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.

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

Gongbao Li

Abstract

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