Lichnerowicz-type equations with sign-changing nonlinearities on complete manifolds with boundary (Q2407135)

The paper deals with the behaviour of positive solutions to the Lichnerowicz-type equations \[ \begin{cases} \Delta u+a(x)u-b(x)u^\sigma+c(x)u^\tau=0 & \text{on int }\,M,\\ \partial_\nu u-g(x,u)=0 & \text{on }\, \partial M \end{cases} \] over complete Riemannian manifolds \(M\) with a nonempty, possibly noncompact boundary \(\partial M\). Here, \(\Delta\) is the Laplace-Beltrami operator on \(M\), \(\partial_\nu\) is the outer normal derivative, \(\tau<1<\sigma\) are constants, the coefficients \(a,b\) and \(c\) are continuous functions in \(M\), and \(g\in C^0(\partial M\times \mathbb{R}^+)\). By using a generalization of the classical method of sub/supersolutions, the authors prove existence of positive solutions to the above problem.