## Duke Mathematical Journal

### Singular behavior of least-energy solutions of a semilinear Neumann problem involving critical Sobolev exponents

#### Article information

Source
Duke Math. J. Volume 67, Number 1 (1992), 1-20.

Dates
First available in Project Euclid: 20 February 2004

http://projecteuclid.org/euclid.dmj/1077294269

Digital Object Identifier
doi:10.1215/S0012-7094-92-06701-9

Mathematical Reviews number (MathSciNet)
MR1174600

Zentralblatt MATH identifier
0785.35041

#### Citation

Ni, Wei-Ming; Pan, Xing-Bin; Takagi, I. Singular behavior of least-energy solutions of a semilinear Neumann problem involving critical Sobolev exponents. Duke Math. J. 67 (1992), no. 1, 1--20. doi:10.1215/S0012-7094-92-06701-9. http://projecteuclid.org/euclid.dmj/1077294269.

#### References

• [AM] Adimurthi and G. Mancini, The Neumann problem for elliptic equations with critical non-linearity, preprint.
• [AY1] Adimurthi and S. L. Yadava, On a conjecture of Lin-Ni for the semilinear Neumann problem, preprint.
• [AY2] Adimurthi and S. L. Yadava, Existence and nonexistence of positive radial solutions for the Sobolev critical exponent problem with Neumann boundary conditions, preprint.
• [AR] A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis 14 (1973), 349–381.
• [BN] H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (1983), no. 4, 437–477.
• [BKP] C. Budd, M. C. Knaap, and L. A. Peletier, Asymptotic behavior of solutions of elliptic equations with critical exponents and Neumann boundary conditions, preprint.
• [CGS] L. A. Caffarelli, B. Gidas, and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math. 42 (1989), no. 3, 271–297.
• [CL] W. X. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J. 63 (1991), no. 3, 615–622.
• [CK1] M. Comte and M. C. Knaap, Solutions of elliptic equations involving critical Sobolev exponents with Neumann boundary conditions, Manuscripta Math. 69 (1990), no. 1, 43–70.
• [CK2] M. Comte and M. C. Knaap, Existence of solutions of elliptic equations involving critical Sobolev exponents with Neumann boundary conditions in general domains, preprint.
• [DN] W.-Y. Ding and W.-M. Ni, On the existence of positive entire solutions of a semilinear elliptic equation, Arch. Rational Mech. Anal. 91 (1986), no. 4, 283–308.
• [E] H. Egnell, Asymptotic results for finite-energy solutions of semilinear elliptic equations, to appear in J. Differential Equations.
• [GNN] B. Gidas, W. M. Ni, and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\bf R\spn$, Mathematical analysis and applications, Part A, Adv. in Math. Suppl. Stud., vol. 7, Academic Press, New York, 1981, pp. 369–402.
• [LN] C. S. Lin and W.-M. Ni, On the diffusion coefficient of a semilinear Neumann problem, Calculus of variations and partial differential equations (Trento, 1986) eds. S. Hildebrandt, D. Kinderlehrer, and M. Miranda, Lecture Notes in Math., vol. 1340, Springer, Berlin, 1988, pp. 160–174.
• [LNT] C.-S. Lin, W.-M. Ni, and I. Takagi, Large amplitude stationary solutions to a chemotaxis system, J. Differential Equations 72 (1988), no. 1, 1–27.
• [N1] W.-M. Ni, Recent progress in semilinear elliptic equations, RIMS Kokyuroku 679 (1989), 1–39.
• [N2] W. M. Ni, On the positive radial solutions of some semilinear elliptic equations on $\bf R\spn$, Appl. Math. Optim. 9 (1983), no. 4, 373–380.
• [N3] W.-M. Ni, Some aspects of semilinear elliptic equations on $\bf R\sp n$, Nonlinear diffusion equations and their equilibrium states, II (Berkeley, CA, 1986) eds. W.-M. Ni, L. A. Peletier, and J. Serrin, Math. Sci. Res. Inst. Publ., vol. 13, Springer, New York, 1988, pp. 171–205.
• [NT] W.-M. Ni and I. Takagi, On the shape of least-energy solutions to a semilinear Neumann problem, Comm. Pure Appl. Math. 44 (1991), no. 7, 819–851.
• [T] I. Takagi, Point-condensation for a reaction-diffusion system, J. Differential Equations 61 (1986), no. 2, 208–249.
• [Tr] N. S. Trudinger, Remarks concerning the conformal deformation of Riemannian structures on compact manifolds, Ann. Scuola Norm. Sup. Pisa (3) 22 (1968), 265–274.
• [W] X.-J. Wang, Neumann problems of semilinear elliptic equations involving critical Sobolev exponents, J. Differential Equations 93 (1991), no. 2, 283–310.