A bifurcation result for a Keller-Segel-type problem

We consider a parametric elliptic problem governed by the spectral Neumann fractional Laplacian on a bounded domain of RN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^N$$\end{document}, N≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N\ge 2$$\end{document}, with a general nonlinearity. This problem is related to the existence of steady states for Keller-Segel systems in which the diffusion of the chemical is nonlocal. By variational arguments we prove the existence of a weak solution as a local minimum of the corresponding energy functional and we derive some qualitative properties of this solution. Finally, we prove a regularity result for weak solutions of the problem under consideration, which is of independent interest.


Keller-Segel systems
In 1970 Keller and Segel proposed the following system as a model of chemotactic aggregation, through chemicals whose concentration is v, of a population u of slime mold amoebae (see [6]).This biological population, together with others (e.g.leukocytes, bacteria, gametes), has the ability to sense spatial differences of chemicals present in its environment, moves towards places of higher concentration, and eventually form aggregates.This oriented movement is known as chemotaxis (cf.also [8,21] for a more detailed description).
In (1), Ω ⊂ R N is a suitable bounded region which does not allow any flux across the boundary, u(x, t) represents the population of amoebae at place x and time t, v(x, t) the concentration of the chemical, D 1 , D 2 , χ are positive constants, φ : (0, +∞) → R is a sensitivity function which is assumed to be smooth and with φ > 0, k : (0, +∞) × (0, +∞) → R models the reactions leading to production and degradation of v and satisfies k u ≥ 0, k v ≤ 0 and u 0 , v 0 ≥ 0 are nonnegative functions defined on Ω which are positive on a set of positive measure.Different choices of the functions φ and k lead to different interpretations of the model.For instance, φ(v) = v means that cells measure directly the gradient of v, φ(v) = log v represents a logarithmic sensitivity, φ(v) = v 2 /(1 + v 2 ) a cooperative binding (see for instance [21] and references therein).
If one considers the stationary version of Keller-Segel system, as done for instance in the noteworthy [8,21], it can be shown that the original system reduces to a single semilinear PDE coupled with Neumann boundary conditions.In [21] the author studied the bifurcation and the stability of solutions for a problem with general nonlinearity, and examined the bifurcation behavior for various choices of the function φ.In [8] it was treated the case φ(v) = log v, k(u, v) = −av + bu, a, b positive constants, and it was showed that the system has positive non-constant steady states if χ > D 1 for N = 1, 2, χ/D 1 ∈ (1, (N + 2)/(N − 2)) if N ≥ 3, and ε := D 2 /a is sufficiently small.The number ε, in particular, is strategical in this setting and plays the role of a perturbation parameter.Still in [8] the authors proved non-existence results for ε large, and L ∞ -boundedness and asymptotic behavior of the positive solutions as ε → 0 + .Recently, in [22] the original model (1) has been modified into the following localnonlocal system, to accomodate a nonlocal diffusion of the chemoattractant v, where the operator (−Δ) 1/2 is the spectral fractional Laplacian with Neumann boundary conditions (see Subsection 2.1 for a precise definition).In particular, as it happens in [8], in [22] system (2) has been studied in the case in Ω. ( As anticipated before, looking for steady state solutions to (3) (as well as solutions to (1)), i.e. solutions to amounts to solving a single scalar equation.Indeed, observe first of all that from the first equation in ( 3), one has

123
thanks to the boundary conditions on u and v in (3).Hence, we get and therefore solutions to (4) solve the system with ū > 0 in given in (5).En passant, we observe that the pair is always a solution to (6), so it makes sense to look for non-constant solutions to (6).
then u = λv χ/D 1 for some constant λ > 0. As a consequence, ( 6) is equivalent to the following scalar problem with Neumann boundary conditions where In recent years elliptic fractional problems involving suitable Neumann boundary conditions have been studied also in connection with interesting probabilistic motivations, see [15,19,23].Furthermore, interesting a priori estimates as well as results of existence and multiplicity of solutions for nonlocal Neumann-type problems have been established in [16][17][18].

Topic and main results of the paper
Motivated by the wide interest in the current literature for elliptic fractional equations and also by their relevance in practical applications (as shown in Subsection 1.1), in this paper we are interested in the following nonlocal fractional problem where Ω is a smooth bounded domain of R N , N ≥ 2, λ is a positive parameter, ν is the outer unit normal to ∂Ω.On the weight function α : Ω → R we assume that there exists a set D ⊆ Ω, with |D| > 0, such that essinf while on the nonlinear term f : R → R we make the following requirements: there exist κ > 0 and q ∈ (1, 2 * − 1) such that lim sup where is the fractional critical Sobolev exponent (see (38) and (39)) and As a prototype for f we can take the function with m ∈ (0, 1).Notice that under our set of assumptions, both α and f can be sign-changing.
The leading operator (−Δ) being {λ k } k∈N 0 and {ϕ k } k∈N 0 , respectively, the eigenvalues and the eigenfunctions of the Laplace operator −Δ on Ω with homogeneous Neumann boundary conditions (see Section 2 for more details).
Computations similar to those of Subsection 1.1 show that problem (8), in the presence of a constant weight α, is related to a Keller-Segel-type system (2) in which, in particular, φ(v) = log g(v), g : [0, +∞) → R is positive together with g , and The non-constancy of α (see (10)) is a result of our demonstration strategy and is needed to exclude constant solutions (cf.Remark 1).We emphasize that, contrary to [22], we consider here a parameterized nonlinear term with a general subcritical growth and obtain existence results -of bifurcation type -by means of completely different techniques, albeit variational.
Our main result about the existence of solutions for problem (8) can be stated as follows: Theorem 1 Let Ω be a smooth bounded domain of R N , N ≥ 2. Assume that α : Ω → R satisfies (9), (10) and (11) and f : R → R is such that (12), ( 13) and (14) hold.
The proof of Theorem 1 relies on different tools.First of all, the original problem (8) is rephrased in terms of a local one in the cylinder C := Ω × (0, +∞) ⊂ R N +1 via the notion of the 1/2-Neumann harmonic extension.The existence part is deduced by means of a constrained local minimum result for parameterized C 1 functionals established in [20].A key point, in this regard, is the choice of the parameter inside a small interval involving a suitable real function ϕ (see Theorem 3).The non-triviality of the solution found is guaranteed by assumptions (9), (10), (11), (14) and the construction of an ad-hoc sequence of test functions which prevents zero from being a local minimizer of the energy functional associated with the problem.
Along the present paper we discuss also the regularity of the solutions of problem (8).In this setting our main result, which is of independent interest, is the next one: Theorem 2 Let Ω be a smooth bounded domain of R N , N ≥ 2. Assume that α : Ω → R satisfies (9) and f : R → R is such that (12) and (13) hold.Let u ∈ H 1/2 (Ω) be a weak solution of problem (8).
For the proof of Theorem 2 we need to consider various cases, depending on the range of q and N , and treat each of them in a different way.The regularity result stems from some estimates for the solutions to Neumann fractional Laplacian equations obtained in [22] and from the classical Moser iteration scheme.
The paper is organized as follows.In Section 2, after recalling some preliminary facts about the spectrum of the Laplace operator with homogeneous Neumann boundary conditions, we define the spectral Neumann fractional Laplacian and state the abstract variational tool Theorem 3. In Section 3 we give the variational setting useful for the study of problem (8).Finally, Section 4 is devoted to the proof of Theorem 1 and some additional comments, while in Section 5 we discuss and prove the regularity result stated in Theorem 2.

Preliminaries
Throughout the paper we denote by L 2 (Ω) the usual Lebesgue space endowed with the scalar product and by H 1 (Ω) the usual Sobolev space endowed with the scalar product and the norm It is known that the eigenvalue problem for −Δ with homogeneous Neumann boundary conditions, i.e. the problem has a sequence of non-negative eigenvalues {λ k } k∈N 0 such that λ k → +∞ as k → +∞, and a corresponding sequence of eigenfunctions Also, the sequence {ϕ k } k∈N 0 is an orthonormal basis of L 2 (Ω) and an orthogonal basis of H 1 (Ω).Furthermore, since any ϕ k is a solution to (17) with λ = λ k , it is easily seen that for any ϕ ∈ H 1 (Ω) and k ∈ N 0 .
The space H 1 (Ω) can be characterized in terms of the orthogonal basis {ϕ k } k∈N 0 as follows For u ∈ H 1 (Ω) the Neumann Laplacian −Δu can be defined as the element in the dual space H −1 (Ω) which acts as follows that is, in terms of the orthogonal basis {ϕ k } k∈N 0 , for every u, v ∈ H 1 (Ω), where •, • denotes the duality pairing between H 1 (Ω) and its dual H −1 (Ω).Notice that −Δ has a non-trivial kernel in H 1 (Ω) represented by the set of constant functions.

The Neumann spectral fractional Laplacian
Following the definition of the operator −Δ with Neumann boundary conditions, we are now in a position to define the spectral fractional operator First of all, we denote by H 1/2 (Ω) the usual fractional Sobolev space Analogously to H 1 (Ω), the space H 1/2 (Ω) can be characterized in terms of the orthogonal basis {ϕ k } k∈N 0 as follows and it turns out to be a Hilbert space under the scalar product which induces the norm in the wake of (21), for u ∈ H 1/2 (Ω) we define the spectral Neumann fractional Laplacian (−Δ) 1/2 u to be the element of H −1/2 (Ω) which acts as for all v ∈ H 1/2 (Ω), or, analogously to (20), by , where (−Δ) 1/4 is defined as in ( 23) by replacing 1/2 with 1/4 and •, • denotes the duality pairing between H 1/2 (Ω) and its dual H −1/2 (Ω).

An abstract critical point theorem
As anticipated in the Introduction, the existence of a critical point of the energy functional associated with (8) or, more precisely, to the extended problem of (8) (see Section 3) stems from the following abstract result established in [20, Theorem 2.1]: Theorem 3 Let X be a reflexive real Banach space, let Φ, Ψ : X → R be two Gâteaux differentiable functionals, with Φ strongly continuous, sequentially weakly lower semicontinuous and coercive in X and Ψ sequentially weakly upper semicontinuous in X .Let J λ := Φ − λΨ , λ ∈ R, and for any r > inf X Φ let Then, for any r > inf X Φ and any λ ∈ (0, 1/ϕ(r )), where the second endpoint of the interval is meant to be +∞ whenever ϕ(r ) = 0, the restriction of J λ to Φ −1 ((−∞, r )) has a global minimum, which is a critical point (i.e., a local minimum) of J λ in X .
Following the seminal work [20], an impressive number of publications have appeared in literature, dedicated to the study of suitable extensions of this variational principle as well as of its consequences; see, for instance, the books [7,10,12] and the references therein.Recent applications of Theorem 3, namely of [20, Theorem 2.1], can be found in [2,4,9,11,13,14].

Variational formulation of the problem
We start off by discussing an extension problem related to the spectral Neumann fractional Laplace operator (−Δ) 1/2 u.Later we will consider the variational structure of this extended problem.This will allow us to give the definition of weak solution to the original problem (8).

An extension problem related to (−1) 1/2
First of all, we recall that in the seminal paper [3], Caffarelli and Silvestre showed that any fractional Laplace operator of order s ∈ (0, 1) acting on a function u defined on the whole R N can be interpreted as the normal derivative of the s-harmonic extension of u to the upper halfspace.
Adopting this viewpoint, as shown in [22], let us consider the problem where The following result (see [22,Theorem 2.1] with ε = 1) gives the existence and uniqueness of the weak solution for problem (25), provided that u has zero mean in Ω, and allows to define (−Δ) 1/2 via such a solution, in the spirit of [3].
is the unique weak solution of problem (25) such that Ω v(x, y)dx = 0 for all y ≥ 0.
In addition, v is the unique minimizer of the functional The function v in (26), solution of problem (25), is called the 1/2-Neumann harmonic extension of u ∈ H 1/2 (Ω) and will be denoted by E(u).
In order to have a similar definition also for functions u ∈ H 1/2 (Ω) such that u Ω = 0, where is the mean value of u in Ω, define Since, as recalled, λ 0 = 0 and ϕ 0 = 1 √ |Ω| , we get that since v does not belong to L 2 (C) (being C unbounded) even if ∇v does.So we consider instead problem (25) with initial datum ũ = u − u Ω and apply Lemma 1 to obtain which is the unique solution of problem (25) with u = ũ, that is ṽ = E( ũ). 123 Now, let us define the 1/2-Neumann harmonic extension of u as By the definitions of ũ, E( ũ) and (31), we have that where, in the third equality, we have used (18) and, in the last equality, the explicit values of λ 0 and ϕ 0 .
Since the fractional Neumann Laplacian has a non-trivial kernel made of constant functions and by ( 27), we deduce that Since, in general, the 1/2-Neumann harmonic extension E(u) defined in (32) does not belong to H 1 (C), we need to consider a new functional space as the energy space.To this end, let us define H 1 (C) to be the completion of H 1 (C) under the norm which is induced by the scalar product Notice that constant functions belong to H 1 (C), so that H 1 (C) ⊂ H 1 (C).As a consequence, the 1/2-Neumann harmonic extension E(u) defined in (32) is in H 1 (C) (but, in general, not in H 1 (C)).
Overall we have that, for a function u ∈ H 1/2 (Ω) the 1/2-Neumann harmonic extension E(u) ∈ H 1 (C) is defined as where v is given in (26) and u Ω in (28).Finally, we recall the following important trace result (see [22,Lemma 2.4] with ε = 1), which states that we can define a trace operator on the whole H 1 (C).
Lemma 2 For all v ∈ H 1 (C) one has Moreover, there exists a unique bounded linear operator T : As a consequence of Lemma 2, we can endow the space H 1 (C) with the norm given by Moreover, taking also account of the following embeddings of H 1/2 (Ω) into the classical Lebesgue spaces (see, for instance, [1,5]) where the fractional critical Sobolev exponent 2 * is given in (15), we deduce that the embedding In the light of (36) and (40), the positive constant is finite for any p ∈ [1, 2 * ).

Weak solutions and energy functional associated with the problem
Thanks to the machinery introduced in Subsection 3.1, now we are able to give the variational formulation of the nonlocal fractional problem (8).
First of all, we define a weak solution u ∈ H 1/2 (Ω) to ( 8) as the trace T v over Ω of a weak solution v ∈ H 1 (C) to the extended problem From now on, therefore, looking for weak solutions u of ( 8) means studying weak solutions v of problem (42).To this purpose we will look for critical points of the energy functional I λ : H 1 (C) → R given by Observe that I λ is well defined and C 1 in H 1 (C), thanks to ( 9), ( 12), ( 13) and (40).

Proof of the existence result consequences
This section is devoted to the proof of Theorem 1, whose ingredients are variational methods, and to some comments on the results obtained in this paper.

Proof of Theorem 1
First of all, set and for all v ∈ H 1 (C), and write I λ as we address separately each point to be proved.

1.
Existence.Having in mind to appeal to Theorem 3, let us show that Ψ is continuous in the weak topology of H 1 (C).To this purpose, take a sequence v j v in H 1 (C) as j → +∞.Since T is a bounded linear operator, then T v j T v as j → +∞ and by the compactness of the embedding T (H almost everywhere in Ω as j → +∞, and there exists η p ∈ L p (Ω) such that for almost every x ∈ Ω and for all j ∈ N, with p ∈ [1, 2 * ).

Some final remarks
We conclude Section 4 by providing some comments on the results illustrated before.

Remark 1
As shown in the proof of Theorem 1, assumption ( 14) comes into play in proving that local minimizers of I λ are not zero.The existence of non-zero solutions can be also obtained, for instance, by dropping (14) and by assuming, besides (13), that f ∈ C(R) and f (0) = 0.
Remark 2 By (52), we get the following estimate from below for the value of the constant Λ in Theorem 1 when q ∈ [2, 2 * ) (we have already shown that where C 2 is the constant defined in (41) with p = 2 and for any r > 0.

Remark 4
Our approach allows us to deduce a monotonicity property for the energies of the extended problem useful to distinguish solutions v λ to (42) corresponding to different λ's.First of all, notice that, by (62), the map Let us show that it is also strictly decreasing in (0, Λ).To this end, for any v ∈ H 1 (C) let us write Let v λ i be the global minimum of I λ i restricted to Φ −1 (−∞, rλ ) , i = 1, 2. By (67) and (69), we have and, moreover, since the infimum is taken on the same set Φ −1 (−∞, rλ ), by (69) we deduce that Therefore, by ( 69) and (70), we obtain i.e., λ → I λ (v λ ) is strictly decreasing in (0, λ).The arbitrariness of λ < Λ shows that λ I λ (v λ ) is strictly decreasing in (0, Λ) and therefore solutions to (42) as v λ 1 and v λ 2 corresponding, respectively, to distinct λ 1 and λ 2 ∈ (0, Λ) need necessarily to be distinct.

Proof of the regularity result
In this section we discuss and prove the regularity of weak solutions of problem (8).
Our proof relies on some estimates for the solutions to Neumann fractional Laplacian equations obtained in [22], which actually provide regularity results on solutions to a Neuman fractional problem which are expressed under regularity assumptions of the nonlinear term, and on the Moser iteration scheme.