Existence and L 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

(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.

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},
     mrnumber = {2542569},
     language = {en},
     url = {http://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/

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