Anisotropic singular Neumann equations with unbalanced growth

We consider a nonlinear parametric Neumann problem driven by the anisotropic $(p,q)$-Laplacian and a reaction which exhibits the combined effects of a singular term and of a parametric superlinear perturbation. We are looking for positive solutions. Using a combination of topological and variational tools together with suitable truncation and comparison techniques, we prove a bifurcation-type result describing the set of positive solutions as the positive parameter $\lambda$ varies. We also show the existence of minimal positive solutions $u_\lambda^*$ and determine the monotonicity and continuity properties of the map $\lambda\mapsto u_\lambda^*$.


Introduction
This paper was motivated by several recent contributions to the qualitative analysis of nonlinear problems with unbalanced growth. We mainly refer to the pioneering contributions of Marcellini [22,23,24] who studied lower semicontinuity and regularity properties of minimizers of certain quasiconvex integrals. Problems of this type arise in nonlinear elasticity and are connected with the deformation of an elastic body, cf. Ball [4,5].
We are concerned with the qualitative analysis of a class of anisotropic singular problems with Neumann boundary condition and driven by a differential operator with unbalanced growth. The features of this paper are the following: (i) the problem studied in the present work is associated to a double phase energy with variable exponents (variational integral with anisotropic unbalanced growth); (ii) the reaction is both singular and anisotropic; (iii) we assume a Neumann boundary condition.
To the best of our knowledge, this is the first paper dealing with the combined effects generated by the above features.

Unbalanced problems and their historical traces
Let Ω be a bounded domain in R N (N ≥ 2) with a smooth boundary. If u : Ω → R N is the displacement and Du is the N × N matrix of the deformation gradient, then the total energy can be represented by an integral of the type where the energy function f = f (z, ξ) : Ω×R N ×N → R is quasiconvex with respect to ξ, see Morrey [26]. One of the simplest examples considered by Ball is given by functions f of the type where det ξ is the determinant of the N × N matrix ξ, and g, h are nonnegative convex functions, which satisfy the growth conditions where c 1 is a positive constant and 1 < p < N . The condition p ≤ N is necessary to study the existence of equilibrium solutions with cavities, that is, minima of the integral (1) that are discontinuous at one point where a cavity forms; in fact, every u with finite energy belongs to the Sobolev space W 1,p (Ω, R N ), and thus it is a continuous function if p > N . In accordance with these problems arising in nonlinear elasticity, Marcellini [22,23] considered continuous functions f = f (x, u) with unbalanced growth that satisfy where c 1 , c 2 are positive constants and 1 ≤ p ≤ q. Regularity and existence of solutions of elliptic equations with p, q-growth conditions were studied in [23].
The study of non-autonomous functionals characterized by the fact that the energy density changes its ellipticity and growth properties according to the point has been continued in a series of remarkable papers by Mingione et al. [6,7,12]. These contributions are in relationship with the work of Zhikov [43,44], which describe the behavior of phenomena arising in nonlinear elasticity. In fact, Zhikov intended to provide models for strongly anisotropic materials in the context of homogenisation. In particular, he considered the following model functional P p,q (u) := Ω (|Du| p + a(z)|Du| q )dz, 0 ≤ a(x) ≤ L, 1 < p < q, where the modulating coefficient a(x) dictates the geometry of the composite made of two differential materials, with hardening exponents p and q, respectively. In the present paper we are concerned with a problem whose energy is of the type defined in (2) but such that the exponents p and q are variable (they depend on the point).
In this problem, we make the following hypotheses for the exponents p(·), q(·), η(·): p, q, η ∈ C 1 (Ω), q − ≤ q + < p − ≤ p + , 0 < η(z) < 1 for all z ∈ Ω, where for every r ∈ C(Ω) we define Also for r ∈ C(Ω) with 1 < r(z) < ∞ for all z ∈ Ω, we denote by ∆ r(z) the r(z)-Laplace differential operator defined by The potential function ξ ∈ L ∞ (Ω) satisfies ξ(z) > 0 for a.a. z ∈ Ω. In the reaction we have two terms. One is the singular term x → x −η(z) with 0 < η(z) < 1 for all z ∈ Ω and the other is a parametric perturbation λf (z, x) with λ > 0 being the parameter. The function f (z, x) is a Carathéodory function, that is, for all x ∈ R the mapping z → f (z, x) is measurable and for a.a. z ∈ Ω the function x → f (z, x) is continuous. We assume that for a.a. z ∈ Ω, the function f (z, ·) exhibits a (p + − 1)-superlinear growth near +∞ with p + = max Ω p, but without satisfying the so-called Ambrosetti-Rabinowitz condition (the AR-condition for short), which is common in the literature when dealing with superlinear problems. Instead, we use a less restrictive condition which incorporates in our framework superlinear nonlinearities with slower growth near +∞. The precise hypotheses on f (z, x) can be found in Section 2 (see hypotheses H 1 ).
We are looking for positive solutions and our aim is to determine how the set of positive solutions changes as the parameter λ > 0 varies. In this direction we prove a bifurcation-type result describing the changes in the set of positive solutions of (P λ ) as the positive parameter λ increases. We also show that if λ > 0 is admissible (that is, problem (P λ ) admits positive solutions), then there is a minimal positive solution u * λ (that is, a smallest solution) and we examine the monotonicity and continuity of the map λ → u * λ . Analogous studies for p-Laplacian equations with constant exponent, were conducted by Giacomoni, Schindler & Takač [18] and Papageorgiou & Winkert [35]. More general equations driven by nonhomogeneous differential operators, were considered recently by Papageorgiou, Rȃdulescu & Repovš [28,29,30,31] [37], and Ragusa & Tachikawa [38]. We should also mention the very recent works of De Filippis & Mingione [13] and Marcellini [25], on the regularity of solutions of double phase problems. This is a very interesting area with several issues remaining open and requiring further investigation. Finally, we mention the work of Bahrouni, Rȃdulescu & Winkert [3] on a class of double phase problems with convection. Singular anisotropic equations driven by the p(z)-Laplacian, were studied by Byun & Ko [9], Zhang & Rȃdulescu [42], and Saudi & Ghanmi [40]. To the best of our knowledge, there are no works on singular anisotropic (p, q)-equations.
Boundary value problems driven by a combination of differential operators (such as (p, q)equations) arise in many mathematical models of physical processes. We mention the historically first such work of Cahn & Hilliard [10], which deals with the process of separation of binary alloys and the more recent works of Benci, D'Avenia, Fortunato & Pisani [8] on quantum physics, of Cherfils & Ilyasov [11] on reaction diffusion systems and of Bahrouni, Rȃdulescu & Repovš [1,2] on transonic flow problems. Boundary value problems involving differential operators with variable exponents, are studied in the book of Rȃdulescu & Repovš [39], while a comprehensive discussion of semilinear singular problems and a rich relevant bibliography can be found in the book of Ghergu & Rȃdulescu [17].

Mathematical background and hypotheses
Although as we already mentioned in the previous section, we require that our exponents p(·), q(·), η(·) are smooth (in order to exploit the existing anisotropic regularity theory), the introduction of the variable exponent spaces does not require such regularity restrictions.
We introduce the following spaces As usual, we identify in M (Ω) two functions which differ only on a set of measure zero. If p ∈ L ∞ 1 (Ω), then we set p − = essinf Ω p and p + = esssup Ω p.
Given p ∈ L ∞ 1 (Ω), the variable exponent Lebesgue space L p(z) (Ω) is defined by We equip this space with the so-called "Luxemburg norm" defined by Variable exponent Lebesgue spaces are similar to the classical Lebesgue spaces. More precisely, they are separable Banach spaces, they are reflexive if and only if 1 < p − ≤ p + < ∞ (in fact, they are uniformly convex). Moreover, simple functions and continuous functions of compact support are dense in L p(z) (Ω).
Suppose that p, q ∈ L ∞ 1 (Ω). Then we have the following property: if and only if q(z) ≤ p(z) for a.a. z ∈ Ω".
Let r ∈ L ∞ 1 (Ω) and consider the Lebesgue space L r(z) (Ω). The modular function for this space is given by This function is basic in the study of L r(z) (Ω) and is closely related to the norm · p(z) introduced above. More specifically, we have the following result.
Proposition 2.2. The map A r(z) : W 1,p(z) (Ω) → W 1,p(z) (Ω) * is bounded (that is, maps bounded sets to bounded sets), continuous, monotone, hence also maximal monotone and of type (S) + , that is, In addition to the variable exponent spaces, we will use the Banach space C 1 (Ω). This is an ordered Banach space with positive cone C + = {u ∈ C 1 (Ω) : u(z) ≥ 0 for all z ∈ Ω}. This cone has nonempty interior given by int C + = {u ∈ C + : u(z) > 0 for all z ∈ Ω}.
We will also use another open cone in C 1 (Ω) given by with n(·) being the outward unit normal on ∂Ω.
Combining the proofs of Proposition 2.5 of [32] and of Proposition 6 in [31] we have the following strong comparison principle.
If X is a Banach space and ϕ ∈ C 1 (X, R), then we denote by K ϕ the critical set of ϕ, that is, the set Also, we say that ϕ ∈ C 1 (X, R) satisfies the "C-condition", if the following property holds: This is a compactness-type condition on the functional ϕ(·). It compensates for the fact that X is not locally compact, being in general infinite dimensional. The C-condition plays a crucial role in the minimax theory of the critical values of the functional ϕ(·). Now we are ready to introduce the hypotheses on the data of (P λ ).
H 1 : f : Ω × R → R is a Carathéodory function (that is, f (z, x) is measurable in z ∈ Ω and continuous in x ∈ R) such that f (z, 0) = 0 for a.a. z ∈ Ω and (iv) for every s > 0, we can find µ s > 0 such that (v) for every ρ > 0, there existsξ ρ > 0 such that for a.a. z ∈ Ω, the function Remarks. Since we are looking for positive solutions and all the above hypotheses concern the positive semiaxis R + = [0, ∞), without any loss of generality, we may assume that On account of hypotheses H 1 (ii), (iii) we see that for a.a. z ∈ Ω, f (z, ·) is (p + − 1)-superlinear. However the superlinearity property of f (z, ·) is not expressed in terms of AR-condition. We recall that in the present anisotropic setting the AR-condition says that there exist ϑ > p + and M > 0 such that In fact this is a unilateral version of the AR-condition due to (3). Integrating (4) and using (5), we obtain the weaker condition So, the AR-condition implies that f (z, ·) eventually has (ϑ − 1)-polynomial growth. Here we replace the AR-condition by the quasimonotonicity hypothesis H 1 (iii). This hypothesis is a slightly more general version of a condition used by Li & Yang [21]. Note that there exists M > 0 such that for a.a. z ∈ Ω Examples. Consider the following two functions Both functions satisfy hypothesis H 1 , but only f 1 satisfies the AR-condition.
By L p(z) (Ω, ξ) we will denote the weighted L p(z) -space with weight ξ(·). Therefore This space is furnished with the norm Note that since by hypothesis ξ(z) > 0 for a.a. z ∈ Ω (see hypothesis H 0 ), the function Proposition 2.4. If hypothesis H 0 holds, then · and | · | are equivalent norms on W 1,p(z) (Ω).
Proof. From the definitions of the two norms, we have We argue indirectly. So, suppose that the claim is not true. Then we can find {u n } n≥1 ⊆ W 1,p(z) (Ω) such that u n p(z) > n|u n | for all n ∈ N.
Normalizing in L p(z) (Ω), we see that we have Evidently {u n } n≥1 ⊆ W 1,p(z) (Ω) is bounded. So, by passing to a suitable subsequence if necessary, we may assume that From (6) and (7), we have u =c ∈ R \ {0} and On account of (6), we have This proves the claim. Using the claim and the definitions of the two norms, we conclude that · and |·| are equivalent.
In what follows, we denote by γ p(z) : For every λ > 0 the energy functional ϕ λ : W 1,p(z) (Ω) → R for problem (P λ ), is given by On account of the third term, this functional is not C 1 and so the minimax theorems from the critical point theory are not directly applicable to this functional. For this reason we use truncation techniques in order to bypass the singularity and have C 1 -functionals on which the critical point theory applies. For this reason in the next section, we deal with a purely singular problem.
Finally, we mention that, as usual, by a solution of (P λ ), we understand a function u ∈ W 1,p(z) (Ω) such that

A purely singular problem
In this section we deal with the following purely singular problem To solve (8), we first consider a perturbation of (8) which removes the singularity. So, we consider the following approximation of problem (8): We solve this problem using a topological approach (fixed point theory). So, given g ∈ L p(z) (Ω), g ≥ 0 and ε ∈ (0, 1], we consider the following problem: For this problem we have the following result. Proposition 3.1. If hypotheses H 0 hold, problem (9) admits a unique solutionũ ε ∈ int C + .
Clearly, a fixed point of this map will be a solution for problem (8 ε ).
We have for all h ∈ W 1,p(z) (Ω), all n ∈ N.
Next we show the uniqueness of this solution. Suppose thatû ε ∈ W 1,p(z) (Ω) is another positive solution of (8). Again we haveû ε ∈ int C + . Also, we have Interchanging the roles of u ε andû ε in the above argument, we also have thatû ε ≤ u ε , therefore u ε =û ε . This proves the uniqueness of the positive solution u ε ∈ int C + of problem (8 ε ).
Since u 1 ≤ u n for all n ∈ N, we have u = 0 and so u ∈ int C + . Moreover, passing to the limit as n → ∞ in (19) and using (20), we conclude that u ∈ int C + is a positive solution of problem (8).
Finally, we show the uniqueness of this positive solution. So, suppose thatũ ∈ W 1,p(z) (Ω) is another positive solution of (8). As in the proof of Proposition 3.2, using the fact that the map x → x −η(z) is strictly decreasing on (0, +∞), we obtaiñ u = u, ⇒ u ∈ int C + is the unique positive solution of (8).
The proof is now complete.
In the next section we will use u ∈ int C + and truncation techniques to bypass the singularity and show that problem (P λ ) has positive solutions for certain values of the parameter λ > 0.

Positive solutions
We introduce the following two sets  Proof. Let u ∈ int C + be the unique positive solution of problem (8) produced in Proposition 3.4.
We consider the following auxiliary problem: From Proposition 3.1, we know that this problem has a unique positive solutionû ∈ int C + . We have Sinceû ∈ int C + , on account of hypothesis H 1 (i), we have 0 ≤ f (·,û(·)) ∈ L ∞ (Ω).
Next, we show that L is, in fact, an interval.
Also we have From Proposition 4.2 we know that u ≤ u λ . So, we can define the following truncation of the reaction of problem (P µ ) This is a Carathéodory function. We setT µ (z, x) = x 0τ (z, s)ds and consider the C 1 -functional As before (see the proof of Proposition 4.1), using the direct method of the calculus of variations and (31), (32), we can find u µ ∈ W 1,p(z) (Ω) such that The proof is now complete. Remark 4.6. As a byproduct of this proof we have that if 0 < µ < λ ∈ L and u λ ∈ S λ ⊆ int C + then we can find u µ ∈ S µ ⊆ int C + such that In fact we can improve this result using the strong comparison principle (see Proposition 2.3).
Proof. From Proposition 4.5 and its proof, we already know that µ ∈ L and we can find u µ ∈ S µ ⊆ int C + such that u µ ≤ u λ , u µ = u λ .
Let ρ = u λ ∞ and letξ ρ > 0 be as postulated by hypothesis H 1 (v). We have (see (34) and hypothesis H 1 (v)) On account of hypothesis H 1 (iv) and since u µ ∈ int C + , we have Hence from (35) and Proposition 2.3, it follows that The proof is now complete.
From (39) and (37) it is clear that we may assume that Indeed, otherwise we already have a whole sequence of positive smooth solutions all bigger than u 0 and so we are done. From (43), (44) and Theorem 5.7.6 of Papageorgiou, Rȃdulescu & Repovš [27, p. 449], we know that we can find ρ ∈ (0, 1) small such that Also hypothesis H 1 (ii) and (37) imply that if u ∈ int C + , then Claim: v λ (·) satisfies the C-condition.
We set y n = u + n u + n for all n ∈ N. Then y n = 1, y n ≥ 0 for all n ∈ N. So, we may assume that y n w → y in W 1,p(z) (Ω) and y n → y in L p(z) (Ω), y ≥ 0.
We have 0 ≤ t n u + n ≤ u + n for all n ∈ N. Hence hypothesis H 1 (iii) implies that for some C 14 > 0, all n ∈ N (see (53)).
So, we may assume that u n w → u in W 1,p(z) (Ω) and u n → u in L r(z) (Ω) as n → ∞.
In (49) we choose h = u n − u ∈ W 1,p(z) (Ω), pass to the limit as n → ∞ and use (63). Then reasoning as in the proof of Proposition 3.2, using Proposition 2.2, we obtain that This proves the claim. On account of (45), (46) and the claim, we can apply the mountain pass theorem and find u ∈ W 1,p(z) (Ω) such thatû From (66), (45) and (37), we obtain u ∈ S λ ⊆ int C + , u 0 ≤û, u 0 =û, which concludes the proof.
The proof is now complete.

Minimal positive solutions
Recall that a set S ⊆ W 1,p(z) (Ω) is said to be downward directed, if for all u 1 , u 2 ∈ S, we can find u ∈ S such that u ≤ u 1 , u ≤ u 2 .
As in the proof of Proposition 18 of Papageorgiou, Rȃdulescu & Repovš [31], we prove that for every λ ∈ L, the solution set S λ ⊆ int C + is downward directed. We will show that S λ has a minimal element.
Proof. On account of Lemma 3.10 of Hu & Papageorgiou [19, p. 176], we can find {u n } n≥1 ⊆ S λ decreasing (since S λ is downward directed) such that inf n≥1 u n = inf S λ .
We haveφ ′ λ (u n ) = 0 for all n ∈ N, u ≤ u n ≤ u 1 for all n ∈ N.
Then as in the proof of Proposition 4.5 and using the fact that {u n } n≥1 is decreasing, we obtain u n → u * λ in W 1,p(z) (Ω), ⇒ u * λ ∈ S λ ⊆ int C + , u * λ = inf S λ , which concludes the proof.
The next theorem summarizes our main contributions in this paper concerning problem (P λ ).