Bernstein-type theorem for ϕ-Laplacian

Abstract

Abstract In this paper we obtain a solution to the second-order boundary value problem of the form ddtΦ′(u˙)=f(t,u,u˙) $\frac{d}{dt}\varPhi'(\dot{u})=f(t,u,\dot{u})$, t∈[0,1] $t\in [0,1]$, u:R→R $u\colon \mathbb {R}\to \mathbb {R}$ with Sturm–Liouville boundary conditions, where Φ:R→R $\varPhi\colon \mathbb {R}\to \mathbb {R}$ is a strictly convex, differentiable function and f:[0,1]×R×R→R $f\colon[0,1]\times \mathbb {R}\times \mathbb {R}\to \mathbb {R}$ is continuous and satisfies a suitable growth condition. Our result is based on a priori bounds for the solution and homotopical invariance of the Leray–Schauder degree.

Keywords

• Φ-Laplacian
• Boundary value problem
• Fixed point
• A priori bounds
• Leray–Schauder degree