Proceedings of the American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Help
ISSN 1088-6826(e) ISSN 0002-9939(p)

Uniqueness of the least-energy solution for a semilinear Neumann problem

Author(s): Massimo Grossi
Journal: Proc. Amer. Math. Soc. 128 (2000), 1665-1672.
MSC (1991): Primary 35J70
Posted: October 18, 1999
MathSciNet review: 1694340
Retrieve article in: PDF
This article is available free of charge

Abstract | References | Similar articles | Additional information

Abstract: We prove that the least-energy solution of the problem

$\begin{displaymath}\left\{ \begin{array}{ll} -d\Delta u+u=u^p\quad&\mbox{ in }B, u>0\quad&\mbox{ in }B, {{\partial u}\over{\partial\nu}}=0\quad&\mbox{ on }\partial B, \end{array}\right.\end{displaymath}$

where is a ball, and $1<p<{{N+2}\over{N-2}}$ if $N\ge 3$ , if , is unique (up to rotation) if is small enough.

References:

[AR]: A. Ambrosetti and P. H. Rabinowitz, ``Dual variational methods in critical point theory and applications'', J Funct. Anal. 14, 349-381, (1973). MR 51:6412
[BDS]: P. Bates, E.N. Dancer and J. Shi, ``Multi-spike stationary solutions on the Cahn-Hilliard equation in higher dimension and instability'', preprint.
[D]: E.N. Dancer, ``On the uniqueness of the positive solution of a singularly perturbed problem'', Rocky Mountain Journal of Mathematics 25 (1995), 957-975. MR 96j:35021
[DY]: E.N. Dancer and S. Yan, ``Multipeak solutions for a singularly perturbed Neumann problem'' (to appear).
[GNN]: B. Gidas, W.N. Ni and L. Nirenberg, ``Symmetry of positive solutions of nonlinear elliptic equation in ${\mathbb R}^N$ '', Advances in Math. Studies 7 A, 369-402, (1981). MR 84a:35083
[GT]: D. Gilbarg and N. Trudinger, ``Elliptic partial differential equations of second order'', Springer Verlag (1983). MR 86c:35035
[Gu]: C.Gui, ``Multipeak solutions for a semilinear Neumann problem'', Duke Math. J. 84, 739-769, (1996). MR 97i:35052
[K]: M.K. Kwong, ``Uniqueness of positive solutions of $\Delta u-u+u^p=0$ in ${\mathbb R}^N$ '', Arch. Rat. Mech. Anal., 105, 243-266, (1989). MR 90d:35015
[LN]: C.S. Lin and W.M. Ni, ``On the diffusion coefficient of a semilinear Neumann problem, Calculus of Variations and Partial Differential Equations'', S. Hildebrandt, D. Kinderlehrer and M. Miranda, eds., Lecture Notes in Math. 1340, Springer-Verlag, Berlin-Heidelberg-New York-Tokyo, 160-174, (1988). MR 90d:35101
[LNT]: C.S. Lin, W.M. Ni and I. Takagi, ``Large amplitude stationary solutions to a chemotaxis system'', J. Diff. Eqns. 72, 1-27, (1988). MR 89e:35075
[NT]: W.M. Ni and I. Takagi, ``On the shape of Least Energy Solutions to a Semilinear Neumann Problem'', Comm. Pure Math. Appl., Vol XLIV, 819-851, (1991). MR 92i:35052
[W1]: Z.Q. Wang, ``On the existence of multiple single-peaked solution for a semilinear Neumann problem'', Arch. Rat. Mech. Anal., 120, 375-399, (1992). MR 93k:35109
[W2]: Z.Q. Wang, ``Nonradial solutions of nonlinear Neumann problems in radially symmetric domains'', Topology in Nonlinear analysis (Warsaw, 1994), 85-96, Banach Center Publications, 35, Polish Acad. Sci., Warsaw, (1996). MR 98e:35075

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|