We will be concerned with the existence of homoclinics for Lagrangian systems in \({\mathbb {R}}^N\) (\(N\ge 3 \)) of the form \(\frac{d}{dt}\left( \nabla \Phi (\dot{u}(t))\right) +\nabla _{u}V(t,u(t))=0\), where \(t\in {\mathbb {R}}\), \(\Phi {:}\,{\mathbb {R}}^N\rightarrow [0,\infty )\) is a G-function in the sense of Trudinger, \(V{:}\,{\mathbb {R}}\times \left( {\mathbb {R}}^N{\setminus }\{\xi \} \right) \rightarrow {\mathbb {R}}\) is a \(C^2\)-smooth potential with a single well of infinite depth at a point \(\xi \in {\mathbb {R}}^N{\setminus }\{0\}\) and a unique strict global maximum 0 at the origin. Under a strong force type condition around the singular point \(\xi \), we prove the existence of a homoclinic solution \(u{:}\,{\mathbb {R}}\rightarrow {\mathbb {R}}^N{\setminus }\{\xi \}\) via minimization of an action integral.

我们将与homoclinics为拉格朗日系统中存在关注\（{\ mathbb {R}} ^ N \） （\（N \ GE 3 \） ）形式的\（\压裂{d} {DT} \ left（\ nabla \ Phi（\ dot {u}（t））\ right）+ \ nabla _ {u} V（t，u（t））= 0 \），其中\（t \ in {\ mathbb { R}} \），\（\ Phi {：} \，{\ mathbb {R}} ^ N \ rightarrow [0，\ infty）\）在Trudinger的意义上是G函数，\（V {： } \，{\ mathbb {R}} \ times \ left（{\ mathbb {R}} ^ N {\ setminus} \ {\ xi \} \ right）\ rightarrow {\ mathbb {R}} \）是\（C ^ 2 \）-在点\（\ xi \ in {\ mathbb {R}} ^ N {\ setminus} \ {0 \} \）上具有无限深度的单孔的平滑势并且在原点有一个唯一的严格全局最大值0。在奇异点\（\ xi \）周围的强力类型条件下，我们证明了同宿解\（u {：} \，{\ mathbb {R}} \ rightarrow {\ mathbb {R}} ^的存在N {\ setminus} \ {\ xi \} \）通过最小化动作积分。