Ground state and non-ground state solutions of some strongly coupled elliptic systems

Abstract

Abstract. We study an elliptic system of the form Lu = |v| p−1 v and Lv = |u| q−1 u in Ω with homogeneous Dirichlet boundary condition, where Lu := −Δu in the case of a bounded domain and Lu := −Δu + u in the cases of an exterior domain or the whole space R N . We analyze the existence, uniqueness, sign and radial symmetry of ground state solutions and also look for sign changing solutions of the system. More general non-linearities are also considered.

Symmetry and symmetry breaking for ground state solutions of some strongly coupled elliptic systems

“…The presentation in Sections 2.1 and 3.1 is close to the one in [5]. The key point is played here by the regularity result in Lemma 2.8, which is in turn based on Proposition 2.1; we can formulate the problem in such a way that existence, regularity, positivity and foliated Schwarz symmetry of the solutions are deduced smoothly.…”

“…The norm in the latter space will be denoted by | | p+1,α . Once our framework is settled as above, one can deduce Theorem 1.1 by arguing exactly as in [5,Section 2]. For completeness, we indicate here the key steps of the argument.…”

“…The idea goes back to P.L. Lions [22], see also [10,11,16,17,21,32], and has been used recently by the authors in [5]. The presence of the weights makes the functional setting more delicate.…”

“…In the case where α = β = 0, the radial symmetry of the ground state solutions was established in [5,Theorem 1.6]. As mentioned above, such a result is no longer expected for large values of the parameters α or β in (1.1).…”

Critical and noncritical regions on the critical hyperbola

“…We recall that the functions S Ä ' ı;a , with ' ı;a as in (6) and Ä D p.p.N 2/ 2/ 2.pC1/ 2 , are precisely the regular positive solutions of…”