Spectral-Threshold and Normalized Solutions for Nonlinear Elliptic Equations on Finite Weighted Graphs

Abstract In this paper, we study nonlinear elliptic equations on finite weighted graphs from two complementary perspectives. First, we investigate a discrete analogue of the classical Yamabe-type problem, focusing on the existence and nonexistence of spectral-threshold solutions. Secondly, we consider a Schrödinger-type equation with prescribed mass and establish the existence and nonexistence of normalized solutions in every $$L^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> -growth regime of the nonlinearity, without any restriction on the mass.