Morse indices and the number of blow up points of blowing-up solutions for a Liouville equation with singular data

Let Ω ⊂ R be a smooth bounded domain and let Γ = {p1, · · · , pN} ⊂ Ω be the set of prescribed points. Consider the Liouville type equation −∆u = λΠj=1|x− pj|jV (x)e in Ω, u = 0 on ∂Ω, where αj (j = 1, · · · , N) are positive numbers, V (x) > 0 is a given smooth function on Ω, and λ > 0 is a parameter. Let {un} be a blowing up solution sequence for λ = λn ↓ 0 having the m-points blow up set S = {q1, · · · , qm} ⊂ Ω, i.e., λnΠ N j=1|x− pj|jV (x)endx ⇀ m ∑ i=1 biδqi in the sense of measures, where bi = 8π if qi / ∈ Γ, bi = 8π(1 + αj) if qi = pj for some pj ∈ Γ. We show that the number of blow up points m is less than or equal to the Morse index of un for n sufficiently large, provided αj ∈ (0, +∞) \ N for all j = 1, · · · , N . This is a generalization of the result [14] in which nonsingular case (αj = 0 for all j) was studied.


Introduction
Let Ω be a smooth bounded domain in R 2 and λ > 0 is a parameter.Motivated by some physical problems in selfdual Gauge Field Theories such as Chern-Simons vortex theories or others (see [12], [15]), some researchers are interested in the analysis of the problem where Γ = {p 1 , • • • , p N } ⊂ Ω is the set of prescribed singular sources (called "vortices"), δ p is a Dirac mass supported at p, and α j > 0.
If we introduce the Green's function of −∆ acting on H 1 0 (Ω): and write G(x, p) = 1 2π log |x − p| −1 + H(x, p), where H(x, p) is the regular part of G, then the problem (1.1) is equivalent to where u = v + 4π N j=1 α j G(x, p j ) and V (x) = e −4π P N j=1 α j H(x,p j ) is a smooth positive function on Ω.By this reason, we are led to consider the problem (1.2) for general smooth positive functions V .In this case, the study of asymptotic behavior of solutions u n for λ = λ n → +0 in (1.2) was done by P. Esposito in [5] (see also [6] [7]), which extends the results of [9], [10] where the regular case (α j = 0, ∀j) was considered.
Theorem 1 (P.Esposito) Let V be a smooth positive function on Ω and set Let {λ n } be a sequence of positive numbers with λ n → 0 and let {u n } be a solution sequence of (1.2) for λ = λ n such that Then the following alternative holds: Ω) for some α ∈ (0, 1) and u n coincides with the unique minimal solution of (1.2).
(ii) If Σ n → L for some L = 0, then (up to subsequence) there exists a nonempty finite set , and Furthermore, as for the location of blow up points in the case (ii), we have the following: where Also, as a vice versa of Theorem 1, Esposito constructed blowing up solutions with a prescribed blow up set S under the additional assumption that α j ∈ (0, +∞) \ N for all j = 1, • • • , N ; see [6].
In the following, let i M (u) denote the Morse index of a solution u of (1.2), i.e., the number of negative eigenvalues of the operator L u = −∆ − λK(x)e u • acting on H 1 0 (Ω).Now, we state the main result of this note, which is a generalization of [13] [14] in this case.
Theorem 2 Let {u n } be a solution sequence of (1.2) As a corollary, we obtain the following assertion.
Proof.By Theorem 2 and the assumption that i M (u n ) = 1 for n large, we see that the number of blow up points S is 0 or 1 for the sequence {u n }.However, if S = 0, then {u n } is uniformly bounded and Σ n → 0. Thus by Theorem 1, u n coincides with the minimal solution u n of (1.2) for n large.It is well known that the minimal solution u n is stable and its Morse index is exactly 0. This contradicts to the assumption i M (u n ) = 1, thus we have S = 1.

Proof of Theorem 2
In this section, we prove Theorem 2 along the line of [13], [14].Analytical tools needed for the study of singular Liouville equations are provided in Tarantello's nice book [12].In the proof, we need a concentrationcompactness alternative result of Bartolucci and Tarantello ( [2], [3], see also [12]: Proposition 5.4.32), which we recall here in the following form.
where α > 0 and Then there exists δ ∈ (0, 1] and a subsequence of v n (denoted by the same symbol), for which only one of the following alternatives hold: Let {u n } be a solution sequence to (1.2) for λ = λ n with Σ n = O(1) as n → ∞.If Σ n → 0, then S = φ and we have nothing to prove.Thus we consider the case (ii) of Theorem 1, and we have a blow up set On the other hand, it is well known that holds; see, for example, the Appendix of [13].Combining these inequalities, we have λ m (L n , Ω) < 0. Therefore by the definition of the Morse index of u n , we have m ≤ i M (u n ).This proves Theorem 2.
In the following, we will prove Claim.
is strictly positive smooth function near any q ∈ S \ Γ, the argument in [14], which uses a concentration-compactness result of [4] [8], works well around q ∈ S \ Γ.Thus we can find r disjoint balls {B l } k l=1 with the desired property.We refer the reader to [14] [13].Next, we consider blow up points in S ∩ Γ = {p j 1 , • • • , p js } and, for simplicity, we relabel S ∩ Γ = {p 1 , • • • , p s }.We choose r > 0 sufficiently small such that B r (p i ) ⊂⊂ Ω, {B r (p i )} s i=1 are disjoint, and p i is the only blow up point of u n in B r (p i ) for all i.
as n → ∞.Now, let us define δ i n > 0 and ũi n : First, we prove where . Note that Ki is a smooth, strictly positive function on B r (p i ).Also, Theorem 1 (1.3), (1.4) in the sense of measures on B r (p i ) and max as n → ∞.Recall the assumption α i / ∈ N for all i.Therefore, we can apply Proposition 5.6.50 and Corollary 5.4.24 in [12] to v n to conclude that sup ≤ e C , and Indeed, assume the contrary that there exists i ∈ {1, • • • , s} and a subsequence (denoted by the same symbol) such that When (i) happens, we see by (2.3) that 1) which leads to a contradiction.
When (ii) happens, again by (2.3), we see max n → p i as n → ∞, this case can happen only when the alternative (a) in Proposition 4 occurs: This again leads to a contradiction and we have proved the claim.Now, since (δ i n ) 2(1+α i ) = 1 e vn(p i ) , we obtain the lemma.
Incidentally, by (2.1), (2.2) and (2.3), we can apply Theorem 5.6.51 in [12], see also [1], to v n to obtain the following pointwise estimate which is equivalent to where c i = Ki (p i ).
Going back to the proof of Theorem 2, we see that ũi The third equation comes from (2.3).
At this stage, we can apply Lemma 5.4.21 in [12] to ũi n to confirm that ũi n is uniformly bounded in L ∞ loc (R 2 ) and along a subsequence, where |y| 2α i e U i dy < +∞.By a classification result of Prajapat and Tarantello [11] and the assumption α i / ∈ N, we have ).This operator is related to L n by the formula where x = δ i n y + p i for x ∈ B r (p i ) and y ∈ B r/δ i n (0).Also for a domain D ⊂ B r (p i ), we have We will prove that ( Li n w R , w R ) L 2 (B R ) < 0 for n ∈ N and R > 0 sufficiently large with B R (0) ⊂ B r/δ i n (0).Indeed, |y| 2α i Ki (δ i n y + p i )e We observe that where o R (1) → 0 as R → ∞.On the other hand, we have be the linearized operator around u n and let λ j (L n , D) denote the j-th eigenvalue of L n acting on H 1 0 (D) for a regular subdomain D ⊂ Ω.Next is the key in the proof of Theorem 2. Claim: There exist m disjoint open balls {B i } m i=1 , each B i ⊂⊂ Ω, such that λ 1 (L n , B i ) < 0 for any i ∈ {1, • • • , m} and for n large .Assuming for the moment the validity of Claim, we prove Theorem 2. Indeed, by Claim, there exist m open balls B 1