Radial and non radial solutions for Hardy–Hénon type elliptic systems

Abstract

We discuss existence and non-existence of positive solutions for the following system of Hardy and Hénon type:where 0 is a bounded domain in R N , N ≥ 3, p, q > 1, and α, β > −N . We also study symmetry breaking for ground states when is the unit ball in R N .

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

“…The assertions (2), (3) and (4) can be proved similarly arguing as in the proof of [8,Proposition 9]. We stress that the vanishing w(0) = 0 at [8, p. 123 …”

“…actually, in [8] it is assumed p > 1, q > 1 and β < (q + 1)N but a close inspection of their proof shows that (3.8) remains valid as long as β α and α is sufficiently large.…”

“…A first attempt to study this phenomenon for (1.1) was done by Calanchi and Ruf [8]. Further contributions on this type of problem, in the case of Hamiltonian systems, were presented by Liu and Yang [18], see also de Figueiredo et al [14].…”

Asymptotic properties of positive solutions of the Hardy–Sobolev type equations

“…The quantitative properties of this type equation are also interesting in critical point theory and nonlinear elliptic equations (cf. [1,2,5,25,27]). …”

On Caffarelli–Kohn–Nirenberg-type Inequalities for the Weighted Biharmonic Operator in Cones

“…We point out that condition (1.4) defines the "weighted critical hyperbola" (case p = 2) introduced in [9] and [17] in the context of solvability of Hardy-Henon-type elliptic systems in bounded domains. See also [5] for related results on symmetry breaking. If n ≥ 5, then the Sobolev embedding theorem implies that a necessary condition to have S q (C Σ ; α) > 0 is that q ≤ 2 * * , where 2 * * is the critical Sobolev exponent:…”