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Hardy-Sobolev critical elliptic equations with boundary singularities. (English) Zbl 1232.35064

Summary: Unlike the non-singular case \(s=0\), or the case when \(0\) belongs to the interior of a domain \(\Omega\) in \(\mathbb R^n\) \((n\geq 3)\), we show that the value and the attainability of the best Hardy-Sobolev constant on a smooth domain \(\Omega\),
\[ \mu_s(\Omega):=\inf\left\{\int_{\Omega}|\nabla u|^2\,dx; u\in H^1_0(\Omega) \text{ and }\;\int_\Omega\frac{|u|^{2^*(s)}}{|x|^s}=1\right\}, \]
when \(0<s<2\), \(2^{*}(s):= \frac{2(n-s)}{(n-2)}\), and when 0 is on the boundary \(\partial\Omega\) are closely related to the properties of the curvature of \(\partial\Omega\) at \(0\). These conditions on the curvature are also relevant to the study of elliptic partial differential equations with singular potentials of the form:
\[ -\Delta u=\frac{|u|^{2^{*}(s)-2} u}{|x|^s} +f(x,u)\quad \text{ in } \Omega, \]
where \(f\) is a lower order perturbative term at infinity and \(f(x,0)=0\). We show that the positivity of the sectional curvature at \(0\) is relevant when dealing with Dirichlet boundary conditions, while the Neumann problems seem to require the positivity of the mean curvature at \(0\).

MSC:

35J61 Semilinear elliptic equations
35B33 Critical exponents in context of PDEs
35J20 Variational methods for second-order elliptic equations
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References:

[1] Adimurthi; Mancini, G.; Ambhrosetti; etal., The Neumann problem for elliptic equations with critical nonlinearity, Nonlinear Analysis, a tribute in honor of G. Prodi, Scoula norm. sup. Pisa, 9-25, (1991) · Zbl 0836.35048
[2] Brezis, H.; Lieb, E., A relation between point convergence of functions and convergence of functionals, Proc. amer. math. soc, 88, 486-490, (1983) · Zbl 0526.46037
[3] Brezis, H.; Nirenberg, L., Positive solutions of nonlinear elliptic equations involving critical exponents, Comm. pure appl. math, 36, 437-477, (1983) · Zbl 0541.35029
[4] Caffarelli, L.; Kohn, R.; Nirenberg, L., First order interpolation inequality with weights, Compositio math, 53, 259-275, (1984) · Zbl 0563.46024
[5] Catrina, F.; Wang, Z., On the caffarelli – kohn – nirenberg inequalities: sharp constants, existence (and nonexistence) and symmetry of extremal functions, Comm. pure appl. math, 2, 229-258, (2001) · Zbl 1072.35506
[6] Comte, M.; Tarantello, G., A Neumann problem with critical Sobolev exponent, Houston J. math, 18, 279-294, (1992) · Zbl 0754.35039
[7] Egnell, H., Positive solutions of semilinear equations in cones, Trans. amer. math. soc, 11, 191-201, (1992) · Zbl 0766.35014
[8] Ekeland, I.; Ghoussoub, N., Selected new aspects of the calculus of variations in the large, Bull. amer. math. soc. (N.S.), 39, 2, 207-265, (2002) · Zbl 1064.35054
[9] Gallot, S.; Hulin, D.; Lafontaine, J., Riemannian geometry, (1987), Springer-Verlag · Zbl 0636.53001
[10] Ghoussoub, N., Duality and perturbation methods in critical point theory, (1993), Cambridge Univ. Press · Zbl 0790.58002
[11] Ghoussoub, N.; Yuan, C., Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents, Trans. amer. math. soc, 12, 5703-5743, (2000) · Zbl 0956.35056
[12] Gilbarg, D.; Trudinger, N., Elliptic partial differential equations of second order, (1983), Springer Berlin · Zbl 0562.35001
[13] E. Jannelli, S. Solimin, Critical behaviour of some elliptic equations with singular potentials, preprint, 1996
[14] Maz’ja, V.G., Sobolev spaces, (1985), Springer-Verlag
[15] Ni, W.-M.; Takagi, I., On the shape of least-energy solutions to a semilinear Neumann problem, Comm. pure appl. math, 44, 819-851, (1991) · Zbl 0754.35042
[16] Pan, X.; Wang, X., Semiliner Neumann problem in exterior domains, Nonlinear anal, 31, 791-821, (1998) · Zbl 0910.35058
[17] Struwe, M., Variational methods, (1990), Springer-Verlag Berlin
[18] Tarantello, G., Nodal solutions of semilinear elliptic equations with critical exponent, Differential integral equations, 5, 25-42, (1992) · Zbl 0758.35035
[19] Wang, X.-J., Neumann problems of semilinear elliptic equations involving critical Sobolev exponents, J. differential equations, 93, 671-684, (1991)
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