Explosive solutions of quasilinear elliptic equations: Existence and uniqueness. (English) Zbl 0793.35028

This paper deals with the quasilinear elliptic equation \[ -\text{div} \biggl( Q \bigl( | \nabla u | \bigr) \nabla u \biggr)+\lambda \beta (u)=f \quad \text{in } \Omega \subset \mathbb{R}^ N,\;N>1; \] more precisely, existence and uniqueness of local solutions satisfying \[ u(x) \to \infty \quad \text{as dist} (x,\partial \Omega) \to 0 \] and other properties are the main goals here. These kinds of functions are called explosive solutions. No behaviour at the boundary to be prescribed is a priori imposed. However, we are going to show that, under an adequate strong interior structure condition on the equation, explosive behaviour near \(\partial \Omega\) cannot be arbitrary. In fact, there exists a unique such singular character governed by a uniform rate of explosion depending only on the terms \(Q\), \(\lambda\), \(\beta\) and \(f\).


35J60 Nonlinear elliptic equations
35B40 Asymptotic behavior of solutions to PDEs
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[1] Lasry, J. M.; Lions, P. L., Nonlinear elliptic equations with singular boundary conditions and stochastic control with state constraints, Math. Annln., 283, 583-630 (1989) · Zbl 0688.49026
[2] Keller, J. B., Electrohydrodynamics I. The equilibrium of a charged gas in a container, J. ration. Mech. Analysis, 4, 715-724 (1956) · Zbl 0070.44207
[3] Loewner, Ch.; Nirenberg, L., Partial differential equations invariant under conformal or projective transformations, (Contributions to Analysis (1974), Academic Press: Academic Press New York), 245-272
[4] Robinson, P. D., Complementary variational principles, Nonlinear Functional Analysis and Applications (1971), Academic Press: Academic Press New York · Zbl 0234.49031
[5] Díaz, J. I., Nonlinear Partial Differential Equations and Free Boundaries, Elliptic Equations. Res. Notes Math., 106 (1985), Pitman · Zbl 0595.35100
[7] Serrin, J., Local behavior of solutions of quasilinear equations, Acta Math., 111, 247-302 (1964) · Zbl 0128.09101
[8] Serrin, J., The problem of Dirichlet for quasilinear elliptic differential equations with many independent variables, Phil. Trans. R. Soc. London, 264, 413-496 (1969) · Zbl 0181.38003
[9] Gilbarg, D.; Trudinger, N. S., Elliptic Partial Differential Equations of Second Order (1983), Springer: Springer Berlin · Zbl 0691.35001
[10] Díaz, G.; Letelier, R., Uniqueness for viscosity solutions of quasilinear elliptic equations in \(R^N\) without conditions at infinity, Diff. Integral Eqns, 5, 999-1016 (1992) · Zbl 0761.35020
[11] Keller, J. B., On solutions of Δ \(u = f(u)\), Communs pure appl. Math., X, 503-510 (1957) · Zbl 0090.31801
[12] Osserman, R., On the inequality Δ \(u\) ≥ \(f(u)\), Pacific J. math., 7, 1641-1647 (1957) · Zbl 0083.09402
[13] Vasquez, J. L., An a priori interior estimate for the solutions of nonlinear problems representing weak diffusion, Nonlinear Analysis, 5, 95-103 (1981)
[14] Díaz, J. I.; Saa, J. E.; Thiel, U., Sobre la ecuación de curvatura media prescrita y otras ecuaciones cuasilineales elípticas con soluciones anulándose localmente. Aparecerá en las Actas del Homenaje a Julio Rey Pastor (1988), Buenos Aires
[15] Sperb, R., Maximum Principles and Their Applications (1981), Academic Press: Academic Press New York · Zbl 0454.35001
[16] Lions, J. L., Quelques Methodes de Resolutions de Problemes aux Limited non Lineares (1969), Dunod · Zbl 0189.40603
[17] Evans, L. C., A new proof of local \(C^{1+α}\) regularity for certain degenerate elliptic PDE, J. diff. Eqns, 45, 356-373 (1982) · Zbl 0508.35036
[18] Di Benedetto, E., \(C^{1+α}\) local regularity of weak solutions of degenerate elliptic equations, Nonlinear Analysis, 7, 827-850 (1983) · Zbl 0539.35027
[19] Tolksdorff, P., Regularity for a more general class of quasilinear elliptic equations, J. diff. Eqns, 51, 126-150 (1984)
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