Abstract:

We consider a class of quasilinear Schrödinger equations of the form

where $u\in H_{0}^{1}(\Omega)$,$\kappa\in (-2,0)\cup (0,+\infty),\ 2\leq p<2^*,\ N\geq 3$ and $\Omega$ is a bounded domain. By using variational approaches, we establish the existence of a solution $(\lambda, u).$ Particularly, we give the accurate $L^{\infty}$ estimate. For instance, if $\kappa\in (-2,0)$ and $|u|_{p}=1,$ we construct the following $L^{\infty}$ estimate of the solution

where $a=\frac{1}{p}-\frac{1}{2^*},p'=\frac{p}{p-1}$,$C_{N}$ is the best Sobolev constant and $\lambda_{1}$ and $\phi_{1}$ are the first eigenvalue and the first eigenfunction of the operator $-\Delta$ respectively.

Key words: schr?dinger equations, $L^{\infty}$ estimate, eigenvalue problem.