Solvability of second-order uniformly elliptic inequalities involving demicontinuous $$\psi $$-dissipative operators and applications to generalized population models

Abstract

Based on a new developed theory for variational inequalities, the purpose of this article is to investigate existence and uniqueness of nonzero positive weak solutions for a class of general second-order uniformly elliptic inequalities with demicontinuous $$\psi $$ -dissipative operators in reflexive smooth Banach spaces and a generalized second-order elliptic inequality, which are often regarded as some conditions allowing the species to survive in biology or ecological context with heterogeneous character. Further, we also discuss existence of eigenvalues for the uniformly elliptic inequality and present a generalized population model arising in ecological context to verify the availability and significance of our main results. Finally, we describe two kinds of open questions, which are future work of our research.