Existence of Solutions for Nonlinear Choquard Equations with (p, q)-Laplacian on Finite Weighted Lattice Graphs - Dandan Yang; Zhenyu Bai; Chuanzhi Bai (Axioms)
In this paper, we consider the <semantics>(p,q)</semantics>-Laplacian Choquard equation on a finite weighted lattice graph <semantics>G=(KN,E,μ,ω)</semantics>, namely for any <semantics>1<p<q<N</semantics>, <semantics>r>1</semantics> and <semantics>0<α<N</semantics>, <semantics>−Δpu−Δqu+V(x)(|u|p−2u+|u|q−2u)</semantics><semantics>=<mfenced separators="" open="(" close=")"><mstyle displaystyle="true"><munder>∑y∈KN,y≠x</munder></mstyle><mstyle scriptlevel="0" displaystyle="true">|u(y)|rd(x,y)N−α</mstyle></mfenced>|u|r−2u,</semantics> where <semantics>Δν</semantics> is the discrete <semantics>ν</semantics>-Laplacian on graphs, and <semantics>ν∈{p.q}</semantics>, <semantics>V(x)</semantics> is a positive function. Under some suitable conditions on r, we prove that the above equation has both a mountain pass solution and ground state solution. Our research relies on the mountain pass theorem and the method of the Nehari manifold. The results obtained in this paper are extensions of some known studies.