Proceedings of the American Mathematical Society

Author(s): Juncheng Wei
Journal: Proc. Amer. Math. Soc. 125 (1997), 3235-3242.
MSC (1991): Primary 35B40, 35B45; Secondary 35J40
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Abstract: We present the exact multiplicity results for some nonlinear elliptic equations in balls of radius $R$

. We prove that there is a critical value $R_{0}$ such that, for $R < R_{0}$ , the equation has no solution; when $R=R_{0}$ , it has exactly one solution; when $R > R_{0}$ , it has exactly two solutions. Our main tool is the bifurcation theorem due to Crandall and Rabinowitz.

1.: D.G. Aronson and H.F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. in Math. 30 (1978), 33-76. MR 80a:35013
2.: M.G. Crandall and P.H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Rational Mech. Anal. 52, 161-180 (1973). MR 49:5962
3.: P. Clement and G. Sweers, Existence and multiplicity results for s emilinear elliptic eigenvalue problem, Ann. Scuo. Norm. Sup. Pisa 14 (1987), 97-121. MR 89j:35053
4.: E. N. Dancer, On the uniqueness of the positive solutions of a singularly perturbed problem, Rocky Mountain J. of Math. 25 (1995), 957-975. MR 96j:35021
5.: Gidas, B., Ni, W.-M., and Nirenberg, L., Symmetry and Related Properties via the Maximum Principle, Comm. Math. Phys. 68 (3) 1979, 209-243. MR 80h:35043
6.: D. Gilbarg and N.S.Trudinger, Elliptic partial differential equons of second order, Second Edition, Springer-Verlag (1983). MR 86c:35035
7.: J. Jang, On spike solutions of singularly perturbed semilinear Dirichlet problems, J. Diff. Eqns 114(1994), 370-395. MR 95i:35099
8.: P. Korman and T. Ouyang, Exact multiplicity results for two classes of boundary problems, Diff. Int. Eqns 6(1993), 1507-1517. MR 95a:34032
9.: P. Korman, Y. Li and T. Ouyang, Exact multiplicity results for boundary value problems with nonlinearities generalizing cubic, Proc. Roy. Soc. Edinburgh Sect. A 126 (1996), 599-616. MR 97c:34038
10.: M. K. Kwong and L. Zhang, Uniqueness of the positive solution of $\Delta u + f(u) =0$ in an annulus, Diff. Int. Eqns 4 (1991), 583-599. MR 92b:35015
11.: R. Gardner and L.A. Peletier, The set of positive solutions of semilinear equations in large balls, Proc. Roy. Soc. Edinburgh 104 A (1986), 53-72. MR 88e:35063
12.: W.-M. Ni and I. Takagi, Locating the peaks of least-energy solutions to a semilinear Neumann problem, Duke Math.J. 70(1993), 247-281. MR 94h:35072
13.: W.-M. Ni and J. Wei, On the location and profile of spike-layer solutions to singularly perturbed semilinear Dirichlet problem, Comm. Pure Appl. Math. 48 (1995), 731-768. MR 96g:35077
14.: W.-M. Ni, I. Takagi and J. Wei, On the location and profile of intermediate solutions to a singularly perturbed semilinear Dirichlet problem, preprint.
15.: J. Smoller and A. Wasserman, Global bifurcation of steady-state solutions, J. Diff. Eqns 39 (1981), 269-290. MR 82d:58056
16.: S.-H. Wang, A correction for a paper by J. Smoller and A. Wasserman, J. Diff. Eqns 77 (1989), 199-202. MR 90f:58136

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 35B40, 35B45, 35J40

Juncheng Wei
Affiliation: Department of Mathematics, Chinese University of Hong Kong, Shatin, Hong Kong
Email: wei@math.cuhk.edu.hk

DOI: 10.1090/S0002-9939-97-04211-1
PII: S 0002-9939(97)04211-1
Keywords: Exact multiplicity, nonlinear elliptic equations
Received by editor(s): December 15, 1995
Communicated by: Jeffrey Rauch
Copyright of article: Copyright 1997, American Mathematical Society

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