Revista Matemática Iberoamericana


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Volume 3, Issue 3, 1987, pp. 371–399
DOI: 10.4171/RMI/55

Published online: 1987-12-31

Eigenvalue Problems of Quasilinear Elliptic Systems on $R^n$

Gongbao Li[1]

(1) Huazhong Normal University, Wuhan, China

In this paper, we get the existence results of the nontrivial weak solution $(\lambda, u)$ of the following eigenvalue problem of quasilinear elliptic systems $$–D_\alpha (a_{\alpha \beta}(x, u)D_\beta u^i) + \frac{1}{2}D_{u^i}a_{\alpha \beta}(x, u)D_\alpha u^jD_\beta u^j + h(x)u^i = \lambda |u|^{p–2}u^i,$$ for $x \in \mathbb R^n$, $1 ≤ i ≤ N$ and $$u = (u^1, u^2, ..., u^N) \in E = \{ \nu = (\nu^1, \nu^2, ..., \nu^N) | \nu^i \in H^1 (\mathbb R^n), 1 ≤ i ≤ N \},$$ where $a_{\alpha \beta} (x, u)$ satisfy the natural growth conditions. It seems that this kind of problem has never been dealt with before.

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Li Gongbao: Eigenvalue Problems of Quasilinear Elliptic Systems on $R^n$. Rev. Mat. Iberoam. 3 (1987), 371-399. doi: 10.4171/RMI/55