Nonnegative solutions to an elliptic problem with nonlinear absorption and a nonlinear incoming flux on the boundary

Nonnegative solutions to an elliptic problem with nonlinear absorption and a nonlinear incoming flux on the boundary J. Garc´ ıa-Meli´ an1, C. Morales-Rodrigo2, J. D. Rossi3and A. Su´ arez2,4, 1. Dpto. de An´alisis Matem´atico, Universidad de La Laguna, C/. Astrof´ısico Francisco S´anchez s/n, 38271 - La Laguna, SPAIN, 2. Dpto. de Ecuaciones Diferenciales y An´alisis Num´erico Fac. de Matem´aticas, Univ. de Sevilla C/. Tarfia s/n, 41012 - Sevilla, SPAIN, 3. Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, 1428 - Buenos Aires, ARGENTINA Instituto de Matem´aticas y F´ısica Fundamental, CSIC, C/. Serrano 123, 28006 - Madrid, SPAIN E-mail addresses: [email protected], [email protected], [email protected], [email protected] Abstract In this paper we perform an extensive study of the existence, uniqueness (or multiplicity) and stability of nonnegative solutions to the semilinear elliptic equation −∆u=λu −upin Ω, with the nonlinear boundary condition ∂u/∂ν =uron ∂Ω. Here Ω is a smooth bounded domain of IRdwith outward unit normal ν,λis a real parameter and p, r > 0. We also give the precise behavior of solutions for large |λ|in the cases where they exist. The proofs are mainly based on bifurcation techniques, sub-supersolutions and variational methods. Key Words. Elliptic equations, Nonlinear boundary conditions. 1 Introduction and main results Consider a bounded domain Ω ⊂IRd,d≥2, with a C2,γ boundary, ∂Ω, 0 < γ < 1. We are interested in the study of positive solutions to the problem    −∆u=λu −upin Ω, ∂u ∂ν =uron ∂Ω, (1.1) 4Corresponding author: Fax number: +34 954552898 2J. Garc´ıa-Meli´an, C. Morales-Rodrigo, J. D. Rossi and A. Su´arez where p, r > 0, λ∈IR will be regarded as a bifurcation parameter and νis the outward normal vector field to ∂Ω. The study of elliptic problems with nonlinear boundary conditions has attracted a great attention in the last decade, see the survey [38] and references therein. In problem (1.1) there is a competition between the absorption term in the equation and the positive flux at the boundary. Thus, it is interesting to look how the linear term, λu, affects the existence of positive solutions to (1.1). Nonlinear boundary conditions appear in a rather natural way in some physical models, see [38]. In the particular case p > 1, problem (1.1) can be given an ecological meaning, since the equation is the well-known logistic equation, which models the diffusion of a single species in the habitat Ω whose density is given by u. The boundary condition means that the individuals are taken outside the habitat once they reach the boundary ∂Ω, at a rate which also depends on uthrough a power, see [12] for a related problem arising from population dynamics with a different nonlinearity on the boundary condition. Problem (1.1) has been studied previously in different papers. In [32] some particular results have been given for p, r > 1 showing that there exists positive solution for all λ≥0 if p > 2r−1 and for λ < Λ0(for some Λ0≥0) if p < 2r−1 and r < d/(d−2). In the particular cases λ= 0, p, r > 1 and the nonlinearity in the equation is −aup when a∈IR varies, problem (1.1) has been analyzed in [14], [15], [29] and [35] (see also references therein). For these specific values of λ,pand r, it is proved that, if p < r or p > 2r−1 there is a positive solution of (1.1) for a > 0. When p=r, there is a positive solution if a > |∂Ω|/|Ω|, and there is no positive solution of (1.1) if a < |∂Ω|/|Ω| (throughout the paper |Ω|and |∂Ω|will denote the d−dimensional and (d−1)−dimensional measures of Ω and ∂Ω, respectively). If r < p < 2r−1 there exists a0>0 such that there exists positive solution when a > a0and no positive solutions for a < a0. Moreover, if r < d/(d−2) then for a.a. a≥a0(1.1) has at least two positive solutions. In fact, a more detailed analysis is made for the cases d= 1 and Ω a ball (see also [31] for a one-dimensional analysis). This study shows that p= 2r−1 is critical in many aspects. In particular, for the corresponding time-dependent problem some solutions blow-up in finite time if and only if p≤2r−1 (and a < r if p= 2r−1). A detailed study is made in [37] even in the case p, r ≤1. Moreover, if p= 2r−1, a=rand d= 1 then there exists a singular positive equilibrium and all positive solutions of its corresponding time-dependent problem are global and tend to this singular solution as t→ ∞, see [17]. Finally, the case r= 1, λ= 0, both for p > 1 and p < 1, and with a parameter in the boundary condition, is considered in [20] and [21]. On the other hand, when instead of a positive flux at the boundary, there is a negative one, the problem has been analyzed in [10] in the case p, r > 1. Also, if a bounded function g(u) appears in the boundary condition instead of ur, it has been studied in [41], and for more general nonlinearities in [42], where a local bifurcation analysis is carried out using a Lyapunov-Schmidt reduction. We again refer to [38] for further information. In this paper we continue the study of (1.1) when p, r > 1, in the cases p > 2r−1 and p < 2r−1 completing and improving the results of [32]. Also we consider with detail the cases r= 1 and p > 0; p= 1 and r > 0; 0 < r < 1< p and 0 < p < 1< r. Observe that in the case p=r= 1, the problem becomes linear, and hence a positive solution exists only for a value of λ, the principal eigenvalue; see Lemma 2.2. We remark that in most cases we are only considering a subcritical exponent, r, that is r < d/(d−2) when d≥3. See Theorems 1.1–1.4 where we summarize the main results. The cases p, r < 1; p= 2r−1 Nonnegative solutions to an elliptic problem 3 and a detailed study of the case d= 1 will be analyzed elsewhere. We remark that all our results are valid also for d= 1. Our main goal is to determine the set of λ’s for which solutions exist, as well as to determine the stability and uniqueness of the positive solutions, according to the values of pand r. We also provide the precise asymptotic behavior of the solutions when |λ| becomes large, in those cases where solutions exist. Since we are only interested in nonnegative solutions to (1.1), we can extend the functions λu −upand urto be zero for negative values of u. In this case, any solution to (1.1) is nonnegative. Moreover, when p≥1 the strong maximum principle implies that any nonnegative and nontrivial solution to (1.1) is positive. In the case p < 1, the solutions could develop a dead core, but we are not analyzing this phenomenon in the present work (see [21] for a related situation). We also remark that weak solutions to (1.1) in H1(Ω) are smooth up to the boundary (see Lemma 2.1). Before proceeding to the statement of the theorems, we need to introduce some notation. Given m∈L∞(Ω), h∈C1(∂Ω) we denote by λ1(−∆ + m, N +h) the principal eigenvalue of the problem    −∆u+m(x)u=λu in Ω, ∂u ∂ν +h(x)u= 0 on ∂Ω, (the notation Nrefers to the Neumann boundary condition). Some important properties of this eigenvalue will be recalled in Section 2 (see Lemma 2.2). We only quote for the moment that for constant mit holds λ1(−∆ + m, N) = m. We are using the principal eigenvalues to characterize the stability of the solutions with respect to the parabolic counterpart problem. We say that a positive solution u0of (1.1) is stable (resp. unstable) if the principal eigenvalue of the linearization of (1.1) around u0 is positive (resp. negative), i. e., λ1(−∆−λ+pup−1 0, N −rur−1 0)>0 (resp. <0). We also say that u0is weakly stable if the eigenvalue is nonnegative, and neutrally stable if it is zero. We are now able to state our results. Theorem 1.1. 1. Assume r= 1 and p6= 1. There exists a nonnegative and nontrivial solution if and only if λ > λ1(−∆, N −1). Moreover, (a) if p > 1, the solution is positive, unique (denoted by uλ), stable and verifies lim λ&λ1(−∆,N−1) kuλk∞= 0,lim λ%+∞kuλk∞= +∞; (1.2) (b) if p < 1, we have for every family of nonnegative solutions {uλ}that lim λ&λ1(−∆,N−1) kuλk∞= +∞,lim λ%+∞kuλk∞= 0.(1.3) 2. Assume p= 1. 4J. Garc´ıa-Meli´an, C. Morales-Rodrigo, J. D. Rossi and A. Su´arez (a) If 1< r < d/(d−2), there exists a positive solution if and only if λ < λ1(−∆+ 1, N). Moreover, all solutions are unstable and for every family of positive solutions {uλ}it holds lim λ%λ1(−∆+1,N)kuλk∞= 0,lim λ&−∞ kuλk∞= +∞.(1.4) (b) If r < 1, there exists a positive solution if and only if λ<λ1(−∆ + 1, N). Moreover, the solution is unique (denoted by uλ), stable and lim λ%λ1(−∆+1,N)kuλk∞=∞,lim λ&−∞ kuλk∞= 0.(1.5) Theorem 1.2. Assume 0< r < 1< p. There exists a positive solution for all λ∈IR. Moreover, the solution is unique (denoted by uλ), stable and lim λ&−∞ kuλk∞= 0,lim λ%+∞kuλk∞= +∞.(1.6) Theorem 1.3. Assume 0< p < 1< r < d/(d−2). There exists a nonnegative and nontrivial solution for all λ∈IR. Moreover, for every family of positive solutions {uλ}: lim λ&−∞ kuλk∞= +∞,lim λ%+∞kuλk∞= 0.(1.7) Theorem 1.4. Assume p, r > 1. 1. If p > 2r−1, there exists λ0<0such that (1.1) has a positive solution if, and only if, λ≥λ0. Moreover, for every family of positive solutions {uλ}it holds lim λ%+∞kuλk∞= +∞.(1.8) 2. If p < 2r−1and r < d/(d−2), there exists Λ0≥0such that (1.1) has a positive solution if λ < Λ0, and no positive solutions for λ > Λ0. Moreover, if Λ0>0, there exist at least two positive solutions for λ∈(0,Λ0)and at least a positive solution for λ= Λ0. In addition, for every family of positive solutions {uλ}we have lim λ&−∞ kuλk∞= +∞.(1.9) 3. If p < r or p=rand |Ω|>|∂Ω|, and r < d/(d−2) then Λ0>0. Moreover, for every λ∈(0,Λ0)there exists a unique positive stable solution to (1.1). In Figure 1 we have represented the bifurcation diagrams in all the cases. We remark that, in cases b), c), f) and h) the solutions need not be unique in spite of the drawings. It is also important to stress that the asymptotic behavior of the solutions when λ%+∞or λ& −∞ in (1.2) through (1.9) is a consequence of a more precise information obtained for the solutions. Concretely, we prove that whenever positive solutions exist for large |λ|, we have estimates of the form C1|λ|θ≤max u≤C2|λ|θ Nonnegative solutions to an elliptic problem 5 Figure 1: Bifurcation diagrams of (1.1): Case a) r= 1 < p; Case b) r= 1 > p; Case c) p= 1 < r < d/(d−2); Case d) p= 1 > r; Case e) 0 < r < 1< p; Case f) 0<p<1< r < d/(d−2); Case g) p, r > 1, p > 2r−1; Case h) p, r > 1, p < 2r−1, r < d/(d−2), Λ0= 0; Case i) p, r > 1, p < 2r−1, r < d/(d−2), Λ0>0. 6J. Garc´ıa-Meli´an, C. Morales-Rodrigo, J. D. Rossi and A. Su´arez for every positive solution to (1.1), where C1and C2are positive constants, and the exponent θis precisely determined in terms of pand r. See Section 5 for the statement and proof of these results. The paper is organized as follows: in Section 2, we collect some preliminaries needed to prove the theorems. Section 3 deals with bifurcations from infinity and from the trivial solution, while Section 4 is devoted to the proof of the existence, nonexistence and multiplicity issues. Finally, in Section 5 we analyze the precise asymptotic behavior for large |λ|. 2 Preliminaries Since it will be necessary when using variational arguments, we begin this section by recalling that positive weak solutions to (1.1) are indeed classical. By a positive weak solution to (1.1) we mean a positive function u∈H1(Ω) such that ZΩ ∇u· ∇ϕ−Z∂Ω urϕ=ZΩ (λu −up)ϕ for every ϕ∈H1(Ω). This fact is a consequence of the general regularity theory for elliptic equations (we refer to [1], [24] and [28]; see also [13] for the C∞setting). Observe that in all our results when r < 1 we are assuming that p≥1, and so by the strong maximum principle any nonnegative and nontrivial solution of (1.1) is positive. Lemma 2.1. Let u∈H1(Ω) be a nonnegative weak solution to (1.1), where p > 0and 0< r < d/(d−2). Then u∈C2,α(Ω), where α= min{γ, p, r}. Proof. We only sketch the main points. First remark that since 0 < r < d/(d−2), a Moser iteration as in [16] gives that u∈L∞(Ω) (the size of pis unimportant since the corresponding term appears with a minus sign). Thus, it follows that uis a weak solution to a problem of the type    −∆u=f(x) in Ω, ∂u ∂ν =g(x) on ∂Ω, where f∈L∞(Ω) and g∈L∞(∂Ω). The Lpestimates of [1] (Theorem 15.2) give that u∈W2,q(Ω) for every q > 1, and the Morrey embedding provides u∈C1,β(Ω) for some β∈(0,1). Now it is easy to conclude u∈C2,α(Ω) with the help of Theorem 6.31 in [24]. We now recall some well-known facts about the eigenvalue problem    −∆u+m(x)u=λu in Ω, ∂u ∂ν +h(x)u= 0 on ∂Ω, (2.1) where m∈L∞(Ω), h∈C1(∂Ω) (actually a little less regularity would be enough for most properties). As usual when dealing with positive solutions to nonlinear problems, we are only interested in principal eigenvalues, i.e., eigenvalues which have an associated positive eigenfunction. Nonnegative solutions to an elliptic problem 7 Lemma 2.2. Assume m∈L∞(Ω) and h∈C1(∂Ω). Then problem (2.1) admits a unique principal eigenvalue, which will be denoted by λ1(−∆+m, N+h). Moreover, this eigenvalue is simple, and any positive eigenfunction, ϕ, verifies ϕ∈C1,γ(Ω) ∩H2(Ω). In addition, λ1(−∆ + m, N +h)is separately increasing in mand hand verifies lim K→−∞ λ1(−∆ + m, N +K) = −∞, lim K→∞ λ1(−∆ + m, N +K) = λ1(−∆ + m, D), (2.2) where λ1(−∆ + m, D)stands for the principal eigenvalue of −∆ + mwith homogeneous Dirichlet boundary conditions. Proof. The existence of the principal eigenvalue and its simplicity are well-known, see for instance Theorem 2.2 in [2] or Lemma 7 in [22]. The fact that λ1(−∆ + m, N +h) is separately increasing in mand h, and (2.2) follow by [11] (see Propositions 3.3 and 3.5 and Theorem 9.1). See also Lemma 8 in [22]. We close this section with some preliminary estimates for solutions to (1.1). The first one is a pointwise lower estimate for all solutions when p > 1 and λ > 0. Lemma 2.3. Assume that p > 1and r > 0. Then, if uis a positive solution to (1.1) with λ > 0, we have u > λ1/(p−1).(2.3) Proof. It is clear that if uis a positive solution to (1.1), then it is supersolution of the problem    −∆v=λv −vpin Ω, ∂v ∂ν = 0 on ∂Ω. (2.4) Moreover, u=ε > 0 is a subsolution of (2.4) for small ε. Since for λ > 0, λ1/(p−1) is the unique positive solution to (2.4), the result follows. Finally, we obtain a priori bounds for the solutions to (1.1). The proof is based on a blow-up argument, as in [23]. We only sketch the proof and refer the reader to the proof of Theorem 5.1, where the details are carried out in a similar situation. Lemma 2.4. Assume that 1≤p < 2r−1and 1< r < d/(d−2). For every compact interval I⊂Jwith J= (−∞, λ1(−∆ + 1, N)) if p= 1 and J= IR if p > 1, there exists a positive constant Csuch that every solution (λ, u)of (1.1) with λ∈Iverifies kuk∞≤C. (2.5) Proof. We first claim that if uis a solution to (1.1) then its maximum in Ω is attained at ∂Ω. Indeed, if p= 1 we have that −∆u= (λ−1)u≤0 in Ω. Assume now that p > 1; if λ≤0 then −∆u≤0 in Ω and if λ > 0 by (2.3) also −∆u≤0 in Ω. So, the claim is shown. Now assume (2.5) does not hold. Then there exists a sequence {λj} ⊂ Iwith corresponding solutions {uj}such that Mj=kujk∞→+∞as j→ ∞. Let xj∈∂Ω be a point where Mjis attained. By the compactness of ∂Ω, we can assume that xj→x0∈∂Ω. Let vj(y) = uj(xj+M1−r jy) Mj , 8J. Garc´ıa-Meli´an, C. Morales-Rodrigo, J. D. Rossi and A. Su´arez defined in Ωj={y∈IRd:xj+M1−r jy∈Ω}. Observe that 0 ≤vj≤1 and vj(0) = 1, while Ωj→IRd +. It is easily seen that    −∆vj=M2(1−r) jλjvj−Mp−2r+1 jvp jin Ωj, ∂vj ∂ν =vr jon ∂Ωj. By elliptic regularity vjis bounded in C2,α loc (IRd +), α∈(0,1). Therefore, passing to the limit through a subsequence we get a solution 0 < v ≤1 of    ∆v= 0 in IRd +, ∂v ∂ν =vron ∂IRd +. Thanks to Theorem 1.2 of [27], this problem does not admit any positive solution. This contradiction proves the validity of (2.5). 3 Bifurcations from zero and infinity We are dealing first with bifurcation from infinity for problem (1.1), see [36]. In [40] a similar result was proved when the nonlinearities are asymptotically linear, and in [7] when the nonlinearity and the bifurcation parameter appear on the boundary (see also [21]). We omit the proof here and refer to those papers for the details. Proposition 3.1. Assume r < 1 = p(resp. p < 1 = r). There exists an unbounded continuum C∞⊂IR ×C(Ω) of nonnegative and nontrivial solutions to (1.1) bifurcating from infinity at λ=λ1(−∆ + 1, N)(resp. at λ=λ1(−∆, N −1)). Moreover, this is the unique bifurcation point from infinity. Furthermore, if δ0>0is small enough and N= [λ1−δ0, λ1+δ0]× {u∈C(Ω) : kuk∞≥1}with λ1=λ1(−∆ + 1, N)(resp. λ1=λ1(−∆, N −1)), then either 1. C∞\N is bounded in IR×C(Ω) in which case C∞\N meets the set {(λ, 0) : λ∈IR}, or 2. C∞\ N is unbounded in IR ×C(Ω). The following result is related to bifurcation from the trivial solution, see [32]. Here, we say that in the bifurcation point (λ1,0) the bifurcation direction is subcritical (resp. supercritical) if for every sequence {(λj, uj)}of positive solutions to (1.1) with λj→λ1 and kujk∞→0 as j→+∞, we have λj< λ1(resp. λj> λ1) for large j. Proposition 3.2. Assume p≥1and r > 1. There exists an unbounded continuum C0⊂IR ×C(Ω) of positive solutions to (1.1) emanating from the trivial solution at λ= λ1(−∆+1, N)when p= 1 or at λ= 0 when p > 1. Moreover, this is the unique bifurcation point from the trivial solution, and with respect to the bifurcation direction: 1. if p= 1 < r, then the bifurcation direction is subcritical; 2. if 1<p<r(resp. p>r) then the bifurcation direction is supercritical (resp. subcritical); Nonnegative solutions to an elliptic problem 9 3. if p=rthen the bifurcation direction is supercritical (resp. subcritical) for |Ω|>|∂Ω| (resp. |Ω|<|∂Ω|). Proof. The existence of the unbounded continuum C0is proved in [32]. We now show the bifurcation direction in Cases 2 and 3 (the remaining case can be proved similarly). Take a sequence of solutions (λj, uj) such that λj→0 and kujk∞→0 as j→+∞. We integrate the equation (1.1), to obtain −Z∂Ω ur j+ZΩ up j=λjZΩ uj. Now we divide by kujkp ∞: −kujkr−p ∞Z∂Ωµuj kujk∞¶r +ZΩµuj kujk∞¶p =λjkujk1−p ∞ZΩ uj kujk∞ ,(3.1) and take into account that uj/kujk∞→1 in C(Ω) (cf. [32]). Thus we deduce that if 1< p < r, then λj>0 for large j, while if p > r,λj<0 for large j, which proves 2. When p=r, the left-hand side of (3.1) converges to −|∂Ω|+|Ω|. Thus sgn(λj) = sgn(|Ω|−|∂Ω|) for large jand |Ω| 6=|∂Ω|, which proves 3. 4 Proof of the main results We now turn to prove our theorems. For the sake of clarity, we include all the stability results in a single preliminary statement. Lemma 4.1. Let u0be a positive solution to (1.1). 1. If p≥1and r≤1and (p, r)6= (1,1), then u0is stable. 2. If p= 1 and r > 1, then u0is unstable. 3. If 1< p ≤rand λ≤0, then u0is unstable. Proof. We have to ascertain the sign of λ1(−∆−λ+pup−1 0, N −rur−1 0). For that, it is well known (see for instance Lemma 2.2 in [18]) that this eigenvalue is positive (resp. negative) if there exists a strict supersolution (resp. subsolution), that is, a positive function vsuch that    (−∆−λ+pup−1 0)v≥0 (resp. ≤0) in Ω, ∂v ∂ν −rur−1 0v≥0 (resp. ≤0) on ∂Ω, and at least one of the inequalities is strict. Observe that taking v=u0we have    −∆u0−λu0+pup 0= (p−1)up 0in Ω, ∂u0 ∂ν −rur 0= (1 −r)ur 0on ∂Ω, whence we deduce the first and second paragraphs. For the last paragraph, take v=uq 0, with 1 < p ≤q≤r. We have that ∂v ∂ν −rur−1 0v= (q−r)uq+r−1 0≤0 on ∂Ω, 16 J. Garc´ıa-Meli´an, C. Morales-Rodrigo, J. D. Rossi and A. Su´arez The first paragraph follows. 2. Assume that there exists a sequence λn% ∞ with corresponding solutions unof (1.1). Consider the parabolic problem          wt−∆w=−wpin Ω ×(0, T), ∂w ∂ν =wron ∂Ω×(0, T), w(x, 0) = w0in Ω. (4.6) We know by Theorem 2.3 in [6], that if p≤2r−1 then all positive solutions to (4.6) blow-up in a finite time T > 0 provided infΩw0is large enough. If we prove that unis a supersolution of (4.6) for large n, then un(x)> w(x, t) for all t∈(0, T) which is clearly a contradiction. Observe that unis supersolution of (4.6) if un> w0. By Lemma 2.3, we have un> λ1/(p−1) n. Thus for large enough n, we may set w0=λ 1 p−1 n, which concludes the proof of the second paragraph. 3. Assume now that p > 2r−1. We want to show that for λnegative enough, there are no positive solutions to (1.1). First, we claim that there exists a positive function U∈C(Ω) such that uλ≤U(4.7) for all family of positive solutions of (1.1) with λ≤0. Assume (4.7). Consider the problem          vt−∆v=λv −vpin Ω ×(0, T), ∂v ∂ν =vron ∂Ω×(0, T), v(x, 0) = v0(x)>0 in Ω, (4.8) and denote v(t;v0) its positive solution. It is clear that uλ=v(t;uλ)≤v(t;U), and so it suffices to prove kv(t, U)k∞→0 as t→ ∞ for negative enough λ. To this aim, it suffices to construct a global supersolution of (4.8) which goes to zero at infinity. Since p > 2r−1, for every initial datum w0∈L∞(Ω), (4.6) has a positive solution w, which is globally bounded (cf. [6]). Consider v:= e−µtwfor some fixed µ > 0. It is not hard to show that vis a supersolution of (4.8) provided that wp−1(e−(p−1)µt −1) −µ≥λin Ω ×(0, T), eµ(r−1)t≥1 on ∂Ω×(0, T). Since wis bounded, there exists λ0<0 such that for λ≤λ0the two inequalities hold. Thus, it remains to prove (4.7). For that, we are going to use Lemma 4.5 and so we need to construct now a family of strict supersolutions of (1.1). For the particular case λ= 0 a different supersolution was used in [29] and [43]. Take uM:= M(φ+M−σ)−β Nonnegative solutions to an elliptic problem 17 where β= 2/(p−1), φis such that    −∆φ=λ1φin Ω, φ= 0 on ∂Ω, and M, σ > 0 are to be chosen. It is clear that uMis a continuous and increasing function in M. After some calculations, we have that uMis strict supersolution of (1.1) provided that Mp−1−β(1 + β)|∇φ|2−β(φ+M−σ)λ1φ−λ(φ+M−σ)2≥0 in Ω, and −β∂φ ∂ν > Mr−1−σ[β(1−r)+1] on ∂Ω. Take into account that −∂φ/∂ν ≥c1>0 on ∂Ω, it is not hard to show that there exists M0>0 (independent of λ) such that for M≥M0and λ≤0 both inequalities are satisfied, provided that (recall p > 2r−1) r−1−σ[β(1 −r) + 1] <0 that is σ > (p−1)(r−1) p−2r+ 1 . On the other hand, given a positive solution uλof (1.1) with λ≤0, there exists a sufficiently large M(λ)> M0such that uλ< uM(λ). So, we can apply Lemma 4.5 and conclude that uλ≤uM0:= U. This proves (4.7) and completes the proof. We are now ready to come to the proof of Theorem 1.4. 4.4.1 Case p > 2r−1 From Proposition 3.2 it follows that there exists an unbounded continuum C0of positive solutions bifurcating at λ= 0 subcritically (observe that p > r in this case). Take I= [Λ2, K], with K > Λ2arbitrary where Λ2is given by Lemma 4.6. We have a continuous map u:I→C1(Ω), λ7→ u(λ) where u(λ) is the strict supersolution of (1.1) which has been constructed above. Moreover, we have a connected set C0such that for λ0small enough uλ0< u(λ0) for (λ0, uλ0)∈ C0. Then by a similar reasoning to the used in [19] we obtain that uλ< u(λ) for all (λ, uλ)∈ C0and λ∈I. This implies that the projection on the real axis of the continuum C0is [λ2,+∞) for some λ2<0. To complete the proof, set λ0:= inf{λ∈IR : (1.1) has a positive solution}. Thanks to Lemma 4.6 we know that −∞ < λ0<0. Now, we want to show that there exists a solution for all λ≥λ0. Indeed, for λ > λ0, we can take λ1∈(λ0, λ) such that the corresponding solution uλ1(which exists thanks to the definition of λ0) is subsolution of (1.1) for this λ. Again, as supersolution we can take u(λ). Thus there exists a solution for every λ > λ0. Finally, we show that there exists a solution for λ=λ0. Take (λj, uj) a sequence of solutions such that 0 > λj> λ0and λj→λ0. Since we have an a priori bound for all solutions, namely uj< U (see (4.7)), it is standard to pass to the limit to obtain that uj→u0with u0a solution to (1.1) for λ=λ0. Since λ0<0, it cannot be a bifurcation point from the trivial solution, and hence u06≡ 0. This completes the proof. 18 J. Garc´ıa-Meli´an, C. Morales-Rodrigo, J. D. Rossi and A. Su´arez 4.4.2 Case p < 2r−1 Thanks to Proposition 3.2, there exists an unbounded continuum C0of positive solutions to (1.1) which emanates from zero at λ= 0, and by Lemma 2.4 the solutions are bounded for bounded λ. Thus, since there are no positive solutions for large λ, we conclude the existence of λ0≥0 such that there exists at least a positive solution to (1.1) for λ<λ0. Moreover, (1.9) follows by Theorem 5.3 in Section 5. Now define Λ0:= sup{λ∈IR : (1.1) has a positive solution}. We already know that 0≤Λ0<+∞, and clearly there are no solutions for λ > Λ0. It remains to show that when Λ0>0 there exist at least two positive solutions for all λ∈(0,Λ0) and one positive solution for λ= Λ0. We first show that a minimal positive solution exists if λ∈(0,Λ0). Fix such a λ. We have that there exists λ∈(λ, Λ0) such a positive solution uλof (1.1) exists. It is clear that uλis a supersolution to (1.1) for all λ≤λ. On the other hand, u=εis a subsolution for small ε > 0. Thus there exists at least a positive solution for every λ∈(0,Λ0). Moreover, we have that any positive solution uλverifies uλ> λ1/(p−1), thanks to Lemma 2.3. Hence, the existence of a minimal solution to (1.1) follows. It will be denoted by uλ. We now show the existence of a second solution when λ∈(0,Λ0). We are proving for this aim that our problem is in the general setting of [3] (we refer there for the definitions to be used in the sequel). Let Pbe the cone of positive functions of C(Ω). With the ordering induced by P,C(Ω) is an ordered Banach space with a normal cone which has nonempty interior, see Example 1.10 in [3]. Consider the interval I= [−1,Λ0+ 1] and let β > sup λ∈I ku(λ)k∞, being u(λ) any solution to (1.1). This is possible since we have a priori bounds for the solutions when λruns in finite intervals (cf. Lemma 2.4). Take K > 0 a constant to be chosen later, so that (1.1) can be rewritten as    (−∆ + K)u= (λ+K)u−upin Ω, ∂u ∂ν =uron ∂Ω. We want to show that solving our problem is equivalent to find fixed points of a nonlinear operator. For that, let K1:Cα(Ω) 7→ C2,α(Ω), α∈(0,1), be the operator such that f7→ u=K1fwhere uis the unique solution to    (−∆ + K)u=fin Ω, ∂u ∂ν = 0 on ∂Ω. This operator can be extended to a linear, compact and strongly positive map, denoted again by K1,K1:C(Ω) 7→ C1(Ω), see Theorem 4.2 in [3]. Consider now the operator K2:C1,α(∂Ω) 7→ C2,α(Ω), g7→ u=K2g, where uis the unique solution to    (−∆ + K)u= 0 in Ω, ∂u ∂ν =gon ∂Ω. Nonnegative solutions to an elliptic problem 19 Now, by [2], K2can be extended to a linear compact map from C(∂Ω) to C(Ω). It is not hard to prove that uis solution to (1.1) if and only if u=F(u, λ) = K1((λ+K)u−up) + K2(γ(ur)), where γ:C(Ω) 7→ C(∂Ω) is the trace operator. Moreover, F:C(Ω) ×IR →C(Ω) is a differentiable operator, which is compact on bounded sets, and it is strongly increasing for fixed λif Kis large enough and uis restricted to bounded sets. In addition, the partial derivatives, ∂uF(u0, λ0)ξ=K1((λ+K)−pup−1 0)ξ+K2(rγ(ur−1 0))γ(ξ) and ∂λF(u0, λ0)µ=µK1u0, are strongly positive if Kis selected large enough. Indeed, observe that since λ∈I, then supλ∈Iku(λ)k∞< β for any positive solution u(λ) of (1.1), and so Kcan be taken large to make the partial derivatives strongly positive. Hence, Fsatisfies hypothesis (H) of [3] page 680, and so we can apply Theorem 20.9 of [3] (see the arguments after Proposition 20.8 and Theorem 7.4 in [4]) and conclude the existence of at least two positive solutions for λ∈(0,Λ0) and at least a positive solution for λ= Λ0. We quote for its use in the next section that, denoting by ρ=r(u0, λ0) the spectral radius of ∂uF(u0, λ0), then ρsatisfies λ1(−∆ + 1 ρ(pup−1 0−λ0), N +r ρur−1 0) = K(1 ρ−1).(4.9) 4.4.3 Case p < r or p=rand |Ω|>|∂Ω| First of all, notice that p < 2r−1 in this case. Thus there exists a solution for every λ < Λ0, for a certain Λ0≥0. Since a supercritical bifurcation takes place at λ= 0 (Proposition 3.2) we have Λ0>0. Thus only the uniqueness of the stable solution for λ∈(0,Λ0) remains to be proved. We adapt the argument used in [25]. The following result provides us with a complete picture of the structure of the set of positive solutions near a stable or neutrally stable solution. Lemma 4.7. Let (λ0, u0)be a positive solution to (1.1) with λ=λ0. 1. If λ1(−∆−λ0+pup−1 0, N −rur−1 0)>0,(4.10) then, there exists ε > 0and a differentiable mapping u:I= (λ0−ε, λ0+ε)7→ P such that u(λ0) = u0and (λ, u(λ)) is a positive solution to (1.1) for each λ∈I. Moreover, the mapping λ7→ u(λ)is increasing and there exists a neighborhood Vof (λ0, u0)in IR×Psuch that if (λ, u)∈ V is a solution to (1.1), then (λ, u) = (λ, u(λ)) for some λ∈I. 2. If λ1(−∆−λ0+pup−1 0, N −rur−1 0) = 0,(4.11) let Φ0be the principal eigenfunction associated with λ1(−∆−λ0+pup−1 0, N −rur−1 0). Then, there exists ε > 0and a twice continuously differentiable mapping (λ, u) : J= 20 J. Garc´ıa-Meli´an, C. Morales-Rodrigo, J. D. Rossi and A. Su´arez (−ε, ε)7→ IR ×Psuch that (λ(0), u(0)) = (λ0, u0)and for each s∈J,(λ(s), u(s)) is a positive solution to (1.1). Moreover, λ0(0) = 0,u(s) = u0+s(Φ0+v(s)) where v∈C1((−ε, ε), C(Ω)) satisfies v(0) = 0, and finally λ00(0) = ZΩ p(p−1)up−2 0Φ3 0−Z∂Ω r(r−1)ur−2 0Φ3 0 ZΩ u0Φ0 ,(4.12) for s≃0. In addition, there exists a neighborhood Wof (λ0, u0)in IR ×Psuch that if (λ, u)∈ W is a solution to (1.1), then (λ, u) = (λ(s), u(s)) for some s∈J. Also, sgn λ0(s) = sgn λ1(−∆−λ(s) + pu(s)p−1, N −ru(s)r−1).(4.13) Proof. By (4.9), if (4.10) holds, 1 is not an eigenvalue of ∂uF(u0, λ0), and so Id − ∂uF(u0, λ0) is a topological isomorphism. Hence we can apply Proposition 20.6 of [3] and conclude the first paragraph. Again by (4.9), if (4.11) holds, 1 is an eigenvalue with positive eigenfunction of ∂uF(u0, λ0), so we can apply Propositions 20.7 and 20.8 of [3]. Finally, to prove (4.13), observe that from Proposition 20.8 of [3] it follows that sgn λ0(s) = sgn(1 −r(u(s), λ(s))). Taking into account (4.9) it is not hard to show that sgn(1 −r(u(s), λ(s))) = sgnλ1(−∆−λ(s) + pu(s)p−1, N −ru(s)r−1). This completes the proof. We now analyze the behavior of the branch of solutions near a point (λ0, u0) such that (4.11) holds. The fact that λ0(0) = 0 shows that this is actually a turning point of the branch of positive solutions (cf. Corollary 4.9 below). We are elucidating in what follows the direction of the turning. The essential ingredient is a Picone’s type identity (see Section 4 in [8] and Lemma 4.1 in [30], for instance). Let u, v ∈C2(Ω) ∩C1(Ω) be such that v/u ∈C(Ω) and Υ : [0,∞)7→ IR an arbitrary C1function. Then ZΩ Υ(v u)(−v∆u+u∆v) = −ZΩ Υ0(v u)u2∇¯¯¯³v u´¯¯¯ 2−Z∂Ω Υ(v u)[v∂u ∂ν −u∂v ∂ν ].(4.14) Then we have the following important result. Proposition 4.8. Assume p≤r. Let (λ0, u0)be a positive solution to (1.1) with λ=λ0, such that (4.11) holds. Then λ00(0) <0, where λ00(0) is defined in (4.12). Proof. To determine the sign of λ00(0), we use the Picone’s identity (4.14) with Υ(t) = t2, v= Φ0and u=u0, to obtain (p−1) ZΩ up−2 0Φ3 0<(r−1) Z∂Ω ur−2 0Φ3 0.(4.15) From (4.15) and as p≤rwe can infer that λ00(0) <0. This concludes the proof. Nonnegative solutions to an elliptic problem 21 As an easy consequence of Lemma 4.7 (in particular relations (4.12) and (4.13)) and Proposition 4.8, we obtain: Corollary 4.9. Let (λ0, u0)be a positive solution to (1.1) with λ=λ0>0, such that λ1(−∆−λ0+pup−1 0, N −rur−1 0) = 0. Then, there exists ε > 0such that for each λ∈(λ0−ε, λ0),(1.1) has two positive solutions, one of them stable and the other one unstable. Moreover, there exists a neighborhood Nof (λ0, u0)in IR ×Psuch that (1.1) does not have positive solutions in Nfor λ > λ0. We are finally ready to prove the uniqueness of the stable solution. Theorem 4.10. Assume that p≤r. Then, the minimal solution is the unique positive stable solution to (1.1) for all λ∈(0,Λ0). Proof. We first show that the minimal solution uλis stable for all λ∈(0,Λ0). It is well known (see Proposition 20.4 in [3]) that the minimal solution is weakly stable, i.e., λ1(−∆−λ+pup−1 λ, N −rur−1 λ)≥0 for all λ∈(0,Λ0).(4.16) On the other hand, in a neighborhood Nof (λ, u) = (0,0), there exists a unique positive solution for fixed λ. Since the minimal solution exists for all λ∈(0,Λ0), the unique solution coincides with the minimal, so by Corollary 4.9 there exists λsuch that for all 0< λ ≤λwe have that λ1(−∆−λ+pup−1 λ, N −rur−1 λ)>0. Now, we can produce this branch to the right to reach a value λ0≤Λ0such that λ1(−∆− λ+pup−1 λ, N −rur−1 λ)>0 for all λ < λ0and λ1(−∆−λ0+pup−1 λ0, N −rur−1 λ0) = 0.(4.17) If λ0= Λ0we have proved that the minimal solution is stable for all λ < Λ0. So assume that λ0<Λ0. Thanks to (4.16) and Corollary 4.9, there exists a value λ1∈(λ0,Λ0) such that λ1(−∆−λ1+pup−1 λ1, N −rur−1 λ1)>0, and by Lemma 4.7, part 1, we can continue the branch to the left of λ1. Denote Γ = {(λ, u(λ)) : λ≤λ1}. Now two possibilities may arise: 1. There exists λ2< λ1such that λ1(−∆−λ2+pu(λ2)p−1, N −ru(λ2)r−1) = 0. 2. The branch Γ can be continued for all λ≤λ1with λ1(−∆−λ+pu(λ)p−1, N − ru(λ)r−1)>0. If the first possibility holds, then Corollary 4.9 is contradicted. In the second possibility, Γ does not reach negative values of λby Lemma 4.1. So, again two situations are possible: 1. The branch Γ meets the real axis {(λ, 0)}. 22 J. Garc´ıa-Meli´an, C. Morales-Rodrigo, J. D. Rossi and A. Su´arez 2. The branch Γ reaches the minimal solution at some point (λ3, uλ3). If Γ meets the axis {(λ, 0)}, since we know that the unique bifurcation point from the trivial solution is λ= 0, then Γ reaches at (0,0). But, as remarked before, in a neighborhood Nof (λ, u) = (0,0) there exists a unique solution, in fact the minimal solution. So, the second possibility occurs. If λ3is such that uλ3satisfies (4.17), Corollary 4.9 leads to a contradiction. However, if λ3is such that uλ3satisfies λ1(−∆−λ3+pup−1 λ3, N − rur−1 λ3)>0, we know that in a neighborhood Mof (λ3, uλ3) there exists a unique solution, a contradiction. This contradiction shows that the minimal solution uλis stable for all λ∈(0,Λ0) and neutrally stable for λ= Λ0. Now, assume that for some λ0∈(0,Λ0) there exists a second stable solution v0> uλ0. We argue as in the first part of the proof. By Lemma 4.7, part 1, there exists a branch, say Γ0, of stable solutions of the form (λ(s), v(s)), s∈I, with λ(0) = λ0,v(0) = v0. Moreover, we can continue this branch to the left until there exists a value λ∗in which it is noncontinuable. Since, by Lemma 4.1, part 3, all solutions are unstable for λ≤0, it follows that λ∗≥0. If λ∗>0, we would have thanks to Lemma 4.7, part 1, that λ1(−∆−λ∗+pvp−1 λ∗, N − rvr−1 λ∗) = 0, and we arrive at a contradiction with Corollary 4.9. Hence λ∗= 0. Moreover, the branch Γ0has to degenerate at (0,0), otherwise we could continue it thanks to Lemma 4.7, part 1. However, this contradicts the uniqueness of solutions for λ∼0, and the uniqueness of the stable solution is proved. 5 Behavior of solutions for large |λ| This section is devoted to obtain the behavior of all positive solutions to (1.1) when λ% ∞ or λ& −∞. All the proofs are based on the well-known blow-up argument of Gidas and Spruck, [23]. An essential role in them is played by a nonexistence result for problems with nonlinear boundary conditions in a half-space obtained in [27]. We begin by considering the behavior of the positive solutions for λ→+∞in the case p < 1≤r, assuming that ris subcritical. Theorem 5.1. Assume that 0<p<1≤r < d/(d−2). For every λ0>0, there exist positive constants C1,C2such that, for every nonnegative solution uto (1.1) with λ≥λ0, we have C1λ−1 1−p≤max Ωu≤C2λ−1 1−p.(5.1) Proof. We are using as in [21] a blow-up argument. Since this argument will also be used in the next theorems, we detail it in this case. Assume that the right-hand side inequality in (5.1) does not hold. Then there exist sequences λn% ∞, and un∈C2,α(Ω) solutions to (1.1) with λ=λnsuch that λ 1 1−p nMn% ∞,(5.2) where Mnstands for the maximum of un. Take a point xn∈Ω where unattains its maximum and assume with no loss of generality that xn→x0∈Ω. We need to distinguish two cases: x0∈Ω or x0∈∂Ω. Case 1. x0∈Ω. Introduce the scaled functions vn(y) = un(xn+λ−1/2 ny) Mn , Nonnegative solutions to an elliptic problem 23 which verify vn(0) = 1, 0 ≤vn≤1 and −∆vn=vn−1 λnM1−p n vp nin Ωn, where Ωn=λ1/2 n(−xn+ Ω). It is easily seen that Ωn→IRdas n→ ∞. Since vnis bounded, it is standard to obtain bounds in C1,α loc (IRd), which then provide with bounds in C2,α loc (IRd) ([24]). Then, passing to a subsequence, we have that vn→vin C2 loc(IRd), where vis a solution to −∆v=vin IRd,(5.3) with 0 ≤v≤1, v(0) = 1. We claim that this is impossible. Indeed, let λ1(R) be the principal eigenvalue of −∆ under homogeneous Dirichlet boundary conditions in the ball BRof radius Rcentered at the origin, with associated positive eigenfunction φR. If we multiply (5.3) by φRand integrate in BR, we have (λ1(R)−1) ZBR vφR≥0 since ∂φR/∂ν < 0 on ∂BR. Taking into account that λ1(R)→0 as R→ ∞, we arrive at v= 0 in BRfor large R, which is impossible, as v(0) = 1. Hence Mnλ1/(1−p) n→ ∞ is impossible, and the right-hand inequality of (5.1) is proved in this case. Case 2. x0∈∂Ω. As usual, before introducing the scaling, we need to straighten the boundary of Ω near x0. Without loss of generality, we may assume that x0= 0, and that ν(x0) = −ed, the last vector of the canonical basis of IRd. Since Ω is C2,γ, there exist R > 0 and ϕ∈C2,γ(B(0, R)∩ {xd= 0}) verifying ϕ(0) = 0, ∇ϕ(0) = 0 and writing x= (x0, xd), we have Ω∩B(0, R) = {x:xd> ϕ(x0)}and ∂Ω∩B(0, R) = {x:xd=ϕ(x0)}. Then the diffeomorphism y=h(x) given by y0=x0,yd=xd−ϕ(xd) maps B(0, R) onto a neighborhood Vof y= 0 in IRd, while it maps Ω ∩B(0, R) onto V+=V∩IRd +and ∂Ω∩B(0, R) onto V∩∂IRd +. Then problem (1.1) gets transformed into:        −∆u+ d−1 X i=1 ai(y)uyiyd− |∇ϕ(y0)|2uydyd+b(y)uyd=λu −upy∈V+, ∇u·ν1(y) = ury∈V∩∂IRd +, where ai= 2ϕxi, b(y) = ∆ϕ, ν1= (ν0,−ν0∇ϕ+νd), and all functions are evaluated at x=h−1(y). At this point, we claim that λnM−2(r−1) n→ ∞. Since this is clear in the particular case r= 1, we may assume for the moment r > 1. If we had λnM−2(r−1) nbounded for some subsequence, we can assume λnM−2(r−1) n→c, with c≥0. Let yn=h(xn), and introduce the functions vn(y) = un(yn+M−(r−1) ny) Mn 24 J. Garc´ıa-Meli´an, C. Morales-Rodrigo, J. D. Rossi and A. Su´arez (observe that we must have in this case Mn→ ∞). Then vnare solutions to                −∆v+ d−1 X i=1 ai,n(y)vyiyd−ad,n(y)vydyd+M−(r−1) nbn(y)vyd =λn M2(r−1) n v−1 M2(r−1)+1−p n vpy∈Un, ∇v·ν1,n(y) = vry∈∂Un∩IRd +, where Un:= Mr−1 n(B(0, R)∩IRd +)−yn,ai,n =ai(yn+M−(r−1) ny), 1 ≤i≤d−1, ad,n(y) = |∇ϕ(y0 n+M−(r−1) ny0)|2,bn(y) = b(yn+M−(r−1) ny), ν1,n(y) = ν1(yn+M−(r−1) ny). We need to distinguish two subcases: (a) λ1/2 ndist(xn, ∂Ω) is unbounded. Passing to a subsequence, we may assume that λ1/2 ndist(xn, ∂Ω) →+∞. In this case Un→IRd +, and the contradiction is reached as in Case 1, the limit problem being (5.3). (b) λ1/2 ndist(xn, ∂Ω) is bounded. Passing to a subsequence, λ1/2 ndist(xn, ∂Ω) →γ≥0, so that Un→ {yd≥ −γ}. Since 0 ≤vn≤1, we can obtain C2,α loc (IRd +) bounds (cf. [28]), and then pass to the limit through a subsequence to obtain that vn→v, which is a solution to    −∆v=cv in {yd>−γ}, ∂v ∂ν =vron yd=−γ, (5.4) which in addition verifies 0 ≤v≤1, v(0) = 1. With a further translation (which does not change the equation) we may assume problem (5.4) is posed in IRd +. If c= 0, this contradicts Theorem 1.2 in [27]. If c6= 0, we can multiply (5.4) by the eigenfunction φRassociated to the first eigenvalue of −∆ under homogeneous Dirichlet boundary conditions in a ball of radius Rcontained in IRd +to obtain as before that λ1(R)≥ c, which is clearly impossible for large R. Hence we arrive at a contradiction which shows the claim. We now introduce the functions wn(y) = un(yn+λ−1/2 ny) Mn , which are solutions to                −∆w+ d−1 X i=1 ai,n(y)wyiyd−ad,n(y)wydyd+λ−1/2 nbn(y)wyd =w−1 λnM1−p n wpy∈Un, ∇w·ν1,n(y) = Mr−1 nλ−1/2 nwry∈∂Un∩∂IRd +, where now Un:= λ1/2 n(B(0, R)∩IRd +), ai,n =ai(yn+λ−1/2 ny), 1 ≤i≤d−1, ad,n(y) = |∇ϕ(y0 n+λ−1/2 ny0)|2,bn(y) = b(yn+λ−1/2 ny), ν1,n(y) = ν1(yn+λ−1/2 ny). Passing to the limit as before we arrive at wn→w, which either solves −∆w=win IRd +, Nonnegative solutions to an elliptic problem 25 or    −∆w=win IRd +, ∂w ∂ν = 0 on ∂IRd +. Extending wto all of IRdas an even function in the second case, we obtain a nonnegative nontrivial solution to −∆w=win IRdin both cases, which has been shown to be impossible, and hence the right-hand side of (5.1) is proved. Since the arguments used to prove the left-hand inequality in (5.1) are similar, we are only sketching the proof. Thus assume Mnλ1/(1−p) n→0 for a certain sequence λn% ∞ with corresponding solutions un. In Case 1, we set vn(y) = un(xn+M 1−p 2 ny) Mn , and passing to the limit we arrive at a solution vto −∆v=−vpin IRd,(5.5) with 0 ≤v≤1, v(0) = 1, which is clearly impossible, since vattains its maximum at zero, and then −∆v(0) ≥0. In Case 2, we set vn(y) = un(yn+M 1−p 2 ny) Mn , and in the limit obtain equation (5.5) or    −∆v=−vpin IRd +, ∂v ∂ν = 0 on ∂IRd +. In the second case, the function vcan then be extended as an even function to all of IRd, verifying (5.5), which as we have seen is impossible. This completes the proof of (5.1). A similar proof as that of the upper estimate in (5.1) can be made for p > 1 in the cases where positive solutions exist for large λ, that is r≤1 or r > 1 and p > 2r−1 (we recall that no solutions exist for large λif p= 1). This, together with (2.3), leads to: Theorem 5.2. Assume that p > 1and r≤1or 1< r < d/(d−2) and p > 2r−1. For every λ0>0, there exists a positive constant Csuch that, for every nonnegative solution uto (1.1) with λ≥λ0, we have λ−1 1−p≤max Ωu≤Cλ−1 1−p.(5.6) Proof. The lower inequality in (5.6) is (2.3) in Lemma 2.3. To prove the upper inequality, assume it does not hold, that is, there exist sequences λn% ∞ and solutions unverifying (5.2). It is easily seen that the proof is identical to that of Theorem 5.1 provided we show that λnM−2(r−1) n% ∞. We introduce the functions vn(y) = un(yn+M 1−p 2 ny) Mn ,