On some p-Laplacian equation with electromagnetic fields and critical nonlinearity in ℝN

Abstract

In this paper, we consider the existence and multiplicity of solutions for p-Laplacian equation with electromagnetic fields and critical nonlinearity in ℝN: −εpΔp,Au+V(x)up−2u=up*−2u+h(x,up)up−2u for x ∈ ℝN, where Δp,Au(x)≔div(∇u+iA(x)up−2(∇u+iA(x)u). By using Lions’ second concentration compactness principle and concentration compactness principle at infinity to prove that the (PS)c condition holds locally and by variational method, we show that this equation has at least one solution provided that ε<E, for any m ∈ ℕ, it has m pairs of solutions if ε<Em, where E and Em are sufficiently small positive numbers.