The Maximum Principle and the Existence of Principal Eigenvalues for Some Linear Weighted Boundary Value Problems

Abstract

In this work we deal with the problem of the existence and uniqueness of principal eigenvalues for some linear weighted boundary value problems associated to a general second order uniformly elliptic operator. For a large class of sign definited weights, we characterize whether the boundary value problem admits a principal eigenvalue or not.

Stability and uniqueness for cooperative degenerate Lotka–Volterra model

“…is a pair of sub-supersolutions of (21) and (22). From Theorem 6 follows the existence of (u(t, x), v(t, x)) positive solution of (21) …”

“…We refer to [21] to a biological interpretation of (21) and (22). Observe that the second terms of (21) and (22) satisfy the assumptions imposed in the previous sections.…”

“…The main result of this section is: (21) and at least a positive solution (u s , v s ) of (22). In addition, if some of the following options occurs:…”

“…is the unique positive solution of (22) and it is globally asymptotically stable. , x), v(t, x)) of (21) and at least a positive solution (u s , v s ) of (22).…”

“…, x), v(t, x)) of (21) and at least a positive solution (u s , v s ) of (22). Moreover, a positive solution (u s , v s ) of (22) u(t, x), v(t, x)) of (21) and at least a positive solution (u s , v s ) of (22). If m = 2 there exists a unique global positive solution (u(t, x), v(t, x)) of (21) and not bounded.…”