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

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

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

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topGomes, 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 ${\mathbb{R}}^{N}$. 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).

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