A note on a second order PDE with critical nonlinearity

In this work, we are interested in a nonlinear PDE of the form: −∆u = K(x)u n+2 n−2 , u > 0 on Ω and u = 0 on ∂Ω, where n ≥ 3 and Ω is a regular bounded domain of Rn. Following the results of [K. Sharaf, Appl. Anal. 96(2017), No. 9, 1466– 1482] and [K. Sharaf, On an elliptic boundary value problem with critical exponent, Turk. J. Math., to appear], we provide a full description of the loss of compactness of the problem and we establish a general index account formula of existence result, when the flatness order of the function K at any of its critical points lies in (1, ∞).


Introduction and main results
In this work, we consider the existence of smooth solutions of where n ≥ 3, Ω is a regular bounded domain of R n and K is a given function on Ω.
Equation (1.1) can be expressed as a variational problem in H 1 0 (Ω).However, the variational structure presents a loss of compactness since the exponent n+2 n−2 is critical and The first contributions to (1.1) concern the case K = 1, where Bahri-Coron and Pohozaev proved that the resolution of (1.1) depends on the topology of the domain Ω, see [4] and [17].For K = 1, many conditions on K were provided to ensure existence of solutions of (1.1), see for example [6,13,14,[18][19][20][21].
(B) Assume that the least eigenvalue ρ(τ p ) of M(τ p ) is not zero.
The last assumption is Thus, it becomes of interest to study the equation (1.1) in the mixed case situation; that is when there exists some critical points y of K having β(y) < n − 2 and other having β(y) ≥ n − 2 and therefore get global compactness and existence results under The first result of this paper describes the loss of compactness and the concentration phenomenon of the problem (1.1).For a ∈ Ω and λ 1, let Pδ (a,λ) be the almost solution of the Yamabe-type problem defined in the next section.Pδ (y j ,∞) , where (y 1 , . . ., The index of (y 1 , . . ., y p ) ∞ is i(y 1 , . . . ,y p ) The above characterization allow us to derive a global index formula of existence.

Theorem 1.2.
Let Ω be a regular bounded domain of R n , n ≥ 3 and let K : Ω → R be a given function satisfying (A), (B) and then (1.1) has a solution.
Remark 1.3.For an explicit example of function K satisfying the hypotheses of Theorem 1.2, let Ω be the unit ball B n of R n , n ≥ 4 and let β be a real larger than n − 2. For any X ∈ R n , we define For any integer k 0 ≥ 2, we denote y k 0 = 1 k 0 , 0, . . ., 0 .Let θ be the cut-off function defined by: Observe that K admits three critical points y k 0 , −y k 0 and 0 R n .By construction K satisfies ( f ) β condition near its critical points with According to the result of Theorem 1.1, 0 R n does not give a critical point at infinity since However y k 0 and −y k 0 correspond to two critical points at infinity (y k 0 ) ∞ and (−y k 0 ) ∞ respectively.In addition, the pair (y k 0 , −y k 0 ) corresponds to a critical point at infinity if and only if ρ(y k 0 , −y k 0 ) > 0 where ρ is the least eigenvalue of the matrix It is easy to see that ρ(y k 0 , −y k 0 ) > 0 if and only if Thus for k 0 large enough, ρ(y k 0 , −y k 0 ) < 0. Therefore, the only critical points at infinity in our statement are, (y k 0 ) ∞ and (−y k 0 ) ∞ with i( It follows that the function K satisfies the index formula of Theorem 1.2 and the assumption (B).Concerning the assumption (A), observe that outside Our argument follows the critical points at infinity theory of A. Bahri [2].In the next section, we will state the general framework of the variational structure of (1.1).After that we will characterize the critical points at infinity and prove Theorems 1.1 and 1.2.

General framework
Equation (1.1) is equivalent to finding the critical points of the following functional Here and It is known that J fails the Palais-Smale condition.The sequences which violate the (P.S) condition has been analyzed as follows.For a ∈ Ω and λ > 0, define where c 0 is a fixed positive constant.The family δ a,λ , a ∈ Ω and λ > 0 are the only solutions of Define Pδ a,λ on Ω be the unique solution of By the maximum principal and regularity arguments, Pδ a,λ is smooth and positive on Ω.
The following parametrization of V(p, ε) was given in [4].For any u ∈ V(p, ε), u can be written as •, • denotes the inner product on H 1 0 (Ω) associated to the norm • , and ᾱi , āi , λi , i = 1, . . ., p are the unique solution of min In the following, we show that the v-part of u is negligible with respect to the concentration phenomenon.See [2,4].
Moreover, there exists a change of variables v − v → V such that The following definition is extracted from [2].

Definition 2.1 ([2]
).A critical point at infinity of J is a limit of a non-compact flow line u(s) of the gradient vector field (−∂J).By the above argument, u(s) can be written as: Denoting by y i = lim s→+∞ a i (s) and α i = lim s→+∞ α i (s), we then denote by such a critical point at infinity.

Critical points at infinity
In this section we prove Theorems 1.1 and 1.2.We start by the following result which describes the concentration phenomenon of the variational structure associated to the problem (1.1).
Theorem 3.1.Under the assumptions (A), (B) and ( f ) β , β > 1.There exists a decreasing bounded pseudo-gradient W in V(p, ε), p ≥ 1, satisfying the following: There exists c > 0 such that for any u = ∑ Moreover, the only case where λ i (t), i = 1, . . ., p, tends to ∞ is when a i (t) goes to y i , ∀i = 1, . . ., p such that (y 1 , . . ., are defined in the first section.Before presenting the proof of Theorem 3.1, we recall the following result which describes the concentration phenomena of the problem when β ∈ (1, n − 2), see [19,Section 3].

Theorem 3.2 ([19]
).Under the assumptions of Theorem 3.1 with β ∈]1, n − 2[, there exists a decreasing bounded pseudo-gradient W 1 satisfying (i) of Theorem 3.1, for any u = ∑ p i=1 α i Pδ a i ,λ i ∈ V(p, ε) and the only case where λ i (t) goes to +∞, i = 1, . . ., p is when a i (t) goes to y i with (y 1 , . . ., Notice that the case of β = n − 2 was handled also in [19]. Recently we proved the following result which describes the concentration phenomena in the case where β ∈ [n − 2, +∞).

Theorem 3.3 ([21]
).Under the assumptions of Theorem 3.1 with β ∈ [n − 2, ∞), there exists a decreasing bounded pseudo-gradient W 2 satisfying (i) of Theorem 3.1, for any u = ∑ p i=1 α i Pδ a i , λ i ∈ V(p, ε) and the only case, where λ i (t), i = 1, . . ., p goes to +∞ is when a i (t) goes to y i with (y 1 , . . ., The complete construction of the required pseudo-gradient W 2 in V(p, ε) was given in [21].We provide in the next the construction of W 2 in a specific region R δ >n−2 (p, ε) where Here δ is a small positive constant.
Case 1: If ρ(y 1 , . . . ,y p ) > 0. We use the expansion (3.1) below.Since J(u) H(y i , y i ) Observe that as δ small we have, Moreover, since |a i − a j | ≥ ρ 0 , ∀i = j, we have Therefore , Thus where the coefficients m(y i , y j ), 1 ≤ i, j ≤ p are defined in the first section.
For any i = 1, . . ., p we set λi = λ i .The corresponding pseudo-gradient is From the latest expansion, W 2 satisfies , since ρ(y 1 , . . . ,y p ) is the least eigenvalue of M(y 1 , . . . ,y p ).Using the fact that and for any i = j, we have Case 2: If ρ(y 1 , . . ., y p ) < 0. This is the opposite situation od the case 1.Thus satisfies the requirement of Theorem 3.3.
Following the above two results, the only case that we will consider here is when u can be written as where 1 ≤ q < p and Let us denote by W 1 the pseudo-gradient given by Theorem 3.2 and W 2 the pseudo-gradient given by Theorem 3.3.In order to construct the required pseudo-gradient W of Theorem 3.1, we distinguish three cases.Let δ be a fixed positive constant small enough.
According to the construction of [19] and [21], the vector fields W 1 and W 2 in these regions are defined as follows: Observe that all the components λ i of the corresponding flow lines satisfies the differential equation λ = λ i , ∀i = 1, . . ., p.
In this case, we set where W 1 (u) := W 1 (u 1 ) and W 2 (u) := W 2 (u 2 ).Following the computation of [19] and [20], we have Since we have |a i − a j | ≥ ρ 0 , ∀i = j, we obtain Using the fact that K satisfies ( f ) β assumption around each y i , we derive We therefore have and For any 1 ≤ i ≤ q and for any q + 1 ≤ j ≤ p we claim that Indeed, since |a i − a j | ≥ ρ 0 , we have , as M large.
• Case 2. Three possibilities may occur.Either there exists i 0 , 1 ≤ i 0 ≤ q such that λ i 0 |a i 0 − y i 0 | ≥ δ or there exists i 1 , 1 ≤ i 1 ≤ q such that − ∑ n k=1 b k (y i 1 ) < 0 or there exist i = j such that y i = y j .
In all possibilities, it was constructed in [19], section 3, a pseudo-gradient W 1 along which the max 1≤i≤q (λ i (s)) remains bounded and satisfies Therefore, for W 1 (u) = W 1 (u 1 ), we set Denote by i 1 an index such that and let us denote It is easy to see that we can appear all − 1 λ β i , i ∈ L in the upper bound of (3.8).In order to make appear all − |∇K(a i )| λ i , i ∈ L, let us recall the following estimate obtained in [19,Section 3].
For each index i ∈ L \ {1, . . ., q}, we move the concentration point a i with respect to Y i .Using (3.9), the corresponding variation of J is given by: For any j = 2, . . ., [β], we claim that Hence (3.11) follows.From this and (3.3), (3.10) becomes 12) The inequalities (3.8) and (3.12) with the estimates (3.6) yield for m > 0 small enough We now appear − ∑ i =j, i,j∈L ε ij in the last upper bound.For this, we decrease all λ i such that i ∈ L \ {1, . . ., q}.Define Without loss of the generality, we can assume that if i < j then λ i ≤ λ j .In that case we have Therefore, The inequalities (3.13) and (3.14) with the estimate (3.11) yield for m > 0 and small To add the left indices, we denote Observe that the max 1≤i≤p λ i (s) does not move along W 2 , since it acts only on the indices i ∈ L. Using the above techniques (see also [21] for more details), we have • Case 3.