This paper is mainly concerned with the following nonlinear p-Laplacian equation-△pu(x)+(λa(x)+1)|u|^(p-2)(x)u(x)=f(x,u(x)),in V on a locally finite graph G=(V,E)with more general nonlinear term,whereΔp is the disc...This paper is mainly concerned with the following nonlinear p-Laplacian equation-△pu(x)+(λa(x)+1)|u|^(p-2)(x)u(x)=f(x,u(x)),in V on a locally finite graph G=(V,E)with more general nonlinear term,whereΔp is the discrete pLaplacian on graphs,p≥2.Under some suitable conditions on f and a(x),we can prove that the equation admits a positive solution by the Mountain Pass theorem and a ground state solution uλvia the method of Nehari manifold,for anyλ>1.In addition,asλ→+∞,we prove that the solution uλconverge to a solution of the following Dirichlet problem{-△pu(x)+|u|^(p-2)(x)u(x)=f(x,u(x)),inΩ,u(x)=0,onδΩwhereΩ={x∈V:a(x)=0}is the potential well and δΩ denotes the the boundary ofΩ.展开更多
文摘This paper is mainly concerned with the following nonlinear p-Laplacian equation-△pu(x)+(λa(x)+1)|u|^(p-2)(x)u(x)=f(x,u(x)),in V on a locally finite graph G=(V,E)with more general nonlinear term,whereΔp is the discrete pLaplacian on graphs,p≥2.Under some suitable conditions on f and a(x),we can prove that the equation admits a positive solution by the Mountain Pass theorem and a ground state solution uλvia the method of Nehari manifold,for anyλ>1.In addition,asλ→+∞,we prove that the solution uλconverge to a solution of the following Dirichlet problem{-△pu(x)+|u|^(p-2)(x)u(x)=f(x,u(x)),inΩ,u(x)=0,onδΩwhereΩ={x∈V:a(x)=0}is the potential well and δΩ denotes the the boundary ofΩ.
