Non-local Degenerate Diffusion Coefficients Break Down the Components of Positive Solutions

1 Introduction

We are interested in the structure of the set of positive solutions of the equation

where Ω⊂ℝN, N≥1, is a bounded and smooth domain, p>0, f:ℝ×Ω×ℝ→ℝ is a nonlinear function that will be detailed along the paper, and the non-local diffusion coefficient a:ℝ+→[0,∞) is a continuous function, with a⁢(0)≥0, that verifies the following hypothesis (which will be assumed throughout the work):

1. There exists s1>0 such that a⁢(s1)=0 and a⁢(s)>0 for all s>0 and s≠s1.

Equation (1.1) has received a lot of attention in the last years, especially when the function a is far away from zero, see, for example, [5, 6, 8, 14] and references therein. In some papers, for instance, [2] and [9], the hypotheses on a were weakened and it was allowed that

Recently, in [1] and [12], it was assumed that a can vanish in several points. In [12], by considering the same sort of equations as in (1.1), with p>1, and a reaction term similar (but not equal) to a logistic equation, Gasiński and Santos Júnior prove the existence of at least 2⁢K positive solutions if the function a vanishes in K positive points and satisfies some appropriate additional conditions. The approach in [1] is based on fixed point techniques, sub-super solution arguments and variational methods.

In this paper, we study (1.1) for each of the following reaction terms:

1. Linear case: f⁢(λ,x,u)=f⁢(x), where f is a bounded, non-negative and non-trivial function.

2. Eigenvalue problem: f⁢(λ,x,u)=λ⁢u, where λ∈ℝ.

3. Sublinear case: f⁢(λ,x,u)=λ⁢uq, where λ∈ℝ and q∈(0,1).

4. Logistic case: f⁢(λ,x,u)=λ⁢u-u2, where λ∈ℝ.

We start with the first three cases. In the local case, that is, a≡1 in (1.1), the problems are well understood. In the linear case, there exists a unique positive solution. In the eigenvalue case, there exists a increasing sequence {λn}n∈ℕ of eigenvalues with λn→+∞; the first one, λ1, called principal eigenvalue, is the only one having an associated positive eigenfunction. Finally, it is also well known that the sublinear problem possesses a positive solution if and only if λ>0. In the case λ>0, there exists at most one positive solution.

In the non-local case, in the first three cases, basically we can compute directly the solutions. In these cases, the main result can be summarized as follows: the study of equation (1.1) is equivalent to the study of the following nonlinear real equation in R>0:

1. a ⁢ ( R ) ⁢ R 1 / p = A in the linear case,

2. a ⁢ ( R ) = λ λ 1 in the eigenvalue problem,

3. a ⁢ ( R ) ⁢ R 1 - q p = λ ⁢ B in the sublinear case,

where A and B are positive constants related to f and q, respectively (see Section 2 for their concrete values) and λ1 is the principal eigenvalue associated to -Δ under homogeneous Dirichlet boundary conditions. Moreover, the solutions have the following expressions, respectively:

where e and ω are the unique positive solutions of the following equations, respectively:

φ 1 is the positive eigenfunction associated to λ1 such that ∫Ωφ1p=1 and R0 is any positive solutions of the above real equations.

Therefore, in view of the above results, in the linear case, there exists at least one positive solution if A=a⁢(R)⁢R1/p for some R>0.

Now, we focus our attention on the eigenvalue and sublinear cases. Our aim is to show the results with respect to the real parameter λ. In these cases, the results depend on the behavior of the functions a⁢(R) and

respectively. In Figures 1–4 we present a number of admissible bifurcation diagrams for the eigenvalue problem and the sublinear case. We have represented in the vertical axis the quantity ∫Ωup⁢(x)⁢𝑑x of the positive solutions of (1.1).

Figure 1 shows both bifurcation diagrams in the case where a⁢(R)→∞ as R→∞ and a⁢(0)>0, as well as the graph of an admissible graph of a⁢(s).

Figure 2 represents the case a⁢(R)→∞ as R→∞ and a⁢(0)=0. In this case, we have drawn only the bifurcation diagram of the eigenvalue problem due to the fact that the sublinear case is similar to the one already sketched in Figure 1.

Figure 3 represents the eigenvalue problem in the case a⁢(R)→a∞>0 as R→∞. Observe that in this case, there does not exist any positive solution for λ large. The sublinear case is again similar to the one shown in Figure 1.

In Figure 4, we have drawn the case a⁢(R)→0 and b⁢(R)→0 as R→∞. Observe that in this case a bifurcation to infinity scenario occurs as λ→0.

Figure 1

Bifurcation diagrams in the case that a⁢(R)→∞ as R→∞ and a⁢(0)>0. On the left-hand side, the graph of a;in the center and on the right hand side the bifurcation diagrams of the eigenvalue and sublinear cases, respectively.

Figure 2

Bifurcation diagram of the eigenvalue problem in the case a⁢(R)→∞ as R→∞ and a⁢(0)=0.

Figure 3

Bifurcation diagram of the eigenvalue problem in the case a⁢(R)→a∞>0 as R→∞ and a⁢(0)>0.

Figure 4

Bifurcations diagrams of the eigenvalue problem and sublinear case, respectively, when a⁢(R)→0 and b⁢(R)→0as R→∞.

The logistic case is more difficult and interesting. We recall that in the local case, that is, a≡1 in (1.1), there exists a positive solution if and only if λ>λ1. In the case λ>λ1, there exists at most one positive solution. Moreover, from the trivial solution emanates at λ=λ1 an unbounded continuum in ℝ×C⁢(Ω¯) of positive solutions. In fact, from another point of view, we have the following result (see [3, 15] and references therein): given α>0, there exists a unique positive solution (λ,u)=(λ⁢(α),uα)∈ℝ×C2⁢(Ω¯) of (1.1) with ∥uα∥2=α. Moreover, α↦λ⁢(α) is an increasing map of class C1 and λ⁢(α)→λ1 as α→0 and λ⁢(α)→∞ as α→∞. Actually, this result is also true with ∥⋅∥p for any p∈[1,+∞].

In the non-local case, we cannot calculate directly the solution or applying any classical methods due to the degeneracy of the diffusion. Hence, we apply a fixed point argument to prove the following results:

1. Fix α>0 and α≠s1. Then there exists a unique positive solution of (1.1), (λ,u)=(λα,uα)∈ℝ×C2⁢(Ω¯), such that

   ∫ Ω u α p ⁢ ( x ) ⁢ 𝑑 x = α .

2. The map α∈(0,s1)∪(s1,∞)↦λα is continuous and it verifies

   lim α → 0 + ⁡ λ α = a ⁢ ( 0 ) ⁢ λ 1 , lim α → s 1 ⁡ λ α = Λ 1 , lim α → ∞ ⁡ λ α = ∞ ,

   where

   Λ 1 := ( s 1 | Ω | ) 1 / p .

3. Equation (1.1) cannot admit a positive solution, u, such that

   ∫ Ω u p ⁢ ( x ) ⁢ 𝑑 x = s 1 .

Roughly speaking, it can be said that the continuum of positive solutions obtained in the local case is broken in the non-local one. Indeed, if we consider the bifurcation diagram (λ,∫Ωup), then the continuum of positive solutions “breaks” at ∫Ωup=s1 (see Figure 5).

Figure 5

Bifurcations diagrams in the logistic case.

As a consequence of the above results, we provide a more accurate description of the structure of the set of λ in ℝ such that (1.1) possesses at least one positive solution. We obtain the following results:

1. Equation (1.1) possesses at least one positive solution for every

   λ ∈ ( min ⁡ { a ⁢ ( 0 ) ⁢ λ 1 , Λ 1 } , ∞ ) ∖ { Λ 1 } .

   In fact, (1.1) possesses at least one positive solution uλ for each

   λ ∈ ( min ⁡ { a ⁢ ( 0 ) ⁢ λ 1 , Λ 1 } , max ⁡ { a ⁢ ( 0 ) ⁢ λ 1 , Λ 1 } ) , with  ⁢ ∫ Ω u λ p ⁢ ( x ) ⁢ 𝑑 x < s 1 .

   In addition, (1.1) possesses at least one positive solution wλ for all

   λ ∈ ( Λ 1 , ∞ ) , with  ⁢ ∫ Ω w λ p ⁢ ( x ) ⁢ 𝑑 x > s 1 ⁢  for all  λ > Λ 1 .

2. As a consequence, if Λ1<a⁢(0)⁢λ1, then (1.1) possesses at least two positive solutions for every

   λ ∈ ( Λ 1 , a ⁢ ( 0 ) ⁢ λ 1 ) .

In Figure 5 we have represented the bifurcation diagrams in both cases: a⁢(0)⁢λ1<Λ1 and Λ1<a⁢(0)⁢λ1.

It is worth pointing out that all the above results are true in the case a⁢(0)=0. In such a case, the bifurcation point λ=a⁢(0)⁢λ1 transforms into λ=0.

Let us mention that all previous results can be extended to the case where a vanishes in a finite number of points. In Figure 6 we show an example where a possesses two zeros s1<s2. We have represented various diagrams depending on the relative position of a⁢(0)⁢λ1 with respect to Λ1<Λ2.

Figure 6

Bifurcations diagrams when a vanishes in two points.

An outline of the work is as follows. In Section 2 we analyze the linear, eigenvalue and sublinear cases. The logistic equation is studied in Section 3.

2 Computable Cases

In this section we analyze different cases of reaction functions where we can characterize the set of positive solutions of (1.1). Basically, we study the linear and sublinear cases and the eigenvalue problem.

The main result is the following.

3 The Logistic Equation

In this section we study the more interesting case f⁢(λ,x,u)=λ⁢u-u2, λ∈ℝ. Before proving the main result for this case, we recall some well-known results about the local logistic equation

where d>0 and λ∈ℝ. Basically, the following results can be found in [4, 10, 11, 13].

In the following result, we complete the singular perturbation problem (3.3) (see [7] for a wide generalization of (3.3)).

To prove the main result of this section, we are going to use a fixed point argument. The following result will be crucial in our argument.

Now, we are ready to establish the main result:

As a direct consequence of Theorem 3.4, we obtain the following corollary.