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

José Maria Gomes

ESAIM: Control, Optimisation and Calculus of Variations (2009)

  • Volume: 15, Issue: 3, page 499-508
  • ISSN: 1292-8119

Abstract

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We prove the existence of a positive solution to the BVP ( Φ ( t ) u ' ( t ) ) ' = f ( t , u ( t ) ) , u ' ( 0 ) = u ( 1 ) = 0 , imposing some conditions on Φ and f. In particular, we assume Φ ( 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 bound for the solution is provided by the L norm of any test function with negative energy.

How to cite

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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

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