A QUASI-LINEAR NONLOCAL VENTTSEL’ PROBLEM OF AMBROSETTI–PRODI TYPE ON FRACTAL DOMAINS

. We investigate the solvability of the Ambrosetti–Prodi problem for the p -Laplace operator ∆ p with Venttsel’ boundary conditions on a two-dimensional open bounded set with Koch-type boundary, and on an open bounded three-dimensional cylinder with Koch-type fractal boundary. Using a priori estimates, regularity theory and a sub-supersolution method, we obtain a necessary condition for the non-existence of solutions (in the weak sense), and the existence of at least one globally bounded weak solution. Moreover, under additional conditions, we apply the Leray-Schauder degree theory to obtain results about multiplicity of weak solutions.

1. Introduction and the main result.Let Ω 2 ⊆ R 2 be a bounded domain with a Koch-type fractal boundary Γ 2 := ∂Ω 2 , and let Ω 3 ⊆ R 3 be a bounded cylinder with a Koch-type fractal boundary where Γ 3 := Γ 2 × I, for I := [0, 1] (see section 2 for more details on the construction of these sets).Given p ∈ (1, ∞), denote by λ N (•) the usual N -dimensional Lebesgue measure in Ω N (N ∈ {2, 3}), and by µ(•) := H d (•) the d-dimensional Hausdorff measure, for d the Hausdorff dimension of the fractal boundary Γ i (i ∈ {2, 3}).In order to pose our problem of interest, because the notion of the normal derivative (in the classical sense) may not make sense for non-Lipschitz domains, we need to define the notion of extended normal derivative (e.g.[49,Definition 4.1]).Definition 1.1.Let µ be a Borel measure supported on Γ := ∂Ω, and let u ∈ W 1,1 loc (Ω) be such that |∇u| p−2 ∇u • ∇v ∈ L 1 (Ω, dx) for all v ∈ C 1 (Ω).If there exists a function f ∈ L 1 loc (R N , dx) such that for all v ∈ C 1 (Ω), then we say that µ is the p-generalized normal derivative of u, and we denote Note that if Ω is "sufficiently regular", for instance, a bounded Lipschitz domain, taking µ the (N − 1)-dimensional Hausdorff measure H N −1 , which in such case coincides with the classical surface measure on Γ, it follows that the above notion of generalized normal derivative coincide with the classical definition of the normal derivative.
We consider the solvability of the quasi-linear nonlocal elliptic problems of Ambrosetti-Prodi type, formally given by and (1.2) Given D ⊆ R N arbitrary, we consider In this paper, D will be either Ω, or Γ.Then, we assume that φ, h ∈ L ∞ (Ω) with φ > 0 a.e. in Ω, f : Ω × R → R is a Carathéodory function satisfying the conditions lim inf for dL (p) (•, •) the p-Lagrangian form on Γ 2 (see section 3 for complete details about the p-energy form).
When we consider the corresponding eigenvalue problem associated with the Laplace operator ∆ p , such eigenvalue problem is formulated by subject to the general dynamical boundary conditions described in the equations (1.1) and (1.2).The above structure of the Then one has the first eigenvalue, given by the zero eigenvalue λ 1 = 0, from where we see that the condition (1.4) can be regarded as a sort of of an eigenvalue crossing of f .Henceforth, one can find constants M, η 0 > 0 such that f (x, s) ≥ η 0 |s| p−2 s − M, if s > 0, (1.9) and f (x, s) ≥ −η 0 |s| p−2 s − M, if s < 0.
(1.15) Then we define the function space: (1. 16) One has that W p (Ω) is a Banach space with respect to the norm for u ∈ W p (Ω), where We next introduce the notion of weak solutions for our boundary value problems (1.1) and (1.4).
for v, w ∈ W p (Ω), where σ N is defined as in (1.18).
We now make the main assumptions that will be crucial for the main results.
We recall that if p ≥ N , by the Sobolev embedding theorem, all the discussions and estimates of the paper become much easier, with many simplifications (see section 2).Therefore, when deriving a priori estimates, we will concentrate on the critical case, namely, when p < N .Now we state the first main result of this paper, namely, an existence and nonexistence result.
Furthermore, the weak solutions of the equations (1.1) and (1.2) are globally bounded over Ω.
In the preceding theorem, if p ∈ (1, N ], it is unknown (up to the present time) whether weak solutions of either problem (1.1), or problem (1.4), are globally continuous over Ω.Consequently, because of the general structure of the elliptic equations (1.1) and (1.4), we have not been able to deduce further results, such as multiplicity results, for the case p ∈ (1, N ].However, when p > N , then one can refine more the above result, as we see in our second main result of the paper. Theorem 1.5.Assume (in addition to the conditions in Theorem 1.4) that p > N , Then there exists a parameter ξ 0 ∈ R, such that (1) problems (1.1) and (1.2) are not solvable (over W p (Ω)) for all ξ > ξ 0 ; (2) for each ξ ≤ ξ 0 , problems (1.1) and (1.2) admit respective minimal solution (in the weak sense), which are Hölder continuous over Ω.Moreover, when f is locally Hölder continuous in R, and uniformly a.e. on Ω, one gets that (3) there exists ξ 1 ≤ ξ 0 such that each of the problems (1.1) and (1.2) have at least two distinct solutions, whenever ξ < ξ 1 . ( The literature related to Ambrosetti-Prodi-type problems is extensive, and has mostly concentrated in the Dirichlet problem.The motivation of this problem comes from the pioneering paper by Ambrosetti and Prodi [3].The results in [3] opened the door to many generalizations, and further investigation of problems of this type, but in different frameworks ( structures and boundary conditions).In particular, the Ambrosetti-Prodi problem for the p-Laplace operator and Dirichlet boundary conditions has been considered in [2,4,5,25,37,39,40], among many others.It is important to mention that for the Dirichlet problem, the regularity theory is not addressed,since the domain is assumed to be smooth (and thus the regularity results are standard and known).Complications related to the Dirichlet problem of Ambrosetti-Prodi-type arise when proving the non-existence of solutions, mainly because in this case, the constant functions are not in the corresponding function space W 1,p 0 (Ω) where the Dirichlet problem is posed, and also because the first eigenvalue for the Dirichlet problem is strictly positive.To obtain non-existence results for the Dirichlet problem, some key estimates, such as the positivity of the first eigenvalue and Picone's identity, are required.The smoothness of the domain and the regularity of the solution play a crucial role in the application of such identity.
The Ambrosetti-Prodi problem for other boundary conditions is less known.On smooth domains, the Ambrosetti-Prodi problem with (local) Neumann boundary conditions has been addressed in [19,20,44,46], and recently a generalization of the (local) Neumann problem of Ambrosetti-Prodi type to a large class of non-smooth domains has been considered in [47].Furthermore, a nonlocal version of the Neumann problem was investigated in [48], where the domain was assumed to be a bounded Lipschitz domain, and in the same paper, the author introduced for the first time the Venttsel' problem (also denoted by Wentzell problem) of Ambrosetti-Prodi type (on bounded Lipschitz domains).To our knowledge, there is no literature concerning the Venttsel' problem of Ambrosetti-Prodi-type, other than the results in [48], where the Venttsel' operator is given by the p-Laplace-Beltrami operator over Γ It is important to point out that in this case, unlike the case of the Dirichlet problem, non-existence results are easier to be established, but on the other hand, other results and a priori estimates become much harder to be handled.
In the present paper, we turn our attention to the solvability of the nonlocal Ambrosetti-Prodi type problem with Venttsel' boundary conditions, on the domain Ω with fractal-like boundary Γ, given as in (1.3).Such boundary value problem has never been investigated before, and to our knowledge, there is no literature regarding the nonlocal Venttsel' problem Ambrosetti-Prodi-type on fractal domains.In fact, the only known results for (local) Ambrosetti-Prodi problems on fractal domains appears in [47] for the Neumann problem.Furthermore, when Ω = Ω 3 ⊆ R 3 , problem (1.2) is a mixed nonlocal Venttsel' -Neumann boundary value problem.To our knowledge, there are no results in the literature concerning Ambrosetti-Prodi problems with mixed boundary conditions.We also point out that problems (1.1) and (1.2), have a nonlocal term both in the interior of the domain and on the boundary.
We point out that in order to consider Venttsel' boundary conditions we introduce a suitable "surrogate" of the p-Laplacian on the boundary Γ of Ω.This operator has been introduced in the case p = 2 in [30,29,28], and recently generalized to the quasi-linear case in [16,27] (for the case when Ω = Ω 2 ⊆ R 2 ).Fractal boundaries and fractal layers are of great interest for those applications in which the surface effects are enhanced with respect to the surrounding volume (see [13,14,15] for details and motivations).Therefore, the interpretation of the equations (1.1) and (1.2) in a suitable sense represents substantial results for the theory of quasi-linear boundary value problems, as well as for the analysis on fractal domains.
We outline the plan of the paper.In section 2 we provide a brief construction of the domains under consideration, fix the notations, definitions, and state some intermediate well-know results that will be applied in the subsequent sections.In section 3, we define the p-energy functional related to the Ventssel'-type operator on Ω 2 and Ω 3 .Section 4 concerns the regularity theory for weak solutions to problems (1.1) and (1.2).Under the general conditions outlined in Assumption 1.3, assuming that the boundary value problems (1.1) and (1.2) are solvable, we perform a priori estimates for weak solutions to both problems, as well as a priori estimates for the difference of weak solutions.As a consequence, we show that weak solutions of both problems (1.1) and (1.2) are globally bounded.For bounded Lipschitz domains and for the particular case q = p (in (1.12)), these a priori estimates have been obtained in [48], and the same results for optimal growth condition (1.12) have been established for the (local) Neumann problem in [47].Our approach will be inspired by the ones employed in [48,47] but since we are dealing with more general conditions and assumptions, there are also substantial differences and generalizations in the present problem.Such variants will be addressed in detail.In section 5, we establish an alternative version of a sub-supersolution method for the boundary value problems (1.1) and (1.2), which will be a key tool for the establishment of the main results of the paper.To conclude, in section 6 we prove the main results of the paper, namely, Theorem 1.4 and Theorem 1.5.In this section we present some important (well-known) definitions, fix the notations that will be carried out in the subsequent sections, and state some known results that will be used in the later sections.All the arguments will be given under the conditions of Assumption 1.3.

2.1.
Geometry.We denote by |P − P 0 | the Euclidean distance in R n and by B(P 0 , r) = {P ∈ R n : |P − P 0 | < r}, P 0 ∈ R n , r > 0, the euclidean ball.By the Koch snowflake Γ 2 , we denote the union of three com-planar Koch curves K 1 , K 2 and K 3 .We assume that the junction points A 1 , A 3 and A 5 are the vertices of a regular triangle with unit side length, i.e.
K 1 is the uniquely determined self-similar set with respect to a family Ψ 1 of four suitable contractions ψ 4 , with respect to the same ratio 1  3 .Let V (1) 0 h .It holds that K 1 = V (1) .Now let K 0 denote the unit segment whose endpoints are A 1 and A 3 .We set In a similar way, it is possible to approximate K 2 , K 3 by the sequences (V h ) h≥0 , and denote their limits by V (2) , V (3) .
In order to approximate Γ 2 , we define the increasing sequence of finite sets of points The Hausdorff dimension of the Koch snowflake is given by d 2 = log(4) log(3) .One can define, in a natural way, a finite Borel measure µ| Γ 2 supported on Γ 2 by where µ i denotes the normalized d 2 -dimensional Hausdorff measure, restricted to Further, for any n ≥ 1, we define a discrete measure on n by: where δ {p} denotes the Dirac measure at the point p.We have the following result: Proposition 2.1.(see [31]) The sequence (µ i n ) n≥1 is weakly convergent (i.e. in C(K i ) ) to the measure µ i .
The measure µ| Γ 2 is a d 2 -measure (see [24]), that is, there exist two positive constants c 1 , c 2 , such that where d 2 = log (4)  log (3) .In the following we denote by the closed polygonal curve approximating Γ 2 at the (h + 1)-th step.We define we denote the open bounded set having as boundary F h .We denote by Q h the three-dimensional cylindrical domain having S h as "lateral surface" and the sets Ω h × {0} and Ω h × {1} as bases.
In an analogous way, we define the cylindrical-type surface Γ 3 = Γ 2 × I and we denote by Ω 2 the open bounded two-dimensional domain with boundary Γ 2 .Also, by Ω 3 we denote the open cylindrical domain having Γ 3 as lateral surface, and the sets Ω 2 × {0} and Ω 2 × {1} as bases.We denote the points of Γ 3 and S h by the couple P = (x, y), where x = (x 1 , x 2 ) are the coordinates of the orthogonal projection of P on the plain containing Γ 2 and F h respectively (for Γ 3 and S h ) and s is the coordinate of the orthogonal projection of P on the interval (0, 1), that is (x 1 , x 2 ) ∈ Γ 2 (or (x 1 , x 2 ) ∈ F h for the pre-fractal case) and s ∈ I.
We introduce on Γ 3 the measure where ds is the one-dimensional Lebesgue measure on I.It follows that the measure µ| Γ 3 is also a d 3 -measure for d 3 := 1 + log( 4) log (3) .Then we set and µ is considered as in 1.3.
• There exists a constant c 3 > 0 such that for every u ∈ W 1,p (Ω) such that E u dx = 0 for a measurable set E ⊆ Ω with λ N (E) > 0.
• If p ∈ (1, N ) and s ∈ (0, 1), then there exists a linear continuous mapping from W 1,p (Ω) into B p s (Ω) and a constant c 4 > 0 such that u (2.10) We complete this section by collecting some well-known analytical results that will be applied throughout the subsequent section.
Lemma 2.3.(see [26]) Let u be an integrable function over a bounded open set D, such that for arbitrary k ≥ k 0 > 0, where γ, α, are constants such that > 0 and 0 ≤ α ≤ 1 + .Then there exists a constant C, depending on γ, α, , k 0 , and and also in this case there is a constant ) (2.16) Proposition 2.5.(see [6]) Let ξ, ς, c, τ, ∈ [0, ∞) and r ∈ [1, ∞), and assume that the parameter ξ ≤ τ − ς + r c for all > 0. Then one has (2.17) 3. The p-forms on Koch-type domains.In [11] the p-energy forms on the Koch curve have been constructed.For f : it has been shown that the sequence n [f ] is non-decreasing, and by defining for f : the set does not degenerate to a space containing only constant functions.Each can be uniquely extended in C(K i ).We denote this extension on K i still by f and we define the space where i ) is a non-negative energy functional in L p (K i , µ i ) and the following result holds.Theorem 3.1.(see [11]) The following properties hold.
3.1.p-Lagrangians on the Koch curve and on the Koch snowflake.In this subsection, we recall the main properties of the p-Lagrangian on the Koch curve.For the concept of Lagrangians on fractals, i.e. the notion of a measure valued local energy, we refer to [22] and [42].
We also have the following: Proposition 3.2.(see [10]) Let A be any subset of K i .For every u ∈ D E (p) i , the sequence of measures given by converges to a positive semidefinite additive Borel measure the so-called p-Lagrangian measure on K i .Moreover it holds that for every p > 1, there exists in the weak * topology of M (the set of Radon measures) the following limit where we set We note that by straightforward calculations it holds that: ), so proceeding as in [10] we have Finally by proceeding as in [21, Section 4.1-4.2]one can define on Γ 2 a p-Lagrangian (L (p) , D(E (p) )) and a p-energy form (E (p) , D(E (p) )).More precisely In particular it holds ( see [21,Thoerem 4.6]) that We recall that from [12, Theorem 4.1], it follows that D(E (p) ) can be characterized in terms of Lipschitz spaces with equivalent norms: ), for every α < d.
(3.11)We now define the energy form on Γ 3 : with domain D(Γ 3 ) defined as where We now give an embedding result for the domain D(Γ 3 ).Unlike the two dimensional case where there is a characterization of the functions in D(E (p) ) in terms of the so-called Lipschitz space, for D(Γ 3 ) we do not have such characterization, but the following result holds.Proposition 3.3.(see [17]) D(S) → B p β (Γ 3 , µ| Γ 3 ) for any 0 < β < 1.From the above results it follows in particular that the spaces V p ( Ω2 ) and V p ( Ω3 ) are non trivial.
Finally, we recall that for the remaining of the paper, we will consider the sets Ω, Γ, and the measure µ on Γ, in accordance with (1.3).

4.
Global regularity for weak solutions.The following section is devoted to establish global regularity results for weak solutions of both Eq.(1.1) and (1.2), under the conditions of Assumption 1.3.
Given u ∈ W p (Ω) and k ≥ 0 a fixed real number, we put 2) Then, for each D ⊆ R N such that D ∩ Ω = ∅, taking into account (4.1) and (4.2), we write 3) We stress that throughout the remaining of the paper, the measurable set D will be either the interior Ω, or the boundary Γ := ∂Ω.Finally, we put Next, in view of the above notations, we now derive L ∞ -type estimates for weak solutions of both Eq.(1.1) and Eq.(1.2), and a priori estimates for the difference of weak solutions of problem (1.1), and problem (1.2).
for w, v ∈ D(E (p) ), where for κ i := i 1 , . . ., i n .Then, one sees easily that ∂ and thus passing to thee limit, we deduce that From the results for the two dimensional case and the monotonicity of the integral it follows that For u, v ∈ W p (Ω), we define for σ N defined as in (1.18).Then, (1.20) becomes Then we have the following key result.
Theorem 4.2.Suppose that ξ belongs to a bounded interval.If u ∈ W p (Ω) is a weak solution of either (1.1), or (1.2), then there exists a constant C ξ ≥ 0 large enough (and independent of u) such that u ∞,Ω ≤ C ξ .
Proof.We will prove both cases at the same time.Indeed, let u ∈ W p (Ω) be a weak solution of either (1.1), or (1.2).We will show that the L ∞ -norm of both u − and u + are bounded by C ξ .
Then, since λ N (Ω + k ) k→∞ −→ 0 (proof similar as previous case), we select k 1 > 0 large enough, such that ∇u Applying Hölder's inequality and recalling (2.7), one sees that In view of (4.18), (4.16) and (4.17), we have In the same way as before, we can find a constant k 2 > 0 sufficiently large (and independent of u), such that (4.20) Also, we select k 3 > 0 such that Then, for each k ≥ k 0 := max{k 1 , k 2 , k 3 }, we apply Hölder's inequality in (4.19) to obtain that , where we recall that the constant p N is given by (4.4).This last calculation together with Lemma 2.3 entail that u + ∞,Ω is bounded by a constant depending on p, N, γ 0 , ξ, φ ∞,Ω , h ∞,Ω , and u + 1,Ω + k .To complete the proof, we by a constant independent of u.In fact, noticing that k 0 := max{k 1 , k 2 , k 3 }, using (4.17) and (1.12), and applying Hölder's inequality together with Young's inequality, we have the following calculation.
Next, we establish that weak solutions of the Ambrosetti-Prodi problem (1.1) are globally bounded.Such result is automatically true when p > N (by virtue of the Sobolev inequality).Thus we will prove it for the critical case, namely, when 1 < p < N (the case p ≥ N follows in an even simpler way).Furthermore, an a priori estimate for the difference of weak solutions is established.
be Carathéodory functions satisfying the conditions (1.4), (1.11), and (1.12), let φ 1 , φ 2 , h 1 , h 2 ∈ L ∞ (Ω) with φ 1 , φ 2 > 0 a.e. in Ω, and let ξ 1 , ξ 2 ∈ R be parameters (a) If u ∈ W p (Ω) is a weak solutions of either (1.1), or (1.2), then there exists a constant c 6 (ξ) > 0 such that are weak solutions of either (1.1), or (1.2), related to f 1 , φ 1 , h 1 , ξ 1 and f 2 , φ 2 , h 2 , ξ 2 , respectively, then there exists a constant c 7 (ξ 1 , ξ 2 ) > 0 such that (4.28) Proof.We only prove part (b), for part (a) follows similarly (and even in a simpler way).Again we deal both cases at once.As the arguments run similarly as in [48, Theorem 3.2] and [49, Theorem 5.1], we will only sketch the main steps of the proof.Let f 1 , f 2 and ξ 1 , ξ 2 be as in the theorem.Recall that we are assuming all the conditions of Assumption 1.3.Now, we consider the following nonlinear form Λ for u, v ∈ W p (Ω), where the form Λ p (•, •) is given by (1.20), and the last term is defined in (4.6).By the Dominated Convergence Theorem, one has for all u, v ∈ W p (Ω).Then, we define the following sets: Here we recall that Thus in virtue of (4.29) we see that Applying Proposition 2.5 with we get from (2.17) that one gets from (4.31) that using this in (4.32), passing to the limit, and applying the Dominated Convergence Theorem, we arrive at .
• Assume now that p ∈ [2, ∞).As mentioned before (in section 1), it suffices to consider the critical case, namely, when p < N (since for p ≥ N the embedding results are much more sharp, and much better).The proof of this case follows as in the previous one (and even in a simpler way), we will only sketch the main steps of it.Given u 1 , u 2 ∈ W p (Ω) weak solutions of either (1.1), or (1.2), related to f 1 , ξ 1 and f 2 , ξ 2 , respectively, let w := u 1 − u 2 , let k ≥ k 0 be a real number (for k 0 > 0 defined as in the previous case), and let ŵk be defined by (4.1).A direct calculation (similar to the previous case) shows that for some constants ς 0 , ς 0 > 0. Proceeding exactly as in the previous case, one sees that for some constant M ξ1,ξ2 > 0 (that can be computed similarly as in the previous case), for h > k, where we recall that one can proceed in the same way as before to conclude that |u − v| ≤ C ξ1,ξ2 a.e. on Ω, ( for some constant C ξ1,ξ2 > 0. Therefore, (4.44) leads to (4.28) when p ∈ [2, N ), and completes the proof of the theorem.
5. Sub-supersolution method for nonlocal equations.In this part we will derive an alternative sub-supersolution method for the problems (1.1) and (1.2), which will be very useful in the establishment of the main results of the paper.Although some arguments will follow similar approaches as in [33,34], to our knowledge, there is no sub-supersolution method for nonlocal equations of type (1.1) and (1.2).Recall that all the arguments will be carried out under the assumptions (1.4), (1.11), and (1.12).To begin our discussion, given F ⊆ W p (Ω) closed and convex (where we recall that W p (Ω) is defined by (1.16)), we consider the following variational inequality where we recall that Λ p (•, •) is defined by (1.20).Before giving the corresponding definitions for subsolutions and supersolutions of the Eq.(5.1), we will fix some additional notations that will be frequently used in this section.Indeed, for u, v ∈ W p (Ω) and D, F ⊆ W p (Ω), we denote u ∨ v := max{u, v}, u ∧ v := min{u, v}, and where denotes either ∨ or ∧.
6. Proof of the main result.In this section we focus our attention in establishing the main results of this paper, namely, Theorem 1.4 and Theorem 1.5, assuming all the conditions in Assumption 1.3.Our approach will be follow arguments similar as in [48] (with some ideas motivated by the results in [19]).Thus, the proofs of some intermediate results will be omitted, but as our problem under consideration is much more general and several results require generalizations and modifications, more details will be given when needed.In particular, we will give proofs to most of the results here, and outline the main steps, especially when the generalizations come into play.We begin by providing a non-existence result.
then the problems (1.1) and (1.2) have no weak solution over W p (Ω).
Proof.For simplicity, assume that Ω = Ω 3 ∈ R 3 (the other case is analogous).Given c ξ defined as in (6.1), by virtue of (1.10) we have Selecting ϕ ∈ W p (Ω) + arbitrarily, multiplying the Eq. ( 6.2) by ϕ, and integrating over Ω, we get Thus from (6.3), we get proving that c ξ is a weak subsolution of the problem (1.2).Furthermore, if follows immediately that if c is a constant less than c ξ , then c becomes a strict subsolution of the Eq.(1.2).It remains to show that c ξ is a lower bound for any weak solution u ∈ W p (Ω) of (1.2).We suppose that the contrary holds, that is, assume that there exists a constant 0 > 0 such that the function (c ξ − 0 − u(x 0 )) + is strictly positive for some x 0 ∈ Ω.Given ∈ (0, 0 ), we set Notice that (Θ Ω u)u ,ξ ≤ 0, (Θ Γ u)u ,ξ ≤ 0, and also in view of (4.5) and (4.6), we deduce that K (p) (u, u ,ξ ) ≤ 0. As ∇u ,ξ = −∇u, testing (1.19) with the function u ,ξ and applying (1.9), we obtain Recalling the definition of c ξ and the selection ∈ (0, 0 ), the above estimate yields ∇u p p,Ω ,ξ < 0, a clear contradiction.Consequently, u(x) ≥ c ξ for all x ∈ Ω, completing the proof.Now we establish the first main result of this article.
Proof.[Theorem 1.4] Assume the condition of the theorem.From Lemma 6.1 one has the statements (1).To prove the other assertion, for simplicity (as in the previous lemma) we assume that Ω = Ω 3 ∈ R 3 .Given ξ, ξ ∈ R with ξ ≤ ξ 0 , we notice that the zero function 0 is a weak supersolution of (1.2) if and only if ξφ(x) ≤ −f (x, 0) − h(x) for a.e.x ∈ Ω.Thus, set and consequently 0 is a weak supersolution of (1.2) for all ξ ≤ ξ 0 .On the other hand, by Lemma 6.2, the negative constant c ξ given by (6.1) is a weak subsolution of (1.2) for all ξ.Hence, Theorem 5. Next we establish the second main result of the paper.
2and Remark 1 imply that (1.2) is solvable for all ξ ≤ ξ .Moreover, if (1.2) is solvable for some parameter ξ, and u ∈ W p (Ω) is the corresponding weak solution, as φ > 0 a.e. in Ω, it follows that u is a weak supersolution of (1.2) corresponding to the parameter ζ, for ζ < ξ.