A non-variational system involving the critical Sobolev exponent. The radial case

Abstract

In this paper we consider the non-variational system

A fountain of positive bubbles on a Coron's problem for a competitive weakly coupled gradient system

“…Clapp and Pistoia in [10] proved that system (1.1) in any dimension has infinitely many fully nontrivial solutions, which are not conformally equivalent. Gladiali, Grossi and Troestler in [13,14] obtained radial and nonradial solutions to some critical systems like (1.1) using bifurcation methods. Remark 1.6.…”

Existence and phase separation of entire solutions to a pure critical competitive elliptic system

“…Guo, Li and Wei studied the critical system (1.1) in dimension N = 3 for λ < 0 and they established the existence of positive solutions with k peaks for k sufficiently large in [12]. In [10,11] Gladiali, Grossi and Troestler obtained radial and nonradial solutions to some critical systems using bifurcation methods.Here we focus our attention to the competitive case, i.e., to λ < 0. In this case, the system (1.1) does not have a least energy fully nontrivial solution; see Proposition 2.2 below.…”

Symmetry and monotonicity properties of singular solutions to some cooperative semilinear elliptic systems involving critical nonlinearities

“…If Γ reduces to a single point we find the result contained in [44], while if Γ = ∅ then system (1.5) reduces to the classical Sobolev equation (see [11]). For existence results of radial and nonradial solutions for (1.3), we refer to some interesting papers [24,25]. We want to remark that in [24,25] the authors treat the general case of a matrix A in which its entries a ij are not necessarily positive and this fact implies that it is not possible to apply the maximum principle.…”

“…For existence results of radial and nonradial solutions for (1.3), we refer to some interesting papers [24,25]. We want to remark that in [24,25] the authors treat the general case of a matrix A in which its entries a ij are not necessarily positive and this fact implies that it is not possible to apply the maximum principle. As remarked above the natural assumption is u i ∈ H 1 loc (R n \ Γ) ∀i = 1, ..., m and thus the system is understood in the following sense:…”