Semilinear elliptic equations involving critical Sobolev exponents. (English) Zbl 0674.35033

The author considers the semilinear elliptic boundary value problem with variable coefficients: \(Eu=bu^ p+\lambda hu\) in \(\Omega\), \(u>0\) in \(\Omega\) and \(u=0\) on \(\partial \Omega\), where \(Eu\equiv -\partial_ i(a_{ij} \partial_ ju)\) is a symmetric uniformly elliptic operator, b and h are nonnegative nontrivial bounded functions. Furthermore, \(\Omega\) is a bounded domain in \({\mathbb{R}}^ n\), \(n\geq 3\) and \(p=(n+2)/(n-2)\) is the critical exponent. Under certain conditions of \(a_{ij}\), b and h, the author obtains some existence and nonexistence results. The proof is standard. However, some interesting examples are included.
Reviewer: C.F.Wang


35J65 Nonlinear boundary value problems for linear elliptic equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
Full Text: DOI


[1] F. V. Atkinson & L. A. Peletier, Emden-Fowler Equations Involving Critical Exponents. Nonlinear Analysis, Theory, Methods & Applications 10, 755-776, 1986. · Zbl 0662.34024
[2] H. Brezis & E. Lieb, A Relation Between Pointwise Convergence of Functions and Convergence of Functionals. Proceedings of the American Mathematical Society 88, 486-490, 1983. · Zbl 0526.46037
[3] H. Brezis & L. Nirenberg, Positive Solutions of Nonlinear Elliptic Equations Involving Critical Sobolev Exponents. Communications on Pure and Applied Mathematics 36, 437-477, 1983. · Zbl 0541.35029
[4] H. Brezis, Elliptic Equations with Limiting Sobolev Exponents?The Impact of Topology. Communications on Pure and Applied Mathematics S, S17?S39, 1986.
[5] H. Egnell, Extremal Properties of the first Eigenvalue fo a Class of Elliptic Eigenvalue problems. Annali della Scuola Normale Superiore di Pisa 14, 1-48, 1987. · Zbl 0649.35072
[6] B. Gidas, W. M. Ni, & L. Nirenberg, Symmetry and Related Properties via the Maximum Principle. Communications in Mathematical Physics 68, 209-243, 1979. · Zbl 0425.35020
[7] D. Gilbarg & N. S. Trudinger, Elliptic Partial Differential Equations of Second Order. Springer-Verlag 1977. · Zbl 0361.35003
[8] G. H. Hardy, J. E. Littlewood, & G. Pólya, Inequalities. Cambridge University Press 1934. · Zbl 0010.10703
[9] J. Kazdan & F. Warner, Remarks on some Quasilinear Elliptic Equations. Communication on Pure and Applied Mathematics 28, 567-597, 1975. · Zbl 0325.35038
[10] W. M. Ni & J. Serrin, Nonexistence Theorems for Singular Solutions of Quasilinear Partial Differential Equations. Communications on Pure and Applied Mathematics 39, 379-399, 1986. · Zbl 0602.35031
[11] S. I. Pohozaev, Eigenfunctions of the Equation 56-01. Soviet Mathematics Doklady 6, 1408-1411, 1965. · Zbl 0141.30202
[12] N. Trudinger, Remarks Concerning the Conformal Deformation of Riemannian Structures on Compact Manifolds. Annali della Scuola Normale Superiore di Pisa 22, 265-274, 1968. · Zbl 0159.23801
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.