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### Open Access Journals

DCDS

In this paper, we study the following quasilinear elliptic equation with critical Sobolev exponent:

$ -\Delta u +V(x)u-[\Delta(1+u^2)^{\frac 12}]\frac {u}{2(1+u^2)^\frac 12}=|u|^{2^*-2}u+|u|^{p-2}u, \quad x\in {{\mathbb{R}}^{N}}, $ |

which models the self-channeling of a high-power ultra short laser in matter, where 14 ], we obtain the existence of positive ground state solutions for the given problem.

*N*≥ 3; 2 <*p*< 2^{*}= $\frac{{2N}}{{N -2}}$ and*V*(*x*) is a given positive potential. Combining the change of variables and an abstract result developed by Jeanjean in [
CPAA

This paper is concerned with the existence of soliton solutions for a quasilinear Schrödinger equation in $R^N$
with critical exponent, which appears from modelling the self-channeling of a high-power ultrashort
laser in matter. By working with a perturbation approach which was initially proposed in [26],
we prove that the given problem has a positive ground state solution.

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