Using the Mountain Pass Theorem we show that the problem \begin{equation*} \begin{cases} \frac{d}{dt}\mathcal{L}_v(t,u(t),\dot u(t))=\mathcal{L}_x(t,u(t),\dot u(t))\quad \text{ for a.e. }t\in[a,b]\\ u(a)=u(b)=0 \end{cases} \end{equation*} has a solution in anisotropic Orlicz-Sobolev space. We consider Lagrangian $\mathcal{L}=F(t,x,v)+V(t,x)+\langle f(t), x\rangle$ with growth condition determined by anisotropic G-function and some geometric condition of Ambrosetti-Rabinowitz type.

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具有一般各向异性算子的欧拉-拉格朗日方程的山口解

使用山口定理，我们证明问题 \begin{equation*} \begin{cases} \frac{d}{dt}\mathcal{L}_v(t,u(t),\dot u(t)) =\mathcal{L}_x(t,u(t),\dot u(t))\quad \text{ for ae }t\in[a,b]\\ u(a)=u(b)= 0 \end{cases} \end{equation*} 在各向异性 Orlicz-Sobolev 空间有解。我们考虑拉格朗日 $\mathcal{L}=F(t,x,v)+V(t,x)+\langle f(t), x\rangle$ 的增长条件由各向异性 G 函数和一些几何条件决定Ambrosetti-Rabinowitz 型。