Bifurcation results of positive solutions for an elliptic equation with nonlocal terms

: In this paper, we investigate the local and global nature for the connected components of positive solutions set of an elliptic equation with nonlocal terms. The local bifurcation results of positive solutions are obtained by using the local bifurcation theory, Lyapunov-Schmidt reduction technique, etc. Under suitable conditions, we show two proofs of priori estimates by using blow-up technique, upper and lower solution method, etc. Finally, the global bifurcation results of positive solutions are obtained by using priori bounds, global bifurcation theory.


Introduction
In this paper, we consider the elliptic problem with nonlocal terms −∆u = λm(x)u + h(x)u p + Ω u β , x ∈ Ω, ∂u ∂n = 0, where Ω is a bounded domain of R N with a smooth boundary, N ≥ 2; n is the outward unit normal to ∂Ω; m(x), h(x) ∈ C α (Ω) for some α ∈ (0, 1) and m(x), h(x) may change sign in Ω; p, β > 1 and p < N+2 N−2 for N ≥ 3, λ ∈ R is a parameter. Many physical phenomena were formulated into nonlocal mathematical models [1-3, 11, 12] and studied by many authors. For example, J. Bebernes and A. Bressan [11] studied an ignition model similar to (1.1) for a compressible reactive gas which is a nonlocal reaction-diffusion equation. In [11], u is the temperature perturbation of the gas and nonlocal term is due to the compressibility of the gas.
Subsequently, some researchers [2,3,12] discussed the parabolic problems related to the equation This type of problem is frequently encountered in nuclear reaction process, where it is known that the reaction is very strong, say like f (x, u) = λm(x)u + h(x)u p with p > 1 and constant functions m(x) and h(x), but the rate with respect to this power is unknown, say like g Ω u β = Ω u β . The above mathematical problem can also be used to population dynamics and biological science where the total mass is often conserved or known, but the growth of a certain cell is known to be of some form (see [12]). Thus, the problem (1.1) is worthy to be considered. Mathematically, the problem (1.1) combines local and nonlocal terms. It is well known [2,3] that the authors discussed that the case m(x) = 0, h(x) ≡ h 0 < 0, β > 1 and p ≥ 1. In two articles, the parabolic problem related to the equation was studied and the authors showed that the value p = β represents a critical blow-up exponent. They proved that if β > p or β = p and h 0 > − |Ω|, the blow-up phenomenon can occur in finite time. If β < p or β = p and h 0 ≤ − |Ω|, all the solutions are global and bounded. The authors also proved the existence of positive solution for h 0 small in the particular case h 0 < 0, p > β > 1. In [1], F. Corrêa and A. Suárez made a further study for the problem and proved the existence, uniqueness, stability and asymptotic properties of positive solutions for some values of p ≥ 1 and β > 0.
We want to further consider the global bifurcation structure of the positive solutions set of the problem (1.1) when the function m(x) and h(x) are nonconstant functions because for general function h(x), especially sign-changing function, comparing with the local elliptic equation, we see that many methods that prove the boundedness of positive solutions cannot be used in nonlocal elliptic equation, such as the extremum principle, parameter control. Finally, we can obtain the existence, multiplicity and nonexistence of the positive solution for the problem (1.1) when a bounded connected branch of the positive solutions set is established by the global bifurcation theory.
In an early paper, K. J. Brown [4] studied the local and global bifurcation of the semilinear elliptic boundary value problem where 1 < γ < N+2 N−2 , m(x), b(x) may change sign in Ω. The cases where Ω m(x)dx 0 and Ω m(x)dx = 0 were discussed respectively and the author concluded that there are continua of positive solutions of (1.2) connecting λ = 0 to the other principal eigenvalue for Ω m(x)dx 0 when m(x) and b(x) are under suitable conditions. It was also showed that the closed loops of positive solutions occur naturally and properties of these loops are investigated.
In this paper, we are interested in the problem (1.1), namely, the problem (1.2) added a nonlocal term. We want to investigate whether the local and global structures of positive solutions set for the problem (1.1) have similar properties to the problem (1.2). We also investigate sufficient conditions for a bounded continuum of positive solutions. In Theorem 2.2, we see that the direction of bifurcation curve is related to β and p. In Theorem 3.1, we get a priori bound of positive solution when m(x) is under suitable conditions and β > max {p, N(p − 1)/2} by using upper and lower solution method, blow-up technique and boot-strapping method. Moreover, another way of proving boundedness shows that the priori bound still exists when β > max {p, N(p − 1)/2} vanishes.
Before proceeding to the study of local and global nature of positive solutions, we need to introduce some notations. If u > 0 in Ω, we say u is a positive solution of the problem (1.1). (λ, u) is called a nonnegative solution of the problem (1.1) if u is a nonnegative solution of the problem (1.1) with λ. Obviously, (λ, 0) is a nonnegative solution of the problem (1.1), we say it is a trivial solution.
To investigate the bifurcation of problem (1.1) at the trivial solution (λ, 0), we discuss the linear eigenvalue problem where m(x) changes sign in Ω. According to [5], we have the following results. The usual norms of the space L p (Ω) for p ∈ [1, ∞) and C(Ω) are, respectively, Let . We use the following hypothesis.
If m(x) changes sign in Ω h + and (H 1 ) holds, then the equation x ∈ ∂Ω h + has unique positive principal eigenvalue [5,6]. We have the following main global results of positive solutions in two cases where Ω m < 0 and Ω m > 0 by using priori bounds, global bifurcation theory.
The assertions of Theorem 1.1 may be illustrated by the bifurcation diagram shown in Figure 1.
(ii) Comparing with the results of [4], we see that directions of bifurcation curve depend on not only the sign of Ω m and hϕ p+1 1 but also the relationship of β and p. In Theorem 1.1, we only show the case of β > p, while the cases of β < p and β = p can also be listed.
Moreover, if m(x) changes sign in Ω h + and (H 1 ) holds, then the connected components C + of positive solutions set containing (λ(s), u(s)) satisfies the claims (i), (ii) of Theorem 1.1 and the closure C The assertions of Theorem 1.2 may be illustrated by the bifurcation diagram shown in Figure 2. Next, we consider the case Ω m = 0. We shall then study the global bifurcation of positive solutions for the problem (1.1) by the approximation method. Let m ε (x) = m(x) − ε for ε > 0, we have the following conclusions.
then there exists a connected components C + of positive solutions set for the problem (1.1), which bifurcates from the origin and backs to the origin in λ-norm plane, namely, the closure C + of C + in R × C(Ω) is a closed loop.

Remark 1.2.
For the case Ω m = 0, the hypotheses for the Crandall and Rabinowitz theorem are no longer satisfied, and we use the Lyapunov-Schmidt technique to investigate how bifurcation occurs at (λ, u) = (0, 0). Moreover, comparing with the global bifurcation results of [4], we see that the relationship of p and nonlocal term power β also has influence on the continuum.
The assertions of Theorem 1.3 may be illustrated by the bifurcation diagram shown in Figure 3. The rest of this article is organized as follows. In Section 2, we discuss the local properties of positive solutions set in the cases where Ω m 0 and Ω m = 0 by using the local bifurcation theory, Liapunov-Schmidt reduction technique. In Section 3, we show that a priori estimate of positive solutions by using the blow-up technique, global bifurcation theory and upper and lower solution. In Section 4, we complete the proof of main results in two cases.

Local bifurcation results
Let's investigate local bifurcation in the cases where Ω m 0 and Ω m = 0, respectively.

Local bifurcation when
Let ϕ 1 be the positive eigenfunction of λ 1 . If λ 1 = 0, ϕ 1 is a constant and we take ϕ 1 = 1. Let 0 < ε 1 be a constant. We have the following result.
Hence, we get (2.2). By virtue of the Crandall-Rabinowitz local bifurcation theory, we obtain Theorem 2.1.
Next, we discuss the direction of bifurcation.
is a bifurcation curve of positive solutions obtained by Theorem 2.1, then we have the following conclusions. ( If |Ω| 2 + Ω h and Ω m have same (resp. opposite) sign, then the bifurcation curve at (λ 1 , 0) is subcritical (resp. supercritical).
Proof. Since (λ(s), u(s)) is a positive solution of the problem (1.1), we have Multiplying the Eq (2.3) by ϕ 1 , integrating in Ω, and using the Green formula, it follows that Since Ω mϕ 2 1 and Ω m have opposite sign, then we get Theorem 2.2.

Local bifurcation when Ω m = 0
If Ω m = 0, (2.2) doesn't work, then the hypotheses for the Crandall and Rabinowitz theorem are no longer satisfied. However we can use the Lyapunov-Schmidt technique to investigate how bifurcation occurs.
Assume u ∈ X is the solution of the problem (1.1), let u = s + w, where s is a constant and Ω w = 0.
Let w m be the solution of the problem We have the following conclusions.
by using the Morse lemma, we see that for any s > 0, ψ(λ, s) = 0 has a unique solution s = s(λ) in a neighborhood of (0, 0) and we have (1)) .

So we get the conclusion.
For (2) (iii), since using a similar argument as that of (2) (ii), we get (1)) .

A priori estimate
We first prove that under the suitable conditions, the problem (1.1) has no positive solution for any sufficiently large |λ|. More precisely, we have the following results.
Assume m(x) changes sign in Ω h + , (H 1 ) holds. For any λ ∈ λ − 1 , λ + 1 , e λ is the unique positive solution of the equation x ∈ ∂Ω h + . We have the following lemma.
We will use the method of Gidas-Spruck [7] to discuss the priori estimate of positive solutions. Proof. If (λ k , u k ) is a positive solution of the problem (1.1), then It is seen in this case that Ω k → R N as k → ∞. Hence, for any compact subset K 1 , we have K 1 ⊂ Ω k for sufficiently large k. Since 0 < v k ≤ 1, there exist a positive constant C 2 , such that By using the regularity theory of the elliptic equation, we know that, there exists a subsequence of {v k }, still denoted by itself, such that v k → v in C 1 (K 1 ), k → +∞, where v ∈ C 1 (K 1 ). Since K 1 ⊂⊂ Ω k is arbitrarily given, by a diagonal process, we can choose a subsequence, still denoted by {v k }, such that Note that v(0) = 1, by (3.3) and a linear change of coordinates, we find that there exists a nontrivial non-negative function w ∈ C 2 (R N ) satisfying −∆w = w p , which contradicts [7].
By an additional change of coordinates, we can assume that a neighborhood of x 0 in ∂Ω h + is a hyperplane x N = 0 andΩ h + ⊂ H = {x ∈ R N , x N > 0}. Hence, given R > 0, there exists k R such that for k ≥ k R , v k is well defined on We may argue exactly as in Case 1.
is not bounded from below. Assume without loss of generality that M p−1 2 Arguing as in Case 1, there exists v ∈ C 2 (H) such that v ≥ 0, v(0) = 1, and v satisfies This contradicts Corollary 2.1 of [7].
We can proceed as in (ii) and there exists v ∈ C 2 loc (H s ) such that v ≥ 0, v(0) = 1, and v satisfies Taking a change of variable through y N = −s, we have that v ∈ C 2 (H), v ≥ 0, v(0) = 1, and v satisfies Lemma 3.3. Assume m(x) changes sign in Ω h + and (H 1 ) holds, then there exists a constant C > 0 such that u C(Ω h + ) ≤ C for any positive solution (λ, u) of the problem (1.1). Proof. If (λ k , u k ) is a positive solution of the problem (1.1), then λ k ∈ λ − 1 , λ + 1 . Assume where t k = Ω u β k . Moreover, we have t k → ∞. But by using Lemma 3.1, we have u k ≥ e λ k t k , a contradiction. Therefore, there exists a constant C > 0 such that u C(Ω h + ) ≤ C. Proof. Let f (u) = λm(x)u + h(x)u p + Ω u β . By Lemma 3.3, there exists a constant C 1 > 0, such that u ≤ u C(Ω h + ) ≤ C 1 , x ∈ Ω h + . It follows that Ω u β is bounded by Lemma 3.1, so f (u) is bounded in L β/p (Ω). Thus, u is bounded in W 2,β/p (Ω). By using boot-strapping method [1], it follows that there exists a constant C > 0, such that u C(Ω) ≤ C.
On Γ 1 (Ω\Ω h + ), since min Ω w > 0, we know kw ≥ M; on Γ 2 (Ω\Ω h + ), we have ∂kw ∂n = 0. Thusū = kw is a positive strict upper solution. Moreover, by the relationship between the strict upper solution and the principal eigenvalues, we have If (λ, u) ∈ S then it follows that v =ū − u satisfies Thus by result (1) we have By the relationship between the principal eigenvalues and the strong maximum principle, we have u ≤ū. The proof is completed. Next, we prove (iii). We assume by contradiction that (λ k , u k ) are positive solutions of the problem (1.1), and (λ k , u k ) → (γ, 0) in R × C(Ω), where γ λ + and 0. Let v k = u k u k C(Ω)

Proof of main results
, by the problem (1.1), we have By virtue of the regularity theory of the elliptic equation, we know that there is a subsequence, still denoted by {v k }, such that v k → v 0 in C 2 (Ω), v 0 is a solution of the equation So γ = λ + or 0, a contradiction. Moreover, by using Rabinowitz global bifurcation theory, we see that C + bifurcates from (λ + , 0) and backs to (0, 0).
Similarly, we obtain the global bifurcation results of the problem (1.1) for Ω m > 0.
We now investigate C + ε as ε → 0. Although it seems likely in Figure 3 that C + ε approaches a closed loop joining the origin to itself as ε → 0, this seems difficult to establish. We can, however, prove that C + ε does not shrink to a point. For sets E n , n ∈ N, we define lim n→∞ inf E n = {x : there exists N 0 ∈ N such that any neighborhood of x intersects E n for all n ≥ N 0 }, lim n→∞ sup E n = {x : any neighborhood of x intersects E n for infinitely many n }.
According to [8], if ∪ n≥1 E n is precompact in M and lim n→∞ E n ∅, then lim n→∞ E n is non-empty, closed and connected. Here, {E n } is a sequence of connected sets in a complete metric space M.
Therefore, under the conditions of the Theorem 1.3, there exists a connected components C + of positive solutions set such that its closure C + includes lim ε→0 C + ε , which bifurcates from the origin and backs to the origin, namely, C + is a closed loop.