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Semilinear elliptic equations with uniform blow-up on the boundary. (English) Zbl 0802.35042

Summary: We prove the existence and the uniqueness of a solution \(u\) of \(-Lu + h | u |^{\alpha - 1} u = f\) in some open domain \(G \subset\mathbb R^ d\), where \(L\) is a strongly elliptic operator, \(f\) a nonnegative function, and \(\alpha>1\), under the assumption that \(\partial G\) is a \(C^ 2\) compact hypersurface, \(\lim_{x \to \partial G} (\text{dist} (x, \partial G))^{2 \alpha/(\alpha - 1)} f(x) = 0\), and \(\lim_{x \to \partial G} u(x) = \infty\).

MSC:

35J61 Semilinear elliptic equations
35B44 Blow-up in context of PDEs
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[1] Agmon, S.; Douglis, A.; Nirenberg, L., Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, I, Commun. Pure Appl. Math., 12, 623-727 (1959) · Zbl 0093.10401
[2] Aleksandrov, A. D., Uniqueness conditions and estimates for the solution of the Dirichlet problem, Vestnik Leningrad Univ., 18, 5-29 (1963) · Zbl 0139.05702
[3] Aviles, P., A study of isolated singularities of solutions of a class of non-linear elliptic partial differential equations, Commun. Partial Differ. Equ., 7, 609-643 (1982) · Zbl 0495.35036
[4] Bandle, C.; Marcus, M., Sur les solutions maximales de problèmes elliptiques non linéaires: bornes isopérimétriques et comportement asymptotique, C. R. Acad. Sci. Paris, 311, 91-93 (1990) · Zbl 0726.35041
[5] E. B. Dynkin,A probabilistic approach to one class of nonlinear differential equations, to appear. · Zbl 0722.60062
[6] Gilbarg, D.; Trudinger, N. S., Elliptic Partial Differential Equations of Second Order (1982), Berlin: Springer-Verlag, Berlin · Zbl 0691.35001
[7] Iscoe, I., On the support of measure-valued critical branching Brownian motion, Ann. Probab., 16, 200-221 (1988) · Zbl 0635.60094
[8] Keller, J. B., On solutions of Δu=f(u), Commun. Pure Appl. Math., 10, 503-510 (1957) · Zbl 0090.31801
[9] Krylov, N. V., Nonlinear Elliptic and Parabolic Equations of the Second Order (1987), Dordrecht: Reidel, Dordrecht · Zbl 0619.35004
[10] Lasry, J. M.; Lions, P. L., Nonlinear elliptic equations with singular boundary conditions and stochastic control with state constraints, Math. Ann., 283, 583-630 (1989) · Zbl 0688.49026
[11] C. Loewner and L. Nirenberg,Partial differential equations invariant under conformal or projective transformations, inContributions to Analysis (L. Ahlfors et al., eds.), 1974, pp. 245-272. · Zbl 0298.35018
[12] Osserman, R., On the inequality Δu≧f(u), Pacific J. Math., 7, 1641-1647 (1957) · Zbl 0083.09402
[13] Veron, L., Comportement asymptotique des solutionsd’équations elliptiques semi-linéaires dans R^N, Ann. Mat. Pura. Appl., 127, 25-50 (1981) · Zbl 0467.35013
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