On coupled multiparameter nonlinear elliptic systems

Cantrell, Robert Stephen

doi:10.1090/S0002-9947-1986-0819947-4

On coupled multiparameter nonlinear elliptic systems
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Trans. Amer. Math. Soc. 294 (1986), 263-285 Request permission

Abstract:

This paper considers the system of nonlinear Dirichlet boundary value problems \[ \left \{ \begin {gathered} Lu(x) = \lambda f(u(x),v(x)) \hfill \\ Lv(x) = \mu g(u(x),v(x)) \hfill \\ \end {gathered} \right \},\qquad x \in \Omega ,\] a bounded domain in ${{\mathbf {R}}^n}$. Here $L$ is a strongly, uniformly elliptic linear partial differential operator, $\lambda$, $\mu$ are real parameters, and $f$, $g:{{\mathbf {R}}^2} \to R$ are smooth with \[ f(0,0) = 0 = g(0,0).\] A detailed analysis of the solution set to the system is given from the point of view of several parameter bifurcation theory.

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