POSITIVE SOLUTIONS FOR SINGULAR ( p, q ) -LAPLACIAN EQUATIONS WITH NEGATIVE PERTURBATION

. We consider a nonlinear Dirichlet problem driven by the ( p,q )- Laplacian and with a reaction consisting of a singular term plus a negative perturbation. Using regularization of the singular term and truncation and comparison techniques, we show that the problem has a unique positive smooth solution.


Introduction
Let Ω ⊆ R N be a bounded domain with a C 2 -boundary ∂Ω.In this paper we study the following singular Dirichlet (p, q)-equation u ∂Ω = 0, 1 < q < p, 0 < η < 1, u > 0. ( For r ∈ (1, +∞) by ∆ r we denote the r-Laplace differential operator defined by ∆ r u = div(|∇u| r−2 ∇u) for all u ∈ W 1,r 0 (Ω).Equation (1.1) is driven by the sum of two such operators with different exponents (double phase problem).Therefore the differential operator of our problem is not homogeneous.In the reaction (right hand side), there is a singular term u −η and a perturbation −f (z, u), with f (z, x) 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) with values in R + = [0, +∞) (that is, f ≥ 0).So, in problem (1.1) the perturbation of the singular term is negative.This is in contrast with most earlier works on singular elliptic equations, where the perturbation is positive.We refer to works of Sun-Wu-Long [17], Haitao [8], Ghergu-Rȃdulescu [3] (semilinear equations), Giacomoni-Schindler-Takáč [5], Papageorgiou-Winkert [14] (equations driven by the p-Laplacian), Mukherjee-Sreenadh [10] (equations driven by the fractional p-Laplacian) and of Papageorgiou-Rȃdulescu-Repovš [12] (equations driven by a general nonlinear nonhomogeneous differential operator).Singular equations with a negative perturbation were investigated by Godoy-Guerin [7] (semilinear equations driven by the Laplacian) and by Saoudi [16] (nonlinear equations driven by the p-Laplacian).In both works the negative perturbation of the singular term is a power of u.Here we allow a more general perturbation.In both papers the approach is based on the direct method of the calculus of variations.In Godoy-Guerin [7], the notion of weak solution is more restrictive, since they require that the test functions belong in W 1,p 0 (Ω) ∩ L ∞ (Ω).On the other hand Saoudi [16] considers a parametric problem with the parameter λ > 0 multiplying the singular term.The author shows that there exists Λ * ≥ 0 (notation in [16]) such that for all λ > Λ * the problem has a solution.In fact as we explain in Remark 3.5 Λ * = 0 and so the existence theorem is valid for all parameters λ > 0 and so there is no need to introduce a parameter to the problem.Finally we mention that in [7] the equation is driven by the Laplacian (semilinear equation), while in [16] is driven by the p-Laplacian.
The fact that the perturbation is negative, makes it difficult to generate a lower solution for the problem which is helpful in bypassing the singularity and dealing with C 1 -functionals.The solution of the purely singular problem can not serve as a lower solution as is the case in problems with positive perturbation (see for example Papageorgiou-Rȃdulescu-Repovš [12]).Our approach is different and uses upper solutions and regularizations of the singular term.

Mathematical background -hypotheses
The main spaces in the analysis of problem (1.1) are the Sobolev space W 1,p 0 (Ω) and the Banach space C 1 0 (Ω) = {u ∈ C 1 (Ω) : u ∂Ω = 0}.On account of the Poincaré inequality the norm of W 1,p 0 (Ω) is given by u = ∇u p for all u ∈ W 1,p 0 (Ω).The space C 1 0 (Ω) is an ordered Banach space with positive (order) cone given by C + = {u ∈ C 1 0 (Ω) : u(z) ≥ 0 for all z ∈ Ω}.This cone has a nonempty interior int C + = u ∈ C + : u(z) > 0 for all z ∈ Ω, ∂u ∂n ∂Ω < 0 , with n(•) being the outward unit normal on ∂Ω and ∂u ∂n = (∇u, n) R N .Also ordered Banach space is the Lebesgue space L ∞ (Ω) with positive (order) cone We mention that from all the Lebesgue spaces L p (Ω), 1 ≤ p ≤ +∞ (all of which are ordered Banach spaces with the pointwise order), only L ∞ (Ω) has positive cone with a nonempty interior.This is a consequence of the fact that only the norm of L ∞ (Ω) is defined in a pointwise fashion. For We know (see Gasiński-Papageorgiou [2, p. 279]) that A r (•) is bounded (that is, maps bounded sets to bounded sets), continuous, strictly monotone (thus maximal monotone too) and of type (S) + , which means that it has the following property and it has the following properties: • V (•) is continuous, strictly monotone (thus maximal monotone too); 0 (Ω).Our hypotheses on the perturbation f (z, x) are the following: is the critical Sobolev exponent corresponding to p. Since we look for positive solutions and the above hypotheses concern the positive semiaxis R + = [0, +∞), without any loss of generality we may assume that f (z, x) = 0 for a.a.z ∈ Ω, all x ≤ 0.

By a solution of (1.1) we mean a function
for all h ∈ W 1,p 0 (Ω).

Positive solutions
First we consider the purely singular problem Next let ε > 0 and consider the following regularized version of problem (1.1), Proof.Consider the Carathéodory function From (3.3) it is clear that ψ ε (•) is coercive.Also using the Sobolev embedding theorem, we see that ψ ε (•) is sequentially weakly lower semicontinuous.So, by the Weierstrass-Tonelli theorem, we can find (3.4) In (3.4) by using the test function Next in (3.4) we choose the test function h = ( u ε − u) + ∈ W 1,p 0 (Ω).We obtain which implies u ε ≤ u (from the monotonicity of V (•)).So, we have proved that The nonlinear regularity theory by Lieberman [9] implies that Hypotheses (H1) imply that there exists c 1 > 0 such that [15] (pp.111, 120)).
Now we show the uniqueness of this positive solution.To this end we consider the integral functional j : We define dom j = {u ∈ L 1 (Ω) : j(u) < +∞} (the effective domain of j(•)).Also let 0 : R + → R + be the function defined by 0 (t) = 1 p t p + 1 q t q for all t ≥ 0.
The function 0 (•) is strictly increasing, strictly convex and since τ ∈ (1, q] (see hypotheses (H1)), we see that t → 0 (t 1/τ ) is convex on R + .We define (y) = 0 (|y|) for all y ∈ R N .Then : Suppose that v ε (•) is another positive solution of problem (3.2).Again we have v ε ∈ int C + .For δ > 0, we set . From (3.5) it follows that for t ∈ (0, 1) small, we have ( u δ ε ) τ + th ∈ dom j, ( v δ ε ) τ + th ∈ dom j.Then the convexity of j(•) implies that the directional derivatives of j(•) at ( u δ ε ) τ and at ( v δ ε ) τ in the direction h exist and using the chain rule and Green's identity (see [11, p. 35]), we have The convexity of j(•) implies the monotonicity of the directional derivative.So, we have We let δ → 0 and use the dominated convergence theorem.Then on account of hypotheses (H1), we obtain This proves the uniqueness of the positive solution Next we show a monotonicity property of the map ε → u ε .
Finally we pass to the limit as ε → 0 + to produce a positive solution for problem (1.1).Consider the Dirichlet problem From Papageorgiou-Rȃdulescu-Zhang [13] (see the proof of Proposition 3.3), we know that this problem has a unique solution u ε ∈ int C + and u ε ↑ u in C 1 0 (Ω) as ε → 0 + .Moreover, since f ≥ 0, as in the proof of Proposition 3. Proof.Let ε n → 0 + and let u n = u εn ∈ int C + as in Proposition 3.2.We have (3.17) In (3.11) we choose h = ( u n − u) ∈ W 1,p 0 (Ω) and pass to the limit as n → +∞.We obtain lim n→+∞ V ( u n ), u n − u = 0,