Multiplicity results for a class of quasilinear equations with exponential critical growth

Abstract

In this work, we prove the existence and multiplicity of positive solutions for the following class of quasilinear elliptic equations $$ \left \{ \begin{array}{lll} -\epsilon^{N}\Delta_{N} u + \left(1+\mu A(x) \right)\left| u\right|^{N-2}u= f(u)\,\,\,\, \mbox{ in } \, \RR^{N},\\ u > 0 \,\,\,\, \mbox{ in } \, \RR^{N}, \end{array} \right. $$ \\ where $\Delta_{N}$ is the N-Laplacian operator, $N \geq 2$, $f$ is a function with exponential critical growth, $\mu$ and $\epsilon$ are positive parameters and $A$ is a nonnegative continuous function verifying some hypotheses. To obtain our results, we combine variational arguments and Lusternik-Schnirelman category theory .