Positive Solutions for the Neumann p-Laplacian with Superdiffusive Reaction

Bulletin of the Malaysian Mathematical Sciences Society, Aug 2015

We consider a generalized logistic equation driven by the Neumann p-Laplacian and with a reaction that exhibits a superdiffusive kind of behavior. Using variational methods based on the critical point theory, together with truncation and comparison techniques, we show that there exists a critical value \(\lambda _*>0\) of the parameter, such that if \(\lambda >\lambda _*\), the problem has at least two positive solutions, if \(\lambda =\lambda _*\), the problem has at least one positive solution and it has no positive solution if \(\lambda \in (0,\lambda _*)\). Finally, we show that for all \(\lambda \geqslant \lambda _*\), the problem has a smallest positive solution.

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Positive Solutions for the Neumann p-Laplacian with Superdiffusive Reaction

Positive Solutions for the Neumann p-Laplacian with Superdiffusive Reaction Leszek Gasin´ ski 0 1 Nikolaos S. Papageorgiou 0 1 Mathematics Subject Classification 0 1 0 Department of Mathematics, National Technical University , Zografou Campus, 15780 Athens , Greece 1 Faculty of Mathematics and Computer Science, Jagiellonian University , ul. Łojasiewicza 6, 30-348 Kraków , Poland We consider a generalized logistic equation driven by the Neumann pLaplacian and with a reaction that exhibits a superdiffusive kind of behavior. Using variational methods based on the critical point theory, together with truncation and comparison techniques, we show that there exists a critical value λ∗ > 0 of the parameter, such that if λ > λ∗, the problem has at least two positive solutions, if λ = λ∗, the problem has at least one positive solution and it has no positive solution if λ ∈ (0, λ∗). Finally, we show that for all λ λ∗, the problem has a smallest positive solution. Communicated by Rosihan M. Ali. p-Laplacian; Superdiffusive reaction; Local minimizers; Mountain pass theorem; Comparison principle; Bifurcation-type theorem - 35J25 · 35J92 1 Introduction Let ⊆ RN be a bounded domain with a C 2-boundary ∂ . In this paper, we study the following nonlinear parametric Neumann problem: ⎩ ∂n = 0 on ∂ , λ > 0, u > 0, ⎨⎧ −∂u pu(z) + β(z)u(z) p−1 = λg z, u(z) − f z, u(z) in , ( P)λ with β ∈ L∞( )+, β = 0. Here defined by p denotes the p-Laplace differential operator, pu = div ∇u p−2∇u ∀u ∈ W 1, p( ), with p ∈ (1, +∞). Also n(·) denotes the outward unit normal on ∂ . When the reaction in ( P)λ has the particular form λζ q−1 − ζ r−1, with q < r , then the resulting equation is the p-logistic equation (or simply the logistic equation when p = 2). The logistic equation is important in mathematical biology (see Gurtin and Mac Camy [ 21 ] and Afrouzi and Brown [ 1 ]) and describes the dynamics of biological populations whose mobility is density dependent. There are three different types of the p-logistic equation, depending on the value of the exponent q with respect to p. More precisely, we have • the “subdiffusive” type, when q < p < r ; • the “equidiffusive” type, when q = p < r ; • the “superdiffusive” type, when p < q < r . The subdiffusive and equidiffusive cases are similar, but the superdiffusive case differs essentially and it exhibits bifurcation phenomena (see Takeuchi [ 29,30 ] and Filippakis et al. [7], where the Dirichlet problem is studied). The aim of this work, is to prove a bifurcation-type theorem for the positive solutions of ( P)λ as the parameter λ > 0 varies in (0, +∞) and the reaction ζ −→ λg(z, ζ ) − f (z, ζ ) (which is more general than the standard p-logistic equation; see Afrouzi and Brown [ 1 ]), exhibits a superdiffusive kind of behavior. To the best of our konwledge, the Neumann p-logistic equation has not been studied. There is only the recent work of Marano-Papageorgiou [ 25 ], where the equidiffusive case is examined. Our approach is variational based on the critical point theory, combined with suitable truncation and comparison techniques. In the next section, for the convenience of the reader we recall main mathematical tools which we will use in the sequel. This work is the outgrowth of a remark made by the referee of [ 19 ]. In that paper, the authors deal with the parametric equation − pu(z) = λ f z, u(z) u|∂ = 0 on ∂ in and some analogous bifurcation-type results were proved. It was pointed out by the referee that in mathematical biology, the Neumann model is a more realistic one. For some other recent results on nonlinear Neumann boundary value problems involving p-Laplacian, we refer to Gasin´ski and Papageorgiou [ 11–17 ]. 2 Mathematical Background Let X be a Banach space and let X ∗ be its topological dual. By ·, · we denote the duality brackets for the pair (X, X ∗). Let ϕ ∈ C 1(X ). We say that ϕ satisfies the Palais–Smale condition, if the following holds: “Every sequence {xn}n 1 ⊆ X , such that ϕ(xn) n 1 ⊆ R is bounded and ϕ (xn) −→ 0 in X ∗ as n → +∞, admits a strongly convergent subsequence.” Using this compactness-type condition on ϕ, we can state the following theorem, known in the literature as the “mountain pass theorem”. Theorem 2.1 If X is a Banach space, ϕ ∈ C 1(X ) satisfies the Palais–Smale condition, x0, x1 ∈ X , 0 < < x0 − x1 , max ϕ(x0), ϕ(x1) < inf ϕ(x ) : x − x0 = = η , c = γin∈f 0mtax1 ϕ γ (t ) , where = γ ∈ C [ 0, 1 ]; X : γ (0) = x0, γ (1) = x1 , then c η and c is a critical value of ϕ (i.e., there exists x ∈ X , such that ϕ (x ) = 0 and ϕ(x ) = c). In the study of problem ( P)λ, we will use the Sobolev space W 1, p( ) and the ordered Banach space C 1( ). The positive cone of the latter is C+ = u ∈ C 1( ) : u(z) 0 for all z ∈ This cone has a nonempty interior, given by int C+ = u ∈ C+ : u(z) > 0 for all z ∈ The next result relates local minimizers in W 1, p( ) with local minimizers in the smaller Banach space C 1( ). A result of this type was first proved for the Dirichlet Laplacian by Brézis and Nirenberg [ 5 ] and was later extended to the p-Laplacian by García Azorero et al. [ 8 ] and Guo and Zhang [ 20 ] (in the latter, for p 2). Extensions to the Neumann p-Laplacian or Neumann p-Laplacian-like operators can be found in Motreanu et al. [26] and Motreanu and Papageorgiou [28]. So let f0 : ×R −→ R be a Carathéodory function (i.e., for all ζ ∈ R, the function z −→ f0(z, ζ ) is measurable and for almost all z ∈ , the function ζ −→ f0(z, ζ ) is continuous), which exhibits subcritical growth in ζ ∈ R, i.e., f0(z, ζ ) with a ∈ L∞( )+, c > 0 and 1 < r < p∗, where p∗ = N p N − p +∞ p < N , p N . F0(z, ζ ) ds = f0(z, s) ds if if 0 ζ We set and consider the C 1-functional ψ0 : W 1, p( ) −→ R, defined by ψ0(u) = 1 p ∇u pp − F0 z, u(z) d z ∀u ∈ W 1, p( ). Theorem 2.2 If u0 ∈ W 1, p( ) is a local C 1( )-minimizer of ψ0, i.e., there exists 1 > 0, such that ψ0(u0) ψ0(u0 + h) ∀h ∈ C 1( ), h C1( ) 1, then u0 ∈ C 1( ) and it is a local W 1, p( )-minimizer of ψ0, i.e., there exists 2 > 0, such that ψ0(u0) ψ0(u0 + h) ∀h ∈ W 1, p( ), h 2. Remark 2.3 In [ 26,28 ], the result was stated in terms of Wn1, p( ) = C 1( ) · , where n C 1( ) = n ∂u u ∈ C 1( ) : ∂n = 0 on ∂ Proposition 2.4 The map A : W 1, p( ) −→ W 1, p( )∗ defined by (2.1) is continuous, strictly monotone (hence maximal monotone too) and of type (S)+, i.e., if {un}n 1 ⊆ W 1, p( ) is a sequence, such that un −→ u weakly in W 1, p( ) and lim sup A(un), un − u n→+∞ 0, then un −→ u in W 1, p( ). The next simple lemma, will be useful in our estimations and can be found in Aizicovici et al. [4, Lemma 2]. Recall that by · we denote the norm of the Sobolev space W 1, p( ), i.e., u = u pp + ∇u pp 1p ∀u ∈ W 1, p( ). Lemma 2.5 If β ∈ L∞( ), β(z) exists ξ0 > 0, such that We conclude this section by fixing some notation. By | · |N we denote the Lebesgue measure on RN . For every u ∈ W 1, p( ), we set u± = max{±u, 0}. We know that u± ∈ W 1, p( ), u = u+ − u−, |u| = u+ + u−. Finally for every measurable function h : × R −→ R, we define Nh (u)(·) = h ·, u(·) ∀u ∈ W 1, p( ) (the Nemytskii map corresponding to h). 3 A Bifurcation-Type Theorem The hypotheses on the data of problem ( P)λ are the following: Hg g : × R −→ R is a Carathéodory function, such that g(z, 0) = 0 for almost all z ∈ and (i) we have with a ∈ L∞( )+, c > 0 and p < r < p∗; (ii) there exist ϑ > q > p, such that 0 < ηg lim inf ζ →+∞ ζ q−1 uniformly for almost all z ∈ is nonincreasing on (0, +∞); (iii) we have and for almost all z ∈ g(z,ζ ) , the function ζ −→ ζ ϑ−1 H0 For every λ > 0 and > 0, we can find γ = γ (λ) > 0, such that for almost all z ∈ , the function ζ −→ λg(z, ζ ) − f (z, ζ ) + γ ζ ϑ−1 is nondecreasing on [0, ] (ϑ > q > p as in the hypothesis Hg(i i )). Remark 3.1 Since we are interested in positive solutions and hypotheses Hg, H f and H0 concern the positive semiaxis R+ = [0, +∞), we may (and will) assume that g(z, ζ ) = f (z, ζ ) = 0 for almost all z ∈ Example 3.2 The following functions satisfy hypotheses Hg, H f and H0 (for the sake of simplicity we drop the z-dependence): (a) g(ζ ) = ζζ qs−−11 iiff ζζ ∈> [ 10, 1 ], and f (ζ ) = ζ ϑ−1 for all ζ 0, with p < q < s < ϑ < p∗. (b) g(ζ ) = ζζ qs−−11 −− ζζϑp−−11 iiff ζζ ∈> [ 10, 1 ], and f (ζ ) = ζ ϑ−1 − ζ s−1 for all ζ 0, with p < q < s < ϑ < p∗. Example (a) corresponds to the standard superdiffusive p-logistic reaction (see Afrouzi and Brown [1]). By a positive solution of problem ( P)λ, we understand a function u ∈ W 1, p( ), u = 0, which is a weak solution of ( P)λ. Then u ∈ L∞( ) (see e.g., Gasin´ski and Papageorgiou [ 9,18 ] and Hu and Papageorgiou [23]). Invoking Theorem 2 of Lieberman [ 24 ], we have that u ∈ C+ \ {0}. Let = u ∞ and let γ = γ (λ) > 0 be as postulated by hypothesis H0. We have − pu(z) + β(z)u(z) p−1 + γ u(z)ϑ−1 = λg z, u(z) − f z, u(z) + γ u(z)ϑ−1 (see Motreanu and Papageorgiou [ 27 ]), so pu(z) β ∞ + γ ϑ− p u(z) p−1 for almost all z ∈ and finally u ∈ int C+ (see Vázquez [ 31 ]). So, we see that the positive solutions of problem ( P)λ, if they exist, belong in int C+. Let Y = λ > 0 : problem ( P)λ has a positive solution. Proposition 3.3 If hypotheses Hg, H f and H0 hold, then inf Y > 0. Proof By virtue of hypothesis Hg(i i ), we can find η1 > 0 and M > 0, such that η1ζ q−1 for almost all z ∈ On the other hand, from hypothesis Hg(i i i ), for a given ε > 0, we can find δ ∈ (0, 1) small, such that εζ p−1 for almost all z ∈ , all ζ ∈ [0, δ]. (3.1) (3.2) The function ζ −→ σζ1q(−ζ1) is upper semicontinuous on [δ, M ] and so, we can find ξ ∈ [δ, M ], such that (see hypothesis Hg(i v)). From (3.1), (3.2) and (3.3), it follows that (see (2.1) for the definition of A). On (3.7) we act with uλ and obtain p ∇uλ p + βuλp d z = λg(z, uλ) − f (z, uλ) uλ d z, so using Lemma 2.5 and (3.6), we have ξ0 uλ p 2ε uλ p. recalling that ε ∈ 0, ξ20 , we conclude that uλ = 0, a contradiction. Therefore inf Y λ > 0. If Y = ∅, then inf Y = +∞. In the next proposition, we establish the nonemptiness of Y. Proposition 3.4 If hypotheses Hg, H f and H0 hold, then Y = ∅ and if λ ∈ Y and τ > λ, then τ ∈ Y. (3.3) (3.4) (3.5) (3.7) Proof Let ϕλ : W 1, p( ) −→ by ϕλ(u) = R be the energy functional for problem ( P)λ, defined Since q < ϑ , using Young inequality with ε > 0, from (3.8) we see that for a given ε > 0, we can find c2 = c2(ε) > 0, such that ε(ζ +)ϑ + c2 for almost all z ∈ Also, from hypotheses H f (i ), (i i ), we see that we can find ξ2 > 0 and c3 > 0, such that F (z, ζ ) ξ2(ζ +)ϑ − c3 for almost all z ∈ ξ0 u p + (ξ2 − λε) u+ ϑϑ − c4 ∀u ∈ W 1, p( ), p for some c4 = c4(ε) > 0 (see Lemma 2.5 and (3.9), (3.10)). We choose ε ∈ 0, ξλ2 . Then, from (3.11), it follows that ϕλ is coercive. Also, it is easy to see that ϕλ is sequentially weakly lower semicontinuous. Therefore, by the Weierstrass theorem, we can find uλ ∈ W 1, p( ), such that ϕλ(uλ) = inf u∈W 1,p( ) ϕλ(u) = mλ. Let u ∈ int C+. Then clearly for λ > 0 big, we have ϕλ(u) < 0. Hence ϕλ(uλ) = mλ < 0 = ϕλ(0) ∀λ > 0, big (see (3.12)), so From (3.12), we have so On (3.14) we act with −uλ− ∈ W 1, p( ) and we obtain (see Lemma 2.5), i.e., uλ Then (3.14) becomes 0, uλ = 0 (see (3.13)). uλ solves problem ( P)λ, sϑ−1g z, uλ(z) g z, suλ(z) = g z, u(z) for almost all z ∈ . (3.17) Similarly, using hypothesis H f (i i ), we have f (z, uλ(z)) uλ(z) p−1 f (z, u(z)) u(z) p−1 f (z, u(z)) = s p−1uλ(z) p−1 , s p−1 f z, uλ(z) f z, suλ(z) = f z, u(z) i.e., Y = ∅. Now suppose that λ ∈ Y and τ > λ. We choose s ∈ (0, 1), such that (Wreecasellttuha=tϑsu>λ ∈p ainntdCλ+<.Tτh)e.nSince λ ∈ Y, problem ( P)λ has a solution uλ ∈ int C+. − pu + βu p−1 = s p−1 − uλ + βuλp−1 = s p−1 λg(z, uλ) − f (z, uλ) . (3.16) By virtue of hypothesis Hg(i i ) and since s ∈ (0, 1), we have g(z, uλ(z)) uλ(z)ϑ−1 g(z, u(z)) u(z)ϑ−1 g(z, u(z)) = sϑ−1uλ(z)ϑ−1 , λ = sϑ−1τ (3.15) Returning to (3.16) and using (3.15), (3.17) and (3.18), we have We consider the following truncation of the reaction in problem ( P)τ : − pu(z) + β(z)u(z) p−1 = λs p−1g z, uλ(z) − s p−1 f z, uλ(z) sϑ−1τ g z, uλ(z) − f z, u(z) τ g z, u(z) − f z, u(z) and consider the C 1-functional ψτ : W 1, p( ) −→ R, defined by ψτ (u) = As we did for ϕλ earlier in this proof, we can check that ψτ is coercive and sequentially weakly lower semicontinuous. So, we can find uτ ∈ W 1, p( ), such that On (3.21) we act with (u − uτ )+ ∈ W 1, p( ) and obtain ψτ (uτ ) = u∈Wi n1,fp( ) ψτ (u), ψτ (uτ ) = 0, A(uτ ) + β|uτ | p−2uτ = Nhτ (uτ ). (3.21) A(uτ ), (u − uτ )+ + β|uτ | p−2uτ (u − uτ )+ d z = = hτ (z, uτ )(u − uτ )+ d z τ g(z, u) − f (z, u) (u − uτ )+ d z A(u), (u − uτ )+ + βu p−1(u − uτ )+ d z (see (3.20) and (3.19)), so {u>uτ } + {u>uτ } ∇uτ p−2∇uτ − ∇u p−2∇u, ∇uτ − ∇u Rd z β |uτ | p−2uτ − u p−1 (uτ − u) d z We recall the following elementary inequalities (see e.g., Gasin´ski and Papageorgiou [10, Lemma 6.2.13, p. 740]). If 1 < p 2, then ( p − 1)|y − v|2(1 + |y| + |v|) p−2 ≤ |y| p−2 y − |v| p−2v, y − v RN ∀y, v ∈ RN and if 2 < p, then 1 2 p−2 |y − v| p ≤ |y| p−2 y − |v| p−2v, y − v RN ∀y, v ∈ RN . If 1 < p 2, then from (3.22), (3.23) and since uτ , u ∈ int C+, we have (3.22) (3.23) (3.24) for some c5 > 0, so i.e., u uτ . If 2 < p, then from (3.22) and (3.24), we have p − 1 c5 {u>uτ } ∇uτ − ∇u 2 d z 0 {u > uτ } N = 0, 1 2 p−2 {u>uτ } ∇uτ − ∇u p d z 0, {u > uτ } N = 0, uτ and then (3.21) becomes A(uτ ) + βuτp−1 = τ Ng(uτ ) − N f (uτ ) so i.e., u uτ . So, finally u (see (3.20)), so uτ ∈ int C+ is a positive solution of ( P)λ, i.e., τ ∈ Y. Proposition 3.5 If hypotheses H f , Hg and H0 hold and λ > λ∗, then problem ( P)λ has at least two positive solutions. Proof Let τ ∈ (λ∗, λ) ∩ Y. Then, we can find uτ ∈ int C+, such that = 0 on ∂ . , Proceeding as in the proof of Proposition 3.4, we introduce the following truncation of the reaction: hλ(z, ζ ) = λg z, uλ(z) − f z, uλ(z) λg(z, ζ ) − f (z, ζ ) if ζ uτ (z), if uτ (z) < ζ. This is a Carathéodory function. We set Hλ(z, ζ ) = hλ(z, s) ds 0 ζ and consider the C 1-functional ψ : W 1, p( ) −→ R, defined by ψλ(u) = As we did for ϕλ in the proof of Proposition 3.4, we can check that ψλ is coercive and 0 sequentially weakly lower semicontinuous. So, we can find uλ ∈ W 1, p( ), such that ψλ(u0λ) = u∈Wi n1,fp( ) ψλ(u), ψλ(u0λ) = 0, A(u0λ) + β|u0λ| p−2u0λ = Nhλ (u0λ). From this, as before, acting with (uτ − u0λ)+ ∈ W 1, p( ) and using (3.25) and (3.26), we show that uτ u0λ. Hence, we have A(u0λ) + β(u0λ) p−1 = λNg(u0λ) − N f (u0λ) (see (3.26)), so u0λ ∈ int C+ is a solution of ( P)λ and u0λ uτ . Claim 1 u0λ − uτ ∈ int C+. Let = u0λ ∞. By hypothesis H0, we can find γ = γ (λ) > 0, such that for all z ∈ , the function ζ −→ λg(z, ζ ) − f (z, ζ ) + γ ζ ϑ−1 is nondecreasing on [0, ]. For δ > 0, we set uτ = uτ + δ ∈ int C+. Then − puτ + βuτp−1 + γ uτϑ−1 − puτ + βuτp−1 + γ uτϑ−1 + ξ(δ) = τ g(z, uτ ) − f (z, uτ ) + γ uτϑ−1 + ξ(δ) = λg(z, uτ ) − f (z, uτ ) + (τ − λ)g(z, uτ ) + γ uτϑ−1 + ξ(δ) λg(z, uτ ) − f (z, uτ ) − (λ − τ )σ0(uτ ) + γ uτϑ−1 + ξ(δ) with ξ(δ) → 0 as δ 0 (see hypothesis Hg(i v) and recall that τ < λ). Since uτ ∈ int C+, the function z −→ σ0 uτ (z) is upper semicontinuous on (see hypothesis Hg(i v)). So, we can find z0 ∈ , such that σ0 uτ (z0) = max σ0 uτ (z) > 0. z∈ We use (3.28) in (3.27). Since ξ(δ) 0 and δ 0 and λ > τ , we infer that − puτ + βuτp−1 + γ uτϑ−1 λg(z, uτ ) − f (z, uτ ) + γ uϑ−1 τ λg(z, u0λ) − f (z, u0λ) + γ (u0λ)ϑ−1 = − pu0λ + β(u0λ) p−1 + γ (u0λ)ϑ−1 for almost all z ∈ . for δ > 0 small (see H0 and recall that uτ u0λ). Acting on this inequality with (uτ − u0λ)+ ∈ W 1, p( ) and using the nonlinear Green’s identity (see e.g., Gasin´ski and Papageorgiou [ 9 ]) as above, we obtain so This proves Claim 1. Let [uτ ) = u ∈ W 1, p( ) : uτ (z) u(z) for almost all z ∈ . (3.27) (3.28) From (3.26), we see that ψλ|[uτ ) = ϕλ|[uτ ) + c, (3.29) for some c ∈ R. Then Claim 1 and (3.29) imply that u0 is a local C 1( )-minimizer λ of ϕλ. From Theorem 2.2, it follows that u0λ is a local W 1, p( )-minimizer of ϕ(λ). By virtue of hypotheses Hg(iii) and H f (iii), for a given ε > 0 we can find δ = δ(ε) > 0, such that − εp ζ p for almost all z ∈ So, if u ∈ C 1( ) with u C1( ) δ, then , all ζ ∈ (0, δ]. (3.30) ξ0 u p p − ξ0 − (λ + 1)ε u p p (see Lemma 2.5) and (3.30). Choosing ε ∈ 0, λξ+01 , we infer that ϕλ(u) 0 = ϕλ(0) ∀u ∈ C 1( ), u C1( ) δ, so and thus u = 0 is a local C 1( )-minimizer of ϕλ u = 0 is a local W 1, p( )-minimizer of ϕλ (see Theorem 2.2). Without any loss of generality, we may assume that ϕλ(0) = 0 ϕλ(u0λ) (the analysis is similar if the opposite inequality is true). Moreover, we may assume that both local minimizers u = 0 and u = u0λ are isolated (otherwise it is clear that we have a whole sequence of positive solutions of ( P)λ and so we are done). Reasoning as in Aizicovici et al. [2, Proposition 29], we can find ∈ 0, u0λ small, such that ϕλ(0) = 0 ϕλ(u0λ) < inf ϕλ(u) : 0 u − uλ = = η0λ. (3.31) Recall that ϕλ is coercive (see the proof of Proposition 3.4). Hence it satisfies the Palais– Smale condition. This fact and (3.1) permit the use of the mountain pass theorem (see Theorem 2.1) and so, we obtain uλ ∈ W 1, p( ), such that and η ϕλ(uλ) ϕλ(uλ) = 0. From (3.31) and (3.32), it follows that uλ ∈/ {0, u0λ}. From (3.33), we have Proposition 3.6 If hypotheses H f , Hg and H0 hold, then λ∗ ∈ Y. Proof Let λn > λ∗ for n 1 be such that λn λ∗ and let un = uλn ∈ int C+ be positive solutions for problem ( P)λ for n 1 (see Proposition 3.4). We have A(un) + βunp−1 = λn Ng(un) − N f (un) ∀n 1. (3.34) By virtue of hypothesis Hg(i i ) and since ϑ > q, we have lim ζ →+∞ ζ ϑ−1 = 0 uniformly for almost all z ∈ . This fact combined with hypothesis Hg(i ), implies that for a given ε > 0, we can find c6 = c6(ε) > 0, such that g(z, ζ )ζ (ζ +)ϑ + c6 for almost all z ∈ In a similar fashion, using hypotheses H f (i ) and (i i ), we see that we can find η > 0 and c7 > 0, such that f (z, ζ )ζ (ζ +)ϑ − c7 for almost all z ∈ On (3.34) we act with un ∈ W 1, p( ) and obtain βunp d z = λn g(z, un)un d z − f (z, un )un d z λnε − η ϑ un ϑϑ + c8 ∀n 1, for some c8 > 0 (see (3.35) and (3.36)). (3.32) (3.33) (3.35) (3.36) (3.37) We choose ε ∈ 0, λη1 Lemma 2.5, it follows that (recall that λn λ1 for all n 1). Then from (3.37) and with θ < p∗. On (3.34) we act with un − u∗, pass to the limit as n → +∞ and use (3.38). We obtain so Let From (3.41) and Theorem 2 of Lieberman [ 24 ], we know that we can find α ∈ (0, 1) and M > 0, such that un ∈ C 1,α( ) and un C1,α( ) M ∀n 1. From the compactness of the embedding C 1,α( ) ⊆ C 1( ), we have un −→ u∗ in C 1( ). un un yn = ∀n (3.42) (3.43) (3.44) (3.45) with 0 ζ (z) ζ ∗ for almost all z ∈ . So, if in (3.44) we pass to the limit as n → +∞ and we use (3.45) and (3.46), we obtain A(y∗) + β y∗p−1 = −ζ y∗p−1, So, passing to a subsequence if necessary, we may assume that From (3.34), we have Then so so thus yn 0, yn = 1 ∀n 1. yn −→ y∗ yn −→ y∗ in Lϑ ( ). From hypotheses Hg(i ), (iii) and H f (i ), (iii), it follows that the sequences Ng(un) un p−1 n 1 , N f (un) un p−1 n 1 ⊆ L p ( ) are bounded (where 1p + p1 = 1). Acting on (3.44) with yn − y∗, passing to the limit as n → +∞ and using (3.42), we obtain p ∇ y∗ p + β y∗p d z ζ y∗p d z 0, ξ0 y∗ p − 0 (see Lemma 2.5) and finally, we have that y∗ = 0, which contradicts to (3.45). TheTrheifsorperoλv∗e∈s tYha.t u∗ = 0. Hence u∗ ∈ int C+ is a solution of problem ( P)λ∗ . We show that for every λ solution. λ∗, problem ( P)λ has an extremal (smallest) positive Proposition 3.7 If hypotheses H f , Hg, and H0 hold and λ has a smallest positive solution u∗λ ∈ int C+. λ∗, then problem ( P)λ Proof Let S(λ) be the set of positive solutions for problem ( P)λ. Since λ λ∗, S(λ) = 0 and S(λ) ⊆ int C+. Let C ⊆ S(λ) be a chain (i.e., a nonempty linearly ordered subset of S(λ)). From Dunford and Schwartz [6, p.336], we know that we can find a sequence {un}n 1 ⊆ C , such that inf un = inf C. n 1 Moreover, from Lemma 11.5(a) of Heikkilä and Lakshmikantham [22, p. 15], we know that we may assume that the sequence {un}n 1 is decreasing. We have A(un) + βunp−1 = λNg(un) − N f (un) ∀n 1, (3.47) βunp d z = λg(z, un) − f (z, un) un d z M1 ∀n 1, for some M1 > 0 (see hypotheses Hg(i ), H f (i ) and recall that un So, u1 for all n 1). so so (see Proposition 2.4). (see Lemma 2.5) and thus the sequence {un}n 1 ⊆ W 1, p( ) is bounded. So, passing to a subsequence if necessary, we may assume that with θ < p∗. On (3.47) we act with un − u∗, pass to the limit as n → +∞ and use (3.48). Then un −→ u∗ un −→ u∗ in Lθ ( ), weakly in W 1, p( ), lim n→+∞ u∗ λ ( P )λ. Reasoning as in the proof of Proposition 3.6, we show that u∗ = 0 and so u∗ ∈ int C+ is a positive solution of ( P )λ. Hence u∗ = inf C ∈ S(λ) and since C was an arbitrary chain, from the Kuratowski-Zorn lemma, we infer that S(λ) has a minimum element u∗λ ∈ int C+. But S(λ) is downward directed (i.e., if u, v ∈ S(λ), then there exists y ∈ S(λ), such that y min{u, v}; see Aizicovici et al. [ 3 ]). So, it follows that u for all u ∈ S(λ), i.e., u∗λ ∈ int C+ is the smallest positive solution of problem Summarizing the situation, we have the following bifurcation-type theorem describing the dependence of positive solutions of ( P )λ on the parameter λ > 0. Theorem 3.8 If hypotheses H f , Hg and H0 hold, then there exists λ∗ > 0, such that: (a) for all λ > λ∗, problem ( P )λ has at least two positive solutions u0, u ∈ int C+; (b) for λ = λ∗, problem ( P )λ has at least one positive solution u∗ ∈ int C+; (c) for all λ ∈ (0, λ∗), problem ( P )λ has no positive solution. Moreover, if λ λ∗, then problem ( P )λ has a smallest positive solution u∗λ ∈ int C+. Acknowledgments The authors wish to thank the two referees for their corrections and helpful remarks. 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Leszek Gasiński, Nikolaos S. Papageorgiou. Positive Solutions for the Neumann p-Laplacian with Superdiffusive Reaction, Bulletin of the Malaysian Mathematical Sciences Society, 2017, 1711-1731, DOI: 10.1007/s40840-015-0212-3