×

zbMATH — the first resource for mathematics

The existence of positive solution for semilinear elliptic equations with multiple an inverse square potential and Hardy-Sobolev critical exponents. (English) Zbl 07054491
Summary: Via the concentration compactness principle, delicate energy estimates, the strong maximum principle, and the Mountain Pass lemma, the existence of positive solutions for a nonlinear PDE with multi-singular inverse square potentials and critical Sobolev-Hardy exponent is proved. This result extends several recent results on the topic.
MSC:
35 Partial differential equations
58 Global analysis, analysis on manifolds
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Abdellaoui, B.; Felli, V.; Peral, I., Existence and nonexistence results for quasilinear elliptic equations involving the p-Laplacian, Bollettino della Unione Matematica Italiana-B, 9, 2, 445-484, (2006) · Zbl 1118.35010
[2] Cao, Y.; Kang, D., Solutions of a quasilinear elliptic problem involving a critical Sobolev exponent and multiple Hardy-type terms, Journal of Mathematical Analysis and Applications, 333, 2, 889-903, (2007) · Zbl 1154.35034
[3] Cao, D.; Han, P., Solutions to critical elliptic equations with multi-singular inverse square potentials, Journal of Differential Equations, 224, 2, 332-372, (2006) · Zbl 1198.35086
[4] Ding, L.; Tang, C.-L., Existence and multiplicity of solutions for semilinear elliptic equations with hardy terms and hardy–sobolev critical exponents, Applied Mathematics Letters, 20, 12, 1175-1183, (2007) · Zbl 1137.35360
[5] Felli, V.; Terracini, S., Elliptic equations with multi-singular inverse-square potentials and critical nonlinearity, Communications in Partial Differential Equations, 31, 3, 469-495, (2006) · Zbl 1206.35104
[6] Han, P., Quasilinear elliptic problems with critical exponents and Hardy terms, Nonlinear Analysis: Theory, Methods & Applications, 61, 5, 735-758, (2005) · Zbl 1210.35021
[7] Huang, L.; Wu, X.-P.; Tang, C.-L., Existence and multiplicity of solutions for semilinear elliptic equations with critical weighted hardy–sobolev exponents, Nonlinear Analysis: Theory, Methods & Applications, 71, 5-6, 1916-1924, (2009) · Zbl 1170.35407
[8] Gao, W.; Peng, S., An elliptic equation with combined critical sobolev–hardy terms, Nonlinear Analysis: Theory, Methods & Applications, 65, 8, 1595-1612, (2006) · Zbl 1233.35092
[9] Ambrosetti, A.; Rabinowitz, P. H., Dual variational methods in critical point theory and applications, Journal of Functional Analysis, 14, 349-381, (1973) · Zbl 0273.49063
[10] Caffarelli, L.; Kohn, R.; Nirenberg, L., First order interpolation inequalities with weights, Compositio Mathematica, 53, 3, 259-275, (1984) · Zbl 0563.46024
[11] Azorero, J. P. G.; Alonso, I. P., Hardy inequalities and some critical elliptic and parabolic problems, Journal of Differential Equations, 144, 2, 441-476, (1998) · Zbl 0918.35052
[12] Ghoussoub, N.; Yuan, C., Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents, Transactions of the American Mathematical Society, 352, 12, 5703-5743, (2000) · Zbl 0956.35056
[13] Kang, D.; Peng, S., Positive solutions for singular critical elliptic problems, Applied Mathematics Letters, 17, 4, 411-416, (2004) · Zbl 1133.35358
[14] Chen, J., Existence of solutions for a nonlinear pde with an inverse square potential, Journal of Differential Equations, 195, 2, 497-519, (2003) · Zbl 1039.35035
[15] Lions, P. L., The concentration-compactness principle in the calculus of variations.(the limit case, part i.), Revista Matemática Iberoamericana, 1, 1, 145-201, (1985) · Zbl 0704.49005
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.