Hardy-Sobolev inequalities with distance to the boundary weight functions. (English) Zbl 1526.35188
Summary: In this paper we shall establish some sharp weighted Hardy-Sobolev inequalities whose weights are distance functions to the boundary.
MSC:
| 35J91 | Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian |
| 35J25 | Boundary value problems for second-order elliptic equations |
| 35A23 | Inequalities applied to PDEs involving derivatives, differential and integral operators, or integrals |
| 26D10 | Inequalities involving derivatives and differential and integral operators |
References:
| [1] | Aubin, T., Espaces de Sobolev sur les variétés Riemanniennes. Bull. Sci. Math., 2, 149-173 (1976) · Zbl 0328.46030 |
| [2] | Aubin, T., Problèmes isopérimétriques et espaces de Sobolev. J. Differ. Geom., 4, 573-598 (1976) · Zbl 0371.46011 |
| [3] | Badiale, Ma.; Tarantello, G., A Sobolev-Hardy inequality with applications to a nonlinear elliptic equation arising in astrophysics. Arch. Ration. Mech. Anal., 4, 259-293 (2002) · Zbl 1010.35041 |
| [4] | Bliss, G. A., An integral inequality. J. Lond. Math. Soc., 1, 40-46 (1930) · JFM 56.0434.02 |
| [5] | Brezis, H.; Lieb, E., A relation between pointwise convergence of functions and convergence of functionals. Proc. Am. Math. Soc., 3, 486-490 (1983) · Zbl 0526.46037 |
| [6] | Caffarelli, L.; Gidas, B.; Spruck, J., Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth. Commun. Pure Appl. Math., 3, 271-297 (1989) · Zbl 0702.35085 |
| [7] | Castro, H., Extremals for Hardy-Sobolev type inequalities with monomial weights. J. Math. Anal. Appl., 2 (2021) · Zbl 1469.46028 |
| [8] | Chen, S.; Li, S., Hardy-Sobolev inequalities in half-space and some semilinear elliptic equations with singular coefficients. Nonlinear Anal., 2, 324-348 (2007) · Zbl 1387.35209 |
| [9] | Cheng, X.; Wei, L.; Zhang, Y., Estimate, existence and nonexistence of positive solutions of Hardy-Hénon equations. Proc. R. Soc. Edinb., Sect. A, 2, 518-541 (2022) · Zbl 1489.35135 |
| [10] | Chou, K. S.; Chu, C. W., On the best constant for a weighted Sobolev-Hardy inequality. J. Lond. Math. Soc., 1, 137-151 (1993) · Zbl 0739.26013 |
| [11] | Das, U.; Pinchover, Y.; Devyver, B., On existence of minimizers for weighted \(L^p\)-Hardy inequalities on \(C^{1 , \gamma}\)-domains with compact boundary |
| [12] | Davies, E. B., The Hardy constant. Q. J. Math. Oxf. (2), 417-431 (1995) · Zbl 0857.26005 |
| [13] | Dou, J.; Sun, L.; Wang, L.; Zhu, M., Divergent operator with degeneracy and related sharp inequalities. J. Funct. Anal., 2, 109-294 (2022) |
| [14] | Gagliardo, E., Ulteriori proprietà di alcune classi di funzioni in più variabili. Ric. Mat., 24-51 (1959) · Zbl 0199.44701 |
| [15] | Gazzini, M.; Musina, R., On a Sobolev-type inequality related to the weighted p-Laplace operator. J. Math. Anal. Appl., 99-111 (2009) · Zbl 1173.46015 |
| [16] | Ghoussoub, N.; Robert, F., The effect of curvature on the best constant in the Hardy-Sobolev inequalities. Geom. Funct. Anal., 6, 1201-1245 (2006) · Zbl 1232.35044 |
| [17] | Ghoussoub, N.; Yuan, C., Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents. Trans. Am. Math. Soc., 12, 5703-5743 (2000) · Zbl 0956.35056 |
| [18] | Gidas, B.; Ni, W.; Nirenberg, L., Symmetry and related properties via the maximum principle. Commun. Math. Phys., 3, 209-243 (1979) · Zbl 0425.35020 |
| [19] | Hardy, G. H.; Littlewood, J. E., Notes on the theory of series (XII): on certain inequalities connected with calculus of variations. J. Lond. Math. Soc., 34-39 (1930) · JFM 56.0434.01 |
| [20] | Hardy, G. H.; Littlewood, J. E.; Polya, G., Inequalities (1934), Cambridge University Press: Cambridge University Press Cambridge · JFM 60.0169.01 |
| [21] | Hebey, E.; Vaugon, M., Meilleures constantes dans le thréorème d’inclusion de Sobolev. Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 57-93 (1996) · Zbl 0849.53035 |
| [22] | Lamberti, P. D.; Pinchover, Y., \( L^p\) Hardy inequality on \(C^{1 , \gamma}\) domains. Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5), 3, 1135-1159 (2019) · Zbl 1429.35012 |
| [23] | Lee, J. M.; Parker, T. H., The Yamabe problem. Bull. Am. Math. Soc. (N.S.), 1, 37-91 (1987) · Zbl 0633.53062 |
| [24] | Lewis, R.; Li, J.; Li, Y. Y., A geometric characterization of a sharp Hardy inequality. J. Funct. Anal., 7, 3159-3185 (2012) · Zbl 1243.26008 |
| [25] | Li, Y. Y.; Zhu, M., Uniqueness theorems through the method of moving spheres. Duke Math. J., 383-417 (1995) · Zbl 0846.35050 |
| [26] | Li, Y. Y.; Zhu, M., Sharp Sobolev trace inequality on Riemannian manifolds with boundary. Commun. Pure Appl. Math., 449-487 (1997) · Zbl 0869.58054 |
| [27] | Lieb, E. H., Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities. Ann. Math., 349-374 (1983) · Zbl 0527.42011 |
| [28] | Lions, P. L., The concentration-compactness principle in the calculus of variations. The limit case I. Rev. Mat. Iberoam., 145-201 (1985) · Zbl 0704.49005 |
| [29] | Lions, P. L., The concentration-compactness principle in the calculus of variations. The limit case II. Rev. Mat. Iberoam., 45-121 (1985) · Zbl 0704.49006 |
| [30] | Marcus, M.; Mizel, V.; Pinchover, Y., On the best constant for Hardy’s inequality in \(R^n\). Trans. Am. Math. Soc., 8, 3237-3255 (1998) · Zbl 0917.26016 |
| [31] | Matskewich, T.; Sobolevskii, P. E., The best possible constant in generalized Hardy’s inequality for convex domain in \(\mathbb{R}^n\). Nonlinear Anal., 1601-1610 (1997) · Zbl 0876.46025 |
| [32] | Maz’ya, V., Sobolev Spaces (2011), Springer |
| [33] | Nirenberg, L., On elliptic partial differential equations. Ann. Sc. Norm. Super. Pisa, Cl. Sci. (3), 115-162 (1959) · Zbl 0088.07601 |
| [34] | Opic, P.; Kufner, A., Hardy-Type Inequalities. Pitman Research Notes in Mathematics Series (1990), Longman Scientific & Technical: Longman Scientific & Technical Harlow · Zbl 0698.26007 |
| [35] | Rosen, G., Minimum value for c in the Sobolev inequality \(\| \phi^3 \| \leq c \| \operatorname{\nabla} \phi \|^3\). SIAM J. Appl. Math., 30-32 (1971) · Zbl 0201.38704 |
| [36] | Talenti, G., Best constant in Sobolev inequality. Ann. Mat. Pura Appl., 353-372 (1976) · Zbl 0353.46018 |
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