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

Gomes, José Maria. "Existence and L∞ estimates of some Mountain-Pass type solutions." ESAIM: Control, Optimisation and Calculus of Variations 15.3 (2009): 499-508. <http://eudml.org/doc/250569>.

@article{Gomes2009,
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. },
author = {Gomes, José Maria},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Second order singular differential equation; variational methods; Mountain Pass Theorem; second order singular differential equation; mountain pass theorem; estimates; elliptic problem in annulus},
language = {eng},
month = {7},
number = {3},
pages = {499-508},
publisher = {EDP Sciences},
title = {Existence and L∞ estimates of some Mountain-Pass type solutions},
url = {http://eudml.org/doc/250569},
volume = {15},
year = {2009},
}

TY - JOUR
AU - Gomes, José Maria
TI - Existence and L∞ estimates of some Mountain-Pass type solutions
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2009/7//
PB - EDP Sciences
VL - 15
IS - 3
SP - 499
EP - 508
AB - 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.
LA - eng
KW - Second order singular differential equation; variational methods; Mountain Pass Theorem; second order singular differential equation; mountain pass theorem; estimates; elliptic problem in annulus
UR - http://eudml.org/doc/250569
ER -

References

top

H. Berestycki, P.L. Lions and L.A. Peletier, An ODE approach to the existence of positive solutions for semilinear problems in $ℝ^{N}$ . Indiana Univ. Math. J.30 (1981) 141–157.
L.E. Bobisud and D. O'Regan, Positive solutions for a class of nonlinear singular boundary value problems at resonance. J. Math. Anal. Appl.184 (1994) 263–284.
D. Bonheure, J.M. Gomes and P. Habets, Multiple positive solutions of a superlinear elliptic problem with sign-changing weight. J. Diff. Eq.214 (2005) 36–64.
C. De Coster and P. Habets, Two-point boundary value problems: lower and upper solutions, Mathematics in Science Engineering205. Elsevier (2006).
M. del Pino, P. Felmer and J. Wei, Multi-peak solutions for some singular perturbation problems. Calc. Var. Partial Differential Equations10 (2000) 119–134.
J.M. Gomes, Existence and $L_{\infty}$ estimates for a class of singular ordinary differential equations. Bull. Austral. Math. Soc.70 (2004) 429–440.
L. Malaguti and C. Marcelli, Existence of bounded trajectories via lower and upper solutions. Discrete Contin. Dynam. Systems6 (2000) 575–590.
D. O'Regan, Solvability of some two point boundary value problems of Dirichlet, Neumann, or periodic type. Dynam. Systems Appl.2 (1993) 163–182.
D. O'Regan, Nonresonance and existence for singular boundary value problems. Nonlinear Anal.23 (1994) 165–186.
P.H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conference Series in Mathematics65. American Mathematical Society, Providence, USA (1986).

NotesEmbed ?

top

You must be logged in to post comments.