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

TL;DR

This paper proves the existence of multiple small solutions for a class of nonhomogeneous quasilinear elliptic systems in ^N, using advanced variational methods and iteration techniques without growth restrictions.

Contribution

It introduces a novel approach combining Clark's theorem variant and Moser's iteration for nonhomogeneous operators, expanding solution existence results.

Findings

Multiple small solutions are established under sublinear and symmetric conditions.

The relationship between solution norms is characterized, overcoming nonhomogeneity challenges.

The method avoids growth hypotheses, broadening applicability of elliptic system analysis.

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.