Special Issue in honor of John W. Neuberger Electronic Journal of Differential Equations, Special Issue 02 (2023), pp. 109–124. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu or https://ejde.math.unt.edu RESONANT SOLUTIONS FOR ELLIPTIC SYSTEMS WITH NEUMANN BOUNDARY CONDITIONS BRICEYDA B. DELGADO, ROSA PARDO Honoring the memory of Professor John W. Neuberger Abstract. We consider a sublinear perturbation of an elliptic eigenvalue sys- tem with homogeneous Neumann boundary conditions. For oscillatory non- linearities and using bifurcation from infinity, we prove the existence of an unbounded sequence of turning points and an unbounded sequence of reso- nant solutions. 1. Introduction We consider the nonlinear elliptic system with homogeneous Neumann boundary conditions −∆u1 + u1 = λ(a1u1 + a2u2) + f1 ( x, (u1, u2)), −∆u2 + u2 = λ(a2u1 + a3u2) + f2 ( x, (u1, u2)), in Ω , ∂u1 ∂η = ∂u2 ∂η = 0 on ∂Ω, (1.1) where Ω ⊂ RN (N ≥ 2) is a bounded domain, ∂/∂η := η(x) · ∇ denotes the outer normal derivative on ∂Ω, ai > 0, i = 1, 2, 3 are fixed, and λ ∈ R is a bifurca- tion parameter. The nonlinearity f = (f1, f2), where fi : Ω × R2 → R, i = 1, 2 are Carathéodory functions, that is, fi = fi ( x, s ) are measurable in x ∈ Ω and continuous with respect to s = (s1, s2) ∈ R2. Roughly speaking, we assume that the nonlinearity satisfies (a) f(x, s) = o(|s|) at infinity, and (b) f is oscillatory. Assumption (a), by a mechanism of parametric resonance, produces unbounded branches of solutions when λ approaches certain eigenvalue in the sense of Ra- binowitz [14]. Assumption (b) transfers the oscillatory behavior to the solutions branch, yielding infinitely many turning points and infinitely many resonant solu- tions (see definitions 1.1-1.2). We assume that f satisfies the following hypothesis: 2020 Mathematics Subject Classification. 35J65, 35J61, 35J15. Key words and phrases. Bifurcation from infinity; nonlinear elliptic systems; Neumann boundary conditions; resonant solutions. ©2023 This work is licensed under a CC BY 4.0 license. Published March 27, 2023. 109 110 B. B. DELGADO, R. PARDO EJDE/SI/02 (H1) There exists h ∈ Lr(Ω) with r > N/2 and a continuous function f̃ : R2 → R+ satisfying |fi(x, s)| ≤ h(x)f̃(s), i = 1, 2, with lim |s|→∞ f̃(s) |s| = 0, where |s| = ∣∣(s1, s2) ∣∣ := √ s2 1 + s2 2. (H2) There exists a function B ∈ Lr(Ω) with r > N/2, α < 1 and s0 > 0 such that we have |fi(x, s)| |s|α ≤ B(x), for |s| > s0, x ∈ Ω, and i = 1, 2. (H3) f(x, s) is differentiable in s = (s1, s2), ∂fi ∂sj (·, ·) ∈ C(Ω× R2;R), and for all i, j = 1, 2, sup |s|≥M ∥∥∂fi ∂sj (·, s) ∥∥ L∞(Ω) → 0 as M →∞. By a weak solution to (1.1) we mean a pair (λ, u) ∈ R× ( H1(Ω) )2 such that for all ψ = (ψ1, ψ2) ∈ H1(Ω)2,∫ Ω ∇u1∇ψ1 + u1ψ1 = λ ∫ Ω (a1u1 + a2u2)ψ1 + ∫ Ω f1(x, (u1, u2))ψ1,∫ Ω ∇u2∇ψ2 + u2ψ2 = λ ∫ Ω (a2u1 + a3u2)ψ2 + ∫ Ω f2(x, (u1, u2))ψ2. (1.2) Because of (H1), weak solutions of (1.1) lie in the space ( W 2,r(Ω) )2 , r > N/2, which is continuously embedded in ( C(Ω) )2 . Therefore, we consider R × ( C(Ω) )2 as our underlying space. Throughout this work, we consider the Banach space( C(Ω) )2 equipped with the norm ‖u‖ = ( ‖u1‖2C(Ω) + ‖u2‖2C(Ω) )1/2 . Note that system (1.1) can be rewritten in matrix form as (−∆ + I)u = λAu+ f(x, u), in Ω, ∂u ∂η = 0, on ∂Ω, (1.3) where u = ( u1 u2 ) , A = ( a1 a2 a2 a3 ) , f(x, u) = ( f1 ( x, (u1, u2)) f2 ( x, (u1, u2)) ) , (1.4) I is the 2× 2 identity matrix, and (−∆ + I)u = ( (−∆ + I)u1, (−∆ + I)u2 )T . Note that the matrix A is symmetric. It is straightforward to show that the eigenvalues of the matrix A are the numbers µ± = a1 + a3 2 ± √(a1 − a3 2 )2 + a2 2, (1.5) with µ+ > 0, and µ− ≶ 0 if and only if det(A) ≶ 0. Let us denote by b = ( b1 b2 ) the eigenvector associated to µ+, Ab = µ+b, normalized so that |b| = (b21 + b22)1/2 = 1. (1.6) EJDE-2023/SI/02 RESONANT SOLUTIONS FOR ELLIPTIC SYSTEMS 111 Since A is symmetric and ai > 0, i = 1, 2, 3, then, bTA = µ+b T and b1, b2 > 0. Moreover b is the only eigenvector of A with both components positive. Using a Rabinowitz result [14], we prove that the branch of solutions bifurcating from infinity at λ = 1/µ+, denoted by D ⊂ R× ( C(Ω) )2 , forms a continuum, that is, the closed and connected set {(λ, u) ∈ R× ( C(Ω) )2 : (λ, u) is a weak solution to (1.1)}. D contains large positive solutions (in both components), and large negative solu- tions to (1.1), see Theorem 2.2. Let D+ denote the continuum of positive solutions bifurcating from infinity at λ = 1/µ+ (resp. D− for negative solutions). As a mater of fact, solutions can be expressed as u = tb+ w, where t > 0, w = o(|t|) as t → +∞, and w ∈ (span{b})⊥, see Proposition 3.2. We will see that (λn, un) = (λn, tnb+ wn)→ (1/µ+,∞) whenever tn →∞. Definition 1.1. We say that (λ∗n, u ∗ n) ∈ D+ is a turning point if there is a neigh- borhood of (λ∗, u∗) in R× (C(Ω))2 such that there are no solutions (λ, u) close to (λ∗, u∗) for λ > λ∗ or for λ < λ∗. It is well known that the Neumann Laplacian operator has a discrete spectrum of infinitely many non-negative eigenvalues with no finite accumulation point, so σ(−∆ + I) = {λi : 1 = λ1 < λ2 ≤ · · · ≤ λn . . . , λn →∞ as n→∞}. (1.7) Let b+ = b, and b− be the eigenvectors associated to µ−; observe that if (−∆+I)φ = λµ±φ in Ω, and ∂φ ∂η = 0 on ∂Ω, then (−∆ + I)b±φ = λµ±(b±φ) = λA(b±φ), ∂ ∂η b±φ = 0. In other words, λµ± ∈ σ(−∆ + I), implies that 0 ∈ σ(−∆ + I− λA). In particular, if u is a solution to (1.1) corresponding to λ = 1/µ+, we will say that u is a resonant solution to (1.1). Definition 1.2. If λµ± ∈ σ(−∆ + I), we say that the system (1.1) is resonant. One interesting question is whether the bifurcating branch D+ is subcritical or supercritical. That is, if it is formed only with solutions (λ, u) with λ < 1/µ+ or λ > 1/µ+ respectively. We can easily see how to determine whether the bifurcation of positive solu- tions emanating from the first eigenvalue is sub- or super-critical. To do this, let (λn, un) = (λn, tnb + wn) solve (1.1) for λn → 1/µ+ and tn → +∞. Multiplying (1.3) by bT on the left, integrating the result, using that ∫ Ω bTun = tn|Ω|, and the symmetry of the matrix A, so bTA = µ+b T , we obtain tn|Ω|(1− λnµ+) = ∫ Ω b1f1(·, tnb+ wn) + b2f2(·, tnb+ wn). (1.8) Hence, the sign of 1 − λµ+ is the same as that of the right hand side. Hence, if the right-hand side is greater than 0, the bifurcation of positive solutions will be subcritical and if right-hand side is less than 0, it will be supercritical. Let us define the quantities F+ := ∫ Ω lim inf s→+∞ s ∑2 i=1 bifi(·, s) |s|1+α , F+ := ∫ Ω lim sup s→+∞ s ∑2 i=1 bifi(·, s) |s|1+α . 112 B. B. DELGADO, R. PARDO EJDE/SI/02 From the above, and Fatou’s Lemma, if F+ > 0, then D+ is subcritical, while if F+ < 0, then D+ is supercritical. In this work, we will consider nonlinearities satisfying F+ < 0 < F+. (1.9) The above-mentioned condition means that the bifurcating continuum D+ is neither sub-critical nor super-criticall, and hence Landesman-Lazer type conditions do not hold. The main purpose of this article is to establish the existence of infinitely many resonant solutions at λ = 1/µ+ in the absence of Landesman-Lazer type conditions. We note that condition (1.9) reflects the oscillatory behavior of D+ near infinity around the bifurcation point λ = 1/µ+, yielding infinitely many resonant solutions. Hypothesis (H1) guarantees that the continuum of positive solutions or negative solutions bifurcates from infinity at λ = 1/µ+ in the sense of Rabinowitz [14], see Theorem 2.2. Hypothesis (H2) provides the estimates |λ − 1/µ+| = O(tα−1) and ‖w‖ = O(tα) in Proposition 3.2. We try to unveil the sign of the right-hand side in (1.8), just looking at the signs of lim inf n→+∞ ∫ Ω tn ∑2 i=1 bifi(·, tnb) |tn|1+α , lim sup n→+∞ ∫ Ω tn ∑2 i=1 bifi(·, tnb) |tn|1+α . This is done in Lemma (3.4). Hypothesis (H3) helps establishing the inequalities (3.14)–(3.15). With these tools, in Theorem 1.3 we take two sequences {tn} and {t′n} satisfying (1.10), and from here we obtain the existence of unbounded sequences of sub- and super-critical solutions of (1.1) in D+. In particular, we prove the following result. Theorem 1.3. Let (H1)–(H3) hold. Suppose that there exists two increasing se- quences {tn} and {t′n} tending to +∞ and satisfying −∞ < lim n→∞ t′n ∑2 i=1 bi ∫ Ω fi(·, t′nb) |t′n|1+α < 0 < lim n→∞ tn ∑2 i=1 bi ∫ Ω fi(·, tnb) |tn|1+α < +∞, (1.10) where b is the positive eigenfunction of the matrix A defined in (1.4). Then, the following assertions hold: (i) There exist two sequences { (λn, un) } and { (λ′n, u ′ n) } in D+ approaching to (1/µ+,+∞) as n → ∞, with λn < 1/µ+ (subcritical) and λ′n > 1/µ+ (supercritical), respectively. (ii) There is a sequence of turning points {(λ∗n, u∗n)} in D+ approaching to (1/µ+,+∞) as n→∞. Furthermore, one can choose two sequences of turning points, one of them subcritical λ∗2n+1 < 1/µ+ and the other supercritical λ∗2n > 1/µ+. (iii) There is a sequence of resonant solutions. That is, there are infinitely many solutions {(1/µ+, ûn)} ∈ D+ such that ‖ûn‖C(Ω) →∞ as n→∞. Similar results have been considered for the single equation in [9], and for the single equation case with nonlinear boundary conditions, in [4, 5, 3, 8], and in [7] for bifurcation from zero results. In Section 2 we will see that hypothesis (H1) guarantees that the unbounded continuum of positive and negative bifurcates from infinity at λ = 1/µ+. More precisely, the solutions have the form (2.8) and (2.9), respectively. In Section 3 we will prove a series of technical results involving hypotheses (H1)–(H3), needed for proving Theorem 1.3 in Section 4. EJDE-2023/SI/02 RESONANT SOLUTIONS FOR ELLIPTIC SYSTEMS 113 2. Bifurcation from infinity Let us consider the solid cone P = {v ∈ C(Ω): v ≥ 0 in Ω}, and let K : Lr(Ω)→{ W 2,r(Ω): ∂v ∂η = 0 } be the linear operator defined as K h := v, (2.1) where v is the solution to the Neumann problem (−∆ + I)v = h, in Ω, ∂v ∂η = 0, in ∂Ω, (2.2) and ‖v‖W 2,r(Ω) ≤ C‖h‖Lr(Ω), (see [10, p. 162]). K is known as the resolvent operator for the Neumann problem (2.2). For r > N/2, and by Rellich-Kondrachov theorem on compact embedding of Sobolev spaces [6], W 2,r(Ω) ↪→W 1,q(Ω) if q < r∗ (where 1 r∗ = 1 r − 1 N < 1 N ), moreover W 1,q(Ω) ↪→ C(Ω) if q > N . Consequently, K : Lr(Ω)→ C(Ω) is a compact operator. By the maximum principle [2, Theorem 4.1], for all h ≥ 0, h 6= 0, we have v = K h ∈ P̊ , where P̊ = {u ∈ C(Ω): u > 0 in Ω}. In other words, K is strongly positive. By the Krein-Rutman theorem [1], the spectral radius, denoted by r(K ), and defined as the supremum of the modulus of the elements in the spectrum, is a simple eigenvalue with a positive normalized eigenfunction φ ∈ P̊ . Furthermore, there is no other eigenvalue with a positive eigenfunction. Moreover, r(K ) = 1, and φ = 1. (2.3) Using the operator K , equation (1.1) (or its matrix form (1.3)) can be rewritten as u = λAK u+ K f(x, u), in Ω. (2.4) We prove that Rabinowitz’s bifurcation results [13, 14] remain valid in nonlinear systems with homogeneous Neumann boundary conditions such as (1.1). We start with a technical Proposition, characterizing the behavior of a blowing up sequence of solutions. The bifurcation from infinity point is reached when λ = 1/µ+, and there exists a subsequence of solutions such that, when normalized, converges to b in ( Cγ(Ω) )2 for some 0 < γ < 1. Proposition 2.1. Let (H1) hold. Let (λn, un) be a sequence of solutions of (1.1), where un = (u1,n, u2,n)T are positive in both components, λn → λ0 and ‖un‖(C(Ω))2 →∞. Then λ0 = 1/µ+ and there exists a sub-sequence, again denoted by {un}, such that lim n→∞ un ‖un‖(C(Ω))2 = b, where b 6= ( 0 0 ) is defined by (1.6) in ( Cγ(Ω) )2 for some γ ∈ (0, 1). Proof. Let vn := ( u1,n ‖un‖(C(Ω))2 , u2,n ‖un‖(C(Ω))2 )T . Since un ∈ ( W 2,r(Ω) )2 with r > N/2, by the compact embedding theorem, un ∈ ( Cγ(Ω) )2 for some γ > 0. By (H1), we obtain that ‖vn‖(Cγ(Ω))2 ≤ C. Using the compact embedding, ( Cγ(Ω) )2 ↪→ ( Cγ ′ (Ω) )2 for 0 < γ′ < γ, there exists a convergent sub-sequence, namely vn, such that λn → λ0, and vn → φ = 114 B. B. DELGADO, R. PARDO EJDE/SI/02 (φ1, φ2)T , with φi ≥ 0, in ( Cγ ′ (Ω) )2 . Since vi,n ≥ 0, i = 1, 2, and ‖vn‖(C(Ω))2 = 1, it is easy to see that φ 6≡ (0, 0)T , and since φi ≥ 0, also ( ∫ Ω φ1, ∫ Ω φ2 ) 6= (0, 0). Moreover, vn satisfies (−∆ + I)vn = λnAvn + f(x, un) ‖un‖(C(Ω))2 , in Ω ∂vn ∂η = 0, on ∂Ω. (2.5) By (H1) and the continuity of f̃ , we obtain that lim n→∞ fi(x, un) ‖un‖(C(Ω))2 = 0, i = 1, 2. Taking the limit in the weak formulation of the system (2.5) we obtain (−∆ + I)φ = λ0Aφ, in Ω, ∂φ1 ∂η = ∂φ2 ∂η = 0, on ∂Ω, (2.6) or equivalently λ0AK φ = φ. (2.7) Integrating (2.6) in Ω, and since the divergence theorem, ∫ Ω ∆φi = ∫ ∂Ω ∂φi ∂η = 0, so (λ0a1 − 1) ∫ Ω φ1 + λ0a2 ∫ Ω φ2 = 0, λ0a2 ∫ Ω φ1 + (λ0a3 − 1) ∫ Ω φ2 = 0. From ∫ Ω φi ≥ 0, for i = 1, 2, and ( ∫ Ω φ1, ∫ Ω φ2 ) 6= (0, 0), we have 0 ∈ σ(λ0A − I) with an associated eigenvector of nonnegative components. In other words, λ0 6= 0, 1/λ0 ∈ σ(A), and 1/λ0 = µ+, which implies that (2.7) reduces to AK φ = µ+φ, in Ω. By (2.13), r(AK ) = µ+, and φ = b = ( b1 b2 ) is the first normalized eigenfunction of AK . � Next we prove a bifurcation from infinity result. Theorem 2.2. If f satisfies (H1), then the set of solutions of (1.1) possesses an unbounded component bifurcating from infinity at λ = 1/µ+, namely D ⊂ R × (C(Ω))2, where µ+ is defined by (1.5). Moreover, the set of solutions bifurcating from infinity at λ = 1/µ+ contains large positive solutions or large negative solutions of (1.1), namely D+ and D− respectively. Also we have: (i) There exists a neighborhood O+ of (1/µ+,∞) such that (λ, u) ∈ D+ ∩ O+ and (λ, u) 6= (1/µ+,∞) implies that this solutions can be expressed as (λ, u) = (λ, tb+ w) , (2.8) where t > 0, and w = o(|t|) as t→ +∞. (ii) There exists a neighborhood O− of (1/µ+,∞) such that (λ, u) ∈ D− ∩ O− and (λ, u) 6= (1/µ+,∞) implies that this solutions can be expressed as (λ, u) = (λ, tb+ w) , (2.9) where t < 0, and w = o(|t|) as t→ −∞. EJDE-2023/SI/02 RESONANT SOLUTIONS FOR ELLIPTIC SYSTEMS 115 Proof. Let K be the resolvent operator to the Neumannn problem (2.1)-(2.2). Recall that (2.4) is equivalent to (1.1). To use the Rabinowitz bifurcation from infinity result [14, Theorem 1.6] it is necessary to verify the following four conditions: (a) AK is compact on (C(Ω))2. (b) K f is continuous with K f(u) = o(‖u‖) as ‖u‖ → ∞. (2.10) (c) ‖u‖2K f ( u ‖u‖2 ) is a compact operator. (d) µ+ is a simple eigenvalue of AK and its corresponding normalized eigen- function is b. (a) It follows from the compactness of the operator K . (b) Given u ∈ ( C(Ω) )2 , by the continuity of K : Lr(Ω)→ C(Ω) and using (H1) we have that for each ε > 0 there exist C1, C2 > 0 such that∥∥K f(·, u(·)) ∥∥ (C(Ω))2 ≤ C1 ∥∥f(·, u(·)) ∥∥ (Lr(Ω))2 ≤ C1‖h‖Lr(Ω) ∥∥f̃(u(·)) ∥∥ C(Ω) ≤ C1‖h‖Lr(Ω) ( ε‖u‖+ C2 ) which implies (2.10). (c) Let us define H : Ω× R2 → R2 as follows H (x, u) = ‖u‖2K f ( x, u ‖u‖2 ) . (2.11) Let us consider the closed subset Ω×Bδ with Bδ = { u ∈ ( C(Ω) )2 : ‖u‖(C(Ω))2 ≤ δ } . It is sufficient to prove that the image of Ω×Bδ under H is relatively compact in( C(Ω) )2 for some δ > 0 sufficiently small. Let us consider the map u 7→ f(·, u) from ( C(Ω) )2 to ( Lr(Ω) )2 , r > N/2. Using (H1) and the continuity of f̃ , for each ε > 0 there exists C > 0 such that∫ Ω |fi(x, u)|r ≤ ∫ Ω h(x)rf̃(u)r dx ≤ ∫ Ω h(x)r ( ε|u(x)|+ C )r dx. Thus, ‖f(·, u)‖(Lr(Ω))2 ≤ ‖h‖Lr(Ω) ( ε‖u‖(C(Ω))2 + C ) , ∀u ∈ ( C(Ω) )2 . (2.12) Let v = (v1, v2)T ∈ Bδ. Applying (2.12) to v/‖v‖2 (C(Ω))2 and taking ε = 1, we readily obtain ‖v‖2 ∥∥f(·, v ‖v‖2 )∥∥ (Lr(Ω))2 ≤ ‖h‖Lr(Ω) ( ‖v‖+ C‖v‖2 ) ≤ ‖h‖Lr(Ω)(δ + Cδ2). By the continuity of K : Lr(Ω)→W 2,r(Ω), there exists C3 > 0 such that ‖H (·, v)‖ ≤ C3‖h‖Lr(Ω) (δ + Cδ2), where H is defined in (2.11), which implies that the map H sends closed sets into bounded sets. (d) By (2.3) we know that r(K ) = 1 is a simple eigenvalue of K and its corresponding eigenfunction is equal to 1. Notice that the hypothesis ai > 0 for i = 116 B. B. DELGADO, R. PARDO EJDE/SI/02 1, 2, 3, guarantee that AK is a strongly positive compact operator. Again by the Krein-Rutman theorem [1], there exists a unique positive eigenfunction associated to r(AK ) up to a multiplicative constant. Now, observe that AK b = Ab = µ+b, so b = ( b1 b2 ) is the first normalized eigenfunction of AK . That is, r(AK ) = µ+, and b is its corresponding eigenfunction, (2.13) as desired. � 3. Functional framework and auxiliary results We analyze the associated linear system, the nonhomogeneous Neumann problem (−∆ + I)u = λAu+ g(x), in Ω, ∂u ∂η = 0, on ∂Ω, (3.1) where λ ∈ J :=  ( −∞,min { 1 µ− , λ2 µ+ }) , if µ− > 0,( −∞, λ2 µ+ ) , if µ− = 0,( 1 µ− , λ2 µ+ ) , if µ− < 0, where λ2 is defined in (1.7), and g ∈ ( Lr(Ω) )2 . By the Fredholm alternative, (3.1) has a unique solution u ∈ ( W 2,r(Ω) )2 if λ 6= 1/µ+, see [11]. Moreover, by the compact embedding theorem u ∈ ( C(Ω) )2 , since r > N/2. Let us consider the orthogonal decomposition( Lr(Ω) )2 = span{b} ⊕ { φ = ( φ1 φ2 ) ∈ ( Lr(Ω) )2 : ∫ Ω b1φ1 + b2φ2 = 0 } . (3.2) For g ∈ ( Lr(Ω) )2 , there exists g(1) ∈ ( Lr(Ω) )2 such that g(·) = ã1 b+g(1)(·), where g = ( g1 g2 ) and g(1) = ( g (1) 1 g (1) 2 ) satisfy ã1 = 1 |Ω| ∫ Ω b1g1(x) + b2g2(x) and ∫ Ω b1g (1) 1 (x) + b2g (1) 2 (x) = 0. (3.3) By the Fredholm alternative, the linear elliptic system (3.1) has a unique solution if λ 6= 1/µ+ and does not have a solution if λ = 1/µ+ and ã1 6= 0. Consequently, for λ 6= 1/µ+, the solution u = u(λ) of (3.1) belongs to ( W 2,r(Ω) )2 ⊂ (Lr(Ω) )2 , and u = ã1 1− λµ+ b+ w, with ∫ Ω b1w1(x) + b2w2(x) dx = 0. (3.4) Moreover, w = w(λ) satisfies (−∆ + I)w = λAw + g(1)(x), in Ω, ∂w ∂η = 0, on ∂Ω, (3.5) where g(1) is defined by (3.3). EJDE-2023/SI/02 RESONANT SOLUTIONS FOR ELLIPTIC SYSTEMS 117 If λ = 1/µ+, by the Fredholm alternative there exists a function v ∈ ( W 2,r(Ω) )2 such that v+cb solves (3.5) for any c ∈ R. Fixing c0 such that ∫ Ω b1v1+b2v2+c0 = 0, and defining w(1/µ+) = v + c0b, then, w(1/µ+) ∈ (span{b})⊥. Next, we estimate the norm of the solution to the linear system (3.5) in ( C(Ω) )2 whenever g ∈ ( Lr(Ω) )2 . From now on, throughout this paper C (possibly with indices) denotes several constants independent of the solution u, and which may change from line to line. Lemma 3.1. For each compact set K ⊂ J , there exists a constant C = C(K) > 0, independent of λ ∈ K such that ‖w(λ)‖(C(Ω))2 ≤ C‖g(1)(·)‖(Lr(Ω))2 , where w = (w1, w2)T satisfies (3.5) with ∫ Ω b1w1+b2w2 = 0, and g(1) ∈ (span{b})⊥. Proof. By the above discussion, w = w(λ) satisfying (3.4) and (3.5) is well defined. Suppose that there exists a sequence λn → 1/µ+ such that ‖w(λn)‖(C(Ω))2 →∞. Therefore, w(λn) ‖w(λn)‖(C(Ω))2 → b, which contradicts that ∫ Ω b1w1(λn, ·)+b2w2(λn, ·) = 0. Therefore, there exists C > 0 and δ > 0 such that ‖w(λ)‖(C(Ω))2 < C independent of λ for any |λ− 1/µ+| < δ. Let λ ∈ K \ (1/µ+ − δ, 1/µ+ + δ). By the Fredholm alternative, w(λ) ∈ (W 2,r(Ω))2 is the unique solution to (3.5). Using the Lr-estimate and the em- bedding (W 2,r(Ω))2 ↪→ ( C(Ω) )2 , we have that ‖w(λ)‖(C(Ω))2 ≤ C‖w‖(W 2,r(Ω))2 ≤ C‖g(1)(·)‖(Lr(Ω))2 <∞. Now, let λ ∈ K and let us define a family of bounded operators as follows Tλ : { g1 ∈ ( Lr(Ω) )2 : ∫ Ω b1g (1) 1 + b2g (1) 2 = 0 } → ( C(Ω) )2 , Tλg1 = w(λ), where w(λ) is the solution to (3.5). More precisely, for every λ ∈ K, Tλ is a continuous operator. Moreover, supλ∈K ‖Tλg1‖(C(Ω))2 < ∞. Finally, the Uniform Boundedness Principle implies the result. � Proposition 3.2. Let (H1) and (H2) hold. Then, there exists a neighborhood of (1/µ+,∞)× ( C(Ω) )2 ⊂ R× ( C(Ω) )2 given by O := { (λ, u) ∈ R× ( C(Ω) )2 : |λ− 1/µ+| < δ0, ui > 0, i = 1, 2, ‖u‖(C(Ω))2 > M0 } , for some small δ0 and large M0, such that the following hold: (i) There exist positive constants C1, C2 (independent of λ) such that if (λ, u) ∈ D+ ∩ O and ( λ, ‖u‖(C(Ω))2 ) 6= (1/µ+,∞) then u = tb+ w, where t > 0, and ∫ Ω b1w1 + b2w2 = 0, (3.6) ‖w‖(C(Ω))2 ≤ C1‖B‖Lr(Ω)t α as t→∞, (3.7) |λ− 1/µ+| ≤ C2t α−1 as t→∞. (3.8) 118 B. B. DELGADO, R. PARDO EJDE/SI/02 (ii) There exists t0 > 0 such that for all t ≥ t0 there exists (λ, u) ∈ D+ ∩ O satisfying u = tb+ w with w = (w1, w2) satisfying ∫ Ω b1w1 + b2w2 = 0. Proof. By Theorem 2.2 it is clear that D+ ∩O 6= ∅. Let (λ, u) ∈ D+ ∩O, by (3.2) u can be written as u = tb+ w, where t > 0, ∫ Ω b1w1 + b2w2 = 0. Let (λ, u) = (λ, tb+w) solve (1.1). Multiplying (1.3) by bT on the left, integrating the result, using that ∫ Ω bTu = t|Ω|, and the symmetry of the matrix A, so bTA = µ+b T , we obtain t|Ω|(1− λµ+) = ∫ Ω bT f (·, tb+ w) , or equivalently (1− λµ+) = 1 |Ω| ∫ Ω b1f1(·, tb+ w) + b2f2(·, tb+ w) t . (3.9) Now, using (H1) and since w = o(|t|) as |t| → ∞, we have that |f(x, tb+ w)| |t| = |f(x, tb+ w)| |tb+ w| |b+ w t | → 0 as t→∞. (3.10) By (H1), (3.9)-(3.10), and the Lebesgue dominated convergence theorem, we readily obtain that λ→ 1/µ+ as t→∞. By (H2) we deduce that |f(x, tb+ w)| = |t|α |f(x, tb+ w)| |tb+ w|α ∣∣∣b+ w t ∣∣∣α ≤ |t|αB(x) ∣∣b+ w t ∣∣α. (3.11) Substituting (3.11) into (3.9), |λ− 1/µ+| ≤ |t|α−1 |Ω|µ+ ∫ Ω B(x) ∣∣b+ w t ∣∣α ≤ C‖B‖Lr(Ω)|t|α−1, and assertion (3.8) readily follows. To complete the proof of part (i), it only remains to verify that (3.7) holds. Because of f(·, u(·)) ∈ ( Lr(Ω) )2 , by (3.2) and (3.3) there exists a unique orthogonal decomposition f(x, u) = ã1b+ f (1)(x, u), where ã1 = (1/|Ω|) ∫ Ω b1f1(x, u) + b2f2(x, u) and f (1) = ( f (1) 1 f (1) 2 ) ∈ ( span b )⊥ , that is, ∫ Ω b1f (1) 1 + b2f (1) 2 = 0. By Lemma 3.1, we obtain ‖w‖(C(Ω))2 ≤ C‖f (1)‖(Lr(Ω))2 ≤ C ′‖f‖(Lr(Ω))2 . From estimate (3.11), w = o(|t|), and Hölder inequality, we have ‖w‖(C(Ω))2 ≤ C‖B‖Lr(Ω)|t|α, as t→∞. (ii) Since D+ bifurcates from infinity at λ = 1/µ+, we have that despite of D+∩ O is not necessarily connected, it contains an unbounded connected component, namely G such that if (λ, u) ∈ G ⊂ D+ ∩ O, then u = tb+ w, with ∫ Ω b1w1 + b2w2 = 0 and t = 1 |Ω| ∫ Ω b1u1 + b2u2. (3.12) EJDE-2023/SI/02 RESONANT SOLUTIONS FOR ELLIPTIC SYSTEMS 119 By the continuity of the projection of G on span{b}, the set {t ∈ R : (1.1) has a solution satisfying (3.12)} also contains an unbounded connected set as desired. � A straightforward consequence of Proposition 3.2 is the following result. Corollary 3.3. Assume (H1) and (H2) hold. Let {(λn, un)} ⊂ D+ ∩ O be such that λn → 1/µ+ and un = tnb + wn, with ∫ Ω b1w1,n + b2w2,n = 0 and tn = 1 |Ω| ∫ Ω b1u1,n + b2u2,n →∞. Then lim n→∞ un ‖un‖(C(Ω))2 = b, uniformly in Ω, lim n→∞ un tn = b, uniformly in Ω, lim n→∞ ‖un‖(C(Ω))2 tn = 1, uniformly in Ω. Hypothesis (H3) will be key for proving the following result. Later on, the main theorem of this paper, Theorem 1.3, we will see that the next lemma together with hypothesis (1.10) are sufficient to conclude the existence of subcritical solutions (λ < 1/µ+) and supercritical solutions (λ > 1/µ+), respectively, in the unbounded continuum D+. Lemma 3.4. Let f satisfy (H3). Suppose there exists α < 1 and a function B ∈ L1(Ω) such that for x ∈ Ω, and for all (λ, s) close to the bifurcation point (1/µ+,+∞), we have |f(x, s)| |s|α ≤ B1(x). (3.13) Let λn → 1/µ+ and wn ∈ L∞(Ω) are such that ‖wn‖L∞(Ω) = O(|tn|α) as n → ∞. Then lim inf n→∞ 2∑ i=1 bi ∫ Ω fi(·, tnb+ wn) |tnb+ wn|α ≥ lim inf n→∞ tn ∑2 i=1 bi ∫ Ω fi(·, tnb) |tn|1+α , (3.14) lim sup n→∞ 2∑ i=1 bi ∫ Ω fi(·, tnb+ wn) |tnb+ wn|α ≤ lim sup n→∞ tn ∑2 i=1 bi ∫ Ω fi(·, tnb) |tn|1+α . (3.15) Proof. Let w = (w1, w2) ∈ (L∞(Ω))2, t > 0 and ‖w‖ < t/2. By the Mean Value Theorem, fi(x, tb+w)− fi(x, tb) = ∫ 1 0 ( ∂fi ∂s1 (x, tb+ τw)w1 + ∂fi ∂s2 (x, tb+ τw)w2 ) dτ, (3.16) for all x ∈ Ω and i = 1, 2. Consequently, we can write (3.16) in matrix notation as f(x, tb+ w)− f(x, tb) = (∫ 1 0 Jf(x, tb+ τw) dτ ) w, where Jf := ( ∂f1 ∂s1 ∂f1 ∂s2 ∂f2 ∂s1 ∂f2 ∂s2 ) 120 B. B. DELGADO, R. PARDO EJDE/SI/02 denotes the Jacobian matrix of f = (f1, f2) and the integral of a matrix is taken component-wise. Consequently,∫ Ω |f(·, tb+ w)− f(·, tb)| dx ≤ C‖w‖(L∞(Ω))2 2∑ i,j=1 sup τ∈[0,1] ∥∥∂fi ∂sj (·, tb+ τw) ∥∥ L∞(Ω) . (3.17) If ‖w‖(L∞(Ω))2 = O(|t|α), using (3.17) and (H3), we obtain∫ Ω |f(·, tb+ w)− f(·, tb)| |t|α dx ≤ C 2∑ i,j=1 sup |s|≥M ∥∥∂fi ∂sj (·, s) ∥∥ L∞(Ω) ‖w‖(L∞(Ω))2 |t|α → 0, (3.18) as M →∞. Now, let us consider tn →∞ and wn = O(|tn|α) as n→∞. By (3.18) and the Cauchy-Schwarz inequality, we obtain that lim inf n→∞ ∑2 i=1 bi ∫ Ω fi(·, tnb+ wn) |tn|α ≥ lim n→∞ ∑2 i=1 bi ∫ Ω fi(·, tnb+ wn)− fi(·, tnb) |tn|α + lim inf n→∞ ∑2 i=1 bi ∫ Ω fi(·, tnb) |tn|α = lim inf n→∞ ∑2 i=1 bi ∫ Ω fi(·, tnb) |tn|α . (3.19) Now, note that∑2 i=1 bifi(·, tnb+ wn) |tn|α = ∑2 i=1 bifi(·, tnb+ wn) |tnb+ wn|α ∣∣b+ wn tn ∣∣α. (3.20) Combining (3.19) and (3.20), and since |b| = 1, we readily obtain lim inf n→∞ ∑2 i=1 bi ∫ Ω fi(·, tnb) |tn|α ≤ lim inf n→∞ 2∑ i=1 bi ∫ Ω fi(·, tnb+ wn) |tnb+ wn|α ∣∣b+ wn tn ∣∣α ≤ lim inf n→∞ 2∑ i=1 bi ∫ Ω fi(·, tnb+ wn) |tnb+ wn|α . By (3.13), the right-hand side above is well-defined, which implies (3.14). The proof of (3.15) is analogous. � 4. Proof of Theorem 1.3 Proof. (i) Let (λn, un), (λ′n, u ′ n) ∈ D+ ∩ O such that (λn, ‖un‖), (λ′n, ‖u′n‖) → (1/µ+,∞) provided by Proposition 3.2. Therefore un = tnb+ wn with ∫ Ω b1w1,n + b2w2,n = 0 and tn := 1 |Ω| ∫ Ω b1u1,n + b2u2,n, u′n = t′nb+ w′n with ∫ Ω b1(w′1,n) + b2(w′2,n) = 0 and t′n := 1 |Ω| ∫ Ω b1u ′ 1,n + b2u ′ 2,n. EJDE-2023/SI/02 RESONANT SOLUTIONS FOR ELLIPTIC SYSTEMS 121 By (3.9) we have |Ω| tn ( 1− λnµ+ ) = ∫ Ω b1f1(·, un) + b2f2(·, un). Dividing by ‖un‖α(C(Ω))2 , Corollary 3.3 yields lim inf n→∞ |Ω| ( 1− λnµ+ ) ‖un‖α−1 (C(Ω))2 = lim inf n→∞ ∫ Ω b1f1(·, un) + b2f2(·, un) ‖un‖α(C(Ω))2 . Furthermore, 2∑ i=1 bi ∫ Ω fi(·, un) ‖un‖α (C(Ω))2 = 2∑ i=1 bi ∫ Ω fi(·, un) |un|α ( |un| ‖un‖(C(Ω))2 )α = 2∑ i=1 bi ∫ Ω fi(·, un) |un|α [( |un| ‖un‖(C(Ω))2 )α − 1 ] + 2∑ i=1 bi ∫ Ω fi(·, un) |un|α . Now, by (H3) and Corollary 3.3, 2∑ i=1 bi ∫ Ω ∣∣∣fi(·, un) |un|α [( |un| ‖un‖(C(Ω))2 )α − 1 ]∣∣∣ ≤ ∫ Ω B(x) [( |un| ‖un‖(C(Ω))2 )α − 1 ] → 0, as n→∞. Consequently, lim inf n→∞ |Ω| ( 1− λnµ+ ) ‖un‖α−1 (C(Ω))2 ≥ lim inf n→∞ 2∑ i=1 bi ∫ Ω fi(·, un) |un|α . Now, Lemma 3.4 implies lim inf n→∞ |Ω| ( 1− λnµ+ ) ‖un‖α−1 ∞ ≥ lim inf n→∞ 2∑ i=1 bi ∫ Ω fi(·, tnb+ wn) |tnb+ wn|α ≥ lim inf n→∞ tn ∑2 i=1 bi ∫ Ω fi(·, tnb) |tn|1+α > 0 , where the positivity comes from the hypothesis (1.10). Therefore, λn < 1/µ+ is subcritical for n sufficiently large. Likewise, λ′n > 1/µ+ is supercritical for n sufficiently large. (ii) Let {tn} and {t′n} be sequences such that tn, t ′ n > 0 and tn, t ′ n → ∞ as n→∞. Then, we can choose tn < t′n < tn+1 for all n ≥ 1 and tn, t ′ n ≥ t0, where t0 is as defined in Proposition 3.2, (ii). Then there exists (λn, un), (λ′n, u ′ n) ∈ D+ ∩O such that un = tnb+ wn with ∫ Ω b1w1,n + b2w2,n = 0, u′n = t′nb+ w′n with ∫ Ω b1(w′1,n) + b2(w′2,n) = 0. By part (i) we obtain that λn < 1/µ+ and λ′n > 1/µ+ for n sufficiently large. If (λ, u) ∈ D+ ∩ O and t = 1 |Ω| ∫ Ω b1u1 + b2u2 > t0, Proposition 3.2 (i) yields for t0 122 B. B. DELGADO, R. PARDO EJDE/SI/02 sufficiently large: ‖u‖(C(Ω))2 = ‖tb+ w‖(C(Ω))2 ≤ ( 1 + C‖B‖Lr(Ω)|t0|α−1 ) t ≤ 2t, |λ− 1/µ+| < Ctα−1 ≤ Ctα−1 0 . (4.1) Let us define Kn := { (λ, u) ∈ D+ ∩ O : t = 1 |Ω| ∫ Ω b1u1 + b2u2 and tn ≤ t ≤ tn+1 } . We will show that Kn is a compact set in R × ( C(Ω) )2 . Let {(µk, vk)} ⊂ Kn. If tn ≤ 1 |Ω| ∫ Ω b1v (k) 1 + b2v (k) 2 ≤ tn+1 for all k. By (4.1), we easily obtain that ‖vk‖(C(Ω))2 ≤ 2tn+1 for all k. By [12, Th. 2.4], there exists a constant Cn such that ‖vk‖(Cα(Ω))2 ≤ C1(1 + ‖vk‖(C(Ω))2) ≤ Cn. Using the compact embedding Cα(Ω) ↪→ Cβ(Ω) for some β ∈ (0, α), we infer that there exists u∗ ∈ (Cβ(Ω))2 such that vk → u∗ in (Cβ(Ω))2 up to a subsequence and also (µk, vk)→ (µ∗, u∗). By definition of Kn, we have that (µk, vk) satisfies (−∆ + I)vk = µkAvk + f(x, vk), in Ω, ∂vk ∂η = 0, on ∂Ω, By assumption f is Carathéodory, hence f(µk, ·, vk)→ f(µ∗, ·, u∗) pointwise. Con- sequently, f(µk, ·, vk) → f(µ∗, ·, u∗) in ( Lr(Ω) )2 . Passing to the limit in the weak formulation (1.2), we can see that u∗ is a weak solution to (−∆ + I)u∗ = µ∗Au∗ + f(µk, x, u ∗), in Ω, ∂u∗ ∂η = 0, on ∂Ω. Using the continuity of the projection onto span b implies that t0 ≤ tn ≤ t∗ := 1 |Ω| ∫ Ω b1u ∗ 1 + b2u ∗ 2 ≤ tn+1. Thus, (µ∗, u∗) ∈ Kn, which establishes the compactness of Kn. Since tn < t′n < tn+1, by part (i) there exists (λ′n, u ′ n) ∈ Kn such that u′n = t′nb+ w′n and ∫ Ω b1(w′1,n) + b2(w′2,n) = 0 and λ′n > 1/µ+. We define λ∗n := sup{λ : (λ, u) ∈ Kn} Then, λ∗n ≥ λ′n > 1/µ+ (supercritical). Analogously to the previous limiting argu- ment, there exists u∗n = ( u∗1,n, u ∗ 2,n ) such that (λ∗n, u ∗ n) ∈ Kn. That is, tn ≤ 1 |Ω| ∫ Ω b1u ∗ 1,n + b2u ∗ 2,n ≤ tn+1, recalling that tn and tn+1 are associated to λn < 1/µ+ and λn+1 < 1/µ+, respec- tively. We can observe that there is no solution (λ, u) close to (λ∗n, u ∗ n) with λ > λ∗n. Indeed, if there is a solution (λ, u) close to (λ∗n, u ∗ n) with λ > λ∗n, by the continuity of the projection, we have tn ≤ 1 |Ω| ∫ Ω b1u1 + b2u2 ≤ tn+1, so (λ, u) ∈ Kn, which contradict the definition of λ∗n. Therefore, (λ∗n, u ∗ n) is a supercritical turning point. Similarly, letting K ′n := { (λ, u) ∈ D+ ∩ O : t′ = 1 |Ω| ∫ Ω b1u1 + b2u2 and t′n ≤ t′ ≤ t′n+1 } , EJDE-2023/SI/02 RESONANT SOLUTIONS FOR ELLIPTIC SYSTEMS 123 λ∗,n = inf { λ : (λ, u) ∈ K ′n } . We can guarantee the existence of a sequence {(λ∗,n), u∗,n} ⊂ K ′n, of subcritical turning points with λ∗,n ≤ λn < 1/µ+. Lastly, combining the sequences {λ∗n} and {λ∗,n} and relabeling, we obtain two subsequences of turning points, one subcritical (λ∗2n+1 < 1/µ+) and the other supercritical (λ∗2n > 1/µ+) as we desired. (iii) Now, we prove the existence of a sequence of resonant solutions. That is, solutions u of (1.1) corresponding to λ = 1/µ+. It is sufficient to show that there exists n0 ∈ N such that Kn and K ′n contain resonant solutions of the form (1/µ+, u) for each n ≥ n0. Suppose on the contrary that there exists a sequence of integers numbers nj → ∞ such that Knj does not contain any resonant solutions. Let us define the compact sets K+ nj := { (λ, u) ∈ Knj : λ > 1/µ+ } , which can be rewritten as K+ nj = D+ ∩ O ∩ { (λ, u) ∈ R× ( C(Ω) )2 : λ > 1/µ+, tnj ≤ 1 |Ω| ∫ Ω b1u1 + b2u2 ≤ tnj+1 } . Thus, K+ nj contains at least one nonempty connected component of D+. In view of tn < t′n < tn+1 by construction, we easily obtain that (tnj , tnj+1)∩ (tnj+1, tnj+2) = ∅, then K+ nj ∩K + nj+1 = ∅ for all j ∈ N. Since D+ is a continuum (closed connected set) in R× ( C(Ω) )2 , the contradiction comes from the fact that a continuum cannot contain two nonempty disjoint connected components. A similar argument applied to the sets K ′n also generates a sequence of resonant solutions u corresponding to λ = 1/µ+. � Acknowledgements. Briceyda B. Delgado was supported by Sof́ıa Kovalevskaya grant from the Sof́ıa Kovalevskaya Foundation and the Mexican Mathematical So- ciety. Rosa Pardo was supported by projects PID2019 - 103860GB - I00, MICINN, Spain, and by UCM-BSCH, Spain, GR58/08, Grupo 920894. Part of this material is based on work supported by the National Science Founda- tion under Grant 1440140, while the authors were in residence at the Mathematical Sciences Research Institute, Berkeley, California, during June of 2022. References [1] Nicholas D. 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