Positive solutions for some singular critical growth nonlinear elliptic equations. (English) Zbl 1273.35124
Summary: Let \(\Omega\) be a bounded domain in \(\mathbb{R}^N\) (\(N\geq 4\)) with smooth boundary \(\partial \Omega\) and the origin \(0\in\overline{\Omega}\), \(\mu<((N-2)/2)^2\), \(2^*=2N/(N-2)\), \(K(x)\) is a smooth function on \(\Omega\) and positive somewhere. We obtain existence results of positive solutions to the Dirichlet problem
\[
-\Delta u = \mu\frac{u}{|x|^2} + K(x) |u|^{2^{*}-2} u + f(x,y) \text{ on }\Omega, \quad u=0 \text{ on } \partial\Omega
\]
for various \(K(x)\) and a suitable number \(\mu\).
MSC:
| 35J60 | Nonlinear elliptic equations |
| 35J25 | Boundary value problems for second-order elliptic equations |
| 35B33 | Critical exponents in context of PDEs |
Keywords:
positive solutions; compactness; critical Sobolev exponents; Hardy inequality; singular elliptic equationReferences:
| [1] | Brezis, H.; Lieb, E., A relation between pointwise convergence of functions and convergence of functionals, Proc. Am. Math. Soc., 88, 486-490 (1983) · Zbl 0526.46037 |
| [2] | Brezis, H.; Nirenberg, L., Positive solutions of nonlinear elliptic equations involving critical Sobolev exponent, Commun. Pure Appl. Math., 36, 437-477 (1983) · Zbl 0541.35029 |
| [3] | Caldiroli, P.; Musina, R., On the existence of extremal functions for weighted Sobolev embedding with critical exponent, Calc. Var. Partial A, 8, 365-387 (1999) · Zbl 0929.35045 |
| [4] | Caldiroli, P.; Musina, R., On a variational degenerate elliptic problem, Nonlinear Differential Equations Appl., 7, 187-199 (2000) · Zbl 0960.35039 |
| [5] | Cao, D.; Peng, S., A global compactness result for singular elliptic problems involving critical Sobolev exponent, Proc. Am. Math. Soc., 131, 1857-1866 (2003) · Zbl 1075.35009 |
| [6] | Cao, D.; Peng, S., A note on the sign-changing solutions to elliptic problems with critical Sobolev exponent and Hardy terms, J. Diff. Equats., 193, 424-434 (2003) · Zbl 1140.35412 |
| [7] | Cao, D.; Zhong, X., Multiplicity of positive solutions for Semilinear elliptic equations involving the critical Sobolev exponents, Nonlinear Anal. Theory Methods Appl., 29, 461-483 (1997) · Zbl 0879.35055 |
| [8] | Catrina, F.; Wang, Z. Q., On the Caffarelli-Kohn-Nirenberg inequalitiessharp constants, existence (and nonexistence), and symmetry of extremal functions, Commun. Pure Appl. Math., 54, 229-258 (2001) · Zbl 1072.35506 |
| [9] | Chou, K. S.; Chu, C. W., On the best constant for a weighted Sobolev-Hardy inequality, J. London Math. Soc., 48, 2, 137-151 (1993) · Zbl 0739.26013 |
| [10] | Ekeland, I.; Ghoussoub, N., Selected new aspects of the calculus of variations in the large, Bull. Am. Math. Soc., 39, 207-265 (2002) · Zbl 1064.35054 |
| [11] | Escobar, J. F., Positive solutions for some semilinear elliptic equations with critical Sobolev exponents, Commun. Pure Appl. Math., 40, 623-657 (1987) · Zbl 0635.35033 |
| [12] | Ferrero, A.; Gazzola, F., Existence of solutions for singular critical growth semilinear elliptic equations, J. Differential Equations, 177, 494-522 (2001) · Zbl 0997.35017 |
| [13] | Garcia Azorero, J. P.; Peral Alonso, I., Hardy inequalities and some critical elliptic and parabolic problems, J. Differential Equations, 144, 441-476 (1998) · Zbl 0918.35052 |
| [14] | Ghoussoub, N.; Yuan, C., Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents, Trans. Am. Math. Soc., 352, 5703-5743 (2000) · Zbl 0956.35056 |
| [15] | Han, Z., Asymptotic approach to singular solutions for nonlinear elliptic equations involving critical Sobolev exponent, Ann. Inst. H. Poincare. Anal. Non Linèaire, 8, 159-174 (1992) · Zbl 0729.35014 |
| [16] | Hebey, E., The asymptotic behavior of positive solutions of quasilinear elliptic equations with critical Sobolev growth, Differential Integral Equations, 13, 7-9, 1073-1080 (2000) · Zbl 0976.35022 |
| [17] | Jannelli, E., The role played by space dimension in elliptic critical problems, J. Differential Equations, 156, 407-426 (1999) · Zbl 0938.35058 |
| [18] | W.M. Ni, Recent progress in semilinear elliptic equations, Math. Rep. Minnesota 88-117.; W.M. Ni, Recent progress in semilinear elliptic equations, Math. Rep. Minnesota 88-117. |
| [19] | D. Smets, Nonlinear Schrödinger equations with Hardy potential and critical nonlinearities, Université Catholique de Louvain, Institut de Mathématique Pure et appliquèe, Chemin du Cyclotron 2, Belgium, preprint 2001.; D. Smets, Nonlinear Schrödinger equations with Hardy potential and critical nonlinearities, Université Catholique de Louvain, Institut de Mathématique Pure et appliquèe, Chemin du Cyclotron 2, Belgium, preprint 2001. |
| [20] | Tarantello, G., Nodal solutions of semilinear elliptic equations with critical exponent, Differential Integral Equations, 5, 25-42 (1992) · Zbl 0758.35035 |
| [21] | Terracini, S., On positive solutions to a class equations with a singular coefficient and critical exponent, Adv. Differential Equations, 2, 241-264 (1996) · Zbl 0847.35045 |
| [22] | Wang, Z. Q.; Willem, M., Singular minimization problems, J. Differential Equations, 161, 307-320 (2000) · Zbl 0954.35074 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
