A LOOP TYPE COMPONENT IN THE NON-NEGATIVE SOLUTIONS SET OF AN INDEFINITE ELLIPTIC PROBLEM

. We prove the existence of a loop type component of non-negative solutions for an indeﬁnite elliptic equation with a homogeneous Neumann boun- dary condition. This result complements our previous results obtained in [12], where the existence of another loop type component was established in a diﬀe- rent situation. Our proof combines local and global bifurcation theory, rescaling and regularizing arguments, a priori bounds, and Whyburn’s topological method. A further investigation of the loop type component established in [12] is also provided.

where • λ ∈ R; • 1 < q < 2 < p; • a, b ∈ C α (Ω), α ∈ (0, 1); • a ≡ 0 and b changes sign; • n is the unit outer normal to ∂Ω. By a solution of (P λ ), we mean a classical solution. A solution u of (P λ ) is said to be nontrivial and non-negative if it satisfies u ≥ 0 on Ω and u ≡ 0, whereas it is said to be positive if it satisfies u > 0 on Ω. Note that since b changes sign and 1 < q < 2, the strong maximum principle does not apply and, as a consequence, we can not deduce that nontrivial non-negative solutions of (P λ ) are actually positive solutions, unlike in the case q ≥ 2.
In [12] we have investigated existence, non-existence, and multiplicity of nonnegative solutions as well as their asymptotic behavior as λ → 0. These results led us to analyse the structure of the set of non-negative solutions of (P λ ). In particular, we have proved the existence of a loop type component in this set, under the following conditions (see [12,Theorem 1.6] and Figure 1 (1) We shall assume that a and b are positive in some open ball (see (H 0 ) below), in which case the nonlinearity in (P λ ) has (locally) a concave-convex nature. We refer the reader to [12] for a more general discussion on (P λ ) and related concave-convex problems.
Our purpose is to go further in this investigation, focusing now mostly on the case Before stating our result, let us set Ω a ± = {x ∈ Ω : a ≷ 0}, Ω b ± = {x ∈ Ω : b ≷ 0}. The following conditions shall be assumed in our main result: (H 0 ) a(x 0 ), b(x 0 ) > 0 for some x 0 ∈ Ω; (H 1 ) Ω a + and Ω := Ω \ Ω a + are subdomains of Ω with smooth boundaries, and satisfy either Ω a + ⊂ Ω or Ω ⊂ Ω; (H 2 ) There exist γ > 0 and a function α + which is continuous, positive, and bounded away from zero in a tubular neighborhood U of ∂Ω a + in Ω a + , such that a + (x) = α + (x) dist(x, ∂Ω a + ) γ , x ∈ U, (H 3 ) Ω b ± are subdomains of Ω. These conditions guarantee some a priori bounds in (0, ∞) × C(Ω) for nonnegative solutions. More precisely, (H 0 ) implies that (Q µ, ), a rescaled and regularized version of (P λ ), has no positive solutions for µ sufficiently large, similarly as [12,Proposition 6.1]. On the other hand, (H 1 ) and (H 2 ) provide us with an a priori bound on u C(Ω) for any non-negative solution u of (Q µ, ), see [12,Proposition 6.5]. Let us mention that (H 2 ) goes back to Amann and López-Gómez [2], where the authors have established a priori bounds for positive solutions of indefinite elliptic problems. Finally, (H 3 ) is employed to show that bifurcation from zero for nontrivial non-negative solutions of (P λ ) does not occur at λ = 0, as in [12,Proposition 3.3].
We state now our main result, which gives a positive answer to the open question raised in Subsection 6.1 of [12]. Note that, in contrast with [12, Theorem 1.6], a may be non-negative. Theorem 1. Assume (2), (H 0 ) and (H 3 ). In addition, assume one of the following conditions: (a) a > 0 on Ω, and 2 < p < 2N Then (P λ ) has a bounded component (non-empty, closed, and connected subset in R × C(Ω)) of non-negative solutions C 0 = {(λ, u)}. In addition, C 0 is of loop type, i.e., it is a bounded component that meets a single point on a trivial solution line and joins this point to itself. More precisely, C 0 starts and ends at (0, 0), and has the following properties (see Figure 1(b)): There is no (λ, u) ∈ C 0 with λ < 0, i.e., C 0 bifurcates to the region λ > 0 at (0, 0). (iv) There exist at least two nontrivial non-negative solutions (λ, u 1,λ ), (λ, u 2,λ ) ∈ C 0 , for λ > 0 sufficiently small.
(ii) Condition (b) includes the following cases: (1) a(x) > 0 in Ω, and a(x) vanishes on ∂Ω. This situation is understood as Ω = ∅. (2) {x ∈ Ω : a(x) = 0} = ∅. In particular, it includes the case that a(x) changes sign, as well as the case that a(x) ≥ 0 in Ω. In both cases we need that b(x) > 0 in {x ∈ Ω : a(x) = 0}; Remark 3. The existence of the loop type component C 0 provided by Theorem 1 is consistent with [12, Theorem 1.1]. As a matter of fact, in [12,Theorem 1.1] it is proved that if (2) holds then (P λ ) has two nontrivial non-negative solutions for λ > 0 sufficiently small. Moreover, these solutions converge both to 0 in C 2 (Ω) as λ → 0 + . We believe that these solutions correspond to the upper and lower branches of C 0 .   The existence of bounded (or compact) components in the solution set of nonlinear problems has been investigated by Cingolani and Gámez [6], Cano-Casanova [5], López-Gómez and Molina-Meyer [10], and Brown [3]. A mushroom, i.e. a component connecting two simple eigenvalues of the linearized eigenvalue problem at the trivial solution u = 0, was obtained by Cingolani and Gámez for both the Dirichlet case and Ω = R N , and by Cano-Casanova for a mixed linear boundary condition. In addition to the existence of a mushroom, López-Gómez and Molina-Meyer (for the Dirichlet case) and Brown (for the Neumann case) obtained a loop, i.e. a component that meets a single point on the trivial solution line. Moreover, López-Gómez and Molina-Meyer also proved the existence of an isola, i.e. a component that does not touch the trivial solution line. Finally, we refer to [12,Theorem 1.6], where the existence of a loop type component for (P λ ) was proved in case (1).
Let us remark that the nonlinearities in [6,5,10,3] are C 1 at u = 0, which is not the case for (P λ ). Therefore the standard global bifurcation theory of Rabinowitz [11] (see also López-Gómez [9]) does not apply straightforwardly to (P λ ). To overcome this difficulty, we employ a regularization method around the trivial solution and develop Whyburn's topological analysis [15, (9.12)Theorem] to convert the bifurcation results obtained for the regularized problem to the original problem. In fact, before considering the regularization, we carry out a scaling argument to overcome a difficulty which appears in case (2). Unlike in case (1), it is difficult to study directly (P λ ) and its regularization under (2), since these problems have no positive solutions for λ = 0 [12, Lemma 6.8 (1)]. Even if we can prove the existence of a component of positive solutions for the regularized problem, the non-existence result for λ = 0 may cause the shrinking of the component into the set of trivial solutions when the topological method is employed. It should be emphasized that in order to obtain the loop in case (1) as in Figure 1(a), the following fact was crucial: a component of positive solutions for the -regularized problem of (P λ ) does cut the vertical axis λ = 0, at some point that does not shrink to (0, 0) as → 0.
In order to verify that a component of non-negative solutions of (P λ ) is bounded in (0, ∞) × C(Ω), we shall make good use of a priori bounds for non-negative solutions of (P λ ), as well as for non-negative solutions of (Q µ ) and (Q µ, ) below. We obtain these a priori bounds under either conditions (a) or (b) in Theorem 1, proceeding in the same way just as in [12,Proposition 6.5].
The rest of this article is organized as follows. In Section 2, by the change of variables µ = λ p−2 p−q and v = λ − 1 p−q u, we transform (P λ ) into (Q µ ), and consider a -regularized version of (Q µ ), i.e. (Q µ, ). This regularization scheme enables us to apply the local and global bifurcation theory from simple eigenvalues. We deduce then the existence of a component of bifurcating positive solutions of (Q µ, ) from {(µ, 0)}. Section 3 is devoted to the proof of our main result, Theorem 1. Using Whyburn's topological method, we establish the limiting behavior of the component of (Q µ, ) obtained in Section 2 as → 0 + , and obtain a component of nontrivial nonnegative solutions of (Q µ ) which bifurcates from (0, 0) into the region µ > 0. Finally, by the scaling, we go back to (P λ ), and obtain thus a bounded component of nontrivial nonnegative solutions which is of loop type, joins (0, 0) to itself, and lies in the region λ > 0, as shown in Figure 1(b). In Section 4, we carry out a further analysis for the loop of nontrivial nonnegative solutions of (P λ ) obtained in the case (1) by [12]. The analysis concentrates on the direction of the bifurcation point from which the loop emanates. The main result of this section is Theorem 8.

Scaling and regularization schemes. We set
where µ ≥ 0. We note that the nonlinearity in (Q µ ) is not differentiable at v = 0, so that the local and global bifurcation theory from simple eigenvalues on the trivial line can not be directly applied to (Q µ ). To overcome this difficulty, we shall consider the following regularized version of (Q µ ), where ∈ (0, 1] is fixed: It is understood that (Q µ, ) = (Q µ ) when = 0. It is clear that, in addition to Γ 0 , (Q µ, ) has the trivial line of positive solutions Furthermore, by the strong maximum principle and the boundary point lemma, any nontrivial non-negative solution of (Q µ, ) is positive. First, we discuss bifurcation from Γ 00 . Moreover, (ii) Assume that Ω a = 0. Then, there are no positive solutions v n of (Q µn, ) such that µ n → 0 + , and v n → c in C(Ω) for some constant c > 0. Proof.
(i) The divergence theorem shows that By passing to the limit as n → ∞, it follows that c satisfies (3). Thus, c = c * 0 . Moreover, assertion (4) is verified in a trivial way.
Remark 5. The assertions of Lemma 4 except (4) are also valid for = 0. Now, using [7, Theorem 1.7], we carry out a local bifurcation analysis for (Q µ, ) with > 0 on Γ 0 , where µ is the bifurcation parameter. To this end, we reduce (Q µ, ) to an operator equation in C(Ω). Let M > 0 be fixed.
We introduce the resolvent K : It is well known (cf. [8]) that K is bijective and homeomorphic. It is also well known (cf. [1]) that K can be extended to a compact linear mapping from C(Ω) into C 1 (Ω). In this way, as far as non-negative solutions are concerned, (Q µ, ) is reduced to ∂s (x, 0) = 0, so that F has Fréchet derivatives F v (µ, 0) and F µv (µ, 0) given, respectively, by We consider the eigenvalue problem where is fixed, and µ is the eigenvalue parameter. Since b changes sign and Ω b < 0, this problem has exactly two principal eigenvalues, µ = 0, and µ = µ > 0 (cf. [4]), which are both simple and possess positive eigenfunctions ϕ = 1 and ϕ = ϕ , respectively, both satisfying ϕ ∞ = 1. Hence, the principal eigenvalues µ = 0, and µ = µ satisfy where N (·) and R(·) denote the kernel and range of mappings, respectively. Indeed, the latter two assertions are verified using the conditions that Ω b < 0 and Let w D ∈ C 2 (B) be a positive eigenfunction associated with the first eigenvalue µ D > 0 of (8). We extend w D to the Ω by setting w D = 0 in Ω \ B. Then, Let v be a positive solution of (Q µ, ). By the divergence theorem, we deduce that B ∇ · (v∇w D ) = ∂B v ∂w D ∂n < 0. It follows that On the other hand, by the definition of v, we see that If µ ≥ 1, then we deduce that The rest of the proof follows as in [12, Proposition 6.1]. Indeed, we can show that there exists µ > 0 such that if µ ≥ µ, ∈ (0, 1], x ∈ B and s ≥ 0, then Consequently, µ is bounded from above, uniformly in ∈ (0, 1]. (ii) If Ω a > 0, then, thanks to the previous item, we infer by Lemma 4 (i) that C bifurcates from infinity if it does not meet (0, c * ), so assertion (ii)(a) follows.
Similarly, if Ω a = 0, then the previous item and Lemma 4 (ii) yield that C bifurcates from infinity, so assertion (ii)(b) follows. (iii) This assertion is deduced from the fact that (7) has exactly two principal eigenvalues µ = 0 and µ = µ .
(c) C does not meet Γ 00 , but bifurcation from infinity occurs.  3. Proof of Theorem 1. In the sequel we study the limiting behavior of C as → 0 + . To this end, we recall some definitions. Let X be a complete metric space. Given E n ⊂ X, n ≥ 1, we set where dist (x, A) is the usual distance function for a set A. It is well known from Whyburn [15, (9.12)Theorem] that if {E n } is a sequence of connected sets in X satisfying then lim sup n→∞ E n is nonempty, closed, and connected.
The boundedness of C n ,ρ implies that n C n ,ρ is precompact. Indeed, for any {(µ k , v k )} ⊂ n C εn,ρ the sequence n has a subsequence n k such that (µ k , v k ) ∈ C εn k , where n k ∈ (0, 1]. Then, by elliptic regularity, we deduce that v k ∈ C 2 (Ω), v k C 1 (Ω) is bounded, and Since (µ k , v k ) and n k are bounded, using the compact embedding is nonempty, closed and connected, i.e., it is a nonempty component in [0, Λ] × B ρ . Moreover, we shall show that C 0,ρ consists of non-negative solutions of (Q µ ), and The proof of these facts is similar to the verification of the precompactness of n C n ,ρ . Indeed, given (µ, v) ∈ C 0,ρ , the sequence n → 0 + has a subsequence,  still denoted by the same notation, such that there exist (µ n , v n ) ∈ C εn,ρ satisfying (µ n , v n ) → (µ, v) in R × C(Ω). It follows, by a bootstrap argument based on elliptic regularity, that v n → v in C 1 (Ω), so that v is a non-negative weak solution of (Q µ ) (see (11)), and eventually, a non-negative solution in C 2+θ (Ω) for some θ ∈ (0, 1), by elliptic regularity. Next, we shall prove that C 0,ρ is nontrivial, i.e., we exclude the possibility that C 0,ρ ⊂ Γ 0 ∪ Γ 00 . Let ρ, M be such that 0 < M < c * 0 < ρ. Then, we find from (4) and (13) that C 0,ρ joins (0, 0) to either (0, c * 0 ) or (µ, v) ∈ [0, Λ] × B ρ . Since C 0,ρ is connected, the intermediate value theorem shows the existence of (µ 0 , v 0 ) ∈ C 0,ρ such that v 0 C(Ω) = M . By definition, the sequence n → 0 + has a subsequence, still denoted by the same notation, such that there exist (µ n , v n ) ∈ C n ,ρ with 0 < µ n → µ 0 and v n → v 0 in C(Ω). Assume by contradiction that µ 0 = 0. Then, v 0 = M , so that v n → M in C(Ω). However, applying the divergence theorem to the solution v n of (Q µn, n ), we obtain so that passing to the limit, we deduce that and thus that M = c * 0 , which is impossible. Consequently, µ 0 > 0, and thus, C 0,ρ is nontrivial.
Note added in proof. (i) Regarding (H 3 ), the condition that Ω b − is a subdomain can be removed from Theorem 1. Indeed, although this condition is needed to verify the non-existence of nontrivial non-negative solutions of (P λ ) bifurcating from {(λ, 0)} for λ < 0, such verification is required only for λ > 0 in Theorem 1.
(ii) Let C 0 be a maximal component of nonnegative solutions of (P λ ) that includes the loop type component C 0 provided by Theorem 1 and such that C 0 \ {(0, 0)} consists of nontrivial non-negative solutions. As a further result for Theorem 1, we obtain that C 0 is bounded in R × C(Ω) if, in addition to the hypotheses in Theorem 1, one of the following conditions is assumed.
(a) Ω b + ⊂ Ω, Ω := Ω \ Ω b + is a subdomain, and Ω a + contains a tubular neighborhood of ∂Ω, Ω is a subdomain, and Ω a + ⊂ Ω b + . Indeed, under the additional condition, the strong maximum principle and boundary point lemma show that any nontrivial non-negative solution of (Q µ ) is positive in Ω b + . Consequently, Lemma 6 (i) is also valid for nontrivial non-negative solutions of (Q µ ), and the desired conclusion follows.
According to the arguments developed in [12], this existence result can be verified by considering the regularized version of (P λ ) for u q−1 at u = 0: where > 0 and Ω b− + = ∅. Then (P λ, ) is regular, so that the unilateral global bifurcation theorem by López-Gómez [9, Theorem 6.4.3] may be applied. To this end, we consider the linearized problem at u = 0: Since b − changes sign and Ω (b − ) < 0, this eigenvalue problem has exactly two principal eigenvalues, λ = 0 and λ = λ > 0, which are both simple. We use now the unilateral global bifurcation theory to obtain two components C 0, and C 1, of positive solutions of (P λ, ), bifurcating from (0, 0) and (λ , 0), respectively. Moreover, we can analyze the local nature of these components at the bifurcation points by using the local bifurcation theory proposed by Crandall and Rabinowitz. We can also analyze the global nature of these components by making good use of an a priori bound in R × C(Ω) for positive solutions (λ, u) of (P λ, ). Consequently, C 0, and C 1, are both bounded, so that C 0, = C 1, (=: C ), i.e., C is a mushroom. Finally, based on the fact that λ → 0 as → 0 + (see [12, Lemma 6.6]), we may apply Whyburn's topological method to infer that C 0 = lim sup →0 + C is a nonempty component of non-negative solutions (λ, u) of (P λ ). The limiting features of C 0 mentioned above follow by using some additional results on the set of nonnegative solution of (P λ ). In addition to these features, we shall provide a further result on the direction of bifurcation at (0, 0) for C 0 , using Whyburn's topological method again. We remark that, although properties (i)-(iv) above provide that C 0 is a loop, i.e., C 0 joins (0, 0) to itself passing by (0, u 0 ) for some positive solution u 0 of (P λ ) with λ = 0, Theorem 8 confirms additionally the loop property of C 0 , as follows: Given ρ > 0, we set B ρ ((λ 1 , u 1 )) = {(λ, u) ∈ R × C(Ω) : |λ − λ 1 | + u − u 1 C(Ω) < ρ}.