An elliptic system and the critical hyperbola

Abstract

We consider a nonlinear elliptic system of Lane-Emden type in the whole space ℝn , namely (Formula presented). Our region for (p,q ) covers in particular the critical and supercritical cases with respect to the critical hyperbola (Formula presented) + (Formula presented) + (Formula presented)= (Formula presented). We prove existence of solutions for f ∈ Ld (ℝn), by means of a fixed point technique in the Lebesgue space Lr1 × Lr2. Our results allow unbounded solutions without Hs -regularity. The solutions are shown to be classical and positive when f is smooth enough and positive. Moreover, if f is radial or odd (or even), we prove that the solutions preserve these properties. Also, it is shown that the solutions (u,v ) are nonradial when f is nonradial.

We consider a nonlinear elliptic system of Lane-Emden type in the whole space Rn, namely ( ∆u + v|v|p−1 = 0, x ∈Rn, ∆v + u|u|q−1 + f = 0, x ∈Rn. Our region for (p,q) covers in particular the critical and supercritical cases with respect to the critical hy