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

Gongbao Li

doi:10.4171/rmi/55

JournalsrmiVol. 3, No. 3pp. 371–399

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

Gongbao Li
Huazhong Normal University, Wuhan, China

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Abstract

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.

Cite this article

Gongbao Li, Eigenvalue Problems of Quasilinear Elliptic Systems on $R^n$ . Rev. Mat. Iberoam. 3 (1987), no. 3, pp. 371–399

DOI 10.4171/RMI/55