Method of Lower and Upper Solutions for Elliptic Systems with Nonlinear Boundary Condition and Its Applications

We develop the method of lower and upper solutions for a class of elliptic systems with nonlinear boundary conditions. As its application, an elliptic system modeling a population divided into juvenile and adult age groups is studied, and we find sufficient conditions in terms of the principal eigenvalue of the corresponding linearized system, to guarantee the existence of coexistence states of the above juvenile-adult model.

System (1) with  = 1 arises, in particular, in the study of steady state solutions of nonlinear parabolic equations of the form   +  =  (, ) in Ω × (0, ∞) ,  n +  ()  =  (, ) on Γ × (0, ∞) , where  is some second order, strongly uniformly elliptic differential operator.In this connection, nonlinear boundary conditions seem to be of particular importance.For the study of the stability of the solutions of the parabolic initial-boundary value problem (3), one has to have a good knowledge of the steady states, that is, of the solutions of (1) with  = 1.In the past few decades, the theory of monotone operators has been applied to boundary value problems of the form (3); see, for example, [1][2][3] and the references therein.In all of the above-mentioned papers, the boundary condition is of the special form /v = (), where  is decreasing and v is the conormal with respect to the differential operator .
Besides these results, there are some scattered existence theorems for nonlinear Stecklov problems of the form where  is supposed to be formally self-adjoint such that the homogeneous linear boundary value problem possesses a nontrivial solution; we refer the readers here to [4,5] and the references therein.The stationary version of (3), which covers the above-mentioned several situations, has been studied by several authors; see Amann [6,7] and Hess [8].In particular, Amann [6] studied the stationary version of (3) and obtained a general existence theorem for it, namely, the result that the existence of a subsolution and a supersolution guarantees the existence of a solution.By transforming the stationary version of (3) into an equivalent fixed point equation in (Ω), he gave a new and more elegant proof for the above result.Motivated by the above work, we will develop the method of lower and upper solutions for system (1) under the following assumptions: continuous in the first variable and locally Lipschitz in the second variable;   : Γ × R  → R is locally Lipschitz continuous; (H2) For  = 1, . . ., ,   > 0 in Ω, and   ∈  1+ (Γ) satisfying   > 0 on Γ.

Lower and Upper Solutions Method
has a unique solution Hence the Schauder estimates and the   -estimates take the form respectively.Here and in the following  denotes a positive constant (not necessarily the same in different formulas) which is independent of the functions appearing in these estimates.Hence (8) implies that  : for every ]  , the estimate (10) implies that  has a unique continuous extension for each  ∈ (1, ∞), denoted again by , such that  is also a bounded linear operator from .Now, let  and  be the Nemytskii operators generated by the vector fields ( 1 , . . .,   ) and ( 1 , . . .,   ), respectively.Here and in the following we denote by  an arbitrary, but fixed real number satisfying  > .This implies in particular that where  denotes the usual trace operator.Then  can be considered as a mapping of [, ] into [(Ω)]  .It is obvious that every solution of the system (1) is a fixed point of .

Conversely, if 𝑢 ∈ [𝐶(Ω)]
is a fixed point of , then we can show that  is also a solution of the system (1) by using the same methods as in the proof of Amann Proof of Theorem 2. The regularity assumption (H1) for   and   implies the existence of positive constants   ( = 1, 2, . . ., ) such that Let   =   −   .Then we can easily conclude from ( 12), ( 14), (18), and (19) that From the maximum principle for elliptic boundary value problems it follows that   ≥ 0; that is,   ≤   .Similarly, by using ( 13) and ( 15), we can obtain   ≤   .Consequently,  maps [, ] into itself, and the existence of a fixed point follows from Schauder fixed point theorem.

Application to a Juvenile-Adult Model
In this section, we will apply the method of lower and upper solutions in Section 2 to study the existence of coexistence states of the following elliptic system describing two subpopulations of the same species competing for resources: System (21) arises from population dynamics where it models the steady-state solutions of the corresponding nonlinear evolution problem [12], where  and V represent the concentrations of the adult and juvenile populations, respectively.The function  gives the rate at which juveniles become adults and  corresponds to the death rate of adult population.As adults give birth to juveniles, the function  corresponds to the birth rate of the population.Juveniles are lost both through death and through becoming adults, the function  corresponds to this overall loss.The Laplacian operator shows the diffusive character of  and V within Ω.By using fixed point theory and lower and upper solutions method, several authors have studied the existence of coexistence states of the system (21), subject to Dirichlet or Neumann boundary conditions; see, for example, [13][14][15][16] and the references therein.We consider here the more general model (21), in which the boundary conditions may be interpreted as the conditions that the populations may pass through the boundary of the habitat.This is a mathematical model more closer to the reality.
Proof.We may get the desired results by using Theorem 9 and choosing  0 ≡ V 0 ≡ , where  is any positive constant.