An Elliptic Problem with a Lower Order Term Having Singular Behaviour

Giachetti, Daniela ; Murat, François

Bollettino dell'Unione Matematica Italiana, Tome 2 (2009), p. 349-370 / Harvested from Biblioteca Digitale Italiana di Matematica

Résumé

We prove the existence of distributional solutions to an elliptic problem with a lower order term which depends on the solution $u$ in a singular way and on its gradient $Du$ with quadratic growth. The prototype of the problem under consideration is $$\begin{cases} - \Delta u + \lambda u = \pm \frac{|Du|^{2}}{|u|^{k}} + f \quad & \text{in} \, \Omega, \\ u=0 & \text{on} \, \partial \Omega, \end{cases}$$ where $\lambda > 0$, $k > 0$; $f(x) \in L^{\infty}(\Omega)$, $f(x) \ge 0$ (and so $u \ge 0$). If $0 < k < 1$, we prove the existence of a solution for both the "+" and the "-" signs, while if $k \ge 1$, we prove the existence of a solution for the "+" sign only.

Détail

Publié le : 2009-06-01

MR 2537275 | Zbl 1173.35469

Bibliographie

[1] Arcoya, D. - Barile, S. - Martínez-Aparicio, P. J., Singular quasilinear equations with quadratic growth in the gradient without sign condition, J. Math. Anal. Appl., 350 (2009), 401-408. | MR 2476925 | Zbl 1161.35013

[2] Arcoya, D. - Carmona, J. - Leonori, T. - Martínez-Aparicio, P. J. - Orsina, L. - Petitta, F., Quadratic quasilinear equations with general singularities, J. Differential Equations (2009).

[3] Arcoya, D. - Carmona, J. - Martínez-Aparicio, P. J., Elliptic obstacle problems with natural growth on the gradient and singular nonlinear terms, Adv. Nonlinear Stud., 7 (2007), 299-317. | MR 2308041 | Zbl 1189.35136

[4] Arcoya, D. - Martínez-Aparicio, P. J., Quasilinear equations with natural growth, Rev. Mat. Iberoamericana, 24 (2008), 597-616. | MR 2459205 | Zbl 1151.35343

[5] Boccardo, L., Dirichlet problems with singular and gradient quadratic lower order terms, ESAIM - Control, Optimisation and Calculus of Variations, 14 (2008), 411-426. | MR 2434059 | Zbl 1147.35034

[6] Boccardo, L. - Murat, F. - Puel, J.-P., Existence de solutions faibles pour des équations elliptiques quasi-linéaires à croissance quadratique, in Nonlinear Partial Differential Equations and their Applications, Collège de France Seminar, Vol. IV, ed. by H. Brezis & J.-L. Lions, Res. Notes in Math., 84 (1983), Pitman, Boston, 19-73. | MR 716511

[7] Boccardo, L. - Murat, F. - Puel, J.-P., Résultats d'existence pour certains problèmes elliptiques quasilinéaires, Ann. Scuola Norm. Sup. Pisa, 11 (1984), 213-235. | MR 764943 | Zbl 0557.35051

[8] Boccardo, L. - Murat, F. - Puel, J.-P., Existence of bounded solutions for nonlinear elliptic unilateral problems, Ann. Mat. Pura Appl., 152 (1988), 183-196. | MR 980979 | Zbl 0687.35042

[9] Boccardo, L. - Murat, F. - Puel, J.-P., $L^{\infty}$-estimate for some nonlinear elliptic partial differential equations and application to an existence result, SIAM J. Math. Anal., 23 (1992), 326-333. | MR 1147866 | Zbl 0785.35033

[10] Dall'Aglio, A. - Giachetti, D. - Puel, J.-P., Nonlinear elliptic equations with natural growth in general domains, Ann. Mat. Pura Appl., 181 (2002), 407-426. | MR 1939689 | Zbl 1097.35050

[11] Ferone, V. - Murat, F., Quasilinear problems having quadratic growth in the gradient: an existence result when the source term is small, in Equations aux dérivées partielles et applications, articles dédiés à Jacques-Louis Lions, (1998), Gauthier-Villars, Paris, 497-515. | MR 1648236 | Zbl 0917.35039

[12] Ferone, V. - Murat, F., Nonlinear problems having natural growth in the gradient: an existence result when the source terms are small, Nonlinear Anal. Th. Meth. Appl. Series A, 42 (2000), 1309-1326. | MR 1780731 | Zbl 1158.35358

[13] Ferone, V. - Murat, F., Nonlinear elliptic equations with natural growth in the gradient and source terms in Lorentz spaces, to appear. | MR 3121707 | Zbl 1318.35031

[14] Giachetti, D. - Maroscia, G., Porous medium type equations with a quadratic gradient term, Boll. U.M.I. sez. B, 10 (2007), 753-759. | MR 2351544 | Zbl 1177.35124

[15] Giachetti, D. - Maroscia, G., Existence results for a class of porous medium type equations with a quadratic gradient term, J. Evol. Eq., 8 (2008), 155-188. | MR 2383486 | Zbl 1142.35043

[16] Martínez-Aparicio, P. J., Singular quasilinear equations with quadratic gradient, to appear.

[17] Porretta, A. - Segura De León, S., Nonlinear elliptic equations having a gradient term with natural growth, J. Math. Pures Appl., 85 (2006), 465-492. | MR 2210085 | Zbl 1158.35364