Multiplicity result on a class of nonhomogeneous quasilinear elliptic system with small perturbations in ${\mathbb{R}}^N$

Abstract

We investigate a class of quasilinear elliptic system involving a nonhomogeneous differential operator which is introduced by C. A. Stuart [Milan J. Math. 79 (2011), 327-341] and depends on not only $\nabla u$ but also $u$. We show that the existence of multiple small solutions when the nonlinear term $F(x,u,v)$ satisfies locally sublinear and symmetric conditions and the perturbation is any continuous function with a small coefficient and no any growth hypothesis. Our technical approach is mainly based on a variant of Clark's theorem without the global symmetric condition. We develop the Moser's iteration technique to this quasi-linear elliptic system with nonhomogeneous differential operators and obtain that the relationship between $\Vert u{\Vert }_{\infty }$, $\Vert v{\Vert }_{\infty }$ and $\Vert u{\Vert }_{2^{\ast }}$, $\Vert v{\Vert }_{2^{\ast }}$. We overcome some difficulties which are caused by the nonhomogeneity of the differential operator and the lack of compactness of the Sobolev embedding.