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 $\|u\|_{\infty}$, $\|v\|_{\infty}$ and $\|u\|_{2^{\ast}}$, $\|v\|_{2^{\ast}}$. We overcome some difficulties which are caused by the nonhomogeneity of the differential operator and the lack of compactness of the Sobolev embedding.

Theorems & Definitions (23)

• Remark 1.1
• Theorem 1.2
• Corollary 1.3
• Remark 1.4
• Remark 2.1
• Remark 2.2
• Lemma 2.3
• Remark 2.4
• Lemma 3.1
• Remark 3.2