KIRCHHOFF TYPE EQUATIONS WITH STRONG SINGULARITIES

. An optimal condition is given for the existence of positive solutions of nonlinear Kirchhoﬀ PDE with strong singularities. A byproduct is that − 2 is no longer the critical position for the existence of positive solutions of PDE’s with singular potentials and negative powers of the form: −| x | α ∆ u = u − γ in Ω, u = 0 on ∂ Ω, where Ω is a bounded domain of R N containing 0, with N ≥ 3, α ∈ (0 ,N ) and − γ ∈ ( − 3 , − 1).

where B 1 ⊂ R N , N ≥ 3 is the unit ball, and l is a positive number. It is easily where ω N is the volume of the unit N -sphere. We then obtain the existence of a and l > 0, thanks to (2).
, is just the sufficient and necessary condition derived in [42] for the special case on ∂Ω, admits no positive solution (c.f. [6]). In contrast, it is shown that in presence of negative power nonlinearities −2 is no longer the critical position for the existence of positive solutions. As a remarkable application of Theorem 1.1, we derive our second result as following: admits a positive solution u ∈ H 1 0 (Ω). We then establish the following property of the H 1 0 -solution in Theorem 1.2 which is based on test functions computations.
admits no bounded positive solution.
Notation. In the rest of the paper we make use of the following notation: C, c i , i = 1, 2, · · · denotes (possibly different) constants; We denote by u 2 = Ω |∇u| 2 dx the Dirichlet norm in H 1 0 (Ω).
2. Proof of Theorem 1.1. We first establish two lemmas to establish the validity and connection of the two constrained sets: Lemma 2.1. The constrained sets N i , i = 1, 2 are nonempty.
then define the function U : (0, +∞) → R by Clearly, U is increasing on t > 0, with lim t→+∞ U (t) = +∞ and lim t→0 then it follows that there exists the unique minimizer t(u) > 0 such that that is, I(t(u)u) = min t>0 I(tu). In particular, the assumption (2) of Theorem 1.1 implies the existence of t(u 0 ) > 0 such that t(u 0 )u 0 ∈ N 2 , and hence N 1 (⊃ N 2 ) and N 2 are not empty.
Indeed, thanks to (14), to (15) and to the fact that −α > −N , we have that Then (2) holds. This also ends the proof of Theorem 1.2.