Positive solutions for singular anisotropic $(p,q)$-equations

In this paper we consider a Dirichlet problem driven by an anisotropic $(p,q)$-differential operator and a parametric reaction having the competing effects of a singular term and of a superlinear perturbation. We prove a bifurcation-type theorem describing the changes in the set of positive solutions as the parameter moves. Moreover, we prove the existence of a minimal positive solution and determine the monotonicity and continuity properties of the minimal solution map.


Introduction
Let Ω ⊆ R N be a bounded domain with a C 2 -boundary ∂Ω. In this paper, we deal with the following parametric anisotropic singular (p, q)-equation in Ω, Given r ∈ C(Ω) we define r − = min x∈Ω r(x) and r + = max x∈Ω r(x) and introduce the set For r ∈ E 1 the anisotropic r-Laplace differential operator is defined by ∆ r(·) u = div |∇u| r(x)−2 ∇u for all u ∈ W 1,r(·) 0 (Ω).
This operator is nonhomogeneous on account of the variable exponent r(·). If r(·) is a constant function, then we have the usual r-Laplace differential operator. In problem (P λ ) we have the sum of two such anisotropic differential operators with distinct exponents. So, even in the case of constant exponents, the differential operator in (P λ ) is not homogeneous. This makes the study of problem (P λ ) more difficult. Boundary values problems driven by a combination of differential operators of different nature, such as (p, q)-equations, arise in many mathematical models of physical processes. We mention the works of Benci-D'Avenia-Fortunato-Pisani [1], where (p, 2)-equations are used as a model for elementary particles in order to produce soliton-type solutions. We also mention the works of Cherfils-Il ′ yasov [3], where the authors study the steady state solutions of reaction-diffusion systems and of Zhikov [24,25] who studied problems related to nonlinear elasticity theory.
In the reaction of (P λ ) we have the competing effects of a singular term s → s −η(x) and of a Carathéodory function f : Ω × R → R, that is, x → f (x, s) is measurable for all s ∈ R and s → f (x, s) is continuous for a. a. x ∈ Ω. We assume that f (x, ·) exhibits (p + −1)-superlinear growth uniformly for a. a. x ∈ Ω as s → +∞ but need not satisfy the Ambrosetti-Rabinowitz condition (the AR-condition for short) which is common in the literature when dealing with superlinear problems. The sum of the two terms is multiplied with a parameter λ > 0.
Applying a combination of variational tools from the critical point theory along with truncation and comparison techniques, we prove a bifurcation-type theorem describing the changes in the set of positive solutions as the parameter λ moves on the open positive semiaxis • R + = (0, +∞). We also show that for every admissible parameter λ > 0, problem (P λ ) has a smallest positive solutionũ λ and we determine the monotonicity and continuity properties of the minimal solution map λ →ũ λ .
Boundary values problems driven by the anisotropic p-Laplacian have been studied extensively in the last decade. We refer to the books of Diening-Harjulehto-Hästö-Růžička [4] and Rȃdulescu-Repovš [20] and the references therein. In contrast, the study of singular anisotropic equations is lagging behind. There are very few works on this subject. We mention two such papers which are close to our problem (P λ ). These are the works of Byun-Ko [2] and Saoudi-Ghanmi [21] who examine equations driven by the anisotropic p-Laplacian and the parameter multiplies only the singular term. Moreover, the overall conditions on the data of the problem are more restrictive, see hypotheses (p M ) in [2] and hypotheses (H1)-(H4) in [21]. Finally, we mention the isotropic works of the authors [18,19] on singular equations driven by the (p, q)-Laplacian and the p-Laplacian, respectively.

Preliminaries and Hypotheses
In this section we recall some basic facts about Lebesgue and Sobolev spaces with variable exponents. We refer to the book of Diening-Harjulehto-Hästö-Růžička [4] for details.
Let M (Ω) be the space of all measurable functions u : Ω → R. We identify two such functions when they differ only on a Lebesgue-null set. Given r ∈ E 1 , the anisotropic Lebesgue space L r(·) (Ω) is defined by This space is equipped with the Luxemburg norm defined by The modular function related to these spaces is defined by It is clear that · r(·) is the Minkowski functional of the set {u ∈ L r(·) (Ω) : ̺ r(·) (u) ≤ 1}. The following proposition states the relation between · r(·) and the modular ̺ r(·) : L r(·) (Ω) → R.
For the first, we will need the following ordering notion on M (Ω). So, given y 1 , y 2 : Ω → R two measurable functions, we write y 1 y 2 if for every compact set K ⊆ Ω, we have 0 < c K ≤ y 2 (x) − y 1 (x) for a. a. x ∈ K. Note that if y 1 , y 2 ∈ C(Ω) and y 1 (x) < y 2 (x) for all x ∈ Ω, then y 1 y 2 . The first strong comparison principle is the following one.
In the second strong comparison principle, we strengthen the order condition on y 1 and y 2 but drop the boundary requirements on u and v.
Given a Banach space X and a functional ϕ ∈ C 1 (X), we define being the critical set of ϕ. We say that ϕ satisfies the "Cerami condition", Ccondition for short, if every sequence {u n } n∈N ⊆ X such that {ϕ(u n )} n∈N ⊆ R is bounded and (1 + u n X ) ϕ ′ (u n ) → 0 in X * as n → ∞, admits a strongly convergent subsequence. This is a compactness-type condition on the functional ϕ which compensates for the fact that the ambient space X is not locally compact in general, since it could be infinite dimensional. Using this condition, we can prove a deformation theorem which leads to the minimax theorems of the critical point theory, see, for example, Papageorgiou-Rȃdulescu-Repovš [14,Section 5.4]. Now we are ready to state our hypotheses on the nonlinearity f : Ω × R → R.
Remark 2.5. Without any loss of generality we can assume that f (x, s) = 0 for a. a. x ∈ Ω and for all s ≤ 0 since we are interested in positive solutions of (P λ ).
In most papers in the literature, superlinear problems are treated by using the ARcondition which in the present context has the following form: (AR) + : There exist θ > p + and M > 0 such that This is a unilateral version of the AR-condition since we assumed that f (x, s) = 0 for a. a. x ∈ Ω and for all s ≤ 0. Integrating (2.1) and using (2.2) gives for a. a. x ∈ Ω, for all s ≥ M and for some c 1 > 0. Hence, for a. a. x ∈ Ω and for all s ≥ M , see (2.1). Therefore, the (AR) + -condition dictates that f (x, ·) has at least (θ − 1)-polynomial growth as s → +∞. But this way we exclude superlinear nonlinearities with "slower" growth near +∞ from our considerations. The following example fulfills H 1 , but fails to satisfy the (AR) +condition: Hypotheses When studying singular problems of isotropic and anisotropic type, the presence of the singular term leads to an energy function which is not C 1 and so we cannot apply directly the minimax theorems of the critical point theory on it. We need to find a way to bypass the singularity and deal with C 1 -functionals. To this end, we examine a purely singular problem in the next section. The unique solution of this problem will be helpful in our effort to bypass the singularity of our original problem (P λ ).

An auxiliary purely singular problem
In this section we study the following purely singular anisotropic Dirichlet problem We have the main result in this section.
Proof. For the existence and uniqueness part of the proof we assume for simplicity that λ = 1.
We have (Ω) and for all n ∈ N. We choose h =v n −v ∈ W 1,p(·) 0 (Ω) in (3.5), pass to the limit as n → ∞ and use (3.4) and the fact that Due to the monotonicity of A q(·) (·) we obtain lim sup From this and (3.4) we then conclude that (Ω). (3.6) Passing to the limit in (3.5) as n → ∞ and using (3.6) gives (Ω). Thus,v = K ε (g). Therefore, by the Urysohn criterion for the convergence of sequences, we conclude that for the original sequence we havê Hence, K ε is continuous and this proves Claim 1. Recall (Ω) is bounded, see (3.3). On the other hand, we have the compact embedding W 1,p(·) 0 (Ω) ֒→ L p(·) (Ω). This implies (Ω) such that In fact, this solution is unique. Indeed, suppose that y ε ∈ int C 1 0 (Ω) + is another positive solution of (3.8). Then we have that We obtain u ε ≤ y ε .
Interchanging the roles of u ε and y ε in the argument above also gives y ε ≤ v ε . Hence, u ε = y ε . This proves the uniqueness of the solution u ε ∈ int C 1 0 (Ω) + of problem (3.8).
From (3.14) we then see that for a. a. x ∈ Ω. (Ω). Therefore, if we pass to the limit as n → ∞ in (3.15) and use (3.19)

Positive solutions
We introduce the following two sets L = {λ > 0 : problem (P λ ) has a positive solution} , S λ = {u : u is a positive solution of problem (P λ )} .
First we show that the set L of admissible parameters is nonempty and we determine the regularity properties of the elements of S λ for λ ∈ L.
(Au)' Proposition 4.1. If hypotheses H 0 hold, then problem (Au)' has a unique positive solutionũ ∈ int C 1 0 (Ω) + such that u 1 ≤ũ. Proof. In order to establish the existence of a positive solution, we argue as in the first part of the proof of Proposition 3.1. So, we consider the approximation in Ω, u ∂Ω = 0, n ∈ N.
This problem has a unique solutionũ n ∈ int C 1 0 (Ω) + . Testing the equation with u n we obtain As before, by using the anisotropic Hardy's inequality, we conclude that ̺ p(·) (∇ũ n ) ≤ c 5 ũ n for all n ∈ N and for some c 5 > 0.
In summary we have shown that u λ ∈ [u λ ,ũ] for all λ ∈ (0, 1] small enough. From (4.2) and (4.6) we see that u λ is a solution of our original problem (P λ ), that is, u λ ∈ S λ . This proves the nonemptiness of L.
An immediate consequence of the proof above is the following corollary.
We can improve the conclusion of this corollary.
This is a contradiction to the definition of m 0 > 0. Therefore, λ ∈ L and so λ * ≤λ < ∞.
From this and Proposition 2.2 we conclude that u n → u * in W 1,p(·) 0 (Ω) and u 1 ≤ u * . (4.33) If we now pass to the limit in (4.32) as n → ∞, then, by applying (4.33), we see that u * ∈ S λ * and so λ * ∈ L, that is, L = (0, λ * ]. In summary, we can state the following bifurcation-type result concerning problem (P λ ).

Minimal positive solutions
In this section we are going to show that for every admissible parameter λ ∈ L = (0, λ * ], problem (P λ ) has a smallest positive solution (so-called minimal positive solution)ũ λ ∈ S λ ⊆ int C 1 0 (Ω) + , that is,ũ λ ≤ u for all u ∈ S λ . Moreover, we determine the monotonicity and continuity properties of the minimal solution map L ∋ λ →ũ λ ∈ int C 1 0 (Ω) + . Proposition 5.1. If hypotheses H 0 and H 1 hold and if λ ∈ L ∈ (0, λ * ], then problem (P λ ) has a smallest positive solutionũ λ ∈ int C 1 0 (Ω) + . Proof. As in the proof of Proposition 18 in Papageorgiou-Rȃdulescu-Repovš [16], we show that the set S λ is downward directed, that is, if u, v ∈ S λ , then there exists y ∈ S λ such that y ≤ u and y ≤ v. Invoking Lemma 3.10 of Hu-Papageorgiou [11, p. 178], we can find a decreasing sequence {u n } n∈N ⊆ S λ such that inf S λ = inf n∈N u n and u λ ≤ u n ≤ u 1 for all n ∈ N.
Proof. (a) This is an immediate consequence of Proposition 4.5.