Existence of solutions for $p$-Laplacian equation with electromagnetic fields and critical nonlinearity

Abstract

In this paper, we consider the existence and multiplicity of solutions for perturbed $p$-Laplacian equation problems with critical nonlinearity in $\mathbb{R}^N$: $- \varepsilon^p\Big[g\Big(\displaystyle\int_{\mathbb{R}^N}|\nabla_A u|^pdx\Big)\Big]\Delta_{p,A}u + V(x)|u|^{p-2}u = |u|^{p^\ast-2}u + h(x, |u|^p)|u|^{p-2}u$ for all $(t, x) \in \mathbb{R} \times \mathbb{R}^N$, where $V(x)$ is a nonnegative potential, $\Delta_{p,A}u(x) :=\mathop{\rm div}(|\nabla u+iA(x)u|^{p-2}(\nabla u + iA(x)u)$ and $\nabla_Au := (\nabla + iA)u$. 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 $\varepsilon < \mathcal {E}$, for any $m \in \mathbb{N}$, it has $m$ pairs of solutions if $\varepsilon < \mathcal {E}_m$, where $\mathcal {E}$ and $\mathcal {E}_m$ are sufficiently small positive numbers.