Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

Journals Home Search My Subscriptions Subscribe
Previous issue | This issue | Most recent issue | All issues (1950–Present) | Next issue | Previous article | Articles in press | Recently published articles | Next article
Request Permissions
 
 

 

A remark on least energy solutions in $\mathbf{R}^N$


Authors: Louis Jeanjean and Kazunaga Tanaka
Journal: Proc. Amer. Math. Soc. 131 (2003), 2399-2408
MSC (2000): Primary 35J20; Secondary 35J60, 58E05
DOI: https://doi.org/10.1090/S0002-9939-02-06821-1
Published electronically: November 13, 2002
MathSciNet review: 1974637
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We study a mountain pass characterization of least energy solutions of the following nonlinear scalar field equation in $\mathbf{R}^N$:

\begin{displaymath}-\Delta u = g(u),\, u \in H^1(\mathbf{R}^N), \end{displaymath}

where $N\geq 2$. Without the assumption of the monotonicity of $t\mapsto \frac{g(t)}{t}$, we show that the mountain pass value gives the least energy level.

References [Enhancements On Off] (What's this?)

  • [AT:1] S. Adachi and K. Tanaka, Trudinger type inequalities in ${\mathbf R}^N$ and their best exponents, Proc. Amer. Math. Soc. 128 (2000), 2051-2057. MR 2000m:46069
  • [BL:2] H. Berestycki and P. L. Lions, Nonlinear scalar field equations I, Arch. Rat. Mech. Anal. 82 (1983), 313-346. MR 84h:35054a
  • [BGK:3] H. Berestycki, T. Gallouët and O. Kavian, Equations de Champs scalaires euclidiens non linéaires dans le plan. C. R. Acad. Sci; Paris Ser. I Math. 297 (1983), 5, 307-310 and Publications du Laboratoire d'Analyse Numérique, Université de Paris VI, (1984). MR 85e:35041
  • [CGM:4] S. Coleman, V. Glaser and A. Martin, Action minima among solutions to a class of Euclidean scalar field equations, Comm. Math. Phys. 58 (1978), 211-221. MR 57:8716
  • [DF:5] M. del Pino and P. Felmer, Local mountain passes for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differential Equations 4 (1996), 121-137. MR 97c:35057
  • [JT:6] L. Jeanjean and K. Tanaka, Singularly perturbed elliptic problems with superlinear or asymptotically linear nonlinearities, preprint.
  • [L:7] P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. Part I and II. Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), 109-145 and 223-283. MR 87e:49035a; MR 87e:49035b
  • [NT:8] W.-M. Ni and I. Takagi, On the shape of least-energy solutions to a semilinear Neumann problem, Comm. Pure Appl. Math. 44 (1991), 819-851. MR 92i:35052

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 35J20, 35J60, 58E05

Retrieve articles in all journals with MSC (2000): 35J20, 35J60, 58E05

Additional Information

Louis Jeanjean
Affiliation: Equipe de Mathématiques (UMR CNRS 6623), Université de Franche-Comté, 16 Route de Gray, 25030 Besançon, France
Email: jeanjean@math.univ-fcomte.fr

Kazunaga Tanaka
Affiliation: Department of Mathematics, School of Science and Engineering, Waseda University, 3-4-1 Ohkubo, Skinjuku-ku, Tokyo 169-8555, Japan
Email: kazunaga@mn.waseda.ac.jp, kazunaga@waseda.jp

DOI: https://doi.org/10.1090/S0002-9939-02-06821-1
Keywords: Nonlinear elliptic equations in $\mathbf{R}^N$, least energy solutions, mountain pass theorem
Received by editor(s): March 6, 2002
Published electronically: November 13, 2002
Additional Notes: The second author was partially supported by a Waseda University Grant for Special Research Projects 2001A-098.
Communicated by: David S. Tartakoff
Article copyright: © Copyright 2002 American Mathematical Society