A Lyapunov-Type Inequality for a Laplacian System on a Rectangular Domain with Zero Dirichlet Boundary Conditions

: 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.


Introduction
Consider the second order differential equation subject to the Dirichlet boundary condition where a, b ∈ R, a < b, and p ∈ C([a, b]). It is well known (see, e.g., [1,2]) that if u ∈ C 2 ([a, b]) is a nontrivial solution to Equations (1) and (2), The inequality in Equation (3) is known in the literature as Lyapunov inequality. It has many applications in the study of spectral properties of ODE (see, e.g., [3][4][5][6][7][8]).
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 under Dirichlet boundary conditions, where Ω is an open bounded domain in R N (N ≥ 2), p > 1 and q ∈ L s (Ω), for some s, which depends on p and N. 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 where Ω is an open bounded domain in R N (N ≥ 2), p > 1, 0 < s < 1 and q ∈ L ∞ (Ω). In [12], the authors extended the obtained results in [11] to a fractional p-Laplacian system. In [13], the authors studied the partial differential equation b]) and G γ , γ ≥ 0, is the differential operator given by The authors of [14] extended the obtained results in [13] to the differential operator where g ∈ C([a, b]) and ∆ y is the Laplacian operator with respect to the variable "y". Motivated by the above cited works, in this paper, we consider the two-dimensional coupled system of partial differential equations under the Dirichlet boundary conditions where and Γ is the Gamma function. We derive a Lyapunov-type inequality, which provides a necessary condition for the existence of nontrivial positive solutions to Equations (4) and (5). Note that the used technique in this paper is different to that used in [11,12]. The approach used in this paper is based on an eigenvalue method from Kaplan [15]. Observe that the system in Equation (4) involves the nonlocal where κ ∈ {α, β} and f ∈ {u(·, y), v(·, y)}. Such operators are known in the literature as Riemann-Liouville fractional integrals of order κ. For more details on fractional operators and their applications, see, for example, [16,17].
The rest of the paper is organized as follows. In Section 2, we recall and prove some results on matrices theory that are used in the proof of our main result. In Section 3, we establish a Lyapunov-type inequality for Equations (4) and (5), and we discuss some special cases of Equations (4) and (5).

Preliminaries
Let N ≥ 1 be a natural number. We denote by 0 N the zero vector in R N . Let · N be the Euclidean norm in R N , that is, We define in R N the partial order ≤ N given by It can be easily seen that We denote by M N (R) the set of square matrices of size N with coefficients in R, and by M N (R + ) the subset of M N (R) with positive coefficients. We endow M N (R) with the subordinate matrix norm For a given matrix M ∈ M N (R), let ρ(M) be its spectral radius, i.e., where λ i (M), i = 1, 2, · · · , N, are the (real or complex) eigenvalues of M.
The following result is standard in the theory of matrices.
Proof. From Equation (6) and using the fact that M ∈ M N (R + ), for all natural number n ≥ 1, we have Next, by Lemma 1, we obtain Finally, using Lemma 2, Equation (7) follows.
Proof. Let P M be the characteristic polynomial of the matrix M, that is, Observe that Then, in all cases, we have ρ(M) = λ 1 (M), which proves the desired result.

Lyapunov-Type Inequalities
In this section, a Lyapunov-type inequality is derived for Equations (4) and (5) and some special cases are discussed. (iii) u ≡ 0 and v ≡ 0.

From PDEs to ODEs
In this subsection, we reduce the study of Equation (4) to a coupled system of ordinary differential equations.
Suppose that (u, v) is a nontrivial positive solution to Equations (4) and (5). Let us introduce the functions and v Observe that due to the positivity of the function ϕ in [c, d], (ii) and (iii), we have u ≡ 0 and v ≡ 0. Further, multiplying the first equation in Equation (4) by ϕ(y) and integrating over (c, d), we obtain for all a < x < b. On the other hand, using an integration by parts, we obtain Again, using an integration by parts and Equation (5), we obtain Next, using Fubini's theorem, we have Combining Equations (10), (11) and (12), we obtain Similarly, multiplying the second equation in Equation (4) by ϕ(y) and integrating over (c, d), we obtain Moreover, using the boundary conditions in Equation (5), we have Hence, we have the following result. Proposition 1. Let (u, v) be a nontrivial positive solution to Equations (4) and (5). Then, (u, v) is a nontrivial solution to the coupled system subject to the boundary conditions where u and v are given by Equations (8) and (9).

Main Result
Let us introduce the matrix M = (m ij ) 1≤i,j≤2 ∈ M 2 (R + ) given by Now, we are able to state and prove our main result.
Proof. Let (u, v) be a nontrivial positive solution to Equations (4) and (5). By Proposition 1, we have where u ≡ 0 and Therefore, u is a solution to the integral equation where On the other hand, the arithmetic-geometric-harmonic mean inequality yields Further, let us estimate the term | f (s, u, v)|, a < s < b. We have where Moreover, we have Next, combining Equations (18) and (19), we get Now, combining Equations (16), (17) and (20), we deduce that Similarly, by Proposition 1, we have where v ≡ 0 and Using a similar argument as above, we obtain Using Equations (21) and (22), we deduce that Hence, by Lemma 3, we deduce that Finally, using Lemma 4, Equation (15) follows.

Particular Cases
In this subsection, we discuss some special cases following from Theorem 1.