JournalsrmiVol. 3, No. 3pp. 371–399

Eigenvalue Problems of Quasilinear Elliptic Systems on RnR^n

  • Gongbao Li

    Huazhong Normal University, Wuhan, China
Eigenvalue Problems of Quasilinear Elliptic Systems on $R^n$ cover
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Abstract

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

Dα(aαβ(x,u)Dβui)+12Duiaαβ(x,u)DαujDβuj+h(x)ui=λup2ui,–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 xRnx \in \mathbb R^n, 1iN1 ≤ i ≤ N and

u=(u1,u2,...,uN)E={ν=(ν1,ν2,...,νN)νiH1(Rn),1iN},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 aαβ(x,u)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 RnR^n. Rev. Mat. Iberoam. 3 (1987), no. 3, pp. 371–399

DOI 10.4171/RMI/55