Singularly perturbed $N$-Laplacian problems with a nonlinearity in the critical growth range

Abstract

We consider the following singularly perturbed problem: $$ -\varepsilon^N\Delta_N u+V(x)|u|^{N-2}u= f(u),\quad u(x)>0\quad \mbox{in } \mathbb R^N, $$ where $N\ge 2$ and $\Delta_N u$ is the $N$-Laplacian operator. In this paper, we construct a solution $u_\varepsilon$ which concentrates around any given isolated positive local minimum component of $V$, as $\varepsilon\rightarrow 0$, in the Trudinger-Moser type of subcritical or critical case. In the subcritical case, we only impose on $f$ the Berestycki and Lions conditions. In the critical case, a global condition on the nonlinearity $f$ is imposed. However, any monotonicity of $f(t)/t^{N-1}$ or Ambrosetti-Rabinowitz type conditions are not required.