Homoclinics for singular strong force Lagrangian systems in $${\mathbb {R}}^N$$

Abstract

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