In this talk, our goal is to discuss the existence of at least two distinct conformal metrics with prescribed gaussian curvature and geodesic curvature respectively, $K_{g}= f + \lambda$ and $k_{g}= h + \mu$, where f and h are nonpositive functions and \lambda and \mu are positive constants. Utilizing Struwe's monotonicity trick, we investigate the blowup behavior of the solutions and establish a non-existence result for the limiting PDE, eliminating one of the potential blow-up profiles.