Special Issue in honor of Alan C. Lazer Electronic Journal of Differential Equations, Special Issue 01 (2021), pp. 255–268. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu or https://ejde.math.unt.edu EIGENVALUES AND BIFURCATION FOR NEUMANN PROBLEMS WITH INDEFINITE WEIGHTS MARTA CALANCHI, BERNHARD RUF In memory of Alan Lazer with admiration Abstract. We consider eigenvalue problems and bifurcation of positive solu- tions for elliptic equations with indefinite weights and with Neumann bound- ary conditions. We give complete results concerning the existence and non- existence of positive solutions for the superlinear coercive and non-coercive problems, showing a surprising complementarity of the respective results. 1. Introduction This article concerns the eigenvalues for elliptic equations with an indefinite weight function −∆u = λa(x)u in Ω ⊂ RN Bu = 0 on ∂Ω, (1.1) where Ω ⊂ RN is a bounded domain, a : Ω→ R is a continuous and sign-changing function, and B denotes a homogeneous boundary condition, say Dirichlet or Neu- mann. Eigenvalue problems with indefinite weights have numerous applications in en- gineering, physics, biology, etc.; see the recent work by Sovrano [20] concerning selection-migration models in population genetics. The second order ODE corresponding to (1.1) has been widely studied, begin- ning with the work of Bôcher [2], Hilbert [12], and Richardson [18]. The first work for the Dirichlet boundary value problem in higher dimensions goes back to Holmgren (1904) [13] who considered the equation in bounded domains in two di- mensions, proving the existence of a sequence of positive and a sequence of negative eigenvalues. For recent works on such problems we cite the work by de Figueiredo [7], Hess- Kato [11] and Manes-Micheletti [16] in which the indefinite Dirichlet eigenvalue problem in Ω ⊂ RN was studied. They proved the existence of two sequences of eigenvalues 0 < λ+ 1 < λ+ 2 ≤ · · · → +∞ and 0 > λ−1 > λ−2 ≥ · · · → −∞, and gave a variational min-max characterization for these eigenvalues. The aim of Manes 2010 Mathematics Subject Classification. 35B32, 35B09, 49J35. Key words and phrases. Eigenvalues; indefinite weight; Neumann problems; bifurcation. ©2021 This work is licensed under a CC BY 4.0 license. Published December 14, 2021. 255 256 M. CALANCHI, B. RUF EJDE/SI/01 and Micheletti was to generalize the assumptions for the so-called Ambrosetti- Prodi problem: considering a nonlinearity which crosses asymptotically the first eigenvalue of the Laplacian, Ambrosetti and Prodi (1972) [1] gave in their pioneering result a global description of the solutions structure of the associated Dirichlet problem, characterizing it as a global fold mapping between Banach spaces. To achieve this, the linearization of the nonlinear mapping needs to be controlled in every point of the domain space, which leads to the indefinite eigenvalue problems studied by Manes-Micheletti. For interesting generalizations of these methods, see [4, 5, 6, 19]. Recently, López-Gómez and Rabinowitz [15] studied bifurcation problems asso- ciated to indefinite eigenvalue problems. In particular, for the model problem −∆u = λa(x)u− |u|p−1u in Ω ⊂ RN u = 0 on ∂Ω with 1 < p and a(x) continuous and sign-changing, they showed the existence at least k pairs of solutions for λ > λ+ k , as well as for λ < λ−k , implying that all eigenvalues of equation (1.1) are also bifurcation points. In this article we study the eigenvalue problem with Neumann boundary condi- tions which has been less studied. −∆u = λa(x)u in Ω ⊂ RN ∂u ∂ν = 0 on ∂Ω. (1.2) There is again a positive and a negative sequence of eigenvalues, with the peculiarity that λ+ 1 = 0 if ∫ Ω a(x)dx > 0, and λ−1 = 0 if ∫ Ω a(x)dx < 0; this implies in particular that λ+ 1 = λ−1 = 0 if ∫ Ω a(x)dx = 0. The variational characterization and properties of these eigenvalues are given in section 2. In section 3 we consider the bifurcation of positive solutions from the first eigen- values λ±1 for the problems −∆u = λa(x)u± up in Ω ⊂ RN u > 0 in Ω ∂u ∂ν = 0 on ∂Ω (1.3) We will prove the following results, which show an interesting complementarity between problems (1.3) with (−) and with and (+). Theorem 1.1. Assume that a ∈ C(Ω) and sign-changing, and p > 1. Then equation (1.3) with (−) has • for λ < λ−1 and for λ > λ+ 1 a positive solution, • for λ−1 ≤ λ ≤ λ + 1 no positive solution. Theorem 1.2. Assume that a ∈ C(Ω) and sign-changing, and 1 < p < N+2 N−2 . Then equation (1.3) with (+) has • for λ ≤ λ−1 and for λ ≥ λ+ 1 no positive solution, • for λ−1 < λ < λ+ 1 a positive solution. The complementarity is most striking in the degenerate case ∫ Ω a(x)dx = 0. Then we have λ−1 = λ+ 1 = 0, and hence EJDE-2021/SI/01 NEUMANN PROBLEMS WITH INDEFINITE WEIGHTS 257 Corollary 1.3. If ∫ Ω a(x)dx = 0: • problem (1.3) with (−) has a positive solution for every λ 6= 0 (p > 1); • problem (1.3) with (+) has no positive solution for every λ ∈ R (1 < p < (N + 2)/(N − 2)). 2. Eigenvalue problem with indefinite weights In this section we give a short description of the spectrum, eigenfunctions, and some of their properties for the eigenvalue problem −∆φ = λa(x)φ in Ω ∂φ ∂ν = 0 on ∂Ω (2.1) where Ω ⊂ RN is an open bounded domain, with ∂Ω of class C1, and a = a(x) ∈ L∞(Ω) is a non trivial function. As mentioned in the introduction (see also Manes- Micheletti [16]), if a = a(x) changes sign then there exist two sequences (i) {λ+ j } of positive eigenvalues, with associated eigenfunctions {φ+ j }, (ii) {λ−j } of negative eigenvalues, with associated eigenfunctions {φ−j }. Manes and Micheletti [16] discussed the Dirichlet case (for more general elliptic operators). Here we focus on the Neumann case. We define the bilinear form S(u, v) := ∫ Ω a(x)uv dx Let B+ = {u : S(u, u) = 1}, B− = {u : S(u, u) = −1}. Remark 2.1. Since a = a(x) changes sign, both B+ and B− are nonempty. In what follows, we outline some properties of the eigen-pairs (λ±j , φ ± j ), and rephrase the variational characterization for the eigenvalues given by Manes and Micheletti [16], see also [3]. (a) (Quasi-orthogonality) If λ∗ and λ∗ are two different eigenvalues of (2.1), and resp. φ∗, φ ∗ two associated eigenvectors, then φ∗, φ ∗ are orthogonal∫ Ω ∇φ∗∇φ∗ dx = 0, ∫ Ω a(x)φ∗φ ∗ dx = 0. (b) (First eigenvalues) λ+ 1 = inf u∈B+ ∫ Ω |∇u|2dx ≥ 0, λ−1 = − inf u∈B− ∫ Ω |∇u|2dx ≤ 0 are simple, with associated positive eigenfunctions φ+ 1 and φ−1 . (c) (Higher eigenvalues) For k ≥ 2, λ+ k = inf dimF=k sup u∈B+∩F ∫ Ω |∇u|2dx > 0, λ−k = − inf dimF=k sup u∈B−∩F ∫ Ω |∇u|2dx < 0, or equivalently, using MM characterization, 1 λ+ k = sup dimF=k min u∈F,u6=0 ∫ Ω a(x)u2∫ Ω |∇u|2 , 1 λ−k = − sup dimF=k min u∈F,u6=0 − ∫ Ω a(x)u2∫ Ω |∇u|2 . (2.2) (d) λ+ k → +∞ and λ−k → −∞ as k → +∞. 258 M. CALANCHI, B. RUF EJDE/SI/01 (e) (Positivity of first eigenfunctions) The eigenfunctions corresponding to the first eigenvalues have constant sign. Moreover, the eigenvalues λ 6= λ±1 do not posses a positive eigenfunction. The same results occur in both the Dirichlet and the Neumann case, but a dis- tinction is a must: while for the Dirichlet case the inequality for the first eigenvalues λ±1 is strict, i.e. λ−1 < 0 < λ+ 1 , we have that λ = 0 is always an eigenvalue in the Neumann case. Indeed, for Neumann boundary conditions, when ∫ Ω a(x)dx = 0, then both first eigenvalues λ+ 1 and λ−1 coincide with zero. On the other hand, when the mass of the sign-changing weight a(x) is unbalanced (say, ∫ Ω a(x)dx < 0), we still have that λ−1 = 0 fits in the characterization described above. Roughly speak- ing, if the negative part is dominant, the first “negative” eigenvalue is the trivial one, as stated in the following proposition. Proposition 2.2 (Neumann case). Let a : Ω→ R be a continuous, sign changing function. (1) If ∫ Ω a(x) dx < 0 (resp. > 0), then λ−1 = 0 and λ+ 1 > 0 (resp. λ−1 < 0 and λ+ 1 = 0 ) . (2) If ∫ Ω a(x) dx = 0, then λ−1 = 0 = λ+ 1 . Proof. (1) The first statement is trivial. Indeed, λ−1 := − infu∈B− ∫ Ω |∇u|2dx = 0, since the infimum is attained by the constant function u(x) = α, where α satisfies α2 = − 1∫ Ω a(x) dx . For λ+ 1 we argue by contradiction. If λ+ 1 := infu∈B+ ∫ Ω |∇u|2dx = 0, there is a sequence un = wn + sn, with ∫ Ω wn = 0 and sn ∈ R such that∫ Ω a(x)u2 n = 1, ∫ Ω |∇wn|2dx→ 0, as n→ +∞. Therefore wn → 0 strongly in H1(Ω) and sn is bounded: otherwise we would have (up to subsequences) 1 = ∫ Ω a(x)u2 n = ∫ Ω a(x)(s2 n + 2wnsn + w2 n)dx = s2 n (∫ Ω a(x) dx+ o(1) ) → −∞. Since sn is bounded, up to subsequences, sn → s and un → s strongly, from which we obtain 1 = ∫ Ω a(x)u2 n → s2 ∫ Ω a(x) ≤ 0, which is a contradiction. (2) Let Ω+ = {x ∈ Ω : a(x) > 0} and B ⊂⊂ Ω+ a ball. Let vε(x) = 1 + ε η, where η ∈ C∞0 (Ω) is a positive smooth function with compact support in B, and ε > 0. Then ∫ Ω |∇vε|2∫ Ω a(x)v2 ε = ε2 ∫ Ω |∇η|2∫ Ω a(x)(1 + εη)2 = ε2 ∫ Ω |∇η|2 ε ∫ Ω a(x)(2η + εη2) → 0, as ε→ 0. This proves that λ+ 1 = 0. With a similar argument, taking a smooth function with support in Ω− we obtain that λ−1 = 0. � EJDE-2021/SI/01 NEUMANN PROBLEMS WITH INDEFINITE WEIGHTS 259 3. Superlinear equations - bifurcation of positive solutions We now consider superlinear equations of the form −∆u = λa(x)u± h(u) in Ω u > 0 in Ω ∂u ∂ν = 0 on ∂Ω (3.1) where a : Ω → R is continuous and sign-changing, and h ∈ C(R+,R+) is a super- linear function. As a model function we will consider h(s) = sp, p > 1, but the results will remain valid for a large class of superlinear nonlinearities. We investigate bifurcation results when the parameter λ crosses the eigenvalues λ±1 . We obtain a rather complete picture of existence and non-existence of solutions. We will see (see Figures 1 and 2) that the existence and non-existence results for the equations with −up, resp. +up, have a completely complementary behavior: for λ’s for which there exists a solution for (1.3) with (−) there exists no solution for (1.3) with (+), and vice versa. 3.1. Existence and non-existence of solutions for problem (1.3) with (−). We remark that the corresponding Dirichlet problem has been treated by López- Gómez and Rabinowitz [15], emphasizing on the existence of a growing number of (pairs of) solutions for increasing |λ|. Here we consider the Neumann problem, which presents some peculiarities, and we restrict attention to the existence and non-existence of positive solutions. Let us now consider the model problem (1.3) with (−): −∆u = λa(x)u− up in Ω u > 0 in Ω ∂u ∂ν = 0 on ∂Ω (3.2) First, we prove a non-existence result. Theorem 3.1. Let a = a(x) ∈ C0(Ω), and suppose that a(x) changes sign. By Proposition 2.2, we have that λ−1 ≤ λ + 1 . Then, for every λ ∈ [λ−1 , λ + 1 ], the problem (3.2) has no non-trivial solution. Proof. If ∫ Ω a(x)dx = 0, then [λ−1 , λ + 1 ] = {0} and the assertion is trivial. Let∫ Ω a(x)dx < 0 (the other case is similar); suppose that u is a positive solution of (1.3) with (−). Multiplying by u and integrating we obtain∫ Ω |∇u|2 dx− λ ∫ Ω a(x)u2 + ∫ Ω up+1 dx = 0. Now, if λ ∫ Ω a(x)u2 ≤ 0 the assertion is obvious. If not, ∫ Ω a(x)u2 > 0, and using the characterization of λ+ 1 we obtain 0 = ∫ Ω |∇u|2 dx− λ ∫ Ω a(x)u2 + ∫ Ω up+1 dx ≥ ( 1− λ λ+ 1 ) ∫ Ω |∇u|2 + ∫ Ω up+1 dx > 0. � Next, we show that for λ outside of the interval [λ−1 , λ + 1 ], problem (3.2) has always a positive solution. 260 M. CALANCHI, B. RUF EJDE/SI/01 Theorem 3.2. For every λ > λ+ 1 or λ < λ−1 the Neumann problem (3.2) has at least one positive solution. In particular, if ∫ Ω a(x) = 0, then for every λ 6= 0 the problem has a positive solution. Proof. We proceed by steps. First, we observe that the solutions of (3.2) correspond to critical points of the functional Φλ : H → R, Φλ(u) = 1 2 ∫ Ω |∇u|2 + 1 p+ 1 ∫ Ω up+1 − λ 2 ∫ Ω a(x)u2 (3.3) where H := H1(Ω). Actually, the functional is well defined on H only for 1 < p+ 1 ≤ 2N N−2 . This is not an obstacle. Lemma 3.3 below provides an a priori estimate, which allows to use the following functional instead of (3.3): Φ̃λ : H → R , Φ̃λ(u) = 1 2 ∫ Ω |∇u|2 + ∫ Ω G(u)− λ 2 ∫ Ω a(x)u2. where G(s) =  sp+1 p+1 , if 0 ≤ s ≤ Cλ p p+1C p−1 λ s2 − p−1 p+1C p λs if s > Cλ 0 if s < 0, and Cλ := ( |λ| ‖a‖∞ ) 1 p−1 , as suggested by the next lemma. Lemma 3.3. All solutions of (3.2) and of −∆u = λa(x)u− g(u) in Ω u > 0 in Ω ∂u ∂ν = 0 on ∂Ω (3.4) where g(s) := G′(s), satisfy the estimate 0 ≤ u(x) ≤ ( |λ| ‖a‖∞ ) 1 p−1 := Cλ, x ∈ Ω Proof. If u ∈ H is a weak solution of (3.2), it satisfies∫ Ω ∇u∇v dx+ ∫ Ω ( up − λa(x)u ) v dx = 0 ∀v ∈ H. Take v = (u − Cλ)+, where w+(x) = max{0, w(x)}, and Ω+ λ = {x ∈ Ω : u > Cλ}. We have ∫ Ω+ λ ∇u∇(u− Cλ)+ dx = − ∫ Ω+ λ u ( up−1 − λa(x) ) (u− Cλ)+ dx. i.e. ∫ Ω |∇(u− Cλ)+|2 dx = − ∫ Ω+ λ u ( up−1 − λa(x) ) (u− Cλ)+ dx ≤ 0 since up−1 > ‖a‖∞|λ| and u > 0 on Ω+ λ . This proves that u ≤ Caλ. In a similar way, if u is a weak solution of (3.4), it satisfies∫ Ω ∇u∇v dx+ ∫ Ω (G′(u)− λa(x)u)v dx = 0 ∀v ∈ H. EJDE-2021/SI/01 NEUMANN PROBLEMS WITH INDEFINITE WEIGHTS 261 For v = (u− Cλ)+ it holds∫ Ω |∇(u− Cλ)+|2 dx = − ∫ Ω+ λ [ 2p p+ 1 Cp−1 λ u− p− 1 p+ 1 Cpλ − λa(x)u ] (u− Cλ)+ dx ≤ − ∫ Ω+ λ [ 2p p+ 1 Cp−1 λ u− p− 1 p+ 1 Cp−1 λ u− λa(x)u ] (u− Cλ)+ dx = − ∫ Ω+ λ [Cp−1 λ − λa(x)]u(u− Cλ)+ dx ≤ 0, so that u ≤ Cλ. � Thus, all positive critical points of Φ̃λ will satisfy 0 ≤ u ≤ Cλ, and will hence be also critical points of Φλ, and thus solutions of (3.2). In the next proposition we show that for λ /∈ [λ−1 , λ + 1 ] the functional Φ̃λ has a negative minimum. Proposition 3.4. Let λ > λ+ 1 or λ < λ−1 . Then −∞ < inf u∈H Φ̃λ(u) < 0 (3.5) Proof. We first prove that the functional is coercive (this proves the first inequality). Indeed, we have Φ̃λ(un) ≥ 1 2 ∫ Ω |∇un|2 + ∫ Ω G(un)− λ 2 ∫ Ω a(x)u2 n ≥ 1 2 ∫ Ω |∇un|2 + ∫ [un≥Cλ] G(un)− |λ| 2 ‖a‖∞ ∫ Ω |un|2 = 1 2 ∫ Ω |∇un|2 + p p+ 1 Cp−1 λ ∫ [un≥Cλ] u2 n − p− 1 p+ 1 Cpλ ∫ [un≥Cλ] un − |λ| 2 ‖a‖∞ ∫ Ω u2 n ≥ 1 2 ∫ Ω |∇un|2 + ( p p+ 1 − 1 2 ) |λ| ‖a‖∞ ∫ Ω u2 n − c (∫ Ω u2 n )1/2 − p p+ 1 Cp−1 λ ∫ [0 0 is similar). For λ > λ+ 1 (> 0), it is sufficient to evaluate Φ̃λ on uε = εφ+ 1 , where φ+ 1 is the eigenfunction associated to λ+ 1 and ε > 0 is small. Φ̃λ(uε) = ε2 2 ∫ Ω |∇φ+ 1 |2 + εp+1 p+ 1 ∫ Ω |φ+ 1 |p+1 − ε2λ 2 ∫ Ω a(x)|φ+ 1 |2 = ε2 2 ( 1− λ λ+ 1 )∫ Ω |∇φ+ 1 |2 + o(ε2) < 0, for ε small. For λ < λ−1 = 0 it is sufficient to evaluate Φ̃λ on uε = ε, ε small. The degenerate case ∫ Ω a(x) = 0 requires a special treatment. Recall that in this case λ−1 = λ+ 1 = 0. It is not sufficient to evaluate the functional on the 262 M. CALANCHI, B. RUF EJDE/SI/01 (constant) first eigenfunction to obtain the second inequality in (3.5). Indeed, we need to evaluate Φ̃λ on a more suitable function. Since a is continuous (and sign- changing), it follows that there exists a ball Br(x̄) ⊂ Ω with a(x) > 0 on Br(x̄). Let η ∈ C1 0 (Ω) a positive function with supp η = Br(x̄). First we consider the case λ > 0. We evaluate Φ̃λ on vε = ε ( 1 + ε p−1 2 η ) (with ε > 0 small). Φ̃λ(vε) = ε2 2 ∫ Ω |∇ ( 1 + ε p−1 2 η ) |2 + εp+1 p+ 1 ∫ Ω |1 + ε p−1 2 η|p+1 − ε2λ 2 ∫ Ω a(x)(1 + ε p−1 2 η)2 = εp+1 2 ∫ Ω |∇η|2 + εp+1 p+ 1 ∫ Ω |1 + ε p−1 2 η|p+1 − ε2+ p−1 2 λ ∫ Ω a(x)η − εp+1λ 2 ∫ Ω |η|2 = −ε2+ p−1 2 λ ∫ Ω a(x)η +O(εp+1) < 0 for ε small, since ∫ Ω a(x)η > 0. For the case λ < 0, we change η with −η. � The proof of Theorem 3.2 is now easily completed, observing that the infimum of Φ̃λ < 0 is attained, since Φ̃λ is weakly lower semi-continuous. � We can summarize the solution situation of Theorems 3.1 and 3.2 in the following bifurcation diagrams. We recall that variational methods do not yield continuous branches of solutions, so the figures are (possibly) a simplification. The first plot on the left shows the standard bifurcation diagram when a(x) is a positive weight. The plot in the middle gives the situation when a(x) changes sign (with ∫ Ω a < 0). We see that there is a bounded interval with non-existence of positive solution, while there is existence everywhere else. Finally, the plot on the right illustrates that in the degenerate case, that is for a sign-changing weight a(x) with ∫ Ω a(x) = 0, we have a positive solution for every λ 6= 0. It is interesting to note that from 0 = λ−1 = λ+ 1 emanate two bifurcation branches, albeit the corresponding eigenspace is one-dimensional, spanned by the constant 1. 6 ∃ 0 6 ∃ 0 λ+ 1 0 a(x) definite, a(x) > 0 a(x) indefinite, ∫ Ω a(x) < 0 a(x) indefinite, ∫ Ω a(x) = 0 Figure 1. Bifurcation diagram for equation (3.2) EJDE-2021/SI/01 NEUMANN PROBLEMS WITH INDEFINITE WEIGHTS 263 3.2. Existence and non-existence of solutions for problem (1.3) with (+). We now consider the problem −∆u = λa(x)u+ up in Ω u > 0 on Ω ∂u ∂ν = 0 on ∂Ω (3.6) where Ω ⊂ RN is a bounded domain, a = a(x) is a sign changing continuous function, and p > 1. We first state the following existence result. Theorem 3.5. Assume that a(x) changes sign, and that 1 < p < N+2 N−2 . If λ ∈ (λ−1 , λ + 1 ) := Ia, then problem (3.6) has a positive solution. Remark 3.6. Recall that Ia = (0, λ+ 1 ) if ∫ Ω a(x)dx < 0, Ia = (λ−1 , 0) if ∫ Ω a(x)dx > 0, Ia = ∅ if ∫ Ω a(x)dx = 0 . Proof. We prove the existence result for ∫ Ω a(x) < 0 via a variational approach. The proof for ∫ Ω a(x) > 0 is similar. Let us observe that weak solutions of (3.6) correspond to critical points of the functional Ψλ : H1(Ω)→ R, Ψλ(u) = 1 2 ∫ Ω |∇u|2 − 1 p+ 1 ∫ Ω up+1 − λ 2 ∫ Ω a(x)u2. For λ ∈ (0, λ+ 1 ) we can apply the classical Mountain Pass theorem of Ambrosetti- Rabinowitz, and we first need to prove some geometric estimates. Theorem 3.5 then follows in a standard way, since we have compactness due to the subcritical growth. First, we prove that the functional Ψλ has a mountain-pass geometry. Proposition 3.7 (0 is a local minimum). Assume ∫ Ω a(x)dx < 0 and λ ∈ (0, λ+ 1 ). Then there exist η > 0 and ρ > 0 such that Ψλ(u) ≥ η > 0 ∀u : ‖u‖ = ρ. Proof. It is sufficient to prove that there exists δ > 0 such that J(u) := 1 2 ∫ Ω |∇u|2 − λ 2 ∫ Ω a(x)u2 ≥ δ > 0, ∀u : ‖u‖ = 1. (3.7) Indeed, if (3.7) holds, then, thanks to the compact embedding H1 ⊂⊂ Lp+1, Ψλ(ρu) ≥ δρ2 − Cρp+1 ≥ δ 2 ρ2 := η, for a suitable ρ small. Note first that J(u) ≥ 0 for all u ∈ H1. Indeed, if ∫ Ω a(x)u2dx ≤ 0 this is trivial. Otherwise we use the variational characterization of λ+ 1 : J(u) = 1 2 ∫ Ω |∇u|2 − λ 2 ∫ Ω a(x)u2 ≥ 1 2 ( 1− λ λ+ 1 )∫ Ω |∇u|2 ≥ 0. 264 M. CALANCHI, B. RUF EJDE/SI/01 We now prove (3.7) by contradiction: suppose that there exists a sequence {un} such that ‖un‖ = 1 and J(un)→ 0+. Split un = wn + αn where αn = ∫ Ω a(x)un(x)dx∫ Ω a(x)dx , so that ∫ Ω a(x)wn(x)dx = 0. We have αn → 0, since otherwise, up to subsequence, |αn| ≥ δ > 0, for some positive δ, and J(un) = J(wn) + λα2 n 2 ∣∣∣ ∫ Ω a(x) ∣∣∣ ≥ λδ2 2 ∣∣∣ ∫ Ω a(x) ∣∣∣ Thus we have 1 = ‖un‖2 = ‖wn‖2 + o(1). Then there is η > 0 such that ∫ Ω |∇wn|2 ≥ η > 0. If not, up to subsequences,∫ Ω |∇wn|2 → 0 and (wn is bounded in H1) and wn → w in L2(Ω), and hence there exists w such that wn ⇀ w weakly in H1 and strongly in L2 In particular ‖∇w‖2 = 0, so that w is a constant of norm 1. But this leads to a contradiction, since 0 = ∫ Ω a(x)wn → w ∫ Ω a(x) 6= 0 . Now, since ∫ Ω |∇wn|2 ≥ η > 0, we can use again the argument above to obtain o(1) = J(un) = 1 2 ∫ Ω |∇wn|2 − λ 2 ∫ Ω a(x)w2 n + λα2 n 2 ∣∣∣ ∫ Ω a(x) ∣∣∣ ≥ 1 2 ( 1− λ λ+ 1 ) η, which is a contradiction. Hence, (3.7) holds. � To complete the geometric requirements of the Mountain Pass Theorem, we need to find a function u such that Ψλ(u) < 0. But this is trivial: it is sufficient to evaluate Ψλ on constant functions α: Ψλ(α) = − 1 p+ 1 ∫ Ω αp+1 − λ 2 ∫ Ω a(x)α2 → −∞, |α| → +∞. Finally, we can apply the mountain-pass (MP) theorem of Ambrosetti & Rabi- nowitz and find a non trivial solution of problem (1.3). This completes the proof of Theorem 3.5. � Remark 3.8. We note that by means of minimization arguments concerning the ground state level given by the MP-Theorem, the positivity of the solution is stan- dard (see e.g. [21]). Next, we turn to non-existence results for equation (3.6). Theorem 3.9. Suppose that a(x) changes sign, that 1 < p < N+2 N−2 , and assume that λ /∈ (λ−1 , λ + 1 ). Then equation (3.6) has no positive solution. Proof. First we note that for λ = 0 there is no positive solution in any case. Suppose that ∫ Ω a(x)dx ≤ 0. We show that then the problem (3.6) has no positive solutions for λ ≥ λ+ 1 or λ < 0. (a) Consider first λ > λ+ 1 ≥ 0: suppose by contradiction that u is a positive solution of (P+). We may read u as a positive eigenfunction (associated with the EJDE-2021/SI/01 NEUMANN PROBLEMS WITH INDEFINITE WEIGHTS 265 eigenvalue λ) of the problem −∆ψ = λb(x)ψ, in Ω ∂ψ ∂ν = 0 on ∂Ω, (3.8) where b(x) = a(x) + up−1 λ > a(x) in Ω. If ∫ Ω b(x) dx ≥ 0 (this is the case, for instance, when ∫ Ω a(x)dx = 0) we are done: from property (e), the unique positive eigenfunctions are related to λ−1 (b) and λ+ 1 (b), where λ−1 (b) ≤ λ+ 1 (b) = 0. If ∫ Ω b(x)dx < 0, the unique positive eigenfunctions are related to 0 = λ−1 or λ+ 1 (b), so it must be λ = λ+ 1 (b). But since b(x) > a(x), we have the inclusion B+(a) ⊂ B+(b), where B+(a) = { v ∈ H1 : ∫ Ω a(x)v2 > 0 } , B+(b) = { v ∈ H1 : ∫ Ω b(x)v2 > 0 } . Therefore, λ = λ+ 1 (b) = inf B+(b) ∫ Ω |∇v|2∫ Ω b(x)v2 ≤ inf B+(a) ∫ Ω |∇v|2∫ Ω a(x)v2 = λ+ 1 , so that λ ≤ λ+ 1 . To exclude the case λ = λ+ 1 , observe that both infima are actually minima, and hence λ+ 1 (b) < λ+ 1 . (b) Consider λ < 0: suppose by contradiction that u is a positive solution of (P+). Again we may read u as a positive eigenfunction (associated to the eigenvalue λ) of the problem −∆ψ = λb(x)ψ, in Ω ∂ψ ∂ν = 0 on ∂Ω, (3.9) where b(x) = a(x) + up−1 λ ≤ a(x) in Ω, with ∫ Ω b(x)dx < ∫ Ω a(x)dx ≤ 0. Since ∫ Ω b(x)dx < 0, the unique positive eigenfunctions are related to λ−1 = 0 or λ+ 1 (b) > 0, so it must be λ ≥ 0. The case ∫ Ω a(x)dx > 0 is handled similarly. � Again we can summarize Theorems 3.5 and 3.9 in the following bifurcation dia- grams. 6 ∃ 0 6 ∃ 6 ∃ 0 λ+ 1 6 ∃ 6 ∃ 0 a(x) definite, a(x) > 0 a(x) indefinite, ∫ Ω a(x) < 0 a(x) indefinite, ∫ Ω a(x) = 0 Figure 2. Bifurcation diagram for equation (3.6) In the first plot on the left we show the situation for weights a(x) > 0. We see that it is complementary to the situation in Figure 1: the branch covers now the negative half-line of the λ parameters. The plot in the middle shows the situation for sign-changing weights a(x), with ∫ Ω a(x)dx < 0. Now there exist solutions for every λ between the two first eigenvalues λ−1 = 0 and λ+ 1 , and no solution for all 266 M. CALANCHI, B. RUF EJDE/SI/01 other λ’s. Again, we see that the situation is complementary to the situation in Theorems 3.1 and 3.2. We have drawn the branch as a curve connecting λ−1 and λ+ 1 . This is justified for 1 < p < N N−2 by Theorem 3.10 which gives an a priori bound for all positive solutions for λ in a bounded interval. For N N−2 ≤ p < N+2 N−2 we do not have currently a proof of such a bound, we refer however to the proofs of such bounds for related equations with Dirichlet boundary conditions by de Figueiredo- Lions-Nussbaum [8] and Gidas-Spruck [10]. Finally, for the plot on the right we have the surprising result that for the degenerate case ∫ Ω a(x)dx = 0 we have no positive solution, for any λ ∈ R; again, this is complementary to the situation of Theorems 3.1 and 3.2, where we have existence of a positive solution for all λ 6= 0. 3.3. A priori bound for positive solutions of equation (3.6). In Figure 2 we have drawn a solution curve connecting the first eigenvalues λ−1 and λ+ 1 . This is justified by the following a priori bounds for positive solutions of equation (1.3), in the case that 1 < p < N/(N − 2). Theorem 3.10. Let 1 < p < N N−2 . Then for every Λ > 0 there exists a constant c0 = c0(Λ) such that for |λ| ≤ Λ it holds ‖uλ‖ ≤ c0, for every positive solution uλ of equation −∆u = λa(x)u+ up in Ω u > 0 on Ω ∂u ∂ν = 0 on ∂Ω (3.10) Proof. (a) First we integrate equation (3.10) over Ω and obtain 0 = ∫ Ω −∆u dx = λ ∫ Ω a(x)u(x) dx+ ∫ Ω up dx It follows that ‖u‖pp ≤ |λ| ‖a‖∞ ∫ Ω u(x) dx ≤ d ‖u‖p and hence ‖u‖p ≤ c (3.11) for all positive solutions. (b) Now multiply equation (3.10) by u and integrate,∫ Ω |∇u|2dx = λ ∫ Ω a(x)u2(x) dx+ ∫ Ω up+1dx ≤ c‖u‖p+1 p+1 (3.12) Now we use the well-known Gagliardo-Nirenberg inequality, see Nirenberg [17] which reads: suppose that Ω ⊂ RN is a bounded domain with the cone-property. Then there exist constants c1 and c2 such that for all u ∈Wm,r(Ω) ∩ Lq(Ω) ‖Dju‖p ≤ c1‖Dmu‖ar‖u‖1−aq + c2‖u‖q where 1 p = j N + a (1 r − m N ) + (1− a) 1 q Applying this inequality for j = 0, p→ p+ 1, r = 2, m = 1, q → p, we obtain ‖u‖p+1 ≤ c‖∇u‖a2‖u‖1−ap + c‖u‖p (3.13) where 1 p+ 1 = a (1 2 − 1 N ) + (1− a) 1 p EJDE-2021/SI/01 NEUMANN PROBLEMS WITH INDEFINITE WEIGHTS 267 This condition implies a (1 p + 1 N − 1 2 ) = 1 p − 1 p+ 1 = 1 p(p+ 1) and hence a = 1 p+ 1 2N 2N − (N − 2)p By (3.13) and (3.11) we now have ‖u‖p+1 p+1 ≤ c ‖∇u‖ a (p+1) 2 + c and hence by (3.12) ‖∇u‖22 ≤ c ‖∇u‖ 2N 2N−(N−2)p 2 + c We want that 2N 2N−(N−2)p < 2, which is the case if 1 < p < N N − 2 Then ‖∇u‖2 ≤ c, from which we obtain that ‖u‖ ≤ c. � For p ∈ [ N N−2 , N+2 N−2 ) we have no a priori bound for positive solutions readily available, and so we cannot exclude that the solution branches explode when λ→ 0 = λ−1 or λ → λ+ 1 . 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Marta Calanchi Dip. di Matematica, Università degli Studi di Milano, Via Saldini 50, 20133 Milano, Italy Email address: marta.calanchi@unimi.it Bernhard Ruf Dip. di Matematica, Università degli Studi di Milano, Via Saldini 50, 20133 Milano, Italy Email address: bernhard.ruf@unimi.it 1. Introduction 2. Eigenvalue problem with indefinite weights 3. Superlinear equations - bifurcation of positive solutions 3.1. Existence and non-existence of solutions for problem (??) with (-) 3.2. Existence and non-existence of solutions for problem (??) with (+) 3.3. A priori bound for positive solutions of equation (??) References