Hostname: page-component-76fb5796d-22dnz Total loading time: 0 Render date: 2024-04-27T15:34:11.324Z Has data issue: false hasContentIssue false
  • English
  • Français

Existence and L∞ estimates of some Mountain-Pass type solutions

Published online by Cambridge University Press:  02 July 2009

José Maria Gomes
José Maria Gomes*
Affiliation:
CMAF-Centro de Matemática e Aplicações Fundamentais, 2 Avenida Professor Gama Pinto, 1649-003 Lisboa, Portugal. jgomes@math.ist.utl.pt
Get access

Abstract

We prove the existence of a positive solution to the BVP $$(\Phi(t)u'(t))'=f(t,u(t)),\,\,\,\,\,\,\,\,\,\,\,u'(0)=u(1)=0, $$ imposing some conditions on Φ and f. In particular, we assume $\Phi(t)f(t,u)$ to be decreasing in t. Our method combines variational and topological arguments and can be applied to some elliptic problems in annular domains. An $L_\infty$ bound for the solution is provided by the $L_\infty$ norm of any test function with negative energy.

Keywords

Second order singular differential equationvariational methodsMountain Pass Theorem
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 15 , Issue 3 , July 2009 , pp. 499 - 508
Copyright
© EDP Sciences, SMAI, 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Berestycki, H., Lions, P.L. and Peletier, L.A., ODE, An approach to the existence of positive solutions for semilinear problems in $\mathbb{R}^N$ . Indiana Univ. Math. J. 30 (1981) 141–157. CrossRef
Bobisud, L.E. and O'Regan, D., Positive solutions for a class of nonlinear singular boundary value problems at resonance. J. Math. Anal. Appl. 184 (1994) 263–284. CrossRef
Bonheure, D., Gomes, J.M. and Habets, P., Multiple positive solutions of a superlinear elliptic problem with sign-changing weight. J. Diff. Eq. 214 (2005) 36–64. CrossRef
C. De Coster and P. Habets, Two-point boundary value problems: lower and upper solutions, Mathematics in Science Engineering 205. Elsevier (2006).
del Pino, M., Felmer, P. and Wei, J., Multi-peak solutions for some singular perturbation problems. Calc. Var. Partial Differential Equations 10 (2000) 119–134. CrossRef
Gomes, J.M., Existence and $L_\infty$ estimates for a class of singular ordinary differential equations. Bull. Austral. Math. Soc. 70 (2004) 429–440. CrossRef
Malaguti, L. and Marcelli, C., Existence of bounded trajectories via lower and upper solutions. Discrete Contin. Dynam. Systems 6 (2000) 575–590.
O'Regan, D., Solvability of some two point boundary value problems of Dirichlet, Neumann, or periodic type. Dynam. Systems Appl. 2 (1993) 163–182.
O'Regan, D., Nonresonance and existence for singular boundary value problems. Nonlinear Anal. 23 (1994) 165–186. CrossRef
P.H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conference Series in Mathematics 65. American Mathematical Society, Providence, USA (1986).