×

Quasilinear equations with natural growth. (English) Zbl 1151.35343

Summary: We study the existence of positive solution \(w\in H_0^1(\Omega)\) of the quasilinear equation \(-\Delta w+ g(w)|\nabla w|^2=a(x)\), \(x\in \Omega\), where \(\Omega\) is a bounded domain in \(\mathbb R^N\), \(0\leq a\in L^\infty (\Omega )\) and \(g\) is a nonnegative continuous function on \((0,+\infty)\) which may have a singularity at zero.

MSC:

35J60 Nonlinear elliptic equations
35J65 Nonlinear boundary value problems for linear elliptic equations
35B45 A priori estimates in context of PDEs
PDF BibTeX XML Cite
Full Text: DOI Euclid EuDML

References:

[1] Abdellaoui, B., Dall’Aglio, A. and Peral, I.: Some remarks on elliptic problems with critical growth in the gradient. J. Differential Equations 222 (2006), 21-62. · Zbl 1357.35089
[2] Alaa, N. and Mounir, I.: Weak solutions for some reaction-diffusion systems with balance law and critical growth with respect to the gradient. Ann. Math. Blaise Pascal 8 (2001), 1-19. · Zbl 1099.35512
[3] Bensoussan, A. and Boccardo, L.: Nonlinear systems of elliptic equations with natural growth conditions and sign conditions. Appl. Math. Optim. 46 (2002), 143-166. · Zbl 1077.35046
[4] Bensoussan, A., Boccardo, L. and Murat, F.: On a nonlinear partial differential equation having natural growth terms and unbounded solution. Ann. Inst. H. Poincaré Anal. Non Linéaire 5 (1988), no. 4, 347-364. · Zbl 0696.35042
[5] Boccardo, L., Murat, F. and Puel, J.-P.: Existence de solutions non bornées pour certaines équations quasi-linéaires. Portugal. Math. 41 (1982), 507-534. · Zbl 0524.35041
[6] Boccardo, L., Murat, F. and 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 (Paris, 1981/1982) , 19-73. Res. Notes in Math. 84 . Pitman, Boston, Mass.-London, 1983. · Zbl 0588.35041
[7] Boccardo, L., Murat, F. and Puel, J.-P.: Résultats d’existence pour certains problèmes elliptiques quasilinéaires. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 11 (1984), 213-235. · Zbl 0557.35051
[8] Boccardo, L., Segura de León, S. and Trombetti, C.: Bounded and unbounded solutions for a class of quasi-linear elliptic problems with a quadratic gradient term. J. Math. Pures Appl. (9) 80 (2001), 919-940. · Zbl 1134.35358
[9] Chiacchio, F.: Regularity for solutions of nonlinear elliptic equations with natural growth in the gradient. Bull. Sci. math. 124 (2000), 57-74. · Zbl 0947.35074
[10] Dall’Aglio, A., Giachetti, D. and Puel, J.-P.: Nonlinear elliptic equations with natural growth in general domains. Ann. Mat. Pura Appl. (4) 181 (2002), 407-426. · Zbl 1097.35050
[11] Donato, P. and Giachetti, D.: Quasilinear elliptic equations with quadratic growth in unbounded domains. Nonlinear Anal. 10 (1986), 791-804. · Zbl 0602.35036
[12] Ferone, V. and Murat, F.: Nonlinear problems having natural growth in the gradient: an existence result when the source terms are small. Nonlinear Anal. 42 (2000), 1309-1326. · Zbl 1158.35358
[13] Gilbarg, D. and Trudinger, N. S.: Elliptic partial differential equations of second order . Fundamental Principles of Mathematical Sciences 224 . Springer-Verlag, Berlin, 1983. · Zbl 0562.35001
[14] Grenon, N. and Trombetti, C.: Existence results for a class of nonlinear elliptic problems with \(p\)-growth in the gradient. Nonlinear Anal. 52 (2003), 931-942. · Zbl 1087.35037
[15] Kazdan, J. L. and Kramer, R. J.: Invariant criteria for existence of solutions to second-order quasilinear elliptic equations. Comm. Pure Appl. Math. 31 (1978), 619-645. · Zbl 0368.35031
[16] Keller, J. B.: On solutions of \(\Delta u =f(u)\). Comm. Pure Appl. Math. 10 (1957), 503-510. · Zbl 0090.31801
[17] Ladyzhenskaya, O. A. and Ural’tseva, N. N.: Linear and quasilinear elliptic equations . Academic Press, New York-London, 1968. · Zbl 0164.13002
[18] Landes, R.: On the existence of weak solutions of perturbated systems with critical growth. J. Reine Angew. Math. 393 (1989), 21-38. · Zbl 0664.35027
[19] Miranda, C.: Su alcuni teoremi di inclusione. Ann. Polon. Math. 16 (1965), 305-315. · Zbl 0172.40303
[20] Osserman, R.: On the inequality \(\Delta u\geq f(u)\). Pacific J. Math. 7 (1957), 1641-1647. · Zbl 0083.09402
[21] Porretta, A. and Segura de León, S.: Nonlinear elliptic equations having a gradient term with natural growth. J. Math. Pures Appl. (9) 85 (2006), 465-492. · Zbl 1158.35364
[22] Serrin, J.: The problem of Dirichlet for quasilinear elliptic differential equations with many independent variables. Philos. Trans. Royal Soc. London Ser. A 264 (1969), 413-496. · Zbl 0181.38003
[23] Stampacchia, G.: Équations elliptiques du second ordre à coefficients discontinus . Séminaire de Mathématiques Supérieures, 16 (Été, 1965). Les Presses de l’Université de Montréal, Montreal, Que., 1966 · Zbl 0151.15501
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.