Existence and ${L}_{\infty }$ estimates of some mountain-pass type solutions
ESAIM: Control, Optimisation and Calculus of Variations, Tome 15 (2009) no. 3, pp. 499-508.

We prove the existence of a positive solution to the BVP

 ${\left(\Phi \left(t\right){u}^{\text{'}}\left(t\right)\right)}^{\text{'}}=f\left(t,u\left(t\right)\right),\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{u}^{\text{'}}\left(0\right)=u\left(1\right)=0,$
imposing some conditions on $\Phi$ and $f$. In particular, we assume $\Phi \left(t\right)f\left(t,u\right)$ 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.

DOI : https://doi.org/10.1051/cocv/2009015
Classification : 34B18,  34C11
Mots clés : second order singular differential equation, variational methods, mountain pass theorem
@article{COCV_2009__15_3_499_0,
author = {Gomes, Jos\'e Maria},
title = {Existence and $L\_\infty$ estimates of some mountain-pass type solutions},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {499--508},
publisher = {EDP-Sciences},
volume = {15},
number = {3},
year = {2009},
doi = {10.1051/cocv/2009015},
zbl = {pre05589848},
mrnumber = {2542569},
language = {en},
url = {www.numdam.org/item/COCV_2009__15_3_499_0/}
}
Gomes, José Maria. Existence and $L_\infty$ estimates of some mountain-pass type solutions. ESAIM: Control, Optimisation and Calculus of Variations, Tome 15 (2009) no. 3, pp. 499-508. doi : 10.1051/cocv/2009015. http://www.numdam.org/item/COCV_2009__15_3_499_0/

[1] 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. | MR 600039 | Zbl 0522.35036

[2] 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. | MR 1278388 | Zbl 0805.34019

[3] 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. | MR 2143511 | Zbl pre02189523

[4] C. De Coster and P. Habets, Two-point boundary value problems: lower and upper solutions, Mathematics in Science Engineering 205. Elsevier (2006). | MR 2225284 | Zbl pre05023569

[5] M. Del Pino, P. Felmer and J. Wei, Multi-peak solutions for some singular perturbation problems. Calc. Var. Partial Differential Equations 10 (2000) 119-134. | MR 1750734 | Zbl 0974.35041

[6] J.M. Gomes, Existence and ${L}_{\infty }$ estimates for a class of singular ordinary differential equations. Bull. Austral. Math. Soc. 70 (2004) 429-440. | MR 2103974 | Zbl 1083.34016

[7] L. Malaguti and C. Marcelli, Existence of bounded trajectories via lower and upper solutions. Discrete Contin. Dynam. Systems 6 (2000) 575-590. | MR 1757388 | Zbl 0979.34019

[8] D. O'Regan, Solvability of some two point boundary value problems of Dirichlet, Neumann, or periodic type. Dynam. Systems Appl. 2 (1993) 163-182. | MR 1226995 | Zbl 0785.34025

[9] D. O'Regan, Nonresonance and existence for singular boundary value problems. Nonlinear Anal. 23 (1994) 165-186. | MR 1289125 | Zbl 0827.34010

[10] 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). | MR 845785 | Zbl 0609.58002