Multiplicity result on a class of nonhomogeneous quasilinear elliptic system with small perturbations in $\mathbb{R}^N$
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| Format: | Preprint |
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2025
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| _version_ | 1866910771393331200 |
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| author | Zhang, Xingyong Qi, Wanting |
| author_facet | Zhang, Xingyong Qi, Wanting |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_01602 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Multiplicity result on a class of nonhomogeneous quasilinear elliptic system with small perturbations in $\mathbb{R}^N$ Zhang, Xingyong Qi, Wanting Analysis of PDEs 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. |
| title | Multiplicity result on a class of nonhomogeneous quasilinear elliptic system with small perturbations in $\mathbb{R}^N$ |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2501.01602 |