Positive solutions for the Neumann p-Laplacian with superdiffusive reaction

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(1)Bull. Malays. Math. Sci. Soc. (2017) 40:1711–1731 DOI 10.1007/s40840-015-0212-3. Positive Solutions for the Neumann p-Laplacian with Superdiffusive Reaction Leszek Gasinski ´ 1 · Nikolaos S. Papageorgiou2. Received: 9 February 2015 / Revised: 29 July 2015 / Published online: 21 August 2015 © The Author(s) 2015. This article is published with open access at Springerlink.com. Abstract 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. Keywords p-Laplacian · Superdiffusive reaction · Local minimizers · Mountain pass theorem · Comparison principle · Bifurcation-type theorem Mathematics Subject Classification 35J25 · 35J92. Communicated by Rosihan M. Ali.. B. Leszek Gasiński leszek.gasinski@ii.uj.edu.pl Nikolaos S. Papageorgiou npapg@math.ntua.gr. 1. Faculty of Mathematics and Computer Science, Jagiellonian University, ul. Łojasiewicza 6, 30-348 Kraków, Poland. 2. Department of Mathematics, National Technical University, Zografou Campus, 15780 Athens, Greece. 123.

(2) 1712. L. Gasiński, N. S. Papageorgiou. 1 Introduction Let ⊆ R N be a bounded domain with a C 2 -boundary ∂. In this paper, we study the following nonlinear parametric Neumann problem: ⎧ ⎨ − p u(z) + β(z)u(z) p−1 = λg z, u(z) − f z, u(z) in , ∂u ⎩ = 0 on ∂, λ > 0, u > 0, ∂n. (P)λ. with β ∈ L ∞ ()+ , β = 0. Here p denotes the p-Laplace differential operator, defined by p u = 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 . 123. − p u(z) = λ f z, u(z) in , u|∂ = 0 on ∂.

(3) Positive Solutions for the Neumann p-Laplacian.... 1713. 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 Gasiński and Papageorgiou [11–17].. 2 Mathematical Background Let X be a Banach space and let X ∗ be its topological dual. By

(4) ·, · 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 = inf max ϕ γ (t) , γ ∈

(5) 0t 1. where

(6) =. γ ∈ C [0, 1]; X : γ (0) = x0 , γ (1) = x1 ,. x ∈ X , such that ϕ (. x) = 0 then c η and c is a critical value of ϕ (i.e., there exists. 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. 123.

(7) 1714. L. Gasiński, N. S. Papageorgiou. 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 f 0 : ×R −→ R be a Carathéodory function (i.e., for all ζ ∈ R, the function z −→ f 0 (z, ζ ) is measurable and for almost all z ∈ , the function ζ −→ f 0 (z, ζ ) is continuous), which exhibits subcritical growth in ζ ∈ R, i.e.,

(8)

(9)

(10) f 0 (z, ζ )

(11) a(z) + c|ζ |r −1 for almost all z ∈ , all ζ ∈ R, with a ∈ L ∞ ()+ , c > 0 and 1 < r < p ∗ , where. Np if p < N , ∗ N−p p = +∞ if p N . We set . ζ. F0 (z, ζ ) ds =. f 0 (z, s) ds. 0. and consider the C 1 -functional ψ0 : W 1, p () −→ R, defined by 1 p F0 z, u(z) dz ∀u ∈ W 1, p (). ψ0 (u) = ∇u p − p Theorem 2.2 If u 0 ∈ W 1, p () is a local C 1 ()-minimizer of ψ0 , i.e., there exists 1 > 0, such that ψ0 (u 0 ) ψ0 (u 0 + h) ∀h ∈ C 1 (), hC 1 () 1 , then u 0 ∈ C 1 () and it is a local W 1, p ()-minimizer of ψ0 , i.e., there exists 2 > 0, such that ψ0 (u 0 ) ψ0 (u 0 + h) ∀h ∈ W 1, p (), h 2 . 1, p. Remark 2.3 In [26,28], the result was stated in terms of Wn () = Cn1 () Cn1 () =. ·. , where. ∂u = 0 on ∂ . u ∈ C 1 () : ∂n. Actually, there is no need for this restriction. Let A : W 1, p () −→ W 1, p ()∗ be the nonlinear map defined by ∇u p−2 (∇u, ∇ y)R N dz ∀u, y ∈ W 1, p (). A(u), y = . The next result can be found in Aizicovici et al. [3, Proposition 2].. 123. (2.1).

(12) Positive Solutions for the Neumann p-Laplacian.... 1715. 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 {u n }n 1 ⊆ W 1, p () is a sequence, such that u n −→ u weakly in W 1, p () and lim sup A(u n ), u n − u 0, n→+∞. then u n −→ 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., p p 1 u = u p + ∇u p p ∀u ∈ W 1, p (). Lemma 2.5 If β ∈ L ∞ (), β(z) 0 for almost all z ∈ and β = 0, then there exists ξ0 > 0, such that p β|u| p dz ξ0 u p ∀u ∈ W 1, p (). ∇u p + . We conclude this section by fixing some notation. By | · | N we denote the Lebesgue measure on R N . 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

(13)

(14)

(15) g(z, ζ )

(16) a(z) + c|ζ |r −1 for almost all z ∈ , all ζ ∈ R, with a ∈ L ∞ ()+ , c > 0 and p < r < p ∗ ; (ii) there exist ϑ > q > p, such that 0 < ηg lim inf. ζ →+∞. g(z, ζ ) g(z, ζ ) lim sup q−1 . ηg q−1 ζ ζ →+∞ ζ. 123.

(17) 1716. L. Gasiński, N. S. Papageorgiou. uniformly for almost all z ∈ and for almost all z ∈ , the function ζ −→ is nonincreasing on (0, +∞); (iii) we have lim. ζ →0+. g(z,ζ ) ζ ϑ−1. g(z, ζ ) = 0 ζ q−1. uniformly for almost all z ∈ ; (iv) there exist two functions σ0 , σ1 : (0, +∞) −→ (0, +∞), both upper semicontinuous, such that σ0 (ζ ) g(z, ζ ) σ1 (ζ ) for almost all z ∈ , all ζ > 0. H f f : × R −→ R is a Carathéodory function, such that f (z, 0) = 0 for almost all z ∈ and (i) we have

(18)

(19)

(20) f (z, ζ )

(21) a(z) + c|ζ |r −1 for almost all z ∈ , all ζ ∈ R, with a ∈ L ∞ ()+ , c > 0 and p < r < p ∗ ; (ii) with ϑ > q > p as in hypothesis Hg (ii), we have 0 < η f lim inf. ζ →+∞. f (z, ζ ) f (z, ζ ) lim sup ϑ−1 . ηf ϑ−1 ζ ζ →+∞ ζ. uniformly for almost all z ∈ and for almost all z ∈ , the function ζ −→ is nondecreasing on (0, +∞); (iii) we have 0 lim inf ζ →0+. f (z,ζ ) ζ p−1. f (z, ζ ) f (z, ζ ) lim sup q−1 ζ ∗ q−1 ζ ζ ζ →0+. uniformly for almost all z ∈ ; (iv) there exists a lower semicontinuous function σ2 : (0, +∞) −→ (0, +∞), such that σ2 (ζ ) f (z, ζ ) for almost all z ∈ , all ζ > 0. 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 (ii)). 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 ∈ , all ζ 0.. 123.

(22) Positive Solutions for the Neumann p-Laplacian.... 1717. Example 3.2 The following functions satisfy hypotheses Hg , H f and H0 (for the sake of simplicitywe drop the z-dependence): ζ s−1 if ζ ∈ [0, 1], (a) g(ζ ) = and f (ζ ) = ζ ϑ−1 for all ζ 0, with p < ζ q−1 if ζ > 1 q<s<ϑ < p∗ . ζ s−1 − ζ ϑ−1 if ζ ∈ [0, 1], (b) g(ζ ) = and f (ζ ) = ζ ϑ−1 − ζ s−1 for all ζ 0, ζ q−1 − ζ p−1 if ζ > 1 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., Gasiński 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 − p u(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 0 for almost all z ∈ (see Motreanu and Papageorgiou [27]), so p u(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 (ii), we can find η1 > 0 and M > 0, such that g(z, ζ ) η1 ζ q−1 for almost all z ∈ , all ζ M.. (3.1). On the other hand, from hypothesis Hg (iii), for a given ε > 0, we can find δ ∈ (0, 1) small, such that g(z, ζ ) εζ p−1 for almost all z ∈ , all ζ ∈ [0, δ].. (3.2). 123.

(23) 1718. L. Gasiński, N. S. Papageorgiou. The function ζ −→. ξ ∈ [δ, M], such that. σ1 (ζ ) ζ q−1. is upper semicontinuous on [δ, M] and so, we can find. σ1 (ζ ) ζ) σ1 (. q−1 = η2 (ε) ∀ζ ∈ [δ, M], q−1. ζ ζ so g(z, ζ ) η2 ζ q−1 for almost all ζ ∈ [δ, M]. (3.3). (see hypothesis Hg (iv)). From (3.1), (3.2) and (3.3), it follows that g(z, ζ ) εζ p−1 +. ηζ q−1 for almost all ζ 0,. (3.4). with. η(ε) = max{η1 , η2 } > 0. In a similar fashion, using hypotheses H f (ii), (iii) and (iv), for a given ε > 0, we can find ϑ = ϑ(ε) > 0, such that f (z, ζ ) ϑζ q−1 − εζ p−1 for almost all ζ 0.. (3.5). . λ min 1, ϑ. Let us fix ε ∈ 0, ξ20 (ξ0 > 0 as in Lemma 2.5) and let. η . From (3.4) and (3.5), we have. λg(z, ζ ) − f (z, ζ ) (. λ + 1)εζ p−1 + (. λ. η − ϑ)ζ q−1 2εζ p−1 for almost all z ∈ , all ζ 0.. (3.6). Suppose that for λ ∈ (0,. λ), problem (P)λ has a positive solution (i.e., λ ∈ Y). Then we can find a positive solution u λ ∈ int C+ of (P)λ . Hence p−1. A(u λ ) + βu λ. = λN g (u λ ) − N f (u λ ). (3.7). (see (2.1) for the definition of A). On (3.7) we act with u λ and obtain p p ∇u λ p + λg(z, u λ ) − f (z, u λ ) u λ dz, βu λ dz = . . so using Lemma 2.5 and (3.6), we have ξ0 u λ p 2εu λ p . recalling that ε ∈ 0, ξ20 , we conclude that u λ = 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.. 123.

(24) Positive Solutions for the Neumann p-Laplacian.... 1719. Proof Let ϕλ : W 1, p () −→ R be the energy functional for problem (P)λ , defined by 1 1 p β|u| p dz − λ G(z, u) dz + F(z, u) dz ϕλ (u) = ∇u p + p p for all u ∈ W 1, p (). Evidently ϕλ ∈ C 1 W 1, p () . By virtue of hypotheses Hg (i), (ii), we can find ξ1 > 0 and c1 > 0, such that G(z, ζ ) ξ1 (ζ + )q + c1 for almost all z ∈ , all ζ ∈ R.. (3.8). 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 G(z, ζ ) ε(ζ + )ϑ + c2 for almost all z ∈ , all ζ ∈ R.. (3.9). Also, from hypotheses H f (i), (ii), we see that we can find ξ2 > 0 and c3 > 0, such that (3.10) F(z, ζ ) ξ2 (ζ + )ϑ − c3 for almost all z ∈ , all ζ ∈ R. Then 1 1 p ∇u p + β|u| p dz − λ G(z, u) dz + F(z, u) dz p p ξ0 (3.11) u p + (ξ2 − λε)u + ϑϑ − c4 ∀u ∈ W 1, p (), p. ϕλ (u) =. 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 u λ ∈ W 1, p (), such that ϕλ (u λ ) =. inf. u∈W 1, p (). ϕλ (u) = m λ .. (3.12). Let u ∈ int C+ . Then clearly for λ > 0 big, we have ϕλ (u) < 0. Hence ϕλ (u λ ) = m λ < 0 = ϕλ (0) ∀λ > 0, big (see (3.12)), so u λ = 0.. (3.13). From (3.12), we have ϕλ (u λ ) = 0 ∀λ > 0, big so A(u λ ) + β|u λ | p−2 u λ = λN g (u λ ) − N f (u λ ).. (3.14). 123.

(25) 1720. L. Gasiński, N. S. Papageorgiou. 1, p () and we obtain On (3.14) we act with −u − λ ∈W. . p ∇u − λ p. +. . p β(u − λ ) dz = 0,. so p ξ0 u − λ 0. (see Lemma 2.5), i.e., u λ 0, u λ = 0 (see (3.13)). Then (3.14) becomes p−1. A(u λ ) + βu λ. = λN g (u λ ) − N f (u λ ),. so u λ solves problem (P)λ , i.e., Y = ∅. Now suppose that λ ∈ Y and τ > λ. We choose s ∈ (0, 1), such that λ = s ϑ−1 τ. (3.15). (recall that ϑ > p and λ < τ ). Since λ ∈ Y, problem (P)λ has a solution u λ ∈ int C+ . We set u = su λ ∈ int C+ . Then p−1 = s p−1 λg(z, u λ )− f (z, u λ ) . (3.16) − p u +βu p−1 = s p−1 −u λ +βu λ By virtue of hypothesis Hg (ii) and since s ∈ (0, 1), we have g(z, u λ (z)) g(z, u(z)) g(z, u(z)) = ϑ−1 , ϑ−1 ϑ−1 u λ (z) u(z) s u λ (z)ϑ−1 so s ϑ−1 g z, u λ (z) g z, su λ (z) = g z, u(z) for almost all z ∈ .. (3.17). Similarly, using hypothesis H f (ii), we have f (z, u λ (z)) f (z, u(z)) f (z, u(z)) = p−1 , u λ (z) p−1 u(z) p−1 s u λ (z) p−1 so s p−1 f z, u λ (z) f z, su λ (z) = f z, u(z) for almost all z ∈ .. 123. (3.18).

(26) Positive Solutions for the Neumann p-Laplacian.... 1721. Returning to (3.16) and using (3.15), (3.17) and (3.18), we have − p u(z) + β(z)u(z) p−1 = λs p−1 g z, u λ (z) − s p−1 f z, u λ (z) s ϑ−1 τ g z, u λ (z) − f z, u(z) τ g z, u(z) − f z, u(z) for almost all z ∈ .. (3.19). We consider the following truncation of the reaction in problem (P)τ : h τ (z, ζ ) =. τ g z, u(z) − f z, u(z) τ g(z, ζ ) − f (z, ζ ). if if. ζ u(z), u(z) < ζ.. (3.20). This is a Carathéodory function. We set Hτ (z, ζ ) =. ζ. 0. h τ (z, s) ds. and consider the C 1 -functional ψτ : W 1, p () −→ R, defined by ψτ (u) =. 1 1 p ∇u p + p p. . . β|u| p dz −. . Hτ (z, u) dz ∀u ∈ W 1, p ().. 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 u τ ∈ W 1, p (), such that ψτ (u τ ) =. inf. u∈W 1, p (). ψτ (u),. so ψτ (u τ ) = 0, thus A(u τ ) + β|u τ | p−2 u τ = Nh τ (u τ ). On (3.21) we act with (u − u τ . )+. ∈. W 1, p (). +. . (3.21). and obtain. . β|u τ | p−2 u τ (u − u τ )+ dz A(u τ ), (u − u τ ) + = h τ (z, u τ )(u − u τ )+ dz = τ g(z, u) − f (z, u) (u − u τ )+ dz + A(u), (u − u τ ) + βu p−1 (u − u τ )+ dz . 123.

(27) 1722. L. Gasiński, N. S. Papageorgiou. (see (3.20) and (3.19)), so ∇u τ p−2 ∇u τ − ∇u p−2 ∇u, ∇u τ − ∇u R dz {u>u τ } + β |u τ | p−2 u τ − u p−1 (u τ − u) dz 0. {u>u τ }. (3.22). We recall the following elementary inequalities (see e.g., Gasiński 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−2 v, y − v R N ∀y, v ∈ R N. (3.23). and if 2 < p, then 1 2 p−2. |y − v| p ≤ |y| p−2 y − |v| p−2 v, y − v R N ∀y, v ∈ R N .. (3.24). If 1 < p 2, then from (3.22), (3.23) and since u τ , u ∈ int C+ , we have p−1 c5. {u>u τ }. ∇u τ − ∇u2 dz 0. for some c5 > 0, so

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(29)

(30) {u > u τ }

(31) = 0, N i.e., u u τ . If 2 < p, then from (3.22) and (3.24), we have 1 2 p−2. {u>u τ }. ∇u τ − ∇u p dz 0,. so

(32)

(33)

(34) {u > u τ }

(35) = 0, N i.e., u u τ . So, finally u u τ and then (3.21) becomes A(u τ ) + βu τp−1 = τ N g (u τ ) − N f (u τ ) (see (3.20)), so u τ ∈ 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.. 123.

(36) Positive Solutions for the Neumann p-Laplacian.... 1723. Proof Let τ ∈ (λ∗ , λ) ∩ Y. Then, we can find u τ ∈ int C+ , such that ⎧ ⎨ − p u τ (z) + β(z)u τ (z) p−1 = τ g z, u τ (z) − f z, u τ (z) in , ∂u ⎩ τ = 0 on ∂. ∂n. (3.25). Proceeding as in the proof of Proposition 3.4, we introduce the following truncation of the reaction:. h λ (z, ζ ) =. . λg z, u λ (z) − f z, u λ (z) λg(z, ζ ) − f (z, ζ ). if if. ζ u τ (z), u τ (z) < ζ.. (3.26). This is a Carathéodory function. We set. λ (z, ζ ) = H. . ζ. 0. h λ (z, s) ds. : W 1, p () −→ R, defined by and consider the C 1 -functional ψ. λ (u) = ψ. 1 1 p ∇u p + p p. . β|u| dz − p. . . λ (z, u) dz ∀u ∈ W 1, p (). H. λ is coercive and As we did for ϕλ in the proof of Proposition 3.4, we can check that ψ 0 sequentially weakly lower semicontinuous. So, we can find u λ ∈ W 1, p (), such that. λ (u 0λ ) = ψ. inf. u∈W 1, p (). λ (u), ψ. and. λ (u 0λ ) = 0, ψ so 0 A(u 0λ ) + β|u 0λ | p−2 u 0λ = N. h λ (u λ ).. From this, as before, acting with (u τ − u 0λ )+ ∈ W 1, p () and using (3.25) and (3.26), we show that u τ u 0λ . Hence, we have A(u 0λ ) + β(u 0λ ) p−1 = λN g (u 0λ ) − N f (u 0λ ) (see (3.26)), so u 0λ ∈ int C+ is a solution of (P)λ and u 0λ u τ . Claim 1 u 0λ − u τ ∈ int C+ .. 123.

(37) 1724. L. Gasiński, N. S. Papageorgiou. Let = u 0λ ∞ . 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 u τ = u τ + δ ∈ int C+ . Then − p u τ + βu τp−1 + γ u ϑ−1 τ. − p u τ + βu τp−1 + γ u ϑ−1 + ξ(δ) τ. + ξ(δ) = τ g(z, u τ ) − f (z, u τ ) + γ u ϑ−1 τ. + ξ(δ) = λg(z, u τ ) − f (z, u τ ) + (τ − λ)g(z, u τ ) + γ u ϑ−1 τ + ξ(δ) λg(z, u τ ) − f (z, u τ ) − (λ − τ )σ0 (u τ ) + γ u ϑ−1 τ. (3.27). with ξ(δ) → 0 as δ 0 (see hypothesis Hg (iv) andrecall that τ < λ). Since u τ ∈ int C+ , the function z −→ σ0 u τ (z) is upper semicontinuous on (see hypothesis Hg (iv)). So, we can find z 0 ∈ , such that σ0 u τ (z 0 ) = max σ0 u τ (z) > 0.. (3.28). z∈. We use (3.28) in (3.27). Since ξ(δ) 0 and δ 0 and λ > τ , we infer that − p u τ + βu τp−1 + γ u ϑ−1 τ. λg(z, u τ ) − f (z, u τ ) + γ u ϑ−1 τ. λg(z, u 0λ ) − f (z, u 0λ ) + γ (u 0λ )ϑ−1. = − p u 0λ + β(u 0λ ) p−1 + γ (u 0λ )ϑ−1 for almost all z ∈ . for δ > 0 small (see H0 and recall that u τ u 0λ ). Acting on this inequality with (u τ − u 0λ )+ ∈ W 1, p () and using the nonlinear Green’s identity (see e.g., Gasiński and Papageorgiou [9]) as above, we obtain u τ = u τ + δ u 0λ ∀δ > 0 small, so u 0λ − u τ ∈ int C+ . This proves Claim 1. Let [u τ ) =. 123. . u ∈ W 1, p () : u τ (z) u(z) for almost all z ∈ ..

(38) Positive Solutions for the Neumann p-Laplacian.... From (3.26), we see that. 1725. λ |[u τ ) = ϕλ |[u τ ) +. ψ c,. (3.29). u 0λ is a local C 1 ()-minimizer W 1, p ()-minimizer of ϕ(λ).. for some. c ∈ R. Then Claim 1 and (3.29) imply that of ϕλ . From Theorem 2.2, it follows that u 0λ is a local By virtue of hypotheses Hg (iii) and H f (iii), for a given ε > 0 we can find δ = δ(ε) > 0, such that ε p ε ζ and F(z, ζ ) − ζ p for almost all z ∈ , all ζ ∈ (0, δ]. p p (3.30) So, if u ∈ C 1 () with uC 1 () δ, then G(z, ζ ) . 1 1 p ∇u p + β|u| p dz − λ G(z, u) dz + F(z, u) dz p p ξ0 λ+1 εu + p u p − p p ξ0 − (λ + 1)ε u p p. ϕλ (u) =. ξ0 , we infer that (see Lemma 2.5) and (3.30). Choosing ε ∈ 0, λ+1 ϕλ (u) 0 = ϕλ (0) ∀u ∈ C 1 (), uC 1 () δ, so u = 0 is a local C 1 ()-minimizer of ϕλ and thus 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 (u 0λ ) (the analysis is similar if the opposite inequality is true). Moreover, we may assume that both local minimizers u = 0 and u = u 0λ 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, u 0λ small, such that . ϕλ (0) = 0 ϕλ (u 0λ ) < inf ϕλ (u) : u − u 0λ = = 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. u λ ∈ W 1, p (), such that. 123.

(39) 1726. L. Gasiński, N. S. Papageorgiou. and. η ϕλ (. uλ). (3.32). ϕλ (. u λ ) = 0.. (3.33). / {0, u 0λ }. From (3.33), we have From (3.31) and (3.32), it follows that. uλ ∈ p−1. A(. u λ ) + β. uλ. = λN g (. u λ ) − N f (. u λ ),. so. u λ ∈ int C+ is a solution of (P)λ . uλ ∈ So, we conclude that (P)λ (λ > λ∗ ) has at least two positive solutions u 0λ ,. int C+ . Next we examine what happens in the critical case λ = λ∗ . 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 u n = u λn ∈ int C+ be positive solutions for problem (P)λ for n 1 (see Proposition 3.4). We have p−1. A(u n ) + βu n. = λn N g (u n ) − N f (u n ) ∀n 1.. (3.34). By virtue of hypothesis Hg (ii) and since ϑ > q, we have lim. ζ →+∞. g(z, ζ ) = 0 uniformly for almost all z ∈ . ζ ϑ−1. This fact combined with hypothesis Hg (i), implies that for a given ε > 0, we can find c6 = c6 (ε) > 0, such that ε + ϑ (ζ ) + c6 for almost all z ∈ , all ζ ∈ R. ϑ. g(z, ζ )ζ . (3.35). In a similar fashion, using hypotheses H f (i) and (ii), we see that we can find η > 0 and c7 > 0, such that η + ϑ (ζ ) − c7 for almost all z ∈ , all ζ ∈ R. ϑ. f (z, ζ )ζ . (3.36). On (3.34) we act with u n ∈ W 1, p () and obtain p ∇u n p. +. . p βu n. dz = λn . . g(z, u n )u n dz −. . f (z, u n )u n dz. λn ε − η u n ϑϑ + c8 ∀n 1, ϑ. for some c8 > 0 (see (3.35) and (3.36)).. 123. . . (3.37).

(40) Positive Solutions for the Neumann p-Laplacian.... 1727. We choose ε ∈ 0, λη1 (recall that λn λ1 for all n 1). Then from (3.37) and Lemma 2.5, it follows that ξ0 u n p c8 ∀n 1 and so the sequence {u n }n 1 ⊆ W 1, p () is bounded. So, passing to a subsequence if necessary, we may assume that u n −→ u ∗ weakly in W 1, p (),. (3.38). u n −→ u ∗ in L (),. (3.39). θ. with θ < p ∗ . On (3.34) we act with u n − u ∗ , pass to the limit as n → +∞ and use (3.38). We obtain lim. . n→+∞. A(u n ), u n − u ∗ = 0,. so u n −→ u ∗ in W 1, p (). (3.40). (see Proposition 2.4). So, if in (3.34) we pass to the limit as n → +∞ and use (3.40), we obtain p−1. A(u ∗ ) + βu ∗. = λ∗ N g (u ∗ ) − N f (u ∗ ),. so u ∗ ∈ C+ and it solves problem (P)λ∗ . It remains to show that u ∗ = 0. Arguing by contradiction, suppose that u ∗ = 0. From (3.34), we have ⎧ ⎨ − p u n (z) + β(z)u n (z) p−1 = λg z, u n (z) − f z, u n (z) in , ∂u ⎩ n = 0 on ∂. ∂n. (3.41). From (3.41) and Theorem 2 of Lieberman [24], we know that we can find α ∈ (0, 1) and M > 0, such that u n ∈ C 1,α () and u n C 1,α () M ∀n 1. From the compactness of the embedding C 1,α () ⊆ C 1 (), we have u n −→ u ∗ in C 1 (). Let yn =. un ∀n 1. u n . 123.

(41) 1728. L. Gasiński, N. S. Papageorgiou. Then yn 0, yn = 1 ∀n 1. So, passing to a subsequence if necessary, we may assume that yn −→ y∗ weakly in W 1, p (),. (3.42). yn −→ y∗ in L ().. (3.43). ϑ. From (3.34), we have p−1. A(yn ) + βyn. = λn. N f (u n ) N g (u n ) − ∀n 1. p−1 u n u n p−1. (3.44). From hypotheses Hg (i), (iii) and H f (i), (iii), it follows that the sequences. N g (u n ) u n p−1. . n 1. ,. N f (u n ) u n 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 . lim. n→+∞. so. A(yn ), yn − y∗ = 0,. yn −→ y∗ in W 1, p (). (3.45). (see Proposition 2.4) and so y∗ = 1. Note that by virtue of hypotheses Hg (iii) and H f (iii), we have N g (u n ) N f (u n ) p−1 −→ 0 and −→. ζ y∗ weakly in L p (), p−1 u n u n p−1. (3.46). 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 p−1. A(y∗ ) + βy∗ so p. ∇ y∗ p +. . p. p−1 = −. ζ y∗ ,. . βy∗ dz −. . p. ζ y∗ dz 0,. thus ξ0 y∗ p 0 (see Lemma 2.5) and finally, we have that y∗ = 0, which contradicts to (3.45).. 123.

(42) Positive Solutions for the Neumann p-Laplacian.... 1729. This proves that u ∗ = 0. Hence u ∗ ∈ int C+ is a solution of problem (P)λ∗ . Therefore λ∗ ∈ Y. We show that for every λ λ∗ , problem (P)λ has an extremal (smallest) positive solution. Proposition 3.7 If hypotheses H f , Hg , and H0 hold and λ λ∗ , then problem (P)λ has a smallest positive solution u ∗λ ∈ int C+ . 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 {u n }n 1 ⊆ C, such that inf u n = 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 {u n }n 1 is decreasing. We have p−1. A(u n ) + βu n. = λN g (u n ) − N f (u n ) ∀n 1,. (3.47). so p ∇u n p. +. . p βu n. dz =. . λg(z, u n ) − f (z, u n ) u n dz M1 ∀n 1,. for some M1 > 0 (see hypotheses Hg (i), H f (i) and recall that u n u 1 for all n 1). So, ξ0 u n p M1 ∀n 1 (see Lemma 2.5) and thus the sequence {u n }n 1 ⊆ W 1, p () is bounded. So, passing to a subsequence if necessary, we may assume that u n −→ u ∗ weakly in W 1, p (), u n −→ u ∗ in L θ (),. (3.48) (3.49). with θ < p ∗ . On (3.47) we act with u n − u ∗ , pass to the limit as n → +∞ and use (3.48). Then lim. n→+∞. . A(u n ), u n − u ∗ = 0,. so u n −→ u ∗ in W 1, p () (see Proposition 2.4).. 123.

(43) 1730. L. Gasiński, N. S. Papageorgiou. 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 ∗λ u for all u ∈ S(λ), i.e., u ∗λ ∈ int C+ is the smallest positive solution of problem (P)λ . 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+ . Acknowledgements The authors wish to thank the two referees for their corrections and helpful remarks. The research was supported by the Marie Curie International Research Staff Exchange Scheme Fellowship within the 7th European Community Framework Programme under Grant Agreement No. 295118, the International Project co-financed by the Ministry of Science and Higher Education of Republic of Poland under Grant No. W111/7.PR/2012 and the National Science Center of Poland under Maestro Advanced Project No. DEC-2012/06/A/ST1/00262. Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.. References 1. Afrouzi, G.A., Brown, K.J.: On a diffusive logistic equation. J. Math. Anal. Appl. 225, 326–339 (1998) 2. Aizicovici, S., Papageorgiou, N.S., Staicu, V.: Degree theory for operators of monotone type and nonlinear elliptic equations with inequality constraints. Mem. Am. Math. Soc. 196, 915 (2008) 3. Aizicovici, S., Papageorgiou, N.S., Staicu, V.: Existence of multiple solutions with precise sign information for semilinear Neumann problems. Ann. Math. Pure Appl. 188, 679–719 (2009) 4. Aizicovici, S., Papageorgiou, N.S., Staicu, V.: The spectrum and an index formula for the Neumann p-Laplacian and multiple solutions for problems with a crossing nonlinearity. Discrete Contin. Dyn. Syst. 25, 431–456 (2009) 5. Brézis, H., Nirenberg, L.: H 1 versus C 1 local minimizers. C. R. Acad. Sci. Paris Sér. I Math. 317, 465–472 (1993) 6. Dunford, N., Schwartz, J.T.: Linear Operators. I. General Theory, Volume 7 of Pure and Applied Mathematics. Wiley, New York (1958) 7. Filippakis, M., O’Regan, D., Papageorgiou, N.S.: Positive solutions and biffurcation phenomena for nonlinear elliptic equations of logistic type: the superdiffusive case. Commun. Pure Appl. Anal. 9, 1507–1527 (2010) 8. García Azorero, J., Manfredi, J., Peral Alonso, I.: Sobolev versus Hölder local minimizers and global multiplicity for some quasilinear elliptic equations. Commun. Contemp. Math. 2, 385–404 (2000). 123.

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