An existence result on positive solutions for a class of semilinear elliptic systems. (English) Zbl 1067.35026

Hai, D. D.; Shivaji, R.

Proc. R. Soc. Edinb., Sect. A, Math. 134, No. 1, 137-141 (2004).

Consider the boundary value problem \(-\Delta v= \lambda f(u)\) in \(\Omega\), \(-\Delta u= \lambda f(v)\) in \(\Omega\), \(u= v\) on \(\partial\Omega\), where \(\lambda\) is a positive parameter and \(\Omega\) is a bounded domain in \(\mathbb R^N\). The authors prove the existence of a large positive solution for large \(\lambda\) when \(\lim_{x\to\infty} (f(Mg(x))/x)= 0\) for every \(M> 0\).

Reviewer: Z. G. Sheftel’ (Göttingen)

MSC:

35J55	Systems of elliptic equations, boundary value problems (MSC2000)
35J60	Nonlinear elliptic equations
35B05	Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs

Keywords:

semilinear elliptic system; positive solutions