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

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

$Eigenvalue Problems of Quasilinear Elliptic Systems on $\mathbb R^n$ cover$

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Abstract

In this paper, we get the existence results of the nontrivial weak solution $(λ, u)$ of the following eigenvalue problem of quasilinear elliptic systems

- D_{α} (a_{α β} (x, u) D_{β} u^{i}) + \frac{1}{2} D_{u^{i}} a_{α β} (x, u) D_{α} u^{j} D_{β} u^{j} + h (x) u^{i} = λ ∣ u ∣^{p -2} u^{i},

for $x \in R^{n}$ , $1 \leq i \leq N$ and

u = (u^{1}, u^{2}, ..., u^{N}) \in E = {ν = (ν^{1}, ν^{2}, ..., ν^{N}) ∣ ν^{i} \in H^{1} (R^{n}), 1 \leq i \leq N},

where $a_{α β} (x, u)$ 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