Anisotropic Robin problems with logistic reaction

We consider Robin problems driven by the anisotropic p-Laplace operator and with a logistic reaction. Our analysis covers superdiffusive, subdiffusive and equidiffusive equations. We examine all three cases, and we prove multiplicity properties of positive solutions (superdiffusive case) and uniqueness (subdiffusive and equidiffusive cases). The equidiffusive equation is studied only in the context of isotropic operators. We explain why the more general case cannot be treated.

A feature of the present paper is that in this problem, the exponent of the differential operator is variable, namely p : Ω → R is log-Hölder continuous and 1 < min Ω p. We point out that this regularity assumption is necessary for related Sobolev embeddings (see Diening et al. [4,Section 8.3]); otherwise, p(·) can be assumed only continuous. We denote by Δ p(z) the anisotropic p-Laplacian differential operator defined by Δ p(z) u = div (|Du| p(z)−2 Du) for all u ∈ W 1,p(z) (Ω).
This operator is more difficult to deal with since, in contrast to the isotropic (constant exponent) case, it is not homogeneous. In the reaction (right-hand side of problem (P λ )), there is a parametric term x → λx q(z)−1 , x 0 and a perturbation −f (z, x), with f (·, ·) being a Carathéodory function (that is, for all x ∈ R, z → f (z, x) is measurable and for a.a. z ∈ Ω, x → f (z, x) is continuous). We assume that for a.a. z ∈ Ω, f (z, ·) is (p + − 1)-superlinear as x → +∞, with p + = max Ω p. So, the right-hand side of problem (P λ ) is a generalized logistic reaction. If f (z, x) = x r(z)−1 with r ∈ C(Ω) and p + < r − = min Ω r, then we have a usual logistic reaction with variable exponents.
We mention that in the boundary condition, ∂u ∂n p(z) denotes the variable exponent conormal derivative of u. This directional derivative is interpreted using the nonlinear Green's identity and if u ∈ C 1 (Ω), with n(·) being the outward unit normal on ∂Ω.
Depending on the relation between the exponents q(·) and p(·), we have three types of logistic equations.
(a) If p + < q − , then the equation is "superdiffusive." (b) If q + < p − , then the equation is "subdiffusive." (c) If p(z) = q(z) for all z ∈ Ω, then the equation is "equidiffusive." In this paper, we study cases (a) and (b). Case (c) is difficult to deal with in the context of anisotropic equations, because we do not have a satisfactory spectral analysis of the relevant differential operator. The analysis developed in this paper reveals that cases (a) and (b) are different. More precisely, we show that for the superdiffusive equation, we have multiple positive solutions and, in fact, we prove a bifurcationtype result describing the changes in the set of positive solutions as the parameter λ > 0 moves. In contrast, for the subdiffusive equation, we have uniqueness of the positive solution. The equidiffusive equation is treated only for isotropic problems.
The mathematical analysis of nonlinear problems with variable exponent started after the seminal contributions of Zhikov [37,38], in relationship with phenomena arising in nonlinear elasticity. In fact, Zhikov intended to provide models for strongly anisotropic materials in the context of homogenization. The analysis developed by Zhikov revealed to be important also in the study of duality theory and in the context of the Lavrentiev phenomenon. In particular, Zhikov considered the following model functionals in relationship with the Lavrentiev phenomenon: The functional M is well known, and there is a loss of ellipticity on the set {z ∈ Ω; c(z) = 0}. This functional has been studied in the context of degenerate equations involving Muckenhoupt weights. The functional V has also been the object of intensive interest nowadays, and a huge literature was developed on it. The energy functional defined by V was used to build models for strongly anisotropic materials. More precisely, in a material made of different components, the exponent p(z) dictates the geometry of a composite that changes its hardening exponent according to the point.

Mathematical background
The analysis of problem (P λ ), uses Lebesgue and Sobolev spaces with variable exponents. A comprehensive presentation of these spaces can be found in the books of Diening et al. [6] and Rȃdulescu and Repovš [27].
Let M (Ω) be the space of all Lebesgue measurable functions u : Ω → R. As always we identify two such functions which differ only on a Lebesgue-null subset of Ω. Also, let E 1 = r ∈ C(Ω) : 1 < r − . In what follows for any r ∈ C(Ω), r − = min Ω r, r + = max Ω r. Given r ∈ E 1 , the variable exponent Lebesgue space L r(z) (Ω) is defined by This space is equipped with the so-called Luxemburg norm defined by In the sequel, for simplicity we write Du r(z) = |Du| r(z) . Then, L r(z) (Ω) is a Banach space, which is separable, reflexive (in fact, uniformly convex). Let r ∈ E 1 , be defined by r (z) = r(z) r(z)−1 (that is, 1 r(z) + 1 r (z) = 1 for all z ∈ Ω.) Then, we have L r(z) (Ω) * = L r (z) (Ω) and we have the following version of Hölder's inequality for all u ∈ L r(z) (Ω), h ∈ L r (z) (Ω). If r 1 , r 2 ∈ E 1 and r 1 r 2 , then L r2(z) (Ω) → L r1(z) (Ω) continuously. Using the variable exponent Lebesgue spaces, we can define the corresponding variable exponent Sobolev spaces. So, if r ∈ E 1 , then the variable exponent Sobolev space W 1,r(z) (Ω) is defined by with Du being the gradient of u(·) in the weak sense. The space W 1,r(z) (Ω) is equipped with the following norm In the sequel, for simplicity we write Du r(z) = |Du| r(z) . The space W 1,r(z) (Ω) is a separable, reflexive (in fact, uniformly convex) Banach space. Given r ∈ E 1 , we introduce the following critical exponents: Also, let σ(·) denote the (N − 1)-dimensional Hausdorff (surface) measure on Ω. If r ∈ C 0,1 (Ω) ∩ E 1 and q ∈ C(Ω) with 1 q − , then Similarly, if r ∈ C 0,1 (Ω) ∩ E 1 and q ∈ C(∂Ω) with 1 min Ω q, then using the anisotropic trace theory (see [6, Section 12.1]), we have The following modular function is very useful in the study of the variable exponent spaces Also, for every u ∈ W 1,r(z) (Ω) we write ρ r (Du) = ρ r (|Du|). This modular function is closely related to the Luxemburg norm.

Proposition 2. The operator
maps bounded sets to bounded sets), continuous, monotone (hence maximal monotone, too) and of type (S) + , that is, it has the following property: We will also use the space C 1 (Ω). This is an ordered Banach space with positive (order) cone C + = u ∈ C 1 (Ω) : u(z) 0 for all z ∈ Ω . This cone has a nonempty interior given by We will also use the following open cone in C 1 (Ω): Also, if u, v ∈ int C + , we set From Allegretto and Huang [1], we know that Suppose that X is a Banach space and ϕ ∈ C 1 (X). We set We say that ϕ(·) satisfies the "C-condition", if it has the following property: admits a strongly convergent subsequence.
We have Finally, for notational simplicity, throughout the work, by · we denote the norm of the anisotropic Sobolev space W 1,p(z) (Ω). Recall that Proof. Recall that we have assumed that ξ 0, β 0 and ξ ≡ 0 or β ≡ 0. We first suppose that β ≡ 0. We define and then, we introduce Evidently, | · | is a norm on W 1,p(z) (Ω). We will show that | · | and · are equivalent norms on W 1,p(z) (Ω). Since Next, we show that we can find c 3 > 0 such that Arguing by contradiction, suppose that (3) is not true. We can find We can always assume that Then, from (4), we have From (5) and (6), it follows that {u n } n∈N ⊆ W 1,p(z) (Ω) is bounded. So, by passing to a suitable subsequence if necessary, we may assume that From (6) and (7), it follows that u = 0. Hence, we have which contradicts (5). Therefore, (3) is true and so From (2) and (8), we infer that · and | · | are equivalent norms on W 1,p(z) (Ω).
Now let ξ ≡ 0 and define We set This is also a norm of W 1,p(z) (Ω) and, as above, we show that · and | · | * are equivalent norms on W 1,p(z) (Ω).
Finally, from (9) and (10), we see that we can findĉ,ĉ 0 > 0 such that This proof is now complete.

Superdiffusive equation
In this section, we examine superdiffusive anisotropic logistic equations. As we already mentioned in Introduction, in this case we have multiplicity of positive solutions. The hypotheses on the data of problem (P λ ) are the following.
The hypotheses on the perturbation f (z, x) are the following.

Remark 2.
From hypotheses H a 1 (iii), it is clear that f (z, x) 0 for a.a. z ∈ Ω, all x 0. Also, since we look for positive solutions and all the above hypotheses concern the positive semiaxis R + = [0, +∞), we may assume without any loss of generality that f (z, x) = 0 for a.a. z ∈ Ω, all x 0. If r ∈ C(Ω) with q(z) < r(z) < p * (z) for all z ∈ Ω and f (z, x) = (x + ) r(z)−1 for all z ∈ Ω, all x ∈ R, then hypotheses H a 1 are satisfied. This choice of f (z, x) corresponds to the classical superdiffusive reaction.
Hypotheses H a 0 (i), (iii) imply that we can find c 8 By (13), it follows that in Ω for some c 9 = c 9 ( u λ ) > 0.
Then, from the anisotropic maximum principle of Zhang [36, Theorem 1.2], we have u λ ∈ int C + . We have proved that for λ > 0 big enough we have λ ∈ L, hence L = ∅. Moreover, the arguments in the last part of the proof show that S λ ⊆ int C + .
This proof is now complete.
Suppose thatû = 0. Then, we may assume that u n 1 and u n q(z) 1 for all n ∈ N. We have a contradiction since p + < q − . So,û = 0 and taking the limit as n → ∞ in (15), we have a contradiction. Therefore, λ * > 0. Next, we show that L is connected (an upper half line).
As a by-product of the above proof, we have the following corollary.
We can improve this corollary as follows.
Next, we show that for λ > λ * , we have multiple positive solutions. More precisely, we will show that for λ > λ * problem (P λ ) has at least a pair of positive solutions.
On account of hypothesis H a 1 (iii), we can find δ > 0 such that x q(z)−1 for a.a. z ∈ Ω, all 0 x δ.
We may assume that The analysis is similar if the opposite inequality holds, using (29) instead of (27). It is easy to see that K ϕ λ ⊆ C + . Hence, we may assume that K ϕ λ is finite (otherwise we already have whole sequence of distinct positive solutions in int C + and so we are done). Then, from (27) and Theorem 5.7.6 of Papageorgiou et al. [21, p. 449], we can find ρ ∈ (0, 1) small such that Recall that ϕ λ (·) is coercive (see the proof of Proposition 4). Hence, ϕ λ (·) satisfies the C-condition (see [21, p. 369]). Then, this fact and (30) permit the use of the mountain pass theorem. So, we can find u ∈ W 1,p(z) (Ω) such that (30)), ⇒û ∈ int C + is a second positive solution of problem (P λ ),û = 0. This proof is now complete.
Next, we check the admissibility of the critical parameter λ * .
So, summarizing the situation for the superdiffusive anisotropic logistic equation, we can state the following bifurcation-type result, which describes the changes in the set of positive solutions as the parameter λ > 0 varies. Theorem 11. If hypotheses H a 0 , H a 1 hold, then there exists λ * > 0 such that (a) for every λ > λ * , problem (P λ ) has at least two positive solutions u 0 ,û ∈ int C + , u 0 =û; (b) for λ = λ * , problem (P λ ) has at least one positive solution u * ∈ int C + ; (c) for every λ ∈ (0, λ * ), problem (P λ ) has no positive solution.

Subdiffusive equation
In this section, we examine the subdiffusive equation. As we already mentioned in Introduction, the situation is different from the superdiffusive case and now we have uniqueness of the positive solution.
The next theorem provides a complete picture for the positive solutions of the subdiffusive equation.
Proof. Let ϕ λ : W 1,p(z) (Ω) → R be the energy functional of problem (P λ ) introduced in the proof of Proposition 4. We know that ϕ λ ∈ C 1 (W 1,p(z) (Ω)). Since we deal with the subdiffusive case, we have q + < p − (see hypotheses H b 0 ). This fact in conjunction with hypothesis H b 1 (ii) and Proposition 3, implies that ϕ λ (·) is coercive.
Next, we show the uniqueness of this positive solution. To this end, we introduce the integral functional j : Let dom j = u ∈ L 1 (Ω) : j(u) < +∞ (the effective domain of j(·)). From Theorem 2.2 of Takáč and Giacomoni [30], we know that j(·) is convex.
So, from (39) we conclude that This proof is now complete.

Equidiffusive equation
In the equidiffusive case, we can only deal with the isotropic equation. The reason for this is that in the anisotropic case, there is no satisfactory spectral analysis of the differential operator. More precisely, if we setλ then it can happen thatλ 1 = 0 even if ξ ≡ 0 or β ≡ 0 (see Fan [10]). We are not aware of any reasonable conditions on the exponent p(·) (aside from being constant), which will guarantee thatλ 1 > 0. This prevents us from dealing with the anisotropic equidiffusive equation.