# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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 $G\subset {ℝ}^{d}$, 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, ${lim}_{x\to \partial G}{\left(\text{dist}\left(x,\partial G\right)\right)}^{2\alpha /\left(\alpha -1\right)}f\left(x\right)=0$, and ${lim}_{x\to \partial G}u\left(x\right)=\infty$.
##### MSC:
 35J60 Nonlinear elliptic equations 35A05 General existence and uniqueness theorems (PDE) (MSC2000)
##### References:
 [1] S. Agmon, A. Douglis and L. Nirenberg,Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, I, Commun. Pure Appl. Math.12 (1959), 623–727. · Zbl 0093.10401 · doi:10.1002/cpa.3160120405 [2] A. D. Aleksandrov,Uniqueness conditions and estimates for the solution of the Dirichlet problem, Vestnik Leningrad Univ.18 (1963), 5–29. [3] P. Aviles,A study of isolated singularities of solutions of a class of non-linear elliptic partial differential equations, Commun. Partial Differ. Equ.7 (1982), 609–643. · Zbl 0495.35036 · doi:10.1080/03605308208820234 [4] C. Bandle and M. Marcus,Sur les solutions maximales de problèmes elliptiques non linéaires: bornes isopérimétriques et comportement asymptotique, C. R. Acad. Sci. Paris311 Ser. I (1990), 91–93. [5] E. B. Dynkin,A probabilistic approach to one class of nonlinear differential equations, to appear. [6] D. Gilbarg and N. S. Trudinger,Elliptic Partial Differential Equations of Second Order, 2nd ed., Grundlerhen 224, Springer-Verlag, Berlin, 1982. [7] I. Iscoe,On the support of measure-valued critical branching Brownian motion, Ann. Probab.16 (1988), 200–221. · Zbl 0635.60094 · doi:10.1214/aop/1176991895 [8] J. B. Keller,On solutions of ${\Delta }$u=f(u), Commun. Pure Appl. Math.10 (1957), 503–510. · Zbl 0090.31801 · doi:10.1002/cpa.3160100402 [9] N. V. Krylov,Nonlinear Elliptic and Parabolic Equations of the Second Order, Reidel, Dordrecht, 1987. [10] J. M. Lasry and P. L. Lions,Nonlinear elliptic equations with singular boundary conditions and stochastic control with state constraints, Math. Ann.283 (1989), 583–630. · Zbl 0688.49026 · doi:10.1007/BF01442856 [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. [12] R. Osserman,On the inequality ${\Delta }$u(u), Pacific J. Math.7 (1957), 1641–1647. [13] L. Veron,Comportement asymptotique des solutionsd’équations elliptiques semi-linéaires dans R N, Ann. Mat. Pura. Appl.127 (1981), 25–50. · Zbl 0467.35013 · doi:10.1007/BF01811717