Local gradient estimates and existence of blow-up solutions to a class of quasilinear elliptic equations. (English) Zbl 1284.35190

From the introduction: In this paper we study the existence of minimal and maximal positive blow-up solutions, that is \[ \lim_{_{\substack{ \text{dist}(x,\partial\Omega)\to 0,\\ x\in\Omega}}}u(x)=+\infty, \] to the quasilinear elliptic equation \[ -\Delta u+ H(x,u,\nabla u) = f\quad\text{in}\;\Omega, \tag{1} \] where \(\Omega\) is a smooth bounded domain, \(f\in L^\infty_{\text{loc}}(\Omega)\), and \(H\) is an appropriate function.


35J62 Quasilinear elliptic equations
35B45 A priori estimates in context of PDEs
35B44 Blow-up in context of PDEs
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[1] Bandle, C.; Giarrusso, E., Boundary blow-up for semilinear elliptic equations with nonlinear gradient terms, Adv. differential equations, 1, 133-150, (1996) · Zbl 0840.35034
[2] Bandle, C.; Marcus, M., On second order effects in the boundary behavior of large solutions of semilinear elliptic problems, Differential integral equations, 11, 23-34, (1998) · Zbl 1042.35535
[3] Bernstein, S., Sur la généralization du probléme de Dirichlet II, Math. ann., 69, 82-136, (1910) · JFM 41.0427.02
[4] Bieberbach, L., δu=eu und die automorphen funktionen, Math. ann., 77, 173-212, (1916)
[5] Boccardo, L.; Murat, F.; Puel, J., Existence de solutions faibles pour des équations elliptiques quasilinéaires à croissance quadratique: non linear partial differential equations and their applications, (), 19-73
[6] Del Pino, M.; Letelier, R., The influence of domain geometry in boundary blow-up elliptic problems, Nonlinear anal., 48, 897-904, (2002) · Zbl 1142.35431
[7] Díaz, G.; Letelier, R., Unicidad de soluciones locales en algunas ecuaciones semilineales elípticas, (), 301-305
[8] Dı́az, G.; Letelier, R., Local estimates: uniqueness of solutions to some nonlinear elliptic equations, (), 171-186 · Zbl 0860.35028
[9] Díaz, G.; Letelier, R.; Ortega, J., Existence of a unique solution to a quasilinear elliptic equations on a bounded domain, Panamer. math. J., 6, 1-35, (1996) · Zbl 0874.35036
[10] Dynkin, E.; Kuznetsov, S., Superdiffusions and removable singularities for quasilinear partial differential equations, Comm. pure appl. math., 49, 125-176, (1996) · Zbl 0861.60083
[11] García-Melián, J.; Letelier, R.; Sabina de Lis, J., Uniqueness and asymptotic behaviour for solutions of semilinear problems with boundary blow-up, Proc. amer. math. soc., 129, 3593-3602, (2001) · Zbl 0989.35044
[12] Gilbarg, D.; Trudinger, N., Elliptic partial differential equations of second order, (1983), Springer-Verlag · Zbl 0562.35001
[13] Greco, A.; Porru, G., Asymptotic estimates and convexity of large solutions to semilinear elliptic equations, Differential integral equations, 10, 219-229, (1997) · Zbl 0889.35028
[14] Keller, J., On solutions of δu=f(u), Comm. pure appl. math., 10, 503-510, (1957) · Zbl 0090.31801
[15] Keller, J., Electrohidrodinamics I. the equilibrium of a charged gas in a container, J. rational mech. anal., 4, 715-724, (1956) · Zbl 0070.44207
[16] Kondrat’ev, V.; Nikishkin, V., Asymptotics, near the boundary, of a solution of a singular boundary-value problem for a semilinear elliptic equations, Differ. uravn., Differential equations, 26, 345-348, (1990), English translation: · Zbl 0706.35054
[17] Lair, A.V., A necessary and sufficient condition for existence of large solutions to semilinear elliptic equations, J. math. anal. appl., 240, 205-218, (1999) · Zbl 1058.35514
[18] Lasry, J.; Lions, P.L., Nonlinear elliptic equations with singular boundary conditions and stochastic control with state contrains, Math. ann., 283, 583-630, (1989) · Zbl 0688.49026
[19] Lazer, A.; McKenna, P., On a singular nonlinear elliptic boundary value problems for the monge – ampere operator, J. math. anal. appl., 197, 341-362, (1996) · Zbl 0856.35042
[20] Letelier, R.; Ortega, J., Local gradient estimates and existence of minimal solutions of some nonlinear elliptic equations blowing up on the boundary, Rev. acad. Canaria cienc., 5, 111-124, (1993) · Zbl 0822.35055
[21] Lions, P.L., Quelques remarques sur LES problemes elliptiques quasilineaires du second order, J. anal. math., 45, 234-254, (1985) · Zbl 0614.35034
[22] Lions, P.L., Résolutions de problèmes elliptiques quasilinéaires, Arch. rational mech. anal., 74, 336-353, (1980) · Zbl 0449.35036
[23] Loewner, C.; Nirenberg, L., Partial differential equations invariant under conformal or projective transformations, (), 345-348
[24] Marcus, M.; Verón, L., Uniqueness and asymptotic behavior of solutions with boundary blow-up for a class of nonlinear elliptic equations, Ann. inst. H. Poincaré, 14, 237-274, (1997) · Zbl 0877.35042
[25] McKenna, P.; Reichel, W.; Walter, W., Symmetry and multiplicity for nonlinear elliptic differential equations with boundary blow-up, Nonlinear anal., 28, 1213-1225, (1997) · Zbl 0868.35031
[26] Osserman, R., On the inequality δu⩾f(u), Pacific J. math., 1641-1647, (1957) · Zbl 0083.09402
[27] Pohozaev, S., The Dirichlet problem for the equation δu=u2, Dokl. akad. nauk SSSR, Soviet math., 1, 1143-1146, (1960), English translation: · Zbl 0097.08503
[28] Rademacher, H., Finige besondere probleme partieller differentialgleichungen, (), 838-845
[29] Serrin, J., The problem of Dirichlet for quasilinear elliptic differential equations with many independent variables, Philos. trans. roy. soc. London ser. A, 264, 413-496, (1969) · Zbl 0181.38003
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