Existence and $L_{\infty }$ estimates of some mountain-pass type solutions

Gomes, José Maria

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

$${\left(\Phi \left(t\right)u^{\text{'}}\left(t\right)\right)}^{\text{'}}=f(t,u\left(t\right)),u^{\text{'}}\left(0\right)=u\left(1\right)=0,$$

imposing some conditions on $\Phi$ and $f$. In particular, we assume $\Phi \left(t\right)f(t,u)$ 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.

Classification : 34B18, 34C11
Keywords: second order singular differential equation, variational methods, mountain pass theorem

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