In this paper we obtain a solution to the second-order boundary value problem of the form $\frac{d}{dt}\varPhi'(\dot{u})=f(t,u,\dot{u})$ , $t\in [0,1]$ , $u\colon \mathbb {R}\to \mathbb {R}$ with Sturm–Liouville boundary conditions, where $\varPhi\colon \mathbb {R}\to \mathbb {R}$ is a strictly convex, differentiable function and $f\colon[0,1]\times \mathbb {R}\times \mathbb {R}\to \mathbb {R}$ 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.

中文翻译：

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L -Laplacian的Bernstein型定理

在本文中，我们获得了形式为$ \ frac {d} {dt} \ varPhi'（\ dot {u}）= f（t，u，\ dot {u}）的二阶边值问题的解决方案$，$ t \ in [0,1] $，$ u \冒号\ mathbb {R} \至\ mathbb {R} $具有Sturm–Liouville边界条件，其中$ \ varPhi \冒号\ mathbb {R} \ to \ mathbb {R} $是严格凸的可微函数，并且$ f \ colon [0,1] \ times \ mathbb {R} \ times \ mathbb {R} \到\ mathbb {R} $是连续的并且满足适宜的生长条件。我们的结果基于解决方案的先验边界和Leray-Schauder度的同位不变性。