On a class of singular anisotropic $(p,q)$-equations

We consider a Dirichlet problem driven by the anisotropic $(p,q)$-Laplacian and with a reaction that has the competing effects of a singular term and of a parametric superlinear perturbation. Based on variational tools along with truncation and comparison techniques, we prove a bifurcation-type result describing the changes in the set of positive solutions as the parameter varies.


Introduction
Let Ω ⊆ R N be a bounded domain with a C 2 -boundary ∂Ω. In this paper, we study the following anisotropic Dirichlet problem For r ∈ E 1 , where E 1 is given by we denote by ∆ r(·) the anisotropic r-Laplacian (or r(·)-Laplacian) defined by ∆ r(·) u = div |∇u| r(x)−2 ∇u for all u ∈ W 1,r(·) 0 (Ω).
The differential operator in problem (P λ ) is the sum of two such operators. In the reaction, the right-hand side of (P λ ), we have the competing effects of two terms which are of different nature. One is the singular term s → s −η(x) for s > 0 with η ∈ C(Ω) such that 0 < η(x) < 1 for all x ∈ Ω. The other one is the parametric term s → λf (x, s) with λ > 0 being the parameter and f : Ω × R → R is a Carathéodory function, 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 for a. a. x ∈ Ω as s → +∞ with p + = max x∈Ω p(x). We are looking for positive solutions of problem (P λ ) and our aim is to determine how the set of positive solutions of (P λ ) changes as the parameter λ moves on the semiaxis • R + = (0, +∞). The starting point of our work is the recent paper of Papageorgiou-Winkert [16] where the authors study a similar problem driven by the isotropic p-Laplacian. So, the differential operator in [16] is (p − 1)-homogeneous and this property is exploited in their arguments. In contrast here, the differential operator is both nonhomogeneous and anisotropic.
Anisotropic problems with competition phenomena in the source were recently investigated by Papageorgiou-Rȃdulescu-Repovš [11]. They studied concave-convex problems driven by the p(·)-Laplacian plus an indefinite potential term. In their equation there is no singular term. In fact, the study of anisotropic singular problems is lagging behind. We are aware only the works of Byun-Ko [2] and Saoudi-Ghanmi [20] for Dirichlet as well as of Saoudi-Kratou-Alsadhan [21] for Neumann problems. All the aforementioned works deal with equations driven by the p(·)-Laplacian.
We mention that equations driven by the sum of two differential operators of different nature arise often in the mathematical models of physical processes. We mention the works of Bahrouni-Rȃdulescu-Repovš [1] (transonic flow problems), Cherfils-Il ′ yasov [3] (reaction diffusion systems) and Zhikov [26] (elasticity problems). Some recent regularity and multiplicity results can be found in the works of Ragusa-Tachikawa [19] and Papageorgiou-Zhang [17].
In this paper, under general conditions on the perturbation f : Ω × R → R which are less restrictive than all the previous cases in the literature, we prove the existence of a critical parameter λ * > 0 such that • for every λ ∈ (0, λ * ), problem (P λ ) has at least two positive smooth solutions; • for λ = λ * , problem (P λ ) has at least one positive smooth solution; • for every λ > λ * , problem (P λ ) has no positive solutions.

Preliminaries and Hypotheses
The study of anisotropic equations uses Lebesgue and Sobolev spaces with variable exponents. A comprehensive presentation of the theory of such spaces can be found in the book of Diening-Harjulehto-Hästö-Růžička [4].
Recall that E 1 = {r ∈ C(Ω) : 1 < min x∈Ω r(x)}. For any r ∈ E 1 we define Moreover, let M (Ω) be the space of all measurable functions u : Ω → R. As usual, we identify two such functions when they differ only on a Lebesgue-null set. Then, given r ∈ E 1 , the variable exponent Lebesgue space L r(·) (Ω) is defined as We equip this space with the so-called Luxemburg norm defined by Then (L r(·) (Ω), · r(·) ) is a separable and reflexive Banach space, in fact it is uniformly convex. Let r ′ ∈ E 1 be the conjugate variable exponent to r, that is, We know that L r(·) (Ω) * = L r ′ (·) (Ω) and the following Hölder type inequality holds for all u ∈ L r(·) (Ω) and for all v ∈ L r ′ (·) (Ω). If r 1 , r 2 ∈ E 1 and r 1 (x) ≤ r 2 (x) for all x ∈ Ω, then we have that The corresponding variable exponent Sobolev spaces can be defined in a natural way using the variable exponent Lebesgue spaces. So, if r ∈ E 1 , then the variable exponent Sobolev space W 1,r(·) (Ω) is defined by Here the gradient ∇u is understood in the weak sense. We equip W 1,r(·) (Ω) with the following norm u 1,r(·) = u r(·) + |∇u| r(·) for all u ∈ W 1,r(·) (Ω).
For r ∈ E 1 we introduce the critical Sobolev variable exponent r * defined by for all x ∈ Ω.
In the study of the variable exponent spaces, the modular function is important, that is, for r ∈ E 1 , ̺ r(·) (u) = Ω |u| r(x) dx for all u ∈ L r(·) (Ω).
As before we write ̺ r(·) (∇u) = ̺ r(·) (|∇u|). The importance of this function comes from the fact that it is closely related to the norm of the space. This is evident in the next proposition.
Remark 2.4. Note that in part (a) of Proposition 2.3 we have by the weak comparison principle that u ≤ v, see Tolksdorf [24].
In what follows we will denote by · the norm of the Sobolev space W 1,p(·) 0 (Ω). By the Poincaré inequality we have Suppose that X is a Banach space and let ϕ ∈ C 1 (X). We denote the critical set of ϕ by Moreover, we say that ϕ satisfies the "Cerami condition", C-condition for short, if every sequence {u n } n∈N ⊆ X such that {ϕ(u n )} n∈N ⊆ R is bounded and 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 need not be locally compact being in general infinite dimensional. Applying this condition, one can prove a deformation theorem from which the minimax theorems for the critical values of ϕ follow. We refer to Papageorgiou-Rȃdulescu-Repovš [12, Chapter 5] and Struwe [22, Chapter II].
Given s ∈ (1, +∞) we denote by s ′ ∈ (1, +∞) the conjugate exponent defined by 1 s is a measurable function, then we denote by N f the Nemytskii (also called superposition) operator corresponding to f , that is, Now we are in the position to introduce our hypotheses on the data of problem (P λ ).
for a. a. x ∈ Ω, for all s ≥ 0 and with p + < r < p * − , where ' uniformly for a. a. x ∈ Ω; (iv) for every ρ > 0 there existsξ ρ > 0 such that the function Remark 2.5. Since we are interested in positive solutions and all the hypotheses above concern the positive semiaxis R + = [0, +∞), we may assume without any loss of generality that f (x, s) = 0 for a. a. x ∈ Ω and for all s ≤ 0. Hypotheses However, this superlinearity condition on f (x, ·) is not formulated by using the Ambrosetti-Rabinowitz condition which is common in the literature when dealing with superlinear problems, see Byun-Ko [2], Gasiński-Papageorgiou [6], Saoudi-Ghanmi [20] and Saoudi-Kratou-Alsadhan [21]. Here, instead of the Ambrosetti-Rabinowitz condition, we employ hypothesis H 1 (iii) which is less restrictive and incorporates in our framework nonlinearities with "slower" growth near +∞. For example, consider the function for all x ∈ Ω. This function satisfies hypotheses H 1 , but fails to satisfy the Ambrosetti-Rabinowitz condition, see, for example, Gasiński-Papageorgiou [6].
The difficulty that we encounter when we study a singular problem is that the energy (Euler) functional of the problem is not C 1 because of the presence of the singular term. Hence, we cannot use the results of critical point theory. We need to find a way to bypass the singularity and deal with C 1 -functionals. In the next section, we examine a purely singular problem and the solution of this problem will help us in bypassing the singularity.

An auxiliary purely singular problem
In this section we deal with the following purely singular anisotropic (p, q)equation (Ω) and let 0 < ε ≤ 1. We consider the following Dirichlet problem in Ω, u ∂Ω = 0, u > 0.
This map is continuous and strictly monotone, see Proposition 2.2, hence maximal monotone as well. It is also coercive, see Proposition 2.1. Therefore, it is surjective, The strict monotonicity of V implies the uniqueness of u ε . Thus, we can define the map β : L p(·) (Ω) → L p(·) (Ω) by setting Recall that W 1,p(·) 0 (Ω) ֒→ L p(·) (Ω) is compactly embedded. We claim that the map β is continuous. So, let g n → g in L p(·) (Ω) and let u n ε = β(g n ) with n ∈ N. We have (Ω) and for all n ∈ N.
In the next section we will use this solution to bypass the singularity and deal with C 1 -functionals on which we can apply the results of critical point theory.
We want to determine the regularity of the elements of the solution set S λ . To this end, we first establish a lower bound for the elements of S λ . Proof. Let u ∈ S λ . We introduce the Carathéodory function b : Ω × We consider the following Dirichlet problem As in the proof of Proposition 3.1, using approximations and fixed point theory, we can show that problem (4.9) has a positive solution u 0 ∈ W 1,p(·) 0 (Ω). Applying (4.8), f ≥ 0 and u ∈ S λ yields Therefore, we have Then, (4.10), (4.8), (4.9) and Proposition 3.1 imply that u 0 = u ∈ int C 1 0 (Ω) + . This shows that u ≤ u for all u ∈ S λ , see (4.10).
Using this lower bound, we can now prove the regularity properties of the elements of S λ . Therefore, We consider the linear Dirichlet problem −∆y = g(x) in Ω, y ∂Ω = 0, y > 0.
Using the solution of this problem and reasoning exactly as in the proof of Proposition 4.1, we can show that u ∈ int C 1 0 (Ω) + . Therefore, ∅ = S λ ⊆ int C 1 0 (Ω) + .
From Proposition 4.4 and its proof we have the following corollary.
In the next proposition we are going to improve the assertion of Corollary 4.5.
Next we are going to prove that we have multiple solutions for all λ ∈ (0, λ * ).
This proves the Claim.
It remains to decide whether the critical parameter value λ * > 0 is admissible. Proof. Let {λ n } n∈N ⊆ (0, λ * ) ⊆ L be such that λ n ր λ * as n → ∞. From the proof of Proposition 3.10 we know that we can find u n ∈ S λn ⊆ int C 1 0 (Ω) + such that w λn (u n ) ≤ w λn (u) for all n ∈ N.