Two solutions for a singular elliptic equation by variational methods

Montenegro, Marcelo; Silva, Elves A. B.

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 11 (2012) no. 1, pp. 143-165.

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 $ϵ$-problems with $1/u^{\beta }$ replaced by $u^q/{(u+ϵ)}^{q+\beta }$ with $0<q<p<1$. When the parameter $\lambda >0$ is large enough, we find two critical points of the corresponding $ϵ$-functional which, at the limit as $ϵ\to 0$, 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 $u_{ϵ}$ to a true solution $u$ of the problem.

Classification : 34B16, 35J20, 35B65

[1] A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis 14 (1973), 349–381. | MR | Zbl

[2] H. W. Alt and D. Phillips, A free boundary problem for semilinear elliptic equations, J. Reine Angew. Math. 368 (1986), 63–107. | EuDML | MR | Zbl

[3] H. Brezis and M. Marcus, Hardy’s inequalities revisited, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25 (1997), 217–237. | EuDML | Numdam | MR | Zbl

[4] A. Callegari and A. Nachman, Some singular nonlinear equations arising in boundary layer theory, J. Math. Anal. Appl. 64 (1978), 96–105. | MR | Zbl

[5] A. Canino, Minimax methods for singular elliptic equations with an application to a jumping problem, J. Differential Equations 221 (2006), 210–223. | MR

[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 | Zbl

[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 | Zbl

[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. | EuDML | Numdam | MR | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[13] J. Dávila and M. Montenegro, Positive versus free boundary solutions to a singular elliptic equation, J. Anal. Math. 90 (2003), 303–335. | MR | Zbl

[14] J. Dávila and M. Montenegro, Existence and asymptotic behavior for a singular parabolic equation, Trans. Amer. Math. Soc. 357 (2005), 1801–1828. | MR | Zbl

[15] J. Dávila and M. Montenegro, Radial solutions of an elliptic equation with singular nonlinearity, J. Math. Anal. Appl. 352 (2009), 360–379. | MR | Zbl

[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 | Zbl

[17] W. Fulks and J. S. Maybee, A singular nonlinear equation, Osaka Math. J. 12 (1960), 1–19. | MR | Zbl

[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

[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 | Zbl

[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 | Zbl

[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 | Zbl

[23] K. Perera and E. A. B. Silva, On singular $p$-Laplacian problems, Differential Integral Equations 20 (2007), 105–120. | MR | Zbl

[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 | Zbl

[25] D. Phillips, Hausdorff measure estimates of a free boundary for a minimum problem, Comm. Partial Differential Equations 8 (1983), 1409–1454. | MR | Zbl

[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 | Zbl

[27] J. Shi and M. Yao, On a singular nonlinear semilinear elliptic problem, Proc. Roy. Soc. Edinburgh Sect. A 128 (1998), 1389–1401. | MR | Zbl