The antimaximum principle for the homogeneous Dirichlet problem to the model equation −∆u = λu + f (x) in a smooth bounded domain and with positive f states the existence of a critical value λf > λ1 such that the solution of this problem with λ ∈ (λ1, λf ) is strictly negative. This contrasts with the case λ λ1 in which the solution is strictly positive, i.e., the maximum principle holds.
We will discuss main ideas behind the antimaximum principle, its generalizations, and bounds on λf , some of which are connected with the behavior of energy of solutions in the interval (λ1, λ2). The talk will be based on the works [Bobkov, Dr´aber, Ilyasov, 2020] and [Bobkov, Tanaka, 2023].