Abstract
We consider a coupled system of partial differential equations involving Laplacian operator, on a rectangular domain with zero Dirichlet boundary conditions. A Lyapunov-type inequality related to this problem is derived. This inequality provides a necessary condition for the existence of nontrivial positive solutions.
Highlights
Consider the second order differential equation− u00 ( x ) = p( x )u( x ), a < x < b, (1)subject to the Dirichlet boundary condition u( a) = u(b) = 0, (2)where a, b ∈ R, a < b, and p ∈ C ([ a, b])
We reduce the study of Equation (4) to a coupled system of ordinary differential equations
It would be interesting to study other types of boundary conditions, as well as more general domains
Summary
Where a, b ∈ R, a < b, and p ∈ C ([ a, b]). It is well known (see, e.g., [1,2]) that if u ∈ C2 ([ a, b]) is a nontrivial solution to Equations (1) and (2), . In the multi-dimensional case, there are some important works dealing with Lyapunov-type inequalities for PDEs. In [9], the authors studied the Laplace equation. In [11], the authors studied the fractional p–Laplacian equation (−∆ p )s u = q( x )|u| p−2 u, x ∈ Ω, under the boundary conditions u( x ) = 0, x ∈ R N \Ω, where Ω is an open bounded domain in R N (N ≥ 2), p > 1, 0 < s < 1 and q ∈ L∞ (Ω). ( x, y) ∈ Ω, under Dirichlet boundary conditions, where Ω =] a, b[×O , ( a, b) ∈ R2 , a < b, O is an open bounded subset in R N (N ≥ 1), q ∈ C ([ a, b]) and Gγ , γ ≥ 0, is the differential operator given by.
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