Existence of positive radial solutions for a weakly coupled systems via blow up. (English) Zbl 0965.35058

García-Huidobro, Marta; Guerra, Ignacio; Manásevich, Raúl

Abstr. Appl. Anal. 3, No. 1-2, 105-131 (1998).

The authors prove existence of positive radial solutions of the quasilinear system \[ \text{div} (a_i(|\nabla u_i|)\nabla u_i)+f_i(u_{i+1})=0 \text{ in }\Omega, \quad u_i=0\text{ on }\partial\Omega, \] where \(\Omega=\{x\in\mathbb R^N : |x|<R\}\), \(i=1,2,\dots,n\), \(u_{n+1}=u_1\). Denote \(\phi_i(s)=sa_i(s)\). The functions \(\phi_i,f_i\) are supposed to be continuous, asymptotically homogeneous at zero and at infinity (with suitable powers satisfying several inequalities), \(f_i(0)=0\), \(f_i(s)>0\) for \(s>0\), \(\phi_i:\mathbb R^+\to\mathbb R^+\) are increasing, \(\phi_i(s)\to\infty\) as \(s\to\infty\). The proof is based on the use of the Leray-Schauder degree and on a modification of a priori estimates of B. Gidas and J. Spruck [Commun. Partial Differ. Equations 6, 883-901 (1981; Zbl 0462.35041)].

Reviewer: Pavol Quittner (Bratislava)

MSC:

35J65	Nonlinear boundary value problems for linear elliptic equations
34B15	Nonlinear boundary value problems for ordinary differential equations
35B05	Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs

Keywords:

quasilinear elliptic system; radial solutions; asymptotically homogeneous; a priori estimates