Multiplicity and concentration of positive solutions for a class of quasilinear problems through Orlicz-Sobolev space

In this work, we study existence, multiplicity, and concentration of positive solutions for the following class of quasilinear problems −ΔΦu+V(ϵx)ϕ(u)u=f(u)inRN (N≥2), where Φ(t)=∫0tϕ(s)sds is a N-function, ΔΦ is the Φ-Laplacian operator, ϵ is a positive parameter, V : ℝN → ℝ is a continuous function, and f : ℝ → ℝ is a C1-function.