Discontinuous perturbations of nonhomogeneous strongly-singular Kirchhoff problems

In this paper, we are concerned with a Kirchhoff problem in the presence of a strongly-singular term perturbed by a discontinuous nonlinearity of the Heaviside type in the setting of Orlicz–Sobolev space. The presence of both strongly-singular and non-continuous terms brings up difficulties in associating a differentiable functional to the problem with finite energy in the whole space W 1,Φ 0 (Ω). To overcome this obstacle, we establish an optimal condition for the existence of W 1,Φ 0 (Ω)-solutions to a strongly-singular problem, which allows us to constrain the energy functional to a subset of W 1,Φ 0 (Ω) in order to apply techniques of convex analysis and generalized gradient in the sense of Clarke. Mathematics Subject Classification. 35J25, 35J62, 35J75, 35J20, 35D30, 35B38.


Introduction
In this paper, we are concerned in presenting equivalent conditions for the existence of three solutions for the quasilinear problem which are linked to an optimal compatibility condition between (b, δ) for existence of solution to the strongly-singular problem  ∞) is of Heaviside type, 0 < b ∈ L 1 (Ω), δ > 1, λ, μ > 0 are real parameters. Moreover, −Δ Φ u = −div(a(|∇u|)∇u) stands for the Φ-Laplace operator, where a : (0, ∞) → (0, ∞) is a C 1 -function that defines the increasing homeomorphism φ : R → R given by a(|t|)t if t = 0, 0 ift = 0, whose the associated N-function Φ : R → R is given by Φ(t) = |t| 0 φ(s)ds. The issue about existence of three solutions for a suitable range of parameters λ, μ > 0, for particular forms of Problem (Q λ,μ ), has been considered in the literature recently, principally in the context of non-singular problems (δ < 0) and in the case in which f is continuous, see for instance [3,11,30,33] and references therein. There are few works for singular nonlinearities, we quote for example [12,13,33] who considered Φ(t) = |t| p /p, t > 0, 1 < p < ∞ and M ≡ 1 in (Q λ,μ ). We also refer to [21,24,26] on recent developments to the study of local or nonlocal problems with nonstandard growth.
In [33], a singular problem for low dimensions was studied, while in [12] and [13] a singular problem for high dimensions was treated, but in both cases f has been considered a Carathéodory function with suitable assumptions. More specifically, in [13], the singular perturbation was considered in the weak sense (0 < δ < 1), while in [12] they permitted δ > 1 by balancing the size of this δ with the existence of a 0 < u ∈ C 1 0 (Ω) such that the product bu −δ in L (p * ) (Ω).
In this paper, we establish an optimal condition to the relationship between the power δ > 1 and the potential b(x) > 0 to existence of three solutions to the singular problem (Q λ,μ ), independent of the dimension N , in the presence of both a discontinuous nonlinearity of the Heaviside type and a non-local term. More precisely, we prove how the existence of three solutions to (Q λ,μ ) is associated to the existence of solutions still in W 1,Φ 0 (Ω) to the problem (S). Our approach is based on the existence of positive solution to the problem (S), which provides a non-empty effective domain for the energy functional associated to (Q λ,μ ) and enable us to apply techniques of the generalized gradient in Clarke sense to get a multiplicity result.
Besides this, we prove qualitative results about these three solutions. We highlight how the non-local term M should be to the discontinuity of the function f be effectively attained by the solutions and how the level set of these solutions behaves exactly at the discontinuity point of f . To our knowledge, both the results of equivalent conditions and qualitative information on solutions are new in literature.
As our main results will be obtained via variational methods, we need to introduce the energy functional associated to Problem (Q λ,μ ). To do this, let us denote by W 1,Φ 0 (Ω) the Orlicz-Sobolev space associated to Φ and extend the function f to R as f (x, t) = 0 a.e in Ω and for all t ≤ 0. From these, the functional naturally associated to (Q λ,μ ) is I : x ∈ Ω and t > 0, +∞ for x ∈ Ω and t ≤ 0.
To ease our future references, let us rewrite I as I = Ψ 1 + μΨ 2 , where and The main difficulty in treating strongly-singular problems consists in the fact that the energy functional associated to the equation neither belongs to C 1 , in the sense of Fréchet differentiability, nor is defined in the whole space W 1,Φ 0 (Ω). In fact, when δ > 1 the functional Ψ 2 may not be proper, i.e. it may occur Ψ 2 (u) = ∞, for all u ∈ W 1,Φ 0 (Ω). Another difficulty exploited in this work is the presence of a more general quasilinear operator, which may be even nonhomogeneous. To deal with this situation, we approach the problem (Q λ,μ ) in Orlicz-Sobolev space setting. Below, let us state the assumptions about Φ that we will assume throughout this paper.

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that Ω Φ(|u|)dx < ∞ and it is a Banach space endowed with the Luxemburg norm Associated to the space L Φ (Ω), we can set the Orlicz-Sobolev space and deduce that it is a Banach space with respect to the norm The Orlicz-Sobolev space W 1,Φ 0 (Ω) is naturally defined as the closure of C ∞ 0 (Ω) in W 1,Φ (Ω)-norm, under the hypothesis (φ 1 ). For more information about the Orlicz and Orlicz-Sobolev spaces, we refer [1,17,18].
About M , let us assume . To conclude our assumptions, let us suppose that f : (f 1 ): there exists an odd increasing homeomorphism h from R onto R and nonnegative constants a 1 , a 2 and a 3 such that where H(t) = |t| 0 h(s)ds is a N-function satisfying Δ 2 (H is the its complementary function) such that H ≺≺ Φ * and Before stating the main results, let us clarify what we mean by a solution of (Q λ,μ ).
Our main result on the multiplicity of solutions can be stated as follows.
and (M ) hold. Then, the below claims are equivalents: iii) for each λ > λ * , there exists μ λ > 0 such that for μ ∈ (0, μ λ ], the problem (Q λ,μ ) admits at least three solutions, being two local minima and the other one a mountain pass critical point of the functional I, where Moreover, for each of such solutions the meas{x ∈ Ω : u(x) =ã} = 0. Besides this, u solves (Q λ,μ ) almost everywhere in Ω if in addition bd −δ ∈ LH (Ω) and (iv) either M is non-decreasing and f (x, t) = f (x) for all 0 < t < 1 and a.e.

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for each 0 <ã <ã given. Remark 1.1. About the above theorem, we still highlight the following facts: (i) the equivalency between (i) and (ii) holds true without assuming b ∈ L 2 loc (Ω), (ii) each one of such solutions given by (iii) is such that u(x) ≥ Cd(x) for x ∈ Ω, for some C > 0 dependent on u, (iii) in particular, Theorem 1.2 shows that a perturbation by a Heaviside function is enough to break the uniqueness of solution of the pure singular problem (S) to produce at least three ones for the problem for appropriate parameters λ, μ > 0, where H stands for the Heaviside function. This is new to literature of singular elliptic problems even to Laplacian operator.
In particular, as a consequence of Theorem 1.2 and Corollary 1.1, we have the following.
for some q > 1 and 1 < δ < δ q , then for each λ > λ * given, there exists μ λ > 0 such that for μ ∈ (0, μ λ ] the problem (Q λ,μ ) admits at least three weak solutions with the same properties as those found in item−iii) in Theorem 1.2.
It is worth mentioning that the above theorems improve or complement the related results in the literature both by the presence of the Kirchhoff term, the summability assumption on the potential b, the strongly-singular term and the non-homogeneity of the operator. Our results contribute to the literature principally by: (i) Theorem 1.1 unify some results on Δ p -Laplacian operator, with 1 < p < ∞, to Φ-Laplacian operator, see for instance [2,22]. (ii) Theorem 1.2 establishes necessary and sufficient conditions for existence of multiple solutions for the problem (Q λ,μ ), by connecting and extending the principal result in Yijing [31] to a non-homogeneous operator; (iii) Theorem 1.2 extends the principal result in Faraci et al. [12] and complements the main result in [13], principally by considering a non-homogeneous operator, an optimal condition on the pair (b, δ) to existence of three solutions, a discontinuity of the Heaviside type and including a Kirchhoff term; (iv) Corollary 1.1 gives us an explicit range of variation of δ, in which the existence of solution in W 1,Φ 0 (Ω) for (S) is still guaranteed. In particular, when Φ(t) = |t| p /p and b 0 ≤ b(x) ∈ L ∞ (Ω) for some constant b 0 > 0, the value δ q coincides with the sharp values obtained in [16,19]; (v) Corollary 1.2 complements the principal result in [12] by showing an explicit variation to δ, where the multiplicity is still ensured, namely, To ease the reading, from now on let us assume the assumptions (φ 0 ), (φ 1 ), (φ 2 ), (M ) and gather below some functional that appear throughout the paper.
This paper is organized as follows. In Sect. 2, we present some preliminary knowledge on the Orlicz-Sobolev spaces and some results of non-smooth analysis related to our problem. The Sect. 3 is reserved to prove Theorem 1.1, while in Sect. 4 we prove Theorem 1.2.

Non-smooth analysis for locally Lipschitz functional
In this section, we are going to remember some facts related to non-smooth analysis. However, one of the principal contribution of this section is establishing appropriated assumptions under the N-function Φ, the non-local term M V. D. Rădulescu et al. NoDEA and the discontinuous function f that make possible to approach (ii =⇒ iii), in Theorem 1.2, via Ricceri's multiplicity theorem [28].
Under our hypotheses and the decomposition of the functional I into Ψ 1 plus Ψ 2 , that is, we have written I as a sum of a locally Lipschitz functional Ψ 1 and a convex one Ψ 2 (see (1) and (2)). Below, let us recall few notations and results on the Critical Point Theory for the functional Ψ 1 and Ψ 2 . We refer the reader to Carl, Le & Motreanu [4], Chang [5], Clarke [8] and references therein for more details about this issue. Let us begin by remembering that the generalized directional derivative is a convex function. In particular, ∂Ψ 0 1 (u; 0) is named by the generalized gradient of Ψ 1 at u and denoted by ∂Ψ 1 (u).
About the functional Ψ 2 , its effective domain is defined by Dom( In this context, we say that I satisfies the Palais-Smale condition (the condition (PS) for short) if: In order to prove the next result, let us define the functionals where P is defined by It is well know that, under the hypotheses (φ 0 ) and (φ 1 ), the functional P is sequentially weakly lower semicontinuous and C 1 with (Ω) is a strictly monotonic operator of the type (S + ). Thus, we can rewrite I as where J 1 is C 1 , J 2 is locally Lipschitz and Ψ 2 is a convex functional.
is sequentially weakly lower semicontinuous and Ψ 0 1 is of the type (S + ). Proof. First, we note that the item (i) is an immediate consequence of assumptions on M and properties of P. Next, we present a summary proof of the other items.
The conclusion of the proof is a direct consequence of Theorem 1.1 in [20] and classical Riesz Theorem for Orlicz spaces, see for instance [27].

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(iii) This conclusion is a consequence of item (i) and the fact that P is of the type (S + ). (iv) Let u n u and η n ∈ ∂J 2 (u n ). Since η n ∈ L H (Ω) , the Riesz theorem for Orlicz spaces implies that there exists a unique η n ∈ L H (Ω), still denoted by η n , such that Besides this, by using (f 1 ), H ∈ Δ 2 and Young's inequality, we obtain which leads us to conclude that |η n (u n − u)| ≤ g(x) for some g ∈ L 1 (Ω), after using the compact embedding W 1,Φ 0 (Ω) → L H (Ω) and Lemma 5.3 in [14]. As u n → u a.e in Ω, the first claim follows by Lebesgue's theorem.
To end the proof, it follows from Proposition 2.171 in [4] that there (v) As in the previous item, by using (f 1 ) and dominated convergence the result follows. (vi) This item is a consequence of the continuity and monotonicity ofM and the fact that P is sequentially weakly lower semicontinuous in W 1,Φ 0 (Ω). (vii) By items (i) and (ii) above, we have Ψ 1 ∈ Lip loc (W 1,Φ 0 (Ω); R). Besides this, we get from item (iv) and (f 1 ) that Ψ 1 is sequentially weakly lower semicontinuous. Let u n u such that lim sup n→∞ Ψ 0 1 (u n ; u n − u) ≤ 0. Then, (iii) and (iv) above lead us to lim sup which implies the claimed, after using the (iii). This ends the proof.
The next lemma gives us some properties regarding Ψ 2 .
It follows from the coercivity of I, obtained in the previous lemma, that (u n ) is bounded in W 1,Φ 0 (Ω). Thus, passing to a subsequence if necessary, we may assume that u n u. So, by Lemmas 2.1-(vii) and 2.2, we obtain that I is sequentially weakly lower semicontinuous, which yields Therefore, by using the previous inequality and the lower semicontinuity of Ψ 2 , we get Proof. We just need verify that, under these assumptions, the conditions of Theorem C in [29] are fulfilled. Since W 1,Φ 0 (Ω) is a reflexive and separable space, J 1 and J 2 are sequentially weakly lower semicontinuous and the functional Ψ 1 is coercive (see (9)), we just need to check that to conclude the proof of the proposition, In this direction, let us assume u n u and lim n→∞ inf J 1 (u n ) ≤ J 1 (u). Since J 1 is sequentially weakly lower semicontinuous, we have lim for some subsequence, still denoted by (u n ). Thus, from this fact, continuity and monotonicity ofM in R + , we obtain lim n→∞ P(u n ) = P(u). Therefore, by the hypothesis (φ 1 ) we can apply [10, Theorem 2.4.11 and Lemma 2.4.17] to conclude that u n → u in W 1,Φ 0 (Ω). This ends the proof.
Besides this, if ∇u Φ < , then we have So, it follows from the above information and assumption (f 1 ) that for some > 0 small enough, which shows J 2 (u) ≤ u αφ + αφ+ for all u ∈ W 1,Φ 0 (Ω) with ∇u Φ ≤ . Therefore, we obtain from this fact, hypothesis (M ) and Lemma 5.1 in (8) that holds, whenever ∇u Φ < with > 0 such above, that is, 0 is a strict local minimum of Ψ 1 in the strong topology. Hence, we obtain from Proposition 2.1 that 0 is a local strict minimum of Ψ 1 in the weak topology as well, i.e, there exists a weak neighborhood V w of 0 such that After these information and the assumption that the problem (S) admits a solution in W 1,Φ 0 (Ω), we are able to follow the same strategy of the proof of Theorem 1.1 in [12] to build disjoint open sets D 1 and D 2 , in the strong topology, such that 0 ∈ D 1 , u 0 ∈ D 2 and to findω i ∈ D i such thatω 1 andω 2 are distinct local minima of I. This ends the proof.
By applying Corollary 2.1 of [23] for functional of the type locally Lipschiz plus convex (it is a version of Corollary 3.3 in [32] that considers functional of the type C 1 plus convex), Lemmas 2.4 and 2.5, we have the following property.
In addition, assume that Problem (S) admits a W 1,Φ 0 (Ω)-solution. Then, for each λ > λ * there exists μ λ > 0 such that for μ ∈ (0, μ λ ] the functional I has three critical points, being two of them local minima and the other one a mountain pass point to the functional I.

Proof of Theorem 1.1
Before starting the proof of Theorem 1.1, let us prove the two below lemmas.
Let us prove (ii). It follows from item (i) and the Riesz theorem for Orlicz spaces that there exist a unique element in L H (Ω), still denoted by which implies by (12) that This ends the proof of lemma. Proof of Theorem 1.1-Conclusion. The proof of item (i) is inspired on ideas from [9], while for the proof of (ii) we borrow strategies from [22]. The item (iii)-(v) are consequences of Lemmas 3.1 and 3.2.
Proof of (i): We just consider the case when u is a local minimum for I. Similar arguments work when u is a local maximum for I. In this case, it is readily that (22) holds for any ϕ ∈ W 1,Φ 0 (Ω) and any > 0 given. Below, let us consider two cases. First, fix 0 ≤ ϕ ∈ C ∞ 0 (Ω). So, we obtain from Lebourg's theorem that there exist t 0 (x) ∈ (0, 1) and ξ ∈ ∂F (x, u+t 0 ϕ) such that for each x ∈ Ω.
By using (f 1 ), we are able to estimate ξ by where g ∈ L 1 (Ω) is independent of > 0. Hence, coming back to (23), we obtain for every > 0 small enough. Besides this, the right derivative of F (x, ·) at u is given by because ϕ ≥ 0.

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So, we are in position to apply the Lebesgue theorem, combined with Fatou's lemma and Lemma 3.2, in (22) x ∈ Ω. On the other hand, it follows from (21) and Lemma 2.1-(ii) that x ∈ Ω, due to the fact that u is a critical point of I.
So, as a consequence of (i) and (ii) above, a.e. in Ω. Finally, by applying Lemma 3.1, we have (iii) and (iv), while Lemma 3.2 implies (v). This ends the proof.
As a consequence of our above approach, we proved an abstract result of regularity that is itself important.

Proof of Theorem 1.2
In this section, let us begin proving the equivalences among (i), (ii) and (iii).
To prove (i =⇒ ii), we borrow ideas from Yijing [31], who treated this situation in the context of homogeneous operators. The principal difficulty in doing this is to find appropriated assumptions under the N-function Φ to become possible to obtain compactness results for minimizing sequences on Nehari sets type, while the main obstacles to prove (ii =⇒ iii) were already got over in the last section. The (iii =⇒ i) is immediately. We will end this section ensuring that the discontinuity of the nonlinearity f (x, ·) may be attained. Let us begin by defining the set which turns well-defined the functional Proof. Take u ∈ A. So, it follows from (φ 1 ) and Lemma 5.1 in [14], that and hold for t > 0 large enough. So, we obtain from (28) and (29) that σ (t) → ∞ as t → ∞ and σ (t) → −∞ as t → 0 + . Besides this, we have from (φ 1 ) again that σ (t) > 0 for all t > 0, where and so there exists a unique t * = t * (u) (which is a global minimum of σ) such that σ (t * ) = 0. This shows that t * u ∈ N * . As another consequence of the above information, we have that σ (t) ≥ 0 for all t > 0 large enough, that is, tu ∈ N for all t > 0 large enough. In particular, N is unbounded as well. This ends the proof.
By using similar ideas as done Yijing [31] for the homogeneous case, we are able to prove the below lemma in the context of non-local and nonhomogeneous operator. To complete our basics tools to prove Theorem 1.2, let us prove the below lemma that is interesting itself.
Then the problem has at most one solution in W 1,Φ 0 (Ω). Proof. First, we note that the fact of M being non-increasing implies thatM is convex. With similar arguments together with the hypotheses (φ 0 ), we show that Φ convex as well. These facts and the hypotheses (M ) lead us to infer that the functional is convex as well. Let u, v ∈ W 1,Φ 0 (Ω) be two different solutions of the problem (31). So, it follows from (30) and the convexity of J 1 , that where the last inequality follows from b, δ > 0. This is impossible and so the proof of Lemma 4.3 is done.
Proof of Theorem 1.2-Conclusion. We begin proving the first implication. Proof of (i) =⇒ (ii). First, we note that the assumption (i) implies that A = ∅. So, it follows from Lemmas 4.1 and 4.2 that N is a nonempty complete metric space. Moreover, by Lemmas 2.1 (vi), Lemma 2.2 and the fact that Rădulescu et al. NoDEA we have that J is lower semicontinuous and bounded below. Thus, by the Ekeland Variational Principle there exists a minimizing sequence (u n ) ⊂ N to J constrained to N such that: Besides this, we may assume u n (x) > 0 a.e in Ω, because J(|u n |) = J(u n ) and if we assume that u n = 0 in a measurable set Ω 0 ⊂ Ω, with |Ω 0 | > 0, then we would have from u n ∈ N , b(x) > 0 a.e in Ω and reverse Hölder inequality that which is an absurd. Thus, u n (x) > 0 a.e in Ω.
for all n large enough, which implies that (u n ) is bounded. As a consequence of this, we have that ⎧ ⎨ ⎩ u n u * in W 1,Φ 0 (Ω); u n → u * strongly in L G (Ω) for all N-function G ≺≺ Φ * ; u n → u * a.e in Ω for some u * ∈ W 1,Φ 0 (Ω). By standard arguments, we are able to show that J(u * ) = inf N J, that is, holds. So, as a consequence of (32), Fatou's lemma and Lemma 2.1-(vi), we obtain Thus, it follows from the assumption (φ 1 ), Theorem 2.4.11 and Lemma 2.4.17 in [10] that W 1,Φ 0 (Ω) is uniformly convex. This together with the weak convergence and (33), lead us to conclude that u n → u * in W 1,Φ 0 (Ω). After this strong convergence, we are able to follow similar arguments as done in Yijing [31] in the homogeneous case to prove that holds for any 0 ≤ ϕ ∈ W 1,Φ 0 (Ω) given. Hence, it follows from the same arguments as used to prove Lemma 3.1 that u * is a W 1,Φ 0 (Ω)-solution of (S) such that u * ≥ Cd for some C > 0 independent of u.
Proof of (ii) =⇒ (iii). By Corollary 2.1, there exist three critical points to functional I, being two of them local minima and the other one a mountain pass point to energy functional I. So, by Theorem 1.1 we know that each one of these critical point is a solution for the problem (Q λ,μ ) that satisfy the qualitative properties claimed.
Below, let us prove the items (iv) and (v). We are going to prove (iv) first. Let u = u a be a solution of problem (Q λ,μ ). Assume by contradiction that u ≤ a a.e. in Ω for any a > 0. So, it follows from f (x, t) = f (x) for all 0 < t < 1 and a.e. x ∈ Ω that u a ∈ W 1,Φ 0 (Ω) is a solution of that is, u a is constant in a > 0 by Lemma 4.3.
On the other hand, by taking β > δ > 1, we have that u β a > 0 can be used as a test function in (Q λ,μ ) and this yields the inequality H(a))a β for any a > 0 given. So, by doing a > 0 small enough we get an absurd, because the first term of the above inequality is a positive number that does not depends on a > 0. This ends the proof of this item.
Finally, we are going to prove v). Let u a be a solution of problem (Q λ,μ ). Assume by contradiction that u ≤ a a.e. in Ω for any a > 0 again. So, it follows that u a is a super solution to problem V. D. Rădulescu et al.
On the other hand, we are able to show that the associated-energy functional to Problem (34) is coercive due the assumption α > 1. So, by following standard arguments, we show that there exists a non-trivial 0 ≤ v ∈ W 1,Φ 0 (Ω) solution for the problem (34). That is, we have So, it follows from the hypotheses that M is such that a Comparison Principle holds, that u a ≥ u > 0 for all 0 < a ≤ 1. This fact together with the contradiction assumption lead us to have 0 ≤ u ≤ u a ≤ a for all 0 < a ≤ 1, which is impossible for a > 0 small enough, because u is non-trivial. This ends the proof of item v) and the proof of Theorem 1.2.
Finally, to occur (35) and (4) simultaneously, we have to be able to choose a 0 < θ < 1 satisfying at same time (θ − 1)φ + > −1 and θq(1 − δ) > 1 − q. We can do these by controlling the range of δ. Since if, and only if, 0 < δ < q(2φ we are able to pick a ⊂ (0, 1), whenever δ range as above. This proves that u 0 , defined as above, satisfies the condition of item (i) in Theorem 1.2. This finishes the proof.