μ>0 is large enough.