Uniqueness of the least-energy solution for a semilinear Neumann problem
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Abstract:
We prove that the least-energy solution of the problem \[ \begin {cases} -d\Delta u+u=u^p\quad & \text {in $B$},\\ u>0 & \text {in $B$},\\ \frac {\partial u}{\partial \nu } = 0 & \text {on $\partial B$}, \end {cases} \] where $B$ is a ball, $d>0$ and $1<p<{{N+2}\over {N-2}}$ if $N\ge 3$, $p>1$ if $N=2$, is unique (up to rotation) if $d$ is small enough.References
- Antonio Ambrosetti and Paul H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis 14 (1973), 349–381. MR 0370183, DOI 10.1016/0022-1236(73)90051-7
- P. Bates, E.N. Dancer and J. Shi, “Multi-spike stationary solutions on the Cahn-Hilliard equation in higher dimension and instability”, preprint.
- E. N. Dancer, On the uniqueness of the positive solution of a singularly perturbed problem, Rocky Mountain J. Math. 25 (1995), no. 3, 957–975. MR 1357103, DOI 10.1216/rmjm/1181072198
- E.N. Dancer and S. Yan, “Multipeak solutions for a singularly perturbed Neumann problem” (to appear).
- B. Gidas, Wei Ming Ni, and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\textbf {R}^{n}$, Mathematical analysis and applications, Part A, Adv. in Math. Suppl. Stud., vol. 7, Academic Press, New York-London, 1981, pp. 369–402. MR 634248
- 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, DOI 10.1007/978-3-642-61798-0
- Changfeng Gui, Multipeak solutions for a semilinear Neumann problem, Duke Math. J. 84 (1996), no. 3, 739–769. MR 1408543, DOI 10.1215/S0012-7094-96-08423-9
- 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
- Chang Shou Lin and Wei-Ming Ni, On the diffusion coefficient of a semilinear Neumann problem, Calculus of variations and partial differential equations (Trento, 1986) Lecture Notes in Math., vol. 1340, Springer, Berlin, 1988, pp. 160–174. MR 974610, DOI 10.1007/BFb0082894
- C.-S. Lin, W.-M. Ni, and I. Takagi, Large amplitude stationary solutions to a chemotaxis system, J. Differential Equations 72 (1988), no. 1, 1–27. MR 929196, DOI 10.1016/0022-0396(88)90147-7
- Wei-Ming Ni and Izumi Takagi, On the shape of least-energy solutions to a semilinear Neumann problem, Comm. Pure Appl. Math. 44 (1991), no. 7, 819–851. MR 1115095, DOI 10.1002/cpa.3160440705
- Zhi Qiang Wang, On the existence of multiple, single-peaked solutions for a semilinear Neumann problem, Arch. Rational Mech. Anal. 120 (1992), no. 4, 375–399. MR 1185568, DOI 10.1007/BF00380322
- Zhi-Qiang Wang, Nonradial solutions of nonlinear Neumann problems in radially symmetric domains, Topology in nonlinear analysis (Warsaw, 1994) Banach Center Publ., vol. 35, Polish Acad. Sci. Inst. Math., Warsaw, 1996, pp. 85–96. MR 1448428
Additional Information
- Massimo Grossi
- Affiliation: Dipartimento di Matematica, Università di Roma “La Sapienza", P.le A. Moro 2, 00185, Roma, Italy
- Email: grossi@mat.uniroma1.it
- Received by editor(s): July 9, 1998
- Published electronically: October 18, 1999
- Additional Notes: This research was supported by M.U.R.S.T. (Project “Metodi Variazionali ed Equazioni Differenziali Non Lineari”)
- Communicated by: Lesley M. Sibner
- © Copyright 2000 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 128 (2000), 1665-1672
- MSC (1991): Primary 35J70
- DOI: https://doi.org/10.1090/S0002-9939-99-05491-X
- MathSciNet review: 1694340