Positive solutions for some quasilinear elliptic equations with natural growths

Boccardo, Lucio

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni, Tome 11 (2000), p. 31-39 / Harvested from Biblioteca Digitale Italiana di Matematica

Abstract

We shall prove an existence result for a class of quasilinear elliptic equations with natural growth. The model problem is $$ \begin{cases} - \text{div} ((1+ |u|^{r}) \nabla u) + |u|^{m-2} u |\nabla u|^{2} = f \quad &\text{in} \, \Omega \\ u = 0 &\text{su} \, \partial\Omega. \end{cases} $$

Résumé

È provato un teorema di esistenza di soluzioni per una classe di equazioni ellittiche quasi-lineari, con coefficienti a crescite naturali (come suggerito dal Calcolo delle variazioni). Il problema modello è il seguente $$ \begin{cases} - \text{div} ((1+ |u|^{r}) \nabla u) + |u|^{m-2} u |\nabla u|^{2} = f \quad &\text{in} \, \Omega \\ u = 0 &\text{su} \, \partial\Omega. \end{cases} $$

Détail

Publié le : 2000-03-01

MR 1797052 | Zbl 0970.35061

Bibliographie

[1] Benilan, P. - Boccardo, L. - Gallouët, T. - Gariepy, R. - Pierre, M. - Vazquez, J. L., An $ L^{1} $-theory of existence and uniqueness of solutions of nonlinear elliptic equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci., 22, 1995, 241-273. | MR 1354907 | Zbl 0866.35037

[2] Bensoussan, A. - Boccardo, L. - Murat, F., On a nonlinear partial differential equation having natural growth terms and unbounded solution. Ann. Inst. H. Poincaré Anal. Non Linéaire, 5, 1988, 347-364. | MR 963104 | Zbl 0696.35042

[3] Boccardo, L., Calcolo delle Variazioni. Roma 1 University PhD course, 1996.

[4] Boccardo, L., Some nonlinear Dirichlet problems in $ L^{1} $ involving lower order terms in divergence form. In: A. Alvino et al. (eds), Progress in elliptic and parabolic partial differential equations (Capri, 1994). Pitman Res. Notes Math. Ser., 350, Longman, Harlow 1996, 43-57. | MR 1430139 | Zbl 0889.35034

[5] Boccardo, L. - Gallouët, T., Strongly nonlinear elliptic equations having natural growth terms and $ L^{1} $ data. Nonlinear Anal., 19, 1992, 573-579. | MR 1183664 | Zbl 0795.35031

[6] Boccardo, L. - Gallouët, T. - Orsina, L., Existence and nonexistence of solutions for some nonlinear elliptic equations. J. Anal. Math., 73, 1997, 203-223. | MR 1616410 | Zbl 0898.35035

[7] Boccardo, L. - Murat, F. - Puel, J.-P., Existence de solutions non bornées pour certaines équations quasi-lineaires. Portugal. Math., 41, 1982, 507-534. | MR 766873 | Zbl 0524.35041

[8] 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

[9] Brezis, H. - Browder, F. E., Some properties of higher order Sobolev spaces. J. Math. Pures Appl., 61, 1982, 245-259. | MR 690395 | Zbl 0512.46034

[10] Brezis, H. - Nirenberg, L., Removable singularities for nonlinear elliptic equations. Topol. Methods Nonlinear Anal., 9, 1997, 201-219. | MR 1491843 | Zbl 0905.35027

[11] Dacorogna, B., Direct methods in the calculus of variations. Applied Mathematical Sciences, 78. Springer-Verlag, Berlin-New York1989. | MR 990890 | Zbl 0703.49001

[12] Del Vecchio, T., Strongly nonlinear problems with Hamiltonian having natural growth. Houston J. Math., 16, 1990, 7-24. | MR 1071263 | Zbl 0714.35035

[13] Lions, J.-L., Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod, Paris1969. | Zbl 0189.40603

[14] Porretta, A., Some remarks on the regularity of solutions for a class of elliptic equations with measure data. Preprint, Dip. Mat. Roma 1. | MR 1814734 | Zbl 0974.35032

[15] Porretta, A., Existence for elliptic equations in $ L^{1} $ having lower order terms with natural growth. Preprint, Dip. Mat. Roma 1. | MR 1759814 | Zbl 0963.35068