Isolated singularities for the exponential type semilinear elliptic equation in

Authors:
R. Dhanya, J. Giacomoni and S. Prashanth

Journal:
Proc. Amer. Math. Soc. **137** (2009), 4099-4107

MSC (2000):
Primary 35B32, 35B65, 35J25, 35J60

Published electronically:
July 15, 2009

MathSciNet review:
2538571

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this article we study positive solutions of the equation in a punctured domain in and show sharp conditions on the nonlinearity that enables us to extend such a solution to the whole domain and also preserve its regularity. We also show, using the framework of bifurcation theory, the existence of at least two solutions for certain classes of exponential type nonlinearities.

**1.**Adimurthi,*Existence of positive solutions of the semilinear Dirichlet problem with critical growth for the 𝑛-Laplacian*, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)**17**(1990), no. 3, 393–413. MR**1079983****2.**Adimurthi and S. Prashanth,*Failure of Palais-Smale condition and blow-up analysis for the critical exponent problem in 𝐑²*, Proc. Indian Acad. Sci. Math. Sci.**107**(1997), no. 3, 283–317. MR**1467434**, 10.1007/BF02867260**3.**Adimurthi and S. Prashanth,*Critical exponent problem in 𝑅²-border-line between existence and non-existence of positive solutions for Dirichlet problem*, Adv. Differential Equations**5**(2000), no. 1-3, 67–95. MR**1734537****4.**Adimurthi and Michael Struwe,*Global compactness properties of semilinear elliptic equations with critical exponential growth*, J. Funct. Anal.**175**(2000), no. 1, 125–167. MR**1774854**, 10.1006/jfan.2000.3602**5.**Haïm Brézis and Pierre-Louis Lions,*A note on isolated singularities for linear elliptic equations*, Mathematical analysis and applications, Part A, Adv. in Math. Suppl. Stud., vol. 7, Academic Press, New York-London, 1981, pp. 263–266. MR**634242****6.**Haïm Brezis and Frank Merle,*Uniform estimates and blow-up behavior for solutions of -Δ𝑢=𝑉(𝑥)𝑒^{𝑢} in two dimensions*, Comm. Partial Differential Equations**16**(1991), no. 8-9, 1223–1253. MR**1132783**, 10.1080/03605309108820797**7.**Michael G. Crandall and Paul H. Rabinowitz,*Bifurcation from simple eigenvalues*, J. Functional Analysis**8**(1971), 321–340. MR**0288640****8.**Michael G. Crandall and Paul H. Rabinowitz,*Bifurcation, perturbation of simple eigenvalues and linearized stability*, Arch. Rational Mech. Anal.**52**(1973), 161–180. MR**0341212****9.**Michael G. Crandall and Paul H. Rabinowitz,*Some continuation and variational methods for positive solutions of nonlinear elliptic eigenvalue problems*, Arch. Rational Mech. Anal.**58**(1975), no. 3, 207–218. MR**0382848****10.**E. N. Dancer,*Infinitely many turning points for some supercritical problems*, Ann. Mat. Pura Appl. (4)**178**(2000), 225–233. MR**1849387**, 10.1007/BF02505896**11.**J. Giacomoni, S. Prashanth, and K. Sreenadh,*A global multiplicity result for 𝑁-Laplacian with critical nonlinearity of concave-convex type*, J. Differential Equations**232**(2007), no. 2, 544–572. MR**2286391**, 10.1016/j.jde.2006.09.012**12.**D. D. Joseph and T. S. Lundgren,*Quasilinear Dirichlet problems driven by positive sources*, Arch. Rational Mech. Anal.**49**(1972/73), 241–269. MR**0340701****13.**Takayoshi Ogawa and Takashi Suzuki,*Nonlinear elliptic equations with critical growth related to the Trudinger inequality*, Asymptotic Anal.**12**(1996), no. 1, 25–40. MR**1373480****14.**Paul H. Rabinowitz,*Some global results for nonlinear eigenvalue problems*, J. Functional Analysis**7**(1971), 487–513. MR**0301587**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
35B32,
35B65,
35J25,
35J60

Retrieve articles in all journals with MSC (2000): 35B32, 35B65, 35J25, 35J60

Additional Information

**R. Dhanya**

Affiliation:
Tata Institute of Fundamental Research, Center for Applicable Mathematics, P.B. No. 6503, Sharadanagar, Chikkabommasandra, Bangalore 560065, India

Email:
dhanya@math.tifrbng.res.in

**J. Giacomoni**

Affiliation:
Laboratoire de Mathématiques Appliquées, Université de Pau et des Pays de l’Adour, B.P. 576, 64012 Pau cedex, France

Email:
jgiacomo@univ-pau.fr

**S. Prashanth**

Affiliation:
Tata Institute of Fundamental Research, Center for Applicable Mathematics, P.B. No. 6503, Sharadanagar, Chikkabommasandra, Bangalore 560065, India

Email:
pras@math.tifrbng.res.in

DOI:
https://doi.org/10.1090/S0002-9939-09-09988-2

Keywords:
Isolated singularity,
blow-up,
Laplace equation.

Received by editor(s):
September 30, 2008

Published electronically:
July 15, 2009

Communicated by:
Matthew J. Gursky

Article copyright:
© Copyright 2009
American Mathematical Society