Positive Solutions for Sublinear Equations with Compact Positivity-Improving Resolvent

Abstract

We establish a Brézis-Oswald-type spectral principle for semilinear equations \[ -Lu=f(x,u), \] where $L$ is the generator of a positive $C_0$-semigroup on $L^p$ with compact positivity-improving resolvent. Neither symmetry, variational structure, nor regularizing properties such as ultracontractivity or smoothing are assumed. Let $a_0$ and $a_{\infty }$ denote the asymptotic slopes of the nonlinearity at zero and at infinity, and let ${\lambda }_1(a)$ denote the generalized principal eigenvalue associated with the perturbed operator $-L-a$. We prove that \[ λ_1(a_0)<0<λ_1(a_\infty) \] implies that the equation admits a strictly positive solution. The proof develops a potential-theoretic sub- and supersolution framework based on the order induced by supermedian functions and combines a Deny-type compactness theorem, a Kato-type inequality, and a Doob transform. The construction also yields an order-preserving selection of solutions and allows the lower control on the nonlinearity to be relaxed. If $y\mapsto f(x,y)/y$ is strictly decreasing and $a_0$ is bounded, the spectral condition is necessary as well, and the strictly positive solution is unique.