Maximizers for the variational problems associated with Sobolev type inequalities under constraints

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1 Maximizers for the variational roblems associated with Sobolev tye inequalities under constraints arxiv: v2 [math.fa] 5 May 2018 Van Hoang Nguyen July 6, 2018 Abstract We roose a new aroach to study the existence and non-existence of maximizers for the variational roblems associated with Sobolev tye inequalities both in the subcritical case and critical case under the equivalent constraints. The method is based on an useful link between the attainability of the suremum in our variational roblems and the attainablity of the suremum of some secial functions defined on 0,. Our aroach can be alied to the same roblems related to the fractional Lalacian oerators. Our main results are new in the critical case and in the setting of the fractional Lalacian oerator which was left oen in the work of Ishiwata and Wadade [17,18]. In the subcritical case, our aroach rovides a new and elementary roof of the results of Ishiwata and Wadade [17,18]. 1 Introduction Let N 2 and 1,N]. We denote by the critical exonent in the Sobolev embedding theorem, i.e., = N/N if < N and = if = N. Let > 0andu W 1, R N, we denote u W 1, = u 1 + u, and S N,, = {u W 1, R N : u W 1, = 1}. Institute of Research and Develoment, Duy Tan University, Da Nang, Vietnam, Institut de Mathématiques de Toulouse, Université Paul Sabatier, 118 Route de Narbonne, Toulouse cédex 09, France. van-hoang.nguyen@math.univ-toulouse.fr and vanhoang0610@yahoo.com Mathematics Subject Classification: 46E35, 26D10. Key words and hrases: Sobolev tye embedding, variational roblems, maximizers 1

2 Given α > 0, let us consider the variational roblem D N,,,α,q = su u +α u q q = su J N,,,α,q u, 1.1 u S N,, u S N,, where < q if < N, and N < q < if = N. Such a roblem was recently studied by Ishiwata and Wadade in the subcritical case 1,N] and q N, see [17,18]. It has close relation with the well known Sobolev tye inequality u q S u + u u W1, R N where 1,N] and q, ] if < N and q N, if = N, and S is the best constant deending only on N, and q. The attainability of S is well understood in literature. In the limit case = N, we known that W 1,N R N L R N. In this case, the so-called Moser Trudinger inequality is a erfect relacement. The Moser Trudinger inequality in bounded domains was indeendently roved by Yudovič [33], Pohožaev [28] and Trudinger [32]. It was then sharened by Moser [26] by finding the shar exonent. The Moser Trudinger inequality was generalized to unbounded domains by Adachi and Tanaka [1], by Cao [9] for the scaling invariant form in the subcritical case and by Ruf [29] and by Li and Ruf [20] for the full L N -norm form including the critical case. It takes the following form su Φ u W 1,N R N, N α u N N 1 dx <, 1.2 R N u N + u N dx 1 R N for any α α N := Nω 1/N 1 N 1, where ω N 1 denotes the surface area of the unit shere in R N, and Φ N t = e t N 2 t k k=0. We refer the interest reader to [2 4,13,27] and references k! therein for the recent develoments in the study of the Moser Trudinger inequality. The existence of maximizer for the Moser Trudinger inequality in bounded domains was roved by Carleson and Chang [10] for the unit ball, by Struwe [30] for domains near ball, by Flucher [15]foranydomainindimensiontwo, andbylin[22]foranydomaininr N,N 3. The existence of maximizers for the Moser Trudinger inequality in whole sace R N was roved by Ruf [29] in the dimension two with α = α 2, by Li and Ruf [20] in any dimension N with α = α N, and by Ishiwata [16] for any α 0,α N with N 3 and α 2/B 2,α 2 ] u with N = 2 where B 2 = 4 4. Ishiwata also roved a non-existence su u W 1,2 R 2 \{0} u 2 2 u 2 2 result in dimension two when α > 0 sufficiently small which is different with the bounded domain case. The roof of Ishiwata relies on the careful analysis using the concentration comactness tye argument together with the behavior of the functional J N,α u = u N N + α N u N 2 N 1 N 2 N 1 which aears as two first terms of the Φ R N N α u N N 1 dx. Note that JN,α is exact our functionalj N,N,N,α, N 2 above. The non-existence result even occurs in the higher dimension N 1 2

3 if we relace the condition u N N + u N N = 1 by the condition u a N + u b N = 1 with a, b > 0 see [14, 19]. This non-existence result suggests that the attainability of the Moser Trudinger inequality deends delicately on the choice of normalizing conditions even if conditions are equivalent. This leads to the study of the roblem 1.1. Concerning to the roblem 1.1, it was shown in [17,18] that the existence of maximizers for D N,,,α,q is closed related to the exonent. Esecially, the following quantity α N,,q = inf u S N,, 1 u u q q = inf u S N,, I N,,,q u, 1.3 is the threshold for the attainability of D N,,,α,q. For N 2, 1,N] and q,, let B N,,q denote the best constant in the Gagliardo Nirenberg Sobolev inequality B N,,q = su u W 1, R N,u 0 u c u q q u q c, c = Nq. 1.4 A simle variational argument shows that B N,,q is attained in W 1, R N. Moreover, by a refined Pólya Szegö rincile due to Brothers and Ziemer [8], all maximizers for B N,,q is uniquely determined u to a translation, dilation and multile by non-zero constant by a nonnegative, sherically symmetric and non-increasing function v W 1, R N, i.e., if u is maximizer for B N,,q then ux = cλ 1/q v λ 1/N x x 0 a.e. in R N for some c 0, λ > 0 and x 0 R N. Among their results, Ishiwata and Wadade roved the following theorem on the effect of α and to the attainability of D N,,,α,q. Theorem 1.1. [Ishiwata and Wadade] Let N 2, 1,N], q,, and α, > 0. Then we have 1 If > c then α N,,q = 0 and D N,,,α,q is attained for any α > 0. 2 If = c then α N,,q c = /B N,,q c and D N,,,α,q is attained for any α > α N,,q c while it is not attained for any α α N,,q c. 3 If < c then D N,,,α,q is attained for any α α N,,q while it is not attained for any α < α N,,q. The roof of Theorem 1.1 given in [17,18] is mainly based on the rearrangement argumentwhich enables us to study the roblem1.1 only on radial functions and the subcriticality of the roblem 1.1, i.e., q <. In this case, the embedding W 1, rad RN L q R N is comact, where W 1, rad RN denotes the subsace of W 1, R N of all radial functions. By a careful analysis of the functional J N,,,α,q on normalizing vanishing sequences NVS for short in S N,, a sequence {u n } S N,, is a normalizing vanishing sequence if u n is radial function, u n 0 weakly in W 1, R N and lim lim R n B R u n +α u n q dx = 0, they roved that su{lim suj N,,,α,q u n : {u n } is NVS} = 1. n 3

4 Using this vanishing level, they can exclude the vanishing behavior of any maximizing sequence which can be assumed to be sherically symmetric and non-increasing function for D N,,,α,q and hence obtain the existence and non-existence of maximizers for D N,,,α,q. The critical case q = with 1,N remains oen from the works of Ishiwata and Wadade [17,18]. In this case, we will denote D N,,,α,,α N,,,J N,,,α, and I N,,, by D N,,,α,α N,,J N,,,α and I N,, resectively for sake of simle notation. Our goal of this aer is to treat this remaining oen case. Let us mention here that studying the roblem 1.1 in the critical case is more difficult and more comlicate than studying the one in the subcritical case. The first difficult is that the embedding W 1, rad RN L R N is not comact which is different with the subcritical case where W 1, rad RN L q R N with q < is comact. The second difficult is that the concentrating henomena of the maximizing sequence for D N,,,α can occur a sequence {u n } n S N,, is called a normalizing concentrating sequence NCS for short if u n 0 weakly in W 1, R N and BR u c n N + u n N dx 0 as n for any R > 0. In order to rove the attainability of D N,,,α, the usual way is to exclude both the vanishing and concentrating behavior of the maximizing sequences for D N,,,α by using the concentration comactness rincile due to Lions [23,24] or by comuting exactly the vanishing level defined as above and the concentrating level of J N,,,α, i.e., su{limsu n J N,,,α u n : {u n } is NCS}, then using Brezis Lieb lemma [7] to gain a comact result for this maximizing sequence. It seems that these aroaches are not easy to solve the roblem 1.1 in the critical case. In this aer, we give an elementary roof without using any rearrangement argument for the roblem 1.1 in the critical case. Moreover, our aroach also gives information on the maximizer for D N,,,α if exist. Instead of using the shar Gagliardo Nirenberg Sobolev inequality 1.4, the following shar Sobolev inequality lays an imortant role in our analysis below u S N, u u W 1, R N, 1.5 where S N, is the shar constant which deends only on N and. Noet that the Sobolev inequality 1.5 is exact the critical case of the Gagliardo Nirenberg Sobolev inequality 1.4. The recise value of S N, is well known see, e.g., [5,11,31]. We do not mention its recise value here since it is not imortant in our analysis below. It is also well known that S N, is not attained in W 1, R N unless < N 1/2. However, S N, is attained in a larger sace than W 1, R N, that is, the homogeneous Sobolev sace Ẇ1, R N which is comletion of C 0 RN under the Dirichlet norm u with u C 0 RN. The Sobolev inequality 1.5 also holds true in Ẇ1, R N with the same constant S N, and the equality occurs if ux = au bx x 0 for some a,b > 0, x 0 R N with u x = 1+ x 1 N. 1.6 It is clear that u W 1, R N unless < N 1/2. We refer the interest reader to the aers [5, 11, 31] for more about the shar Sobolev inequality 1.5 and its extremal functions. The main result on the existence and non-existence of maximizers for the roblem 1.1 in the critical case of this aer is the following theorem. 4

5 Theorem 1.2. Let N 2, 1,N and α, > 0. Then D N,,,α is not attained if N 1/2. If 1 < < N 1/2, we then have 1 If > then α N, = 0 and D N,,,α is attained for any α > 0. 2 If = then α N, = / S N, and D N,,,α is attained for any α > α N, while it is not attained for any α α N,. Moreover, if α α N, then D N,,,α = 1. 3 If < < then D N,,,α is attained for any α α N, while it is not attained for any α < α N,. Moreover, if α α N, then D N,,,α = 1. 4 If then α N, = S N, and D N,,,α is not attained for any α > 0. In this case, we have D N,,,α = max{1,αs N, }. Theorem 1.2 together with Theorem 1.1 of Ishiwata and Wadade comletes the study of the variational roblem 1.1. The main key in the roof of Theorem 1.2 is the following exression of D N,,,α and α N,. Theorem 1.3. Let N 2, 1,N and α, > 0. Then it holds and D N,,,α = su 1+t +αs 1+t N, t =: su α N, = 1 1+t inf S 1+t 1 =: N, t 1 S N, f N,,,α t, 1.7 inf g N,,t. 1.8 The equalities 1.7 and 1.8 give us the simle and comutable exressions of D N,,,α and α N,. Their roofs are mainly based on the Sobolev inequality 1.5 and on the estimates on test function constructed from the function u above. Let us exlain the main ideas in the roof of Theorem 1.2. The non-existence result in the case N 1/2 is simly a consequence of the non-attainability of the Sobolev inequality 1.5 in W 1, R N. In the case < N 1/2, the roof is mainly based on the remark that D N,,,α is attained in S N,, if and only if the function f N,,,α attains its maximum value in 0, see Lemma 2.3 below. This remark enables us reduce studying the attainability of D N,,,α to studying theattainabilityofsu f N,,,α, tin0,. Thelatter roblemismoreelementary. We can use some simle differential calculus to solve it. Moreover, we will see from our roof that, whenever su f N,,,α is attained by some t 0 0,, we can construct exlicitly a maximizer for D N,,,α. Our constructed maximizer comes from the maximizer of the Sobolev inequality 1.5. It is interesting that our aroach to rove Theorem 1.2 also gives a new roof of Theorem 1.1. Comaring with the roof of Ishiwata and Wadade, our roof is more simle and more elementary. As in the roof of Theorem 1.2, we will establish the following simle and comutable exression for both D N,,,α,q and α N,,q. 5

6 Theorem 1.4. Let N 2, 1,N], q, and α, > 0. Then it holds D N,,,α,q = su 1+t q +αb N,,q t c 1+t q =: su f N,,,α,q t, 1.9 and α N,,q = 1 B N,,q inf 1+t q 1+t 1 t c =: inf g N,,,qt The equalities 1.9 and 1.10 also will be the key in the roof of Theorem 1.1. Similar with the roof of Theorem 1.2, we also have a remark that D N,,,α,q is attained in S N,, if and only su f N,,,α,q is attained in 0, see Lemma 3.1 below. By this remark, we only have to study the attainability of su f N,,,α,q in 0,. We will see below that the study of su f N,,,α,q t is easier than the one of su f N,,,α t. Indeed, the behavior of f N,,,α,q at infinity does not lay any role in the study of su f N,,,α,q t due to the subcriticality of the roblem it is clear that lim t f N,,,α,q t = 0. Hence, it is enough to study the behavior of f N,,,α,q when t is near 0. Contrary with the subcritical case, the behavior of f N,,,α at infinity is imortant in our analysis in the critical case. In fact, this is corresonding to the aearance of the concentration henomena in studying the maximizing sequence for D N,,,α. Another advantage of our aroach is that it can be alied to the variational roblems of tye 1.1 related to the fractional Sobolev tye inequality i.e., for the fractional Lalacian oerator s with s 0,N/2. It seems that the method of Ishiwata and Wadade can not be alied immediately to these roblems due to the non-locality of the roblems. The results in this setting are given in Theorem 4.1 and Theorem 4.2 below both in subcritical case and critical case. Since the roof of these theorems is comletely similar with the one of Theorem 1.1 and Theorem 1.2. So we only sketch their roof in section 4 below. The rest of this aer is constructed as follows. In the next section, we will rove Theorem 1.2 on the effect of,α and on the existence and non-existence of maximizers for the variational roblem 1.1 in the critical case. Theorem 1.3 is also roved in this section. In section 3, we rove Theorem 1.4 and aly it to give a new roof of Theorem 1.1 following the aroach to Theorem 1.2. In the last section 4, we exlain how our aroach can be alied to the similar roblems for the fractional Lalacian oertors. 2 Proof of Theorem 1.2 This section is devoted to rove Theorem 1.2. We first give a roof of Theorem 1.3 on the exressions of D N,,,α and α N, which will be used in our roof of Theorem 1.2. Proof of Theorem 1.3. We first rove 1.7. For any u S N,,, by Sobolev inequality 1.5 6

7 we get J N,,α, u u +αs N, u = u u + u = su Taking the suremum over all u S N,,, we get 1+ u u u + u 1+t +αs N, 1+ u u +αs 1+t N, t +αs N, u u. u D N,,,α su 1+t +αs 1+t N, t. 2.1 For a reversed inequality of 2.1, we take u to be a maximizer for the Sobolev inequality 1.5 we can take u as in 1.6. Our aim is to construct the test functions to estimate D N,,,α from u. Note that u W 1, R N in general, hence we can not use directly it as a test function for D N,,,α. To roceed, we will truncate the function u as follows. Let ϕ C0 RN be a nonnegative function such that 0 ϕ 1, ϕx = 1 if x 1 and ϕx = 0 if x 2. For any R > 0 define u,r x = u xϕx/r then u,r W 1, R N and lim u,r = u R and lim u,r = u. R Consequently, we have lim R u,r u,r = S N,. For any ǫ > 0, we can choose R > 0 such that u,r S N, ǫ u,r. For any λ > 0 define λ 1 u,r λnx 1 w R,λ x = u,r +λn u,r

8 We have w R,λ S N,,, then it holds D N,,,α J N,,,α w R,λ = u,r u,r +λ N u,r λ u,r +α u,r +λn u,r = u,r +αs N, ǫ u,r +λn u,r 1 ǫ u,r S N, u,r +λn u,r 1 ǫ S N, 1+λ N u,r u +αs N, 1+λ N u,r u λ u,r u,r +λn u,r + αs N, λ u,r u,r +λn u,r λ N u,r u. The receding inequality holds for any λ > 0, hence D N,,,α for any ǫ > 0. Letting ǫ 0 +, we get 1 ǫ su S N, 1+t +αs 1+t N, t, S N,,,α su 1+t +αs 1+t N, t. 2.3 Combining 2.1 and 2.3 together, we obtain 1.7. We next rove 1.8. For any function u S N,,, using the shar Sobolev inequality 1.5, we get I N,, u 1 S N, 1 u u = 1 u + u S N, = 1 S N, u u 1 inf S 1+t N, u + u u u u +1 u u u t We finish our roof by showing a reversed inequality of 2.4. For any ǫ > 0, we choose t 8

9 w R,λ defined by 2.2 as test function. We then have α N, I N,, w R,λ u,r λ 1 u = 1,R u,r +λn u,r u,r +λn u,r 1 u,r λ u 1 S N, ǫ,r u,r +λn u,r u,r +λn u,r 1 = 1+ λ N u,r 1+ λ N u,r u,r 1 S N, ǫ u,r. 2.5 The inequality 2.5 holds for any λ > 0, hence it holds α for any ǫ > 0. Letting ǫ 0, we get 1 inf S N, ǫ 1+t α 1 inf S 1+t N, Combining 2.4 and 2.6 we obtain 1.8. and Recall that f N,,,α t = 1+t g N,, t = 1+t From the roof of 2.1, we see that J N,,,α u f N,,,α u u λ N u,r u,r 1+t 1, t 1+t αs 1+t t N, t 1+t 1. t, u S N,,. 2.7 Moreover, if < N 1/2 then u W 1, R N and u = S N, u. For λ > 0, define w λ x = λ 1 u λnx u +λn u 1 1 9

10 Using the argument in the roof of 2.3 with w λ as test function, we obtain u J N,,,α w λ = f N,,,α λ N u. 2.9 Wefirsthavethefollowingestimatesforsu f N,,,α tandalsoford N,,,α by1.7. Proosition 2.1. Let N 2, 1,N and α, > 0. We have suf N,,,α t max{1,αs N, } Moreover, if > then su f N,,,α t > 1 and if > then su f N,,,α t > αs N,,. Proof. It is easy to see that limf N,,,α t = 1 and lim f N,,,α t = αs t 0 t N,, which then imly If >, we have f N,,,α t = 1 t+αs N, t when t 0. Hence, for t > 0 sufficiently small, we get If >, we have +ot suf N,,,α t f N,,,α t > 1. f N,,,α t = 1+t +αs N, αs N, 1+t 1 +o1+t 1 when t. Consequently, for t > 0 sufficiently large, we get suf N,,,α t f N,,,α t > αs N,. The rest of this section is devoted to rove Theorem 1.2. Our roof is done by a series of lemmatas. Lemma 2.2. Let N 2. If N 1 2 then D N,,α, does not attain for any α, > 0. Proof. Suose that D N,,α, is attained by a function u S N,,, i.e., D N,,,α = J N,,,α u. 10

11 Since N 1/2, then the Sobolev inequality 1.5 does not attains in W 1, R N. Consequently, from the roof of 2.1, we get u J N,,,α u < f N,,,α u. Therefore, we get u D N,,,α = J N,,,α u < f N,,,α u D N,,,α, which is imossible. This finishes our roof. In the sequel, we only consider 1,N 1/2. We have the following relation between the attainability of D N,,,α and the attainability of su f N,,,α t in 0,. Lemma 2.3. Let N 2, 1,N 1/2 and α, > 0. Then D N,,,α is attained in S N,, if and only if su f N,,,α t is attained in 0,. Proof. Suose that D N,,,α is attained by a function u S N,,,α. It follows from 1.7 and 2.7 that u D N,,,α = J N,,,α u f N,,,α u D N,,,α. Consequently, we get u f N,,,α u = D N,,,α, that is f N,,,α attains its maximum value at u / u 0,. Conversely, suose that su f N,,,α t is attained at t 0 0,. Choosing λ > 0 such that the function w λ defined by 2.8 note that u W 1, R N since < N 1/2 satisfies λ N u u = t 0. It follows from 1.7 and 2.9 that u J N,,,α w λ = f N,,,α λ N u = f N,,,α t 0 = suf N,,,α t = D N,,,α, that is D N,,,α is attained by w λ. We next rove a non-existence result when α < α N, as follows. Lemma 2.4. Let N 2, 1,N 1/2 and > 0. If α < α N, then D N,,,α does not attains in S N,,. Moreover D N,,,α = 1. Proof. Since α < α N, then αs N, t / < 1+t / 1+t / for any t > 0. Hence f N,,,α t < 1 = limf N,,,α t suf N,,,α t t > 0. t 0 These estimates imlies that su f N,,,α t does not attains in 0, and hence D N,,,α does not attains in S N,, by Lemma 2.3. Moreover, we get from the above estimates that D N,,,α = su f N,,,α t = 1 by

12 Lemma 2.5. Let N 2, 1,N 1/2 and α, > 0. If >, then α N, = 0 and D N,,,α is attained for any α > 0. Proof. Since >, we then have limg N,, t = 0. Hence α N, = 0. t 0 An easy comutation shows that f N,,,α t = 1+t 2 1+t 1 + αs N, t 1 =: t+1 1+t 2h N,,,αt. Since >, then h N,,,α is strictly decreasing function on 0, with limh N,,,α t = t 0 and lim h N,,,α t =. There exists unique t 0 0, such that h N,,,α t 0 = 0, t h N,,,α > 0 on 0,t 0 and h N,,,α < 0 on t 0,. h N,,,α and f N,,,α has the same sign on 0, which imlies f N,,,α attains it maximum value at unique oint t 0 0,. Our conclusion hence is followed from Lemma 2.3. Lemma 2.6. Let N 2, 1,N 1/2 and α, > 0. If =, then α N, = / S N, and D N,,,α is attained for any α > α N, while it is not attained for any α α N,. Proof. Suose that =, we then have and hence g N,, t = 1+t t 1+t1 t g N,, t = 1 t 1 1+t 2 = 1 t 2 1+t1 +, + 1+t1 t t t 1 By the arithmetic geometric inequality we get g N,, t > 0 for any t > 0. This imlies that g N,, is strictly increasing function on 0, and hence α N, = 1 lim S g N,, t = t N, S N,. and α N, S N, < g N,, t t > Ifα > α N,, thenwecantaket 1 0, such thatαs N, > g N,, t 1whichimlies su f N,,,αt f N,,,αt 1 > 1. Proosition2.1 yields su f N,,,αt > αs N, since = >. Hence f N,,,α attains it maximum value in 0,. By Lemma 2.3, D N,,,α is attained in S N,,. Wenext consider thecase α α N,. It remains torove forα = α N, bylemma 2.4. It follows from 2.11 that α N, S N, < g N,, t for any t > 0 which imlies f N,,,α N, t < 1 su f N,,,α N, t for any t > 0 by Proosition 2.1. Consequently, su f N,,,α N, t is not attained in 0,. By Lemma 2.3, D N,,,α N, is not attained. Moreover, the revious estimate and 1.7 yields D N,,,α N, = 1. 12

13 Lemma 2.7. Let N 2, 1,N 1/2 and α, > 0. If < <, then D N,,,α is attained for any α α N, while it is not attained for any α < α N,. Moreover, we have D N,,,α = 1 if α α N,. Proof. By Lemma 2.4, it remains to consider the case α α N,. Making the change of variable s = t/1+t 0,1, we get and with s 0,1. We have and f N,,,α t = 1 s +αs N, s =: kn,,,α s, g N,, t = s s 1 s =: ln,, s, l N,, s = s 1 + = s s 1 1 s + s 1 s s + 1 s 1 1 =: s 1 m N,, s, m N,,s = 1 s 2 s. Consequently, m N,, = 0 at s 1 = / 0,1, m N,, < 0 on 0,s 1 and m N,, > 0 on s 1,1. Since lim s 0 m N,, s = 0, then there exists unique s 2 s 1,1 such that m N,, s 2 = 0, m N,, < 0 on 0,s 2 and m N,, > 0 on s 2,1. l N,, and m N,, have thesamesign, hence l N,, attainsuniquely itsminimum valueats 2 whichisequivalent that g N,, attainsuniquelyitsminimumvalueatt 2 = s 2 /1 s 2 > s 1 /1 s 1 = /. If α > α N, = S N, l N,,s 2, then we have k N,,,α s 2 = 1 s 2 +αs N, s 2 1 = limk N,,,α s s 0 Let k 1 denote the integer number such that k < / k + 1, i.e, / = k + r for r 0,1]. The i th derivative of k N,,,α is given by i 1 k i N,,,α s = j i 1 1 s i +αs N, j s i. For i = 1,2,...,k we have j=0 i 1 lim s 0 ki N,,,α s = j < 0, limk i s 1 N,,,α s = j=0 13 j=0

14 Combining 2.12 and 2.13, we get that su k i N,,,α s > 0 i = 1,2,...,k s 0,1 Indeed, if this is not true, i.e., there is i such that su s 0,1 k i N,,,α s 0, then using 2.13 consecutively we see that k N,,,α is strictly decreasing on 0,1 which contradicts with The functions k k+1 N,,,α is strictly decreasing on 0,1 with k lim s 0 kk+1 N,,,α s j=0 j > 0, limk i s 1 N,,,α s =. Hence there exists uniquely s k+1 0,1 such that k k+1 N,,,α s k+1 = 0, k k+1 N,,,α > 0 on 0, s k+1 and k k+1 N,,,α < 0 on s k+1,1, i.e., k k N,,,α attains its maximum value at s k+1. Hence k k N,,,α s k+1 > 0. The function k k N,,,α is strictly increasing on 0, s k+1 and strictly decreasing on s k+1,1, hence there are two oints s k 0, s k+1 and s k s k+1,1 such that k k N,,,α s k = kk N,,,α s k = 0, kk N,,,α < 0 on 0, s k s k,1 and kk N,,,α > 0 on s k, s k. Thistogetherwith2.13imliesthatkk 1 N,,,α Obviously, the above argument for k k N,,,α shows that kk 1 attainsitsmaximumvalueonlyat s N,,,α s k > 0, and kk 1 N,,,α k. has only two solutions s k 1 s k, s k and s k 1 s k,1 such that kk 1 N,,,α < 0 on 0, s k 1 s k 1,1 and k k 1 N,,,α > 0 on s k 1, s k 1. Using consecutively this arguement for i = k 2,...1, we conclude that k N,,,α has only two solutions s 1 s 2, s 2 and s 1 s 2,1 such that k N,,,α < 0 on 0, s 1 s 1,1 and kk 1 N,,,α > 0 on s 1, s 1. This together with 2.12 and Proosition 2.1 imlies that k N,,,α attains uniquely its maximum value at s 1 0,1. Suose α = α N,. We have α N, S N, < g N,,t unless t = t 2 which immediately imlies f N,,,αN, t < 1 unless t = t 2. This together with Proosition 2.1 gives D N,,,αN, = suf N,,,αN, t = 1, and su f N,,,αN, t is attained only at t 2 0,1. Let us remark here that when α α N, then suf N,,,α t f N,,,α t 2 f N,,,αN, t 2 = 1 = limf N,, t. t 0 This together with Proosition 2.1 is enough to conclude that su f N,,,αN, t is attained on 0,. Although our roof above is more comlicate, however it shows that su f N,,,αN, t is attained at unique oint in 0,. This will be used to classify all maximizers for D N,,α, below. 14

15 Lemma 2.8. Let N 2, 1,N 1/2 and α, > 0. If 0 <, then α N, = S N, and D N,,,α is not attained for any α > 0. Moreover, we have D N,,,α = max{1,αs N, }.. Proof. Making the change of variable s = t/1+t 0,1 and using again the transformation of functions as in the roof of Lemma 2.7 we have l N,, s = s 1 m N,, s. Since, it is clear that m N,, is a strictly decreasing function of s on 0,1 and lim m N,,s = 0. Hence, m N,, s < 0 for any s 0,1 or equivalently l N,, s < 0 for s 0 any s 0,1. Consequently, we have Hence α N, = S N,. We extend k N,,,α to all [0,1] by inf g N,,t = inf l N,,s = liml N,, s = 1. s 0,1 s 1 k N,,,α 0 = lim s 0 k N,,,α s = 1, and k N,,,α 1 = lim s 1 k N,,,α s = αs N,. Then k N,,,α is continuous on [0,1]. Since < then k N,,,α is strict convex on [0,1]. As a consequence, we have for any s 0,1 that k N,,,α s < sk N,,,α 1+1 sk N,,,α 0 max{1,αs N, }. Hence, for any t > 0 we get t f N,,,α t = k N,,,α 1+t < max{1,αs N, }. The receding inequality together with Proosition 2.1 imlies D N,,,α = suf N,,,α t = max{1,αs N, }. Moreover, weknowthatf N,,,α t < su f N,,,α tforanyt > 0. Hencesu f N,,,α t does not attains in 0, which finishes our roof by Lemma 2.3. Concerning to α N,, we have the following results on its attainability and its nonattainability in S N,,. Proosition 2.9. Let N 2 and 1,N. If N 1/2 then α N, is not attained for any > 0. If < N 1/2 then α N, is attained for any < < while it is not attained for any or. Moreover, the function α N, is continuous on 0, ]. Proof. The non-attainability of α N, for N 1/2 is comletely roved by the method in the roof of Lemma 2.2 by exloiting the non-attainability of the Sobolev inequality 1.5 in W 1, R N. The attainability of α N, for < N 1/2 and, was already roved in the roof of Lemma 2.7. The non-attainability of α N, for < N 1/2 and, follows from the roof of Lemma 2.5, 2.6 and

16 To rove the continuity of α N, on 0, ], it is enough to show that it is continuous on [, ] since α N, = S N, for any. We claim that lim su 0 α N, α N, This is an easy consequence of the continuity of 0, 0, t, g N,, t. Rewriting the function g N,, t as 1+t g N,, t = t t we see that g N,, t is a strictly decreasing function of. Hence α N, is a non-increasing function of > 0 and is strictly decreasing function of, since α N, is attained for,. Then non-increasing roerty of α N, and 2.15 imly lim α N, = α N,. Let t 0, be the unique oint where g N,, attains its minimum value in 0, the uniqueness come from the roof of Lemma 2.7. Also, from the roof of Lemma 2.7, we get t > /, hence t as. Hence lim α N, = 1 S N, lim 1+t t t = S N, = α N,. Hence α N, is continuous at and. For any < a < b < and [a,b] we have and limg N,, t limg N,,b t = > inf g t 0 t 0 N,,at, lim g N,,t lim g N,,b t = 1 > inf g t t N,,at, since a >. Hence there exist 0 < A < B < such that g N,, t > inf g N,,at > inf g N,,t for any t 0,A B, and [a,b]. Thus, we get α N, = S N, inf g N,, t, t [A,B] for any [a,b]. According to the continuity of function [a,b] [A,B],t g N,, t, the function α N, is continuous on [a,b] for any < a < b <. Hence α N, is continuous on,. This comletes our roof. 16

17 To conclude this section, we give some comments on the maximizers for the roblem 1.1 in the critical case. From the roof of Theorem 1.2 more recisely, Lemma 2.3, we see that if the maximizers of the roblem 1.1 in the critical case exists then w λ defined by 2.8 for some suitable λ > 0 determined by the value of D N,,,α is a maximizer. Since our roblem is invariant under the translation and multile by ±1 then ±w λ x 0,x 0 R N is also a maximizer. Conversely, if u is a maximizer of D N,,,α, i.e., u suf N,,,α t = D N,,,α = J N,,,α u f N,,,α u suf N,,,α t, here we used 2.7. This shows that f N,,,α is attained by u / u and u J N,,,α u f N,,,α u. The latter condition imlies that u is a maximizer of the Sobolev inequality 1.5, i.e., u has the form ux = aλ 1 u λnx x 1 0 for some a 0, λ > 0 and x 0 R N where u is given by 1.6. To ensure u S N,,, a must be equal to a = ±1/ u +λ N u Hence ux = ±w λ x x 0 for some x 0 R N and λ > 0. Let t N,,,α 0, be the unique oint where f N,,,α attains its minimum value, then we must have u / u = t N,,,α, which is equivalent to N λ = t N u N,,,α. u Therefore λ is determined uniquely by t N,,,α or equivalent D N,,,α since it is attained at unique oint in 0,. Thus, we have roved the following result. Proosition Let N 2, 1,N 1 2 and α, > 0. If D N,,,α is attained in S N,,, then its maximizers must have the form ±λ 1 u λ 1 Nx x 0 / u + λ N u 1 for some x 0 R N where λ > 0 is determined uniquely by D N,,,α A new roof of Theorem 1.1 This section is devoted to give a new roof of Theorem 1.1. Comaring with the roof of Ishiwata and Wadade in [17], our roof is more simle and elementary. Also, our roof does not use any rearrangement argument as used in [17]. We first rove Theorem 1.4 which is the key in our roof of Theorem 1.1. Proof of Theorem 1.4. Our roof follows the way in the roof of Theorem 1.3. We start by roving 1.9. For any u S N,,, using Gagliardo Nirenberg Sobolev inequality 1.4, 17

18 we get J N,,,α,q u u +αb N,,q u c u q c = u = su u + u q 1+ u u q 1+t q u + u q +αb N,,q u c u q c c u +αb N,,q u 1+ u u q +αb N,,q t c 1+t q Taking the suremum over all u S N,,, we obtain. D N,,,α,q su 1+t q +αb N,,q t c 1+t q. 3.1 For a reversed inequality of 3.1, let us take v W 1, R N \{0} to be a maximizer for the Gagliardo Nirenberg Sobolev inequality 1.4. For any λ > 0, we define We have v λ S N,, and v λ x = λ 1 v λnx v +λn v 1 D N,,,α,q J N,,,α,q v λ = = v v +λ N v v v +λ N v λ q v q q +α v +λn v q λ q v c +αb v q c N,,q v +λn v q = 1+λ N v v q +αb N,,q 1+λ N v v q λ N v v c. 3.3 Since 3.3 holds for any λ > 0, then we get D N,,,α,q su 1+t q +αb N,,q t c 1+t q. 3.4 Combining 3.1 and 3.4 together, we obtain

19 We next rove the inequality For any u S N,,, by using the shar Gagliardo Nirenberg Sobolev inequality 1.4, we get I N,,,q u 1 B N,,q = 1 B N,,q = 1 B N,,q 1 u u c u q c u + u q 1 B N,,q inf 1+ u u q 1+t q u + u u u c 1+ u u u u q c c u 1+t 1 t c Taking the infimum over all u S N,, we obtain α N,,q 1 B N,,q inf 1+t q. 1 1+t 1 t c. 3.5 To finish the rove of 1.10, let us rove a reversed inequality of 3.5. We use again the function v λ defined by 3.2 as test function for α N,,q. Obviously, we have α N,,q I N,,,q v λ v = 1 v +λn v = 1 B N,,q = 1 B N,,q v +λ N v q 1+ λ q N v v α N,,q 1 B N,,q inf 1 λ q v q q v +λn v q v +λn v v λ q v c u q c 1+ λ N v v λ N v v 1 c, 3.6 here we use the equality in 1.4 for v. The inequality 3.6 holds for any λ > 0, then we have 1+t q 1+t 1 Combining 3.7 and 3.5 together, we obtain t c. 3.7

20 and The above roof of Theorem 1.4 shows that J N,,,α,q u f N,,,α,q u u, I N,,,q u 1 B N,,q g N,,,q J N,,,α,q v λ = f N,,,α,q λ N u u, 3.8 u u. 3.9 Recall the definition of functions f N,,,α,q and g N,,,q from 1.9 and It is clear that lim t 0 f N,,,α, t = 1. As a consequence, we get D N,,,α, Similar with Lemma 2.3, the same relation between the attainability of D N,,,α,q and the attainability of su f N,,,α,q t in 0, holds. More recisely, we have Lemma 3.1. Let N 2, 1,N], q, and α, > 0. Then D N,,,α,q is attained in S N,, if and only if su f N,,,α,q t is attains in 0,. Proof. The roof is comletely similar with the one of Lemma 2.3 by using 3.8 and 3.9. Hence we omit it. We next rove Theorem 1.1 Proof of Theorem 1.1. We first rove art 1 under the assumtion > c. In this case, we have lim t 0 g N,,,q t = 0 which then imlies α N,,q = 0. We rewrite f N,,,α,q as hence f N,,,α,q t = 1+t +αbn,,q t c 1+t q, f N,,,α,q q t 1 t = 1+t q q = t 1 h N,,,α,q t +αb N,,q c qt c +αbn,,q c t c 1 Since > c and c < q, then h N,,α,,q is strictly decreasing function on 0,. Note that h N,,,α,q t ast 0 andh N,,,α,q t ast. Consequently, thereisunique t 0 0, such that f N,,,α,q t 0 = 0, f N,,,α,q t > 0 if t 0,t 0 and f N,,,α,q t < 0 if t t 0,. Thus f N,,,α,q attains its maximum value at unique oint t 0 0,. By Lemma 3.1, D N,,,α,q is attained in S N,,. We next rove art 2 under the assumtion that = c. We first show that α N,,q c = / c B N,,q. Indeed, we have q / c = /N < 1 and q/ c > 1, hence the function ht = 1+t q/c 1+t q /c is convex on [0,. Consequently, we get 20

21 ht h0 h 0t which is equivalent to g N,,,q t / c B N,,q for any t > 0. It is clear that lim t 0 g N,,,α = / c B N,,q. These estimates imly α N,,q c = / c B N,,q. As in the roof of art 1, we have q f N,, c,α,qt = t c 1 1+t q c αb N,,q q c t+αb N,,q c c q =: t c 1 h N,,c,α,qt. c Suose that α α N,,q c. We then have f N,, c,α,q t < 0 for any t > 0 hence f N,,,α,q t < limf t 0 N,,c,α,qt = 1 = suf N,,,α,q t, for any t > 0. Hence su f N,,,α,q t is not attained in 0,. By Lemma 3.1, D N,,c,α,q is not attained in S N,,. By 1.9, it is clear that D N,,,α,q = 1. Suose that α > α N,,q c. We have lim t 0 h N,,c,α,qt + αb N,,q c > 0 and lim t h N,,c,α,qt = since c < q. Note that h N,,c,α,q is strictly decreasing function on0,. Hence, there isunique t 0 0, such that f N,, t c,α,q 0 = 0, f N,, c,α,q t > 0 if t 0,t 0 and f N,, t < 0 if t t c,α,q 0,. Consequently, su f N,,c,α,qt is attained at the unique oint t 0 0,. By Lemma 3.1, D N,,c,α,q is attained in S N,,. It remains to rove art 3 under the assumtion < c. Suose that α < α N,,q. We then have αb N,,q < g N,,,α,q t for any t > 0 which yields f N,,,α,q t < 1 for any t > 0. This together with 3.10 imlies that f N,,,α,q t < 1 = suf N,,,α,q t = D N,,,α,q, for any t > 0. Hence su f N,,,α,q t is not attained in 0,. Hence D N,,,α,q is not attained in S N,, by Lemma 3.1. Suose that α α N,,q. Since < c < q, it is clear that limg N,,,q t = = lim g N,,,q t. t 0 t Consequently, there exists t 1 0, such that α N,,q = g N,,,q t 1 and hence Thus we get c αb N,,q t c 1 α N,,q c B N,,q t1 = 1+t 1 q 1+t1 q f N,,,α,q t 1 1 = lim t 0 f N,,,α,q t > 0 = lim t f N,,,α,q t. Consequently, su f N,,,α,q t is attained in 0,. By Lemma 3.1, D N,,,α,q is attained in S N,,.. 21

22 4 Further remarks In this section, we exlain how our method used to rove Theorem 1.1 and Theorem 1.2 can be alied to the maximizing roblems of tye 1.1 for the fractional Lalacian oerators. Let s 0,N/2, the fractional Lalacian oerators s is defined via the Fourier transformation by s uξ = ξ 2s ûξ, where ûξ = e ix ξ uxdx denotes R N the Fourier transformation of u. Let H s R N denote the fractional Sobolev sace of all functions u L 2 R N such that s/2 u L 2 R N. For > 0, we define u H s = u s 2 u 2, u H s R N. For β, > 0 and 2 < q 2 s := 2N/N 2s, we consider the roblem S N,s,,β,q = su u H s R N, u H s =1 u 2 2 +β u q q. 4.1 Let S N,s,q denote the best constant in the fractional Gagliardo Nirenberg Sobolev inequality u q q Nq 2 S N,s,q = su, u H s R N,u 0 2u s s,q 2 u q s,q s,q := s 2 It is well known that if q 2,2 s then S N,s,q is attainedin H s R N. If q = 2 s, 4.2 reduces to the fractional Sobolev inequality whose best constant S N,s,2 s was exlicitly comuted by Lieb [21]. In fact, Lieb comuted the shar constants in the shar Hardy Littlewood Sobolev inequality which is dual form of the shar fractional Sobolev inequality. Moreover, the equality holds in the fractional Sobolev inequality if and only if ux = u s, x := 1+ x 2 2s N/2 u to a translation, dilation and multile by a non-zero constant. Note that u s, H s R N unless s < N/4. Define β N,s,q = inf u H s R N, u H s =1 1 u 2 2 u q. 4.3 q Similar to the roblem 1.1, β N,s,q will lay the role of threshold for the existence of maximizers for the roblem 4.1 as the one of α N,,q. Following the roof of Theorem 1.1 and Theorem 1.2, we can rove the following results for roblem 4.1. Theorem 4.1. [subcritical case q < 2 s] Let N 1, s 0,N/2, q 2,2 s and β, > 0. Then we have 1 If > s,q then β N,s,q = 0 and S N,s,,β,q is attained for any β > 0. 2 If = s,q then β N,s,q s,q = 2/ s,q S N,s,q and S N,s,,β,q is attained for any β > β N,s,q s,q while it is not attained for any β β N,s,q s,q. 3 If < s,q then S N,s,,β,q is attained for any β β N,s,q while it is not attained for any β < β N,s,q. 22

23 In the critical case q = 2 s, we have s,2 s = 2 s. In this case, we denote S N,s,2 s, β N,s,2 s and S N,s,,β,2 s by S N,s, β N,s and S N,s,,β resectively for simlifying notation. We have the following results. Theorem 4.2. [critical case q = 2 s] Let N 1, s 0,N/2 and β, > 0. If s N/4 then S N,s,,β is not attained. If s < N/4, then the following results hold. 1 If > 2 s then β N,s = 0 and S N,s,,β is attained for any β > 0. 2 If = 2 s then β N,s2 s = 2/2 s S N,s and S N,s,,β is attained for any β > β N,s 2 s while it is not attained for any β β N,s 2 s. Moreover, if β β N,s 2 s then S N,s,,β = 1. 3 If 2 < < 2 s then S N,s,,β is attained for any β β N,s while it is not attained for any β < β N,s. Moreover, if β β N,s then S N,s,,α = 1. 4 If 2 then β N,s = S 1 N,s and S N,s,,β is not attained for any β > 0. In this case, we have S N,s,,β = max{1,βs N,s }. The roof of Theorem 4.1 and Theorem 4.2 follows the same way in the roof of Theorem 1.1 and Theorem 1.2. We first can show that and S N,s,,β,q = su β N,s,q = 1 S N,s,q inf 1+t q 2 +βs N,s,q t s,q 1+t q 1+t q 2 1+t 2 1 t s,q =: su f N,s,,β,q t, = 1 S N,s,q inf g N,s,,qt. The functions f N,s,,β,q and g N,s,,q have the same roerties of the functions f N,,,α,q and g N,,,q above also f N,,,α and g N,, resectively. So we can use again the arguments in the roof of Theorem 1.1 and Theorem 1.2 to rove Theorem 4.1 and Theorem 4.2. As a consequence, we obtain the existence results for solutions in H s R N of several non-local ellitic tye equations s 2 u 2 2 s u+ u u 2 2 +βq u q q u = βq u q 2 u 2 u 2 2 +βq u q q under some suitable conditions onq 2,2 s ] and β, > 0. Notethat the revious equation is exact the Euler Lagrange equation with Lagrange multilier of the roblem 4.1. The similar results for the fractional Lalace oerators s where s 0,1 and s < N can be roved by the same way. We refer the reader to the aer [6] and references therein for the definition and roerties of the oerator s. The fractional Sobolev inequality and fractional Gagliardo Nirenberg Sobolev inequality related to s are well known [12, 25]. The existence of maximizers for these inequalities can be roved by using the concentration comactness rincile of Lions [23, 24]. The otimal estimate for the maximizers of the fractional Sobolev inequality was recently roved in [6]. These ingredients are enough to aly our method used to rove Theorem 1.1 and Theorem 1.2 to establish the similar results for s. We leave the details for interesting reader. 23

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