Minimization problems on the Hardy-Sobolev inequality

manuscript No. (will be inserted by the editor) Minimization problems on the Hardy-Sobolev inequality Masato Hashizume Received: date / Accepted: date Abstract We study minimization problems on Hardy-Sobolev type inequality. We consider the case where singularity is in interior of bounded domain R N. The attainability of best constants for Hardy-Sobolev type inequalities with boundary singularities have been studied so far, for example [5] [6] [9] etc.... According to their results, the mean curvature of at singularity affects the attainability of the best constants. In contrast with the case of boundary singularity, it is well known that the best Hardy-Sobolev constant µ s () := { u dx u H 0 (), u (s) x s = is never achieved for all bounded domain if 0. We see that the position of singularity on domain is related to the existence of minimizer. In this paper, we consider the attainability of the best constant for the embedding H () L (s) () for bounded domain with 0. In this problem, scaling invariance doesn t hold and we can not obtain information of singularity like mean curvature. Keywords critical exponent Hardy-Sobolev inequality minimization problem Neumann Mathematics Subject Classification (000) 35J0 Introduction We study the minimization problems for the Hardy-Sobolev type inequalities. Let N 3, is bounded domain in R N, 0, 0 < s <, and (s) := (N s)/(n ). The Hardy- Sobolev inequality asserts that there exists a positive constant C such that C ( u (s (s) u dx () Department of Mathematics, Graduate School of Science, Osaka City University 3-3-38 Sugimoto Sumiyoshi-ku, Osaka-shi, Osaka 558-8585 Japan E-mail: d5san0f06@st.osaka-cu.ac.jp

Masato Hashizume for all u H0 (). For s = 0, the inequality () is called Sobolev inequality and for s =, the inequality () is called Hardy inequality. In the non-singular case (s = 0), it is well known that the best Sobolev constant S is independent of domain and S is never achieved for all bounded domains. But if = R N and H () is replaced by the function space of u L N/(N ) () with u L (), then S is achieved by the function u(x) = c( + x ) ( N)/ and hence the value S = N(N )π[γ (N/)/Γ (N)] /N explicitly (see [], [3] and [6]). In the case of s =, the best constant for the Hardy inequality is [(N )/] and this constant is never achieved for all bounded domains and R N. This fact suggests that it is possible to improve this inequality. For example Brezis and Vazquez [], many people research the optimal inequality of (). In other words, the best remainder term for () is studied actively. In the case of 0 < s <, the best Hardy-Sobolev constant is defined by µ s () := { u dx u H 0 (), u (s) x s =. This constant has some similar properties to these of the best Sobolev constant. Indeed, due to scaling invariance, µ s () is independent of, and thus µ s := µ s () = µ s (R N ) is not attained for all bounded domains. If = R N, then µ s is attained by for some a > 0 and hence y a (x) = [a(n s)(n )] N ( s) (a + x s N s µ s = (N )(N s) ( ω N Γ ( N s s ) s N s s Γ ( (N s) s ) (see [9] and [3]) where ω N is the area of the unit sphere in R N. On the other hand, for 0, the result of the attainability for µ s () is quite different from that in the situation of 0. By Ghoussoub-Robert [6], it has proved that if has smooth boundary and the mean curvature of at 0 is negative, then the extremal of µ s () exists for all N 3. Recently, Lin and Wadade [4] have studied the following minimization problem; µ λ s,p() := inf { u dx + λ ( u dx) p p u H 0 (), u (s) = where λ R and p N/(N ). Furthermore, as related results, Hsia, Lin and Wadade [0] studied the existence of the solution of double critical elliptic equations related with µ s, λ (), that is, they have showed the existence of the solution for { u + λu + u (s) x s = 0, u > 0, in u = 0 on under the appropriate conditions where = N/(N ). To prove these results, we use the theorem of Egnell [4]. He showed that the existence of the extremal for µ s () if is a half space R N + or an open cone. The open cone C is written of the form C := {x R N x =

Minimization problems on the Hardy-Sobolev inequality 3 rθ, θ Σ where Σ is connected domain on the unit sphere S N in R N. By this result, µ s (C ) > µ s (R N ) and there is a positive solution for { u = u (s) x s in C, u = 0 on C, and u(x) = o( x N ) as x. The Neumann case also has been studied. Let has C boundary and the mean curvature of at 0 is positive. Ghoussoub and Kang [5] have showed that there is a least energy solution for { u + λu = u (s) x s in, u ν = 0 on for N 3, λ > 0. Like these results, if 0, we can use the benefit of the mean curvature of at 0 to show the results. However if 0, we cannot obtain the information of singularity such the mean curvature, and the fact causes some technical difficulties. In this paper, we consider the attainability for the following minimization problem µ N s () := inf { The main theorem is as follows. ( u + u )dx u H (), u (s) =. Theorem Let has a smoothness which the Sobolev embeddings hold, then the following statements hold true. (I) If is sufficiently small, then µ N s () is attained. Especially, if satisfies the following; ( ) x s (s) dx µs then µ N s () is attained, where is the N-dimensional Lebesgue measure of domain. (II) There is a positive constant M which depends on only such that µ N s (r) is never attained if r > M. Eventually, the size of domain affects the attainability of µ N s (). The rest of the paper is organized as follows. In Section we introduce three lemmas to prove Theorem. Then in Section 3 we prove Theorem using the lemmas in Section. In Section 4, as an application, we consider the case when singularity is in the boundary of domain. Then we introduce a new result concerning the attainability of µ N s () with boundary singularity. Preparation In this section, we prepare some lemmas to prove Theorem.

4 Masato Hashizume Lemma For r > 0, the value µ N s,r() is defined by µ N s,r() := inf { ( u + ru )dx u H (), u (s) =. We have µ N s,r() = µ N s (r). Proof For r > 0 and u H (), u r is defined by the scaling of u, that is u r (x) := r N u(x/r) H (r). Note that r r u r dx = u r dx = r r u r (s) = With these facts in mind, taking u H () such that u dx, u dx, u (s). u (s) =, ( u + r u )dx µ s,r() N + ε for ε > 0 sufficiently small, we have µ s N (r) ( u r + u r )dx = ( u + r u )dx µ s,r() N + ε. r Hence we have µ N s (r) µ N s,r(). The inverse also holds by replacing with r. Lemma There exists a positive constant C which depends on only such that µ s ( u (s (s) u dx +C u dx (u H ()). () Before beginning the proof, we make a remark. H. Jaber [] has shown that the following theorem. Theorem ([]) If (M, g) is a compact Riemannian manifold without boundary and 0 M, there is a constant C = C(M,g) such that µ s ( M u (s (s) d g (x,0) s dv g u dv g +C u dv g M (u H (M)) where d g is the Riemannian distance on M. Different from Theorem, is bounded domain of R N and therefore has a boundary, thus we can show the inequality () simply.

Minimization problems on the Hardy-Sobolev inequality 5 Proof Let 0 and these two subdomain are taken suitable again later. A cut-off function is defined by ϕ which satisfies ϕ C c (R N ), 0 ϕ in, ϕ = on, ϕ = 0 on \. Here, we construct a partition of unity η, η defined by η := ϕ ϕ + ( ϕ, η ( ϕ) := ϕ + ( ϕ. Note that η, η C () by the definition. We may assume that u C () H () by density. We have ( u (s (s) µ s = µ s u L (s)/ (, x s ) = µ s i u i=η L (s)/ (, x s ) µ s η i u L (s)/ (, x s ) i= = µ s η i u (s) i= = I + I. (s) We estimate I,I for each. For I, since suppη we can use the Hardy-Sobolev inequality. We get that I = µ s η u (s) x s = u η dx + dx (s) (η u) dx (η ) (η u )dx. Since η C () we may integrate by parts the second term and hence we obtain I u η dx (η )η u dx (3) For I, since 0 suppη and taking account to that η = 0 on we have I = µ s η u (s) (s) = µ s u (s) ( ) µ s a η u (s (s) dx \ µ s a \ µ s a \ = µ s a \ (s) (s) (s) \ η ( η u dx \ S(, ) \ (η u) dx S(, ) (η u) dx (s)

6 Masato Hashizume where a := dist(0, ) s/ (s) and { S(, ) := inf u dx u H (), u = 0 on, u =. \ \ Here, let us take 0. It is clearly that a dist(0, 0 ) s/ (s). On the other hand, for u H ( \ ) such that u = 0 on, we define v H ( \ 0 ) by { u in \ v := 0 in \ 0. By identifying u H ( \ ) with v H ( \ 0 ) concerning the calculation of the Sobolev quotient, we may see that {u H ( \ ) u = 0 on {u H ( \ 0 ) u = 0 on 0. Hence we obtain S(, ) S(, 0 ). Consequently, if is sufficiently large, a and S(, ) is bounded from above uniformly. By choosing and close to we obtain I (η u) dx. Therefore I u η dx + η u dx. (4) Here, since η, η C () there is a positive constant C such that This constant depends on only. Consequently (3), (4) and (5) yield that µ s ( max x (η ) C, max x η C. (5) u (s (s) I + I u dx +C u dx. Lemma 3 µ N s () µ s holds (see [9], Lemma.). Furthermore, the following statements hold true; (I) If µ N s () < µ s, then µ N s () is attained. (II) If µ N s () = µ s, then µ N s (r) is not attained for all r >. Firstly, we prove Lemma 3 (I). Proof (Proof of Lemma 3 (I)) Assume {u n n= H () is a minimizing sequence of µ s N (). Without loss of generality, we may assume u n (s) = (6) for all n N and which implies ( u n + u n)dx = µ s N () + o() (n ). (7)

Minimization problems on the Hardy-Sobolev inequality 7 Thus u n is bounded in H (). So we can suppose, up to a subsequence, u n u in H () u n u in L p () ( p < ) u n u in L q (, x s ) ( q < (s)) u n u a.e. in as n. For this limit function u, we show that u 0 a.e. in. Assume that u 0 a.e. in. By the inequality () in Lemma, ( u n (s) µ s (s) holds for all n. Thus (6), (7), (8) and u n u in L () yield u n dx +C u ndx (8) µ s µ N s () + o(). Letting n tend to infinity, we obtain µ s µ N s () and which is contradiction in the assumption of µ N s () < µ s. Consequently u 0. By the theorem of Brezis and Lieb (see [3]), we obtain and it follows that = = u n (s) u (s) = + ( ( ( On the other hand, we have = ( u n (s) (s) u (s) u n u (s) + x s ( u (s (s) + ( u (s (s) + ( u + u )dx µ s N () ( u n + u n)dx µ s N + o() () = + o(). u n u (s) + o() dx (s) u n u (s) + o() (s) (s) + o(). u n u (s) + ( (u n u) + (u n u dx µ s N ()

8 Masato Hashizume Hence there exist a limit and we obtain ( lim n ( = lim =. n u (s) + u n u (s) ( u (s (s) + By the equality condition of the above, we get either (s) u n u (s) (s) u 0 a.e. in or u n u 0 in L (s) (, x s ). Since u 0 we obtain u n u 0 in L (s) (, x s ) and hence this u is the minimizer of µ N s (). Next, we prove Lemma 3 (II). Proof (Proof of Lemma 3 (II)) We assume the existence of the minimizer of µ s N (r) and derive a contradiction. Let u H (r) be a minimizer of µ s N (r), then we have µ s N (r) = ( u + u )dx > ( u + r r r u )dx µ s,/r N (r). By Lemma, the assumption µ N s () = µ s and µ N s (r) µ s, we have This is a contradiction. µ s µ N s (r) > µ N s,/r (r) = µn s () = µ s. 3 Proof of Theorem In this section, we prove Theorem. Proof (Proof of Theorem (I)) We recall that { µ s N () := inf ( u + u )dx u H (), Taking a constant C such that C (s) x s u (s) =. = and u C as a test function, it follows that ( ) µ s N () x s (s). If this C is a minimizer of µ s N (), then by Lagrange multiplier theorem C is a classical solution of { u + u = µ s N () u (s) in ν = 0 This contradicts and therefore ( µ s N () < u x s on. ) x s (s). Combining this estimate and Lemma 3 (I), Theorem (I) holds true.

Minimization problems on the Hardy-Sobolev inequality 9 Proof (Proof of Theorem (II)) Since Lemma, We can define a constant m by m := inf{c > 0 () holds.. M is defined by M := m. In inequality (), C is replaced by M and hence we have µ s ( u + M u )dx ( (s) u (s) (9) for all u H (). Therefore by Lemma we obtain µ s inf ( u H ()\{0 = µ N s,m() = µ N s (M). ( u + M u )dx u (s) (s) Recall that µ N s () µ s holds for all bounded domain and thus µ N s (M) = µ s. Consequently we obtain the result of Theorem (II) by Lemma 3 (II). 4 Singularity on the boundary Throughout this section, assume that 0. If the mean curvature of at 0 is positive, we have obtained the results in Section. However, if the mean curvature of at 0 vanishes, we don t obtain results so far, even if the attainability of µ N s (). In this section, we show the following results by using the strategy in Section and Section 3. Theorem 3 Let R N is bounded domain with smooth boundary, 0 and is flat near the origin. Then the following statements hold; (I) If is sufficiently small, then µ N s () is attained. Especially, if satisfies the following; ( ) x s (s) µ s dx N s s then µ N s () is attained. (II) There is a positive constant M which depends on only such that µ N s (r) is never attained if r > M. This condition of the boundary in this theorem is a special case of vanishing of the mean curvature of at 0. We prove the theorem in the same way as in Section and Section 3. Different from the proof of Theorem, we need the following lemma instead of Lemma. Lemma 4 There is a positive constant C depends on only such that µ s s N s ( u (s (s) u dx +C u dx (u H ()). (0)

0 Masato Hashizume Proof We introduce some notation. B R (0) is an open ball which center is origin and radius is R. R N + is a half space which is defined by R N + := {(x,x N ) R N x n > 0 where x := (x,...,x N ) R N. Since is flat near the origin, by rotating coordinate there is a constant r > 0 such that B r (0) = := B r (0) R N +. For u H () we have ( u (s (s) = ( ( = J + J. u (s) + \ u (s (s) ( u (s (s) + \ u (s (s) For u H (), ũ H (B r (0)) is defined by the even reflection for the direction x N, that is, { ũ(x u(x,x N ) if 0 x N <,x N ) := u(x,x N ) if < x N < 0. Concerning J, by Lemma we have J = = ( u (s (s) ( ( (s) B r (0) ( ( ( = = ( µs (s) µ s (s) µ s s N s ) ( ũ (s (s) ũ dx +C ũ dx B r (0) B r (0) ( u dx +C u dx +C u dx ) ) u dx for some positive constant C depends on only B r (0). Next, we estimate J. Let δ > 0 for sufficiently small. We consider {ϕ i m i= a partition of unity on \ such that ϕ i C and suppϕ i δ for all i. Since x s r s for x \ we have J = ( r s (s) \ m i= u (s (s) ( m ϕ \B + i u (s) dx r (0) i= \ (s). ) ϕ i u (s) x s dx (s)

Minimization problems on the Hardy-Sobolev inequality By Hölder inequalities it follows that ( ϕ \B + i u (s) dx r (0) (s) suppϕ i (s) ϕ i u L (\) δ (s) ϕ i u L (\) for each i N. Since δ is sufficiently small, by using the Sobolev inequalities (If necessary we use the Sobolev inequalities of mixed boundary condition version.) we have Consequently we have ( ) µs J N s s m i= \) (ϕ i u) dx. ( ) ( ) µs J u dx +C N s s u dx. \ \ for some positive constant C depends on only \ B + r (). Combining the estimates of J and J we obtain ( u (s (s) ( ) ( ) µs x s J + J u dx +C u dx N s s for some positive constant C depends on. Acknowledgements The author would like to thank to Prof. Futoshi Takahashi for the suggestion of the problem in this paper and for helpful advices on this manuscript. References. T. Aubin, Problemes isoperimetriques et espaces de Sobolev. J. Differential Geometry (976), no. 4, 573-598.. H. Brezis, J. L. Vazquez, Blow-up solutions of some nonlinear elliptic problems. Rev. Mat. Univ. Complut. Madrid 0 (997), no., 443-469. 3. H. Brezis, E. Lieb, A relation between pointwise convergence of functions and convergence of functionals. Proc. Amer. Math. Soc. 88 (983), no. 3, 486-490. 4. H. Egnell, Positive solutions of semilinear equations in cones. Trans. Amer. Math. Soc. 330 (99), no., 9-0. 5. N. Ghoussoub, X. S. Kang, Hardy-Sobolev critical elliptic equations with boundary singularities. Ann. Inst. H. Poincare Anal. Non Lineaire (004), no. 6, 767-793. 6. N. Ghoussoub, F. Robert, Concentration estimates for Emden-Fowler equations with boundary singularities and critical growth. IMRP Int. Math. Res. Pap. 006, 867, -85. 7. N. Ghoussoub, F. Robert, Elliptic equations with critical growth and a large set of boundary singularities. Trans. Amer. Math. Soc. 36 (009), no. 9, 4843-4870. 8. N. Ghoussoub, F. Robert, The effect of curvature on the best constant in the Hardy-Sobolev inequalities. Geom. Funct. Anal. 6 (006), no. 6, 0-45. 9. N. Ghoussoub, C. Yuan, Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents. Trans. Amer. Math. Soc. 35 (000), no., 5703-5743. 0. C-H. Hsia, C-S. Lin, H. Wadade, Revisiting an idea of Brezis and Nirenberg. (English summary) J. Funct. Anal. 59 (00), no. 7, 86-849.. H. Jaber, Hardy-Sobolev equations on compact Riemannian manifolds. (English summary) Nonlinear Anal. 03 (04), 39-54.

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