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

Jianjun Zhang, João Marcos do Ó, Olímpio H. Miyagaki



We consider the following singularly perturbed problem: $$ -\e^N\Delta_N u+V(x)|u|^{N-2}u= f(u),\quad u(x)> 0\quad \mbox{in } \mathbb R^N, $$ \noindent where $N\ge 2$ and $\Delta_N u$ is the $N$-Laplacian operator. In this paper, we construct a solution $u_\e$ which concentrates around any given isolated positive local minimum component of $V$, as $\e\rg 0$, 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 {\it monotonicity} of $f(t)/t^{N-1}$ or {\it Ambrosetti-Rabinowitz} type conditions are not required.


Nonlinear Schrödinger equations; semiclassical states; potential well; critical growth; penalization

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