On a weighted elliptic equation of N - Kirchho ﬀ type with double exponential growth

: In this work, we study the weighted Kirchho ﬀ problem where B is the unit ball of (cid:2) N , ( ) , the singular logarithm weight in the Trudinger - Moser embedding, and g is a continuous positive function on (cid:2) + . The nonlinearity is critical or subcritical growth in view of Trudinger - Moser inequalities. We ﬁ rst obtain the existence of a solution in the subcritical exponential growth case with positive energy by using minimax techniques combined with the Trudinger Moser inequality. In the critical case, the associated energy does not satisfy the condition of compactness. We provide a new condition for growth, and we stress its importance to check the compactness level.


Introduction
In this article, we consider the following non-local weighted problem: where ( ) = B B 0, 1 is the unit open ball in N , ( ) f x t , is a radial function with respect to x, and the weight ( ) σ x is given by  (2) and → + + g : is a positive continuous function which will be specified later. In 1883, Kirchhoff studied the following parabolic problem: for free vibrations of elastic strings. The parameters in equation (3) have the following meanings: L is the length of the string, h is the area of cross-section, E is the Young's modulus of the material, ρ is the mass density, and P 0 is the initial tension.
These kinds of problems have physical motivations. Indeed, the Kirchhoff operator | | ( ) also appears in the nonlinear vibration equation, namely, which have received the attention of several researchers, mainly as a result of the work of Lions [1].
We mention that non-local problems also arise in other areas, e.g., biological systems where the function u describes a process that depends on the average of itself (e.g., population density), see, e.g., [2,3] and references therein.
In the non-weighted case, i.e., when ( ) ≡ σ x 1 and when = N 2, problem (1) can be seen as a stationary version of the evolution problem (4).
Recently, Xiu et al. [4] studied the following singular nonlocal elliptic problem: that this problem has infinitely many solutions by variational methods and the genus theorem.
In order to motivate our study, we begin by giving a brief survey on Trudinger-Moser inequalities. In the past few decades, Moser gives the famous result about the Trudinger-Moser inequality [5,6]; many applications take place as in conformal deformation theory on manifolds, the study of the prescribed Gauss curvature and mean field equations. After that, a logarithmic Trudinger-Moser inequality was used in a crucial way in [7] to study the Liouville equation of the form where Ω is an open domain of N , ≥ N 2, and λ a positive parameter. Equation (5) has a long history and has been derived in the study of multiple condensate solution in the Chern-Simons-Higgs theory [8,9] and also it appeared in the study of Euler flow [10][11][12][13].
Later, the Trudinger-Moser inequality was improved to weighted inequalities [14,15]. The influence of the weight in the Sobolev norm was studied as the compact embedding [16,17].
When the weight is of logarithmic type, Calanchi and Ruf [18] extended the Trudinger-Moser inequality and gave some applications when = N 2 and for prescribed nonlinearities. After that, Calanchi et al. [19] considered more general nonlinearities and proved the existence of radial solutions.
We point out that recently, in the case ( ) = g t 1, Deng et al. [20] have proved the existence of a nontrivial solution for the following boundary value problem: and has critical double exponential growth. Also recently, de Figueiredo and Severo [21] studied the following problem: where Ω is a smooth-bounded domain in 2 , the nonlinearity f behaves like ( ) → +∞ αt t exp as 2 , for some > α 0. The authors proved that this problem has a positive ground state solution. The existence result was proved by combining minimax techniques and Trudinger-Moser inequalities.
Inspired by the last two works, we investigate our problem by adapting weighted Sobolev space setting. We use the Trudinger-Moser inequality to study and prove the existence of solutions of (1).
Let ⊂ Ω N be a bounded domain and ( ) ∈ σ L Ω 1 be a nonnegative function. We define the weighted Sobolev space as follows: we will limit our attention to radial functions and then consider the subspace, The choice of the weight and the space are motivated by the following exponential inequalities.
Theorem 1.1. [15] Let σ be given by (2), then ( ) and In view of inequalities (6) and (7), we say that f has subcritical growth at +∞, if In this article, we consider problem (1) with subcritical or critical growth nonlinearities ( ) f x t , . Furthermore, we suppose that ( ) f x t , satisfies the following hypotheses: The condition (H 2 ) implies that for any > ε 0, there exists a real > t 0 ε , such that , , u n i f o r m l y i n .
Also, we have that condition ( ) H 3 leads to , 0 for all 0 2 1 uniformly in .
The condition asymptotic ( ) H 4 would be crucial to identify the minmax level of the energy associated with problem (1). We give an example of f . Let ( ) A simple calculation shows that f verifies conditions ( ) We define the function ( ) ( ) The function g is continuous on + and verifies The assumption ( ) Then, one has ( ) ( ) ≤ g t g t 1 for ≥ t 1.
Another consequence of ( ) G 2 is that a simple calculation shows that So, one has A typical example of a function g fulfilling conditions ( ) G 1 and ( ) G 2 is given by Another example is given by ( ) ( ) = + + g t t 1 ln 1 . The major difficulty in this problem lies in the concurrence between the growths of g and f . It will be said that u is a solution to problem (1), if u is a weak solution in the following sense.
The energy functional, also known as the Euler-Lagrange functional associated with (1), is defined by → : It is quite clear that finding weak solutions to problem (1) is equivalent to finding non-zero critical points of the functional over . In the subcritical exponential growth case, we will prove the following result. In the context of the critical double exponential growth, the study of problem (1) becomes more difficult than in the subcritical case. Our Euler-Lagrange function is losing compactness at a certain level. To overcome this lack of compactness, we choose test functions, which are extremal for the Trudinger-Moser inequality (7). Our result is as follows. This article is organized as follows. In Section 2, we give some useful lemmas for the compactness analysis. In Section 3, we prove that the functional satisfies the two geometric properties. Section 4 is devoted to estimate the minmax level of the energy. We conclude with the proofs of Theorems 1.2 and 1.3 in Section 5. We shall use the notation ‖ ‖ u p for the norm in the Lebesgue space ( ) L B p . We will also use the Sobolev weighted space defined by

Preliminaries for the variational formulation
In this section, we will present a number of technical lemmas for our future use. We begin with the radial lemma. . Then, we have The second important lemma is given as follows: where C is a positive constant, then In an attempt to prove a compactness condition for the energy , we need a Lions-type result [23] about an improved TM-inequality when we deal with weakly convergent sequences and double exponential case.
where U is given by: Therefore, Also, we have (6), the last integral is finite. To complete the proof, we have to prove that for every p for some > ε 0 and > q 1. In the following, we suppose that ‖ ‖ < u 1, and in the case of ‖ ‖ = u 1, the proof is similar. When On the other hand, by Brezis-Lieb's lemma [24] we have and so, Now, (12) follows from (7). This completes the proof. □

The mountain pass geometry of the energy
Since the nonlinearity f is critical or subcritical at +∞, there exist > a C , 0 positive constants and there exists > t 1 2 such that So the functional given by (11) is well defined and of class 1 . In order to prove the existence of nontrivial solution to problem (1), we will prove the existence of nonzero critical point of the functional by using the theorem introduced by Ambrosetti and Rabinowitz in [25] (mountain pass theorem) without the Palais-Smale condition.
The number c is called mountain pass level or minimax level of the functional J.
Before starting the proof of the geometric properties for the functional , it follows from the continuous embedding In the next lemmas, we prove that the functional has the mountain pass geometry of Theorem 3.1.

Lemma 4.
Suppose that f has critical or subcritical growth at +∞. In addition, if ( ) ( ) H H , 1 3 , and ( ) Proof. It follows from (9) that there exists > δ 0 Using the continuity of F, we obtain We obtain from (G 1 ), then from Theorem 1.1, we obtain By the following lemma, we prove the second geometric property for the functional .
In particular, for > p N 2 , there exist C 1 and C 2 such that Next, one arbitrarily picks ∈ ū such that ‖ ‖ = ū 1. Thus from (15), for all ≥ t 1, Therefore, We take = e tū, for some > t 0 large enough. So, Lemma 5 follows. □

The minimax estimate of the energy
According to Lemmas 4 and 5, let We are going to estimate the minimax value d of the functional . The idea is to construct a sequence of functions ( ) ∈ v n , and estimate With this choice of ψ n , the sequence ( ) v n is normalized since We have the following elementary crucial result.  The interval [ ] j 1, is then divided as follows: Now, we are going to study the limit of this integral on ⎡ We have also  Therefore, for every Let j be fixed and large enough. A qualitative study conducted on ψ j in [ ) +∞ 1, shows that there exists a unique ( ) ∈ s j 1, j such that the derivative of ( ) ′ = ψ s 0 j j and consequently In addition, from (17) and (18)  Finally, we will study the limit on the interval ⎡ ⎣ ⎤ ⎦ − − j j j , On the other hand, in view of (17) and (18) Furthermore, using the fact that ψ j is convex on ⎡ ⎣ and ( ) , .
Then, by bringing together (19) and (20), we deduce By tending ε to zero, we obtain So, our claim is proved and the lemma follows. □ Finally, we give the desired estimate. We argue by contradiction and suppose that for all ≥ n 1, On the other hand, Now, we claim that the sequence ( ) t n is bounded in ( ) +∞ 0, . Indeed, it follows from ( ) that is, Using the condition (G 2 ), we obtain

(25)
From (25), we obtain for n large enough Now, we are going to estimate the expression in (22). So let a n d ; .
n nn ε n nn ε , , We have  We also have Then using (21) and Lemma 6, we obtain Passing to the limit in (26), we obtain

Proof of main results
First, we begin by some crucial lemmas. Now, we consider the Nehari manifold associated with the functional , namely, 0 , 0, and the number ( ) = ∈ c u inf u . We have the following lemmas.
Lemma 8. Assume that the condition (H 3 ) holds, then for each ∈ x B, . Hence, We have that ( ) ′ = ψ 1 0. We also have by conditions (G 2 ) and ( We have ∈ λ Λ, and hence Since ∈ ū is arbitrary, then ≤ d c. □

Proof of Theorems 1.2 and 1.3
Since possesses the mountain pass geometry, there exists ∈ u n such that and  From (28) and (10), we obtain and we deduce that the sequence ( ) u n is bounded in . As a consequence, there exists ∈ u such that, up to subsequence, Since by Lemma 2, we have , i n a s , then, it follows from ( ) H 2 and the generalized Lebesgue-dominated convergence theorem that So, Next, we are going to make some claims.
It follows that for

(32)
We define for > η 0, the truncation function used in [26] ( ) . By (29) and the Lebesgue-dominated convergence theorem, we obtain Using the well-known inequality, ⟨⋅ ⋅⟩ , is the inner product in N and the fact that From (32) and (33), we deduce that a.e ∈ x B and claim 1 is proved. Claim 2. At this stage, we affirm that ≠ u 0. Indeed, we argue by contradiction and suppose that ≡ u 0.
, and consequently, we obtain First, we claim that there exists > q 1 such that By (28), we have where ′ q is the conjugate of q. Since ( ) u n converges to 0 in ( ) 1 , for all ≥ n n η . Therefore, We choose > ε 0 small enough to obtain hence the second integral is uniformly bounded in view of (7).
. We proceed by contradiction and we suppose that ( ) is positive for t small enough.
Indeed, from (9) and the critical (resp subcritical) growth of the nonlinearity f , for every > ε 0, for every > q N , there exist positive constants C and c such that , .
N q c t 1 N Then, using the condition ( ) G 1 , the last inequality, and the Hölder inequality, we obtain In view of (7) the integral We chose > ε 0, such that − > g Cε 0 0 1 and since > q N , for small t, we obtain ( ) . Using (10), the result of Lemma 7, the semicontinuity of norm, and Fatou's lemma, we obtain  Since the nonlinearity has critical growth at +∞ and from the Trudinger-Moser inequality (7) [16], Theorem 1.9), we can deduce that B 0 is an open and closed set of B. In virtue of the connectedness of B, we reach a contradiction. Hence, Claim 4 is proved.
We affirm that ( ) = u d. Indeed, by Claim 3, (10), and Lemma 8, we obtain Now, using the semicontinuity of the norm and (30), we obtain, Then In addition, which means that  and so, the sequence ( ( )) f x u , n is bounded in L q , > q 1. Using the Hölder inequality, we deduce that Passing to the limit in the last equality, we obtain ( ) ( ) − ‖ ‖ = g ρ ρ g ρ u 0, N N N N therefore ‖ ‖ = u ρ. This is in contradiction with (37). Therefore, ( ) = u d. So, u is a solution of problem (1). The proof of Theorem 1.3 is complete.
Proof of Theorem 1.2. In the subcritical case, since u n is bounded, there exist > M 0 and subsequences such that in weakly in strongly in 1 almost everywhere in .