We find two nontrivial solutions of the equation $-\Delta u=(-\frac{1}{{u}^{\beta}}+\lambda {u}^{p}){\chi}_{\{u>0\}}$ in $\Omega $ with Dirichlet boundary condition, where $0<\beta <1$ and $0<p<1$. In the first approach we consider a sequence of $\u03f5$-problems with $1/{u}^{\beta}$ replaced by ${u}^{q}/{(u+\u03f5)}^{q+\beta}$ with $0<q<p<1$. When the parameter $\lambda >0$ is large enough, we find two critical points of the corresponding $\u03f5$-functional which, at the limit as $\u03f5\to 0$, give rise to two distinct nonnegative solutions of the original problem. Another approach is based on perturbations of the domain $\Omega $, we then find a unique positive solution for $\lambda $ large enough. We derive gradient estimates to guarantee convergence of approximate solutions ${u}_{\u03f5}$ to a true solution $u$ of the problem.
@article{ASNSP_2012_5_11_1_143_0, author = {Montenegro, Marcelo and Silva, Elves A. B.}, title = {Two solutions for a singular elliptic equation~by variational methods}, journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze}, publisher = {Scuola Normale Superiore, Pisa}, volume = {Ser. 5, 11}, number = {1}, year = {2012}, pages = {143-165}, zbl = {1241.35103}, mrnumber = {2953046}, language = {en}, url = {http://www.numdam.org/item/ASNSP_2012_5_11_1_143_0} }
Montenegro, Marcelo; Silva, Elves A. B. Two solutions for a singular elliptic equation by variational methods. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 11 (2012) no. 1, pp. 143-165. http://www.numdam.org/item/ASNSP_2012_5_11_1_143_0/
[1] A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis 14 (1973), 349–381. | MR 370183 | Zbl 0273.49063
[2] H. W. Alt and D. Phillips, A free boundary problem for semilinear elliptic equations, J. Reine Angew. Math. 368 (1986), 63–107. | MR 850615 | Zbl 0598.35132
[3] H. Brezis and M. Marcus, Hardy’s inequalities revisited, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25 (1997), 217–237. | Numdam | MR 1655516 | Zbl 1011.46027
[4] A. Callegari and A. Nachman, Some singular nonlinear equations arising in boundary layer theory, J. Math. Anal. Appl. 64 (1978), 96–105. | MR 478973 | Zbl 0386.34026
[5] A. Canino, Minimax methods for singular elliptic equations with an application to a jumping problem, J. Differential Equations 221 (2006), 210–223. | MR 2193848
[6] A. Canino and M. Degiovanni, A variational approach to a class of singular semilinear elliptic equations, J. Convex Anal. 11 (2004), 147–162. | MR 2159469 | Zbl 1073.35092
[7] Y. S. Choi, A. C. Lazer and P. J. McKenna, Some remarks on a singular elliptic boundary value problem, Nonlinear Anal. 32 (1998), 305–314. | MR 1610645 | Zbl 0940.35089
[8] Y. S. Choi and P. J. McKenna, A singular Gierer-Meinhardt system of elliptic equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 17 (2000), 503–522. | Numdam | MR 1782742 | Zbl 0969.35062
[9] Y. S. Choi and P. J. McKenna, A singular Gierer-Meinhardt system of elliptic equations: the classical case, Nonlinear Anal. 55 (2003), 521–541. | MR 2012446 | Zbl 1140.35408
[10] F. Cîrstea, M. Ghergu and V. Radulescu, Combined effects of asymptotically linear and singular nonlinearities in bifurcation problem of Lane-Emden-Fowler type, J. Math. Pures Appl. 84 (2005), 493–508. | MR 2133126 | Zbl 1211.35111
[11] M. G. Crandall, P. H. Rabinowitz and L. Tartar, On a Dirichlet problem with a singular nonlinearity, Comm. Partial Differential Equations 2 (1977), 193–222. | MR 427826 | Zbl 0362.35031
[12] J. Dávila, Global regularity for a singular equation and local ${H}^{1}$ minimizers of a nondifferentiable functional, Commun. Contemp. Math. 6 (2004), 165–193. | MR 2048779 | Zbl 1048.35022
[13] J. Dávila and M. Montenegro, Positive versus free boundary solutions to a singular elliptic equation, J. Anal. Math. 90 (2003), 303–335. | MR 2001074 | Zbl 1173.35743
[14] J. Dávila and M. Montenegro, Existence and asymptotic behavior for a singular parabolic equation, Trans. Amer. Math. Soc. 357 (2005), 1801–1828. | MR 2115077 | Zbl 1074.35020
[15] J. Dávila and M. Montenegro, Radial solutions of an elliptic equation with singular nonlinearity, J. Math. Anal. Appl. 352 (2009), 360–379. | MR 2499908 | Zbl 1163.35016
[16] J. I. Diaz, J. M. Morel and L. Oswald, An elliptic equation with singular nonlinearity, Comm. Partial Differential Equations 12 (1987), 1333–1344. | MR 912208 | Zbl 0634.35031
[17] W. Fulks and J. S. Maybee, A singular nonlinear equation, Osaka Math. J. 12 (1960), 1–19. | MR 123095 | Zbl 0097.30202
[18] N. Hirano, C. Saccon and N. Shioji, Existence of multiple positive solutions for singular elliptic problems with concave and convex nonlinearities, Adv. Differential Equations 9 (2004), 197–220. | MR 2099611
[19] Y. M. Long, Y. J. Sun and S. P. Wu, Combined effects to singular and superlinear nonlinearities in some singular boundary value problems, J. Differential Equations 176 (2003), 511–531. | MR 1866285 | Zbl 1109.35344
[20] A. Nachman and A. Callegari, A nonlinear singular boundary value problem in the theory of pseudoplastic fluids, SIAM J. Appl. Math. 38 (1980), 275–281. | MR 564014 | Zbl 0453.76002
[21] T. Ouyang, J. Shi and M. Yao, Exact multiplicity and bifurcation of solutions of a singular equation, preprint.
[22] K. Perera and E. A. B. Silva, Existence and multiplicity of positive solutions for singular quasilinear problems, J. Math. Anal. Appl. 323 (2006), 1238–1252. | MR 2260177 | Zbl 1168.35358
[23] K. Perera and E. A. B. Silva, On singular $p$-Laplacian problems, Differential Integral Equations 20 (2007), 105–120. | MR 2282829 | Zbl 1212.34048
[24] D. Phillips, A minimization problem and the regularity of solutions in the presence of a free boundary, Indiana Univ. Math. J. 32 (1983), 1–17. | MR 684751 | Zbl 0545.35013
[25] D. Phillips, Hausdorff measure estimates of a free boundary for a minimum problem, Comm. Partial Differential Equations 8 (1983), 1409–1454. | MR 714047 | Zbl 0555.35128
[26] P. H. Rabinowitz, “Minimax Methods in Critical Point Theory with Applications to Differential Equations”, CBMS Regional Conference Series Math., Vol. 65, Amer. Math. Soc., Providence, 1986. | MR 845785 | Zbl 0609.58002
[27] J. Shi and M. Yao, On a singular nonlinear semilinear elliptic problem, Proc. Roy. Soc. Edinburgh Sect. A 128 (1998), 1389–1401. | MR 1663988 | Zbl 0919.35044