HARDY-SOBOLEV TYPE INEQUALITY AND SUPERCRITICAL EXTREMAL PROBLEM

. This paper deals with Hardy-Sobolev type inequalities involving variable exponents. Our approach also enables us to prove existence results for a wide class of quasilinear elliptic equations with supercritical power-type nonlinearity with variable exponent.

1. Introduction and main results. Let B ⊂ R N , N ≥ 3, be the unit ball, and denote H 1 0,rad (B) as the first order Sobolev space of radial functions, and 2 * = 2N/(N − 2) the corresponding critical Sobolev embeddding exponent. In the recent paper [13], J.M doÓ et al. investigated Sobolev type embeddings for radial functions into variable exponent Lebesgue spaces proving that sup u∈H 1 0,rad (B), ∇u 2=1 B |u| ϕ(x) dx < +∞, where ϕ(r) = 2 * + |x| σ , with σ > 0. Furthermore, it was proved that the supremum (1) is attained if 0 < σ < min {N/2, N − 2}. It is a surprise because the growth of ϕ(x) is strictly larger than 2 * , except in the origin. As we shall see below, the inequality (1) is related with the classical Hardy inequality [15] where p > 1, f ≥ 0 is an integrable function and The Hardy's inequality (2) has many applications in analysis, geometry and in the theory of differential equations see e.g. [1,21,22]. There are a lot of papers which deals with the generalizations, improvements and variations of the Hardy's inequality, see for instance [4,16,19,20,23], and more recently [2,7,25]. For instance, let AC loc (0, R) be the space of all locally absolutely continuous functions on the interval (0, R), 0 < R ≤ ∞; using the Hardy inequality (2), A. Kufner and B. Opic [19] established conditions under the which, for all u ∈ AC loc (0, R) with lim r→R u(r) = 0, the inequality R 0 r θ |u| q dr holds for some positive constant C > 0 depending only on p, q, θ and R, where 1 ≤ p ≤ q < ∞ and θ, α ∈ R.
In fact, in the bounded case R < ∞, the above inequality holds if the following conditions is fulfilled 1 ≤ q ≤ p * , with p * = (θ + 1)p α − p + 1 (4) and α − p + 1 > 0. (Sobolev condition) (5) In the case R = ∞, the inequality (3) it holds only if q = p * , under the Sobolev condition (5). The inequality (3) was used successfully to deal with a wide class of quasilinear elliptic operators. For instance, we cite [6,[8][9][10]12,17,18] and references therein. As will be seen later our approach will be more in the line of [13,19]. Then, inspired by the above results, our goals are the following: • we offer a version of (1) to W 1,p 0,rad (B), p > 2 and "fractional dimentions", • we investigate the associated extremal problem, • we establish improvements for an one-dimensional Hardy-type inequality, • we also apply our results to prove existence of solutions to supercritical elliptic equations including the Hessian operator. In order to be precise our main results, let us introduce briefly some notations and preliminary results. Set X 1,p R (α, β) or simply X R be the space of functions u ∈ AC loc (0, R) with the right boundary condition u(R) = 0 and such that max{ For 0 < R < ∞, p ≥ 1 and α, β > 0 real numbers, up to a completion, X R is a Banach space endowed with the norm Further, as consequence of (3), it is possible to prove the continuous embeddings where p * := p * (α, p, θ) given by (4) is the critical exponent associated with X R , see [6]. Also, the embedding (7) are compact if q < p * . Motivated by (1) and by Hardy-type inequality (3), in this paper we investigate conditions on the function ϕ = ϕ(r) and the real parameters α, p and θ under which the supremum is finite. From now on, we assume the following conditions on p, α and θ: The next result extends the Hardy-type inequality (3) to supercritical growth and variable exponents.
The above theorem has the following easy consequence. For a deepening on this kind of result we refer [11]. Corollary 1. The following embedding is continuous: Denote Σ p the following best constant of Sobolev type for X R (see [6,9]) With this notation we will establish the following main results: Let ϕ(r) = p * + r σ and assume It is well-known the supremum in (10) is not attained, see [6] for more details. The following theorem shows that U ϕ is attained if it is larger than Σ p . Theorem 1.3. Let ϕ(r) = p * + r σ and assume U ϕ > Σ p , then the supremum U ϕ is attained.
We also establish a existence result for a wide class of related elliptic equations with a supercritical nonlinearity: Theorem 1.4. Let α, θ > 0 and p > 1 real numbers satisfying (9). Set ϕ(r) = p * + r σ , with σ under the condition (11). Then the problem has a nontrivial weak solution u ∈ X R .
We emphasize that the class of operators in (13) includes, when acting in symmetric function defined in ball B R ⊂ R N , the Laplacian, p-Laplacian and k-Hessian operators. This kind of problem has received attention of some authors in recent years [6, 8-10, 12, 17, 18]. In [6], Figueiredo et al. consider the Brezis-Nirenberg type problems and, in particular, for ϕ(r) = p * the above problem doesn't admit non-trivial solution. In the above problem (13), following [13], we propose a new kind of perturbation which also allows us to produce existence of solution as well as the classical Brezis-Nirenberg's perburbation [5].
2. Hardy-type inequality. We are going to use the following version of Strauss Radial Lemma [24].
where C is the dependent only on α, p and R. In particular, there is c > 0 such that Proof. For each u ∈ X R , the Hölder inequality gives Now, for any a > 0, we have (1 − t a )/(1 − t) goes to a, as t → 1. Thus, the above inequalities imply for some constant C > 0 depending only on α, p and R.

Proof of Theorem 1.1.
Proof. For u ∈ X R with u = 1, we can write where ρ will be determined later. Since u = 1, the Lemma 2.1 implies We shall estimate each of these two terms in (16) separately. Firstly, Using (17) and the definition of p * , we get Let us denote It easy to show that there are positive constants c 1 and c 2 such that for r near of 0. In particular, g(r) → 0, as r → 0. Hence, given d > 1 there exist Thus, from (18) we obtain Using (19) it easy to check that ρ 0 g(r) r dr < +∞, and thus ρ 0 r θ |u| p * (|u| r σ − 1)dr < +∞.

JOSÉ FRANCISCO DE OLIVEIRA, JOÃO MARCOS DOÓ AND PEDRO UBILLA
In order to estimate the second integral in (16), we proceed analogously. From (17), dr which is finite by continuity.

Proof of Corollary 1.
Proof. Consider the variable exponent Lebesgue space We shall prove that there is a constant C > 0 such that u θ,p * +r σ ≤ C, for u ∈ X R with u = 1. Using Theorem 1.1, there is c > 0 such that Thus, for λ 0 > 1 large enough such that c/λ p * ≤ 1, we can write Then u θ,p * +r σ ≤ λ 0 .
3. Proof of Theorem 1.2. This section is devoted to prove the Theorem 1.2. The proof is based in the modified Bliss function introduced by [6] and follows some ideas in [5,13]. Firstly, for each 0 < R ≤ ∞, we define It is known that S(p * , R) is independent of R, and that it is achieved when R = +∞ (see [6], for more details). Furthermore, for each > 0, the function where where S denotes the value of S(p * , R) for all R = +∞, and then for any for some 0 < r 0 < 2r 0 < R. So we have the following: Let ηu * , for η and u * given by (22) and (25), respectively. Then Proof. From (24), an easy calculation shows that This completes the proof.
Let now where It is easily to check that Finally, using (27) and (28) we have The proof of (12) relies on the following estimate of which the proof shall be postponed.
Claim 2. There exists a constant C > 0 such that for all > 0 small enough Assuming (30) we get which concludes the proof of Theorem 1.2.
Proof of Claim 2. Firstly, we highlight some useful relations on the parameters in (23).
Using e x ≥ 1 + x, for x ≥ 0, we have for suitable C > 0 and > 0 small enough. This proves our claim.

4.1.
Upper bound for the concentrated levels.
Proposition 1. For any (u j ) ⊂ X R normalized concentrating sequence at origin, we have Proof. It suffices to show the following: Given > 0 there are η > 0 and j 0 ∈ N such that for any j ≥ j 0 . In order to prove (a), we first use the Lemma 2.1 to get Noticing that Hence, taking η = η( ) > 0 small enough such that which proves (a).
To get (b) we argue as in Lemma 2.1. For any r ∈ (η( ), R), we obtain where Then we can write for j sufficiently large.

4.2.
Attainability of the best constant. In this section we prove the existence of extremal function stated in Theorem 1.3. In view of Proposition 1, we only need to prove the following: Proposition 2. Let ϕ(r) = p * + r σ . Assume that U ϕ is not attained, then any maximizing sequence for U ϕ is concentrated at origin.
Proof. Assume that U ϕ is not attained. Let (u j ) be a sequence in X R satisfying Then, up to a subsequence, we have Of course, we have u ≤ lim inf u j ≤ 1. Now, we claim that u = 0. If not, we have R 0 r α |u | p dr > 0.
By Brezis-Lieb type argument (see, [3]), we obtain where o(1) → 0, as j → ∞. From (40), if u = 1, we get u j → u strongly in X R and hence U ϕ is attained by continuity. This contradicts our assumption. Hence, we can assume u < 1. Setting w j = u j − u and using (38), (39) and (40), we can write which is contradiction. Thus, we must have u ≡ 0. It remains to show that (u j ) satisfies We claim that the embedding is compact for any q ≥ p. To prove (42), we consider the operator H R : Using [19,Theorem 7.4], for 1 ≤ p ≤ q, the operator H R is compact if and only if An straightforward computation shows that (i) and (ii) are satisfied and, thus H R is compact. Now, the embedding in (42) can be seen as T • H R , where T : Since T is a continuous operator, we conclude the compact embedding (42). Now, under the conditions (9), we have p * > p and with help of (42) and Lemma 2.1, we can write From the Ekeland's principle [14, Theorem 3.1], since (u j ) is a maximizing sequence we have for some multiplier λ j . Choosing v = u j we obtain Thus, we have lim inf j λ j ≥ p * U ϕ . Now, we choose a smooth cut-off function and choose v = ηu j in (44). Thus, we obtain from (43) Using the compact embedding (7) to conclude and thus we get and thus (41) is proved.

5.
Application to a class of supercritical equations. In this section we prove existence of a nontrivial weak solution to problem (13). In order to get this, we apply variational arguments to the functional I : X R → R defined by Precisely, we use the mountain-pass theorem. Initially, we observe the mountain pass geometry of functional. Proof. Of course, (a) holds. Also, there are constants c 1 , c 2 > 0 depending only on p, θ, σ, R and u such that Because p * > p, we get (b) holds. Finally, suppose u ∈ X R , with u < 1. The It follows that with η as in (25) and u * given by (22). Let us consider the mountain pass level Firstly, we analyze the behavior of (t ) >0 . From d dt I(tw )| t=t = 0, we obtain Combining the Claim 1 and above expression we see that where In order to analyze the behavior of A , as in (32), we set It is easy to see that t w (r) ≤ 1, for any r ≥ α . The above expression and (31) give α σ | log | → 0, as → 0.
Claim 3. We have , for any > 0 sufficiently small.
Since R → 0, as → 0, there are positive constants a 1 , a 2 , b 1 and b 2 such that for any 0 ≤ r ≤ R and > 0 small enough. Analogously, we can write Thus, using the Claim 1, the equation (61) and the definition of R , we have where B = R 0 t p * +r σ r θ p * + r σ |w | p * +r σ dr.