Mountain pass solutions to Euler-Lagrange equations with general anisotropic operator

2019/03/17 by Chmara, M., Maksymiuk, J.
#46E30 #46E40 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.1903.07150

Abstract

Using the Mountain Pass Theorem we show that the problem \begincases (d)/(dt)L_v(t,u(t), u(t))=L_x(t,u(t), u(t)) for a.e. t∈[a,b]
u(a)=u(b)=0 \endcases has a solution in anisotropic Orlicz-Sobolev space. We consider Lagrangian L=F(t,x,v)+V(t,x)+⟨ f(t), x⟩ with growth condition determined by anisotropic G-function and some geometric condition of Ambrosetti-Rabinowitz type.

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