## Radially symmetric solutions to a Dirichlet problem involving critical exponents

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- by Alfonso Castro and Alexandra Kurepa
- Trans. Amer. Math. Soc.
**343**(1994), 907-926 - DOI: https://doi.org/10.1090/S0002-9947-1994-1207581-0
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## Abstract:

In this paper we answer, for $N = 3,4$, the question raised in [1] on the number of radially symmetric solutions to the boundary value problem $- \Delta u(x) = \lambda u(x) + u(x)|u(x){|^{4/(N - 2)}}$, $x \in B: = \{ x \in {R^N}:\left \| x \right \| < 1\}$, $u(x) = 0$, $x \in \partial B$, where $\Delta$ is the Laplacean operator and $\lambda > 0$. Indeed, we prove that if $N = 3,4$, then for any $\lambda > 0$ this problem has only finitely many radial solutions. For $N = 3,4,5$ we show that, for each $\lambda > 0$, the set of radially symmetric solutions is bounded. Moreover, we establish geometric properties of the branches of solutions bifurcating from zero and from infinity.## References

- Frederick V. Atkinson, Haïm Brezis, and Lambertus A. Peletier,
*Solutions d’équations elliptiques avec exposant de Sobolev critique qui changent de signe*, C. R. Acad. Sci. Paris Sér. I Math.**306**(1988), no. 16, 711–714 (French, with English summary). MR**944417** - Haïm Brézis and Louis Nirenberg,
*Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents*, Comm. Pure Appl. Math.**36**(1983), no. 4, 437–477. MR**709644**, DOI 10.1002/cpa.3160360405 - Alfonso Castro and Alexandra Kurepa,
*Infinitely many radially symmetric solutions to a superlinear Dirichlet problem in a ball*, Proc. Amer. Math. Soc.**101**(1987), no. 1, 57–64. MR**897070**, DOI 10.1090/S0002-9939-1987-0897070-7 - Alfonso Castro and Alexandra Kurepa,
*Radially symmetric solutions to a superlinear Dirichlet problem in a ball with jumping nonlinearities*, Trans. Amer. Math. Soc.**315**(1989), no. 1, 353–372. MR**933323**, DOI 10.1090/S0002-9947-1989-0933323-8
G. Cerami, - G. Cerami, S. Solimini, and M. Struwe,
*Some existence results for superlinear elliptic boundary value problems involving critical exponents*, J. Funct. Anal.**69**(1986), no. 3, 289–306. MR**867663**, DOI 10.1016/0022-1236(86)90094-7 - Michael G. Crandall and Paul H. Rabinowitz,
*Bifurcation from simple eigenvalues*, J. Functional Analysis**8**(1971), 321–340. MR**0288640**, DOI 10.1016/0022-1236(71)90015-2 - Man Kam Kwong,
*Uniqueness of positive solutions of $\Delta u-u+u^p=0$ in $\textbf {R}^n$*, Arch. Rational Mech. Anal.**105**(1989), no. 3, 243–266. MR**969899**, DOI 10.1007/BF00251502
S. I. Pohozaev, - Patrizia Pucci and James Serrin,
*A general variational identity*, Indiana Univ. Math. J.**35**(1986), no. 3, 681–703. MR**855181**, DOI 10.1512/iumj.1986.35.35036 - Marvin Shinbrot and Robert R. Welland,
*The Cauchy-Kowalewskaya theorem*, J. Math. Anal. Appl.**55**(1976), no. 3, 757–772. MR**492756**, DOI 10.1016/0022-247X(76)90079-2 - Sergio Solimini,
*On the existence of infinitely many radial solutions for some elliptic problems*, Rev. Mat. Apl.**9**(1987), no. 1, 75–86. MR**926233** - Neil S. Trudinger,
*Remarks concerning the conformal deformation of Riemannian structures on compact manifolds*, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3)**22**(1968), 265–274. MR**240748**

*Elliptic equations with critical growth*, College on Variational Problems in Analysis, Lecture Notes SMR 281/24, Internat. Centre for Theoretical Physics, Trieste, Italy, 1988.

*Eigenfunctions of the equation*$\Delta u + \lambda f(u) = 0$, Soviet. Math. Dokl.

**6**(1965), 1408-1411.

## Bibliographic Information

- © Copyright 1994 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**343**(1994), 907-926 - MSC: Primary 35B05; Secondary 35J65
- DOI: https://doi.org/10.1090/S0002-9947-1994-1207581-0
- MathSciNet review: 1207581