On a nonlinear elliptic system: resonance and bifurcation cases. (English) Zbl 1064.35052

Zuluaga, Mario

Commentat. Math. Univ. Carol. 40, No. 4, 701-711 (1999).

The authors consider a class of semilinear elliptic systems \[ \begin{split} -\triangle u = \lambda u + \delta v + g_1(u,v,) - r_1(x), \\ -\triangle v = \theta u + \gamma v + g_2(u,v) - r_2(x) \end{split} \] on a bounded smooth domain \(\Omega \) in \(\mathbb R^N, u,v \in H_0^1(\Omega )\). The assumptions allow to interprete the system as a steady state of a reaction-diffusion problem. The authors generalize to this class some of the results known for a scalar case. Using Lyapunov-Schmidt reduction and fixed point approach they give sufficient conditions for bifurcation and apply the results to a boundary value problem for the biharmonic equation.

Reviewer: Jana Stará (Praha)

MSC:

35J55	Systems of elliptic equations, boundary value problems (MSC2000)
35J65	Nonlinear boundary value problems for linear elliptic equations
35B34	Resonance in context of PDEs
47J15	Abstract bifurcation theory involving nonlinear operators

Keywords:

elliptic systems at resonance; bifurcation points; Lyapunov-Schmidt method