Positive solutions of semilinear equations in cones

Author:
Henrik Egnell

Journal:
Trans. Amer. Math. Soc. **330** (1992), 191-201

MSC:
Primary 35B05; Secondary 35J65

MathSciNet review:
1034662

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Abstract: In this paper we consider the problem of finding a positive solution of the equation in a cone , with zero boundary data. We are only interested in solutions that are regular at infinity (i.e. such that , as ). We will always assume that .

We show that the existence of a solution depends on the sign of and also on the shape of the cone .

**[BC]**A. Bahri and J.-M. Coron,*On a nonlinear elliptic equation involving the critical Sobolev exponent: the effect of the topology of the domain*, Comm. Pure Appl. Math.**41**(1988), no. 3, 253–294. MR**929280**, 10.1002/cpa.3160410302**[BE]**Catherine Bandle and Matts Essén,*On positive solutions of Emden equations in cone-like domains*, Arch. Rational Mech. Anal.**112**(1990), no. 4, 319–338. MR**1077263**, 10.1007/BF02384077**[BaL]**Catherine Bandle and Howard A. Levine,*On the existence and nonexistence of global solutions of reaction-diffusion equations in sectorial domains*, Trans. Amer. Math. Soc.**316**(1989), no. 2, 595–622. MR**937878**, 10.1090/S0002-9947-1989-0937878-9**[BrL]**Haïm Brézis and Elliott Lieb,*A relation between pointwise convergence of functions and convergence of functionals*, Proc. Amer. Math. Soc.**88**(1983), no. 3, 486–490. MR**699419**, 10.1090/S0002-9939-1983-0699419-3**[CO]**Jean-Michel Coron,*Topologie et cas limite des injections de Sobolev*, C. R. Acad. Sci. Paris Sér. I Math.**299**(1984), no. 7, 209–212 (French, with English summary). MR**762722****[E]**E. N. Dancer,*A note on an equation with critical exponent*, Bull. London Math. Soc.**20**(1988), no. 6, 600–602. MR**980763**, 10.1112/blms/20.6.600**[DI]**Wei Yue Ding,*Positive solutions of Δ𝑢+𝑢^{(𝑛+2)/(𝑛-2)}=0 on contractible domains*, J. Partial Differential Equations**2**(1989), no. 4, 83–88. MR**1027983****[DN]**Wei Yue Ding and Wei-Ming Ni,*On the elliptic equation Δ𝑢+𝐾𝑢^{(𝑛+2)/(𝑛-2)}=0 and related topics*, Duke Math. J.**52**(1985), no. 2, 485–506. MR**792184**, 10.1215/S0012-7094-85-05224-X**[EG1]**Henrik Egnell,*Elliptic boundary value problems with singular coefficients and critical nonlinearities*, Indiana Univ. Math. J.**38**(1989), no. 2, 235–251. MR**997382**, 10.1512/iumj.1989.38.38012**[EG2]**Henrik Egnell,*Asymptotic results for finite energy solutions of semilinear elliptic equations*, J. Differential Equations**98**(1992), no. 1, 34–56. MR**1168970**, 10.1016/0022-0396(92)90103-T**[GT]**David Gilbarg and Neil S. Trudinger,*Elliptic partial differential equations of second order*, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR**737190****[LI]**P.-L. Lions,*The concentration-compactness principle in the calculus of variations. The limit case. I*, Rev. Mat. Iberoamericana**1**(1985), no. 1, 145–201. MR**834360**, 10.4171/RMI/6**[KW]**Jerry L. Kazdan and F. W. Warner,*Remarks on some quasilinear elliptic equations*, Comm. Pure Appl. Math.**28**(1975), no. 5, 567–597. MR**0477445****[PA]**Donato Passaseo,*Multiplicity of positive solutions of nonlinear elliptic equations with critical Sobolev exponent in some contractible domains*, Manuscripta Math.**65**(1989), no. 2, 147–165. MR**1011429**, 10.1007/BF01168296**[RA]**Paul H. Rabinowitz,*Minimax methods in critical point theory with applications to differential equations*, CBMS Regional Conference Series in Mathematics, vol. 65, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1986. MR**845785****[ST]**Michael Struwe,*A global compactness result for elliptic boundary value problems involving limiting nonlinearities*, Math. Z.**187**(1984), no. 4, 511–517. MR**760051**, 10.1007/BF01174186

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DOI:
https://doi.org/10.1090/S0002-9947-1992-1034662-5

Article copyright:
© Copyright 1992
American Mathematical Society