×

Brezis-Nirenberg type theorems and multiplicity of positive solutions for a singular elliptic problem. (English) Zbl 1158.35044

The authors study the problem
\[ \begin{cases}\Delta u=u^{-q}+\lambda u^{p},&\text{in }\Omega,\\ u=0, &\text{on }\partial\Omega,\end{cases} \]
where \(\Omega\) is a bounded domain in \(\mathbb R^{N}\), \(q>0\), \(p>1\), and \(\lambda>0\), and obtain existence and multiplicity of positive solution.
Reviewer: Jiaqi Mo (Wuhu)

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
35J20 Variational methods for second-order elliptic equations
35J60 Nonlinear elliptic equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Armitage, D. H.; Gardiner, S. J., Classical Potential Theory (2001), Springer-Verlag: Springer-Verlag London · Zbl 0972.31001
[2] Ambrosetti, A.; Brezis, H.; Cerami, G., Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal., 122, 519-543 (1994) · Zbl 0805.35028
[3] Brezis, H.; Nirenberg, L., Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36, 437-477 (1983) · Zbl 0541.35029
[4] Brezis, H.; Nirenberg, L., \(H^1\) versus \(C^1\) local minimizers, C. R. Acad. Sci. Paris Sér. I Math., 317, 465-472 (1993) · Zbl 0803.35029
[5] Canino, A.; Degiovanni, M., Nonsmooth critical point theory and quasilinear elliptic equations, (Topological Methods in Differential Equations and Inclusions. Topological Methods in Differential Equations and Inclusions, Montreal, PQ, 1994. Topological Methods in Differential Equations and Inclusions. Topological Methods in Differential Equations and Inclusions, Montreal, PQ, 1994, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 472 (1995), Kluwer Acad. Publ.: Kluwer Acad. Publ. Dordrecht), 1-50 · Zbl 0851.35038
[6] Canino, A.; Degiovanni, M., A variational approach to a class of singular semilinear elliptic equations, J. Convex Anal., 11, 147-162 (2004) · Zbl 1073.35092
[7] Coclite, M. M.; Palmieri, G., On a singular nonlinear Dirichlet problem, Comm. Partial Differential Equations, 14, 1315-1327 (1989) · Zbl 0692.35047
[8] Coclite, M. M., On a singular nonlinear Dirichlet problem, II, Boll. Unione Mat. Ital. Sez. B (7), 5, 955-975 (1991) · Zbl 0789.35057
[9] Corvellec, J.-N.; Degiovanni, M.; Marzocchi, M., Deformation properties for continuous functionals and critical point theory, Topol. Methods Nonlinear Anal., 1, 151-171 (1993) · Zbl 0789.58021
[10] Corvellec, J.-N., Quantitative deformation theorems and critical point theory, Pacific J. Math., 187, 263-279 (1999) · Zbl 0939.58016
[11] Degiovanni, M.; Marzocchi, M., A critical point theory for nonsmooth functionals, Ann. Mat. Pura Appl. (4), 167, 73-100 (1994) · Zbl 0828.58006
[12] Crandall, M. G.; Rabinowitz, P. H.; Tartar, L., On a Dirichlet problem with a singular nonlinearity, Comm. Partial Differential Equations, 2, 193-222 (1977) · Zbl 0362.35031
[13] Gilbarg, D.; Trudinger, N. S., Elliptic Partial Differential Equations of Second Order (1983), Springer-Verlag: Springer-Verlag Berlin · Zbl 0691.35001
[14] Haitao, Y., Multiplicity and asymptotic behavior of positive solutions for a singular semilinear elliptic problem, J. Differential Equations, 189, 487-512 (2003) · Zbl 1034.35038
[15] Helms, L. L., Introduction to Potential Theory, Pure Appl. Math., vol. 22 (1969), Wiley-Interscience: Wiley-Interscience New York · Zbl 0188.17203
[16] Hirano, N.; Saccon, C.; Shioji, N., Existence of multiple positive solutions for singular elliptic problems with concave and convex nonlinearities, Adv. Differential Equations, 9, 197-220 (2004) · Zbl 1387.35287
[17] Jabri, Y., The Mountain Pass Theorem, Encyclopedia Math. Appl., vol. 95 (2003), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 1036.49001
[18] Katriel, G., Mountain pass theorems and global homeomorphism theorems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 11, 189-209 (1994) · Zbl 0834.58007
[19] Lazer, A. C.; McKenna, P. J., On a singular nonlinear elliptic boundary-value problem, Proc. Amer. Math. Soc., 111, 721-730 (1991) · Zbl 0727.35057
[20] Malý, J.; Ziemer, W. P., Fine Regularity of Solutions of Elliptic Partial Differential Equations, Math. Surveys Monogr., vol. 51 (1997), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI · Zbl 0882.35001
[21] Marino, A.; Passaseo, D., A jumping behaviour induced by an obstacle, (Progress in Variational Methods in Hamiltonian Systems and Elliptic Equations. Progress in Variational Methods in Hamiltonian Systems and Elliptic Equations, L’Aquila, 1990. Progress in Variational Methods in Hamiltonian Systems and Elliptic Equations. Progress in Variational Methods in Hamiltonian Systems and Elliptic Equations, L’Aquila, 1990, Pitman Res. Notes Math. Ser., vol. 243 (1992), Longman Sci. Tech.: Longman Sci. Tech. Harlow), 127-143 · Zbl 0818.35034
[22] Passaseo, D., Multiplicity of solutions for certain variational inequalities of elliptic type, Boll. Unione Mat. Ital. Sez. B (7), 3, 639-667 (1989), (in Italian) · Zbl 0677.49008
[23] Passaseo, D., Multiplicity of solutions of elliptic nonlinear variational inequalities, Rend. Accad. Naz. Sci. XL Mem. Mat. (5), 15, 19-56 (1991), (in Italian) · Zbl 0829.49009
[24] Sun, Y.; Wu, S.; Long, Y., Combined effects of singular and superlinear nonlinearities in some singular boundary value problems, J. Differential Equations, 176, 511-531 (2001) · Zbl 1109.35344
[25] Struwe, M., Variational Methods (1996), Springer-Verlag: Springer-Verlag Berlin
[26] Stuart, C. A., Existence and approximation of solutions of non-linear elliptic equations, Math. Z., 147, 53-63 (1976) · Zbl 0324.35037
[27] Szulkin, A., Minimax principles for lower semicontinuous functions and applications to nonlinear boundary value problems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 3, 77-109 (1986) · Zbl 0612.58011
[28] Willem, M., Minimax Theorems, Progr. Nonlinear Differential Equations Appl., vol. 24 (1996), Birkhäuser: Birkhäuser Boston · Zbl 0856.49001
[29] Zhang, Z.; Yu, J., On a singular nonlinear Dirichlet problem with a convection term, SIAM J. Math. Anal., 32, 916-927 (2000) · Zbl 0988.35059
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.