A singularity as a break point for the multiplicity of solutions to quasilinear elliptic problems

where Ω ⊂ RN is a bounded smooth domain, 0 μ ∈ L∞(Ω), 0 f ∈ Lp0 (Ω) for some p0 > 2 , 1 < q < 2, α ∈ [0, 1] and λ ∈ R.We establish existence andmultiplicity results for λ > 0 and α < q−1, including the nonsingular case α = 0. In contrast, we also derive existence and uniqueness results for λ > 0 and q − 1 < α ≤ 1. We thus complement the results in [1, 2], which are concernedwith α = q−1, and show that the value α = q−1 plays the role of a break point for the multiplicity/uniqueness of solution.


Introduction
In this paper we deal with the following boundary value problem: in Ω, u = on ∂Ω.
Here, Ω is a bounded domain of R N (N ≥ ) with boundary ∂Ω smooth enough, µ ∈ L ∞ (Ω), f ∈ L p (Ω) for some p > N , < q < , ≤ α ≤ and λ ∈ R. A solution to (P λ ) is a function < u ∈ H (Ω) ∩ L ∞ (Ω) which satis es the equation in (P λ ) in the usual weak sense (we will be more precise about the concept of solution in De nition 3.1 below). Observe that, if α > , then the lower order term presents a singularity as u approaches zero, i.e., as x approaches ∂Ω. Our goal is to study the existence, nonexistence, uniqueness and multiplicity of solutions to (P λ ), specially for λ > .
Hence, roughly speaking, mild singularities at zero do not alter the behavior of the solutions, as far as the multiplicity for λ > is concerned. Nonetheless, the main result in that paper shows that multiplicity fails for < q ≤ and α = q − (see [33] for q = and µ constant). To be precise, the authors prove under natural hypotheses on µ and f that, if α = q − , there exists λ * ∈ ( , λ ] (where λ = inf v∈H (Ω)\{ } Ω |∇v| / Ω v ) such that problem (P λ ) has a solution if and only if λ < λ * , and in this case, the solution is unique (see also [2] for a similar existence result when f and u may change sign). In particular, one has existence and uniqueness for λ > small. Since this result is true for < q ≤ , it is natural to wonder whether α = q − is a break point for the multiplicity of solutions not only in the case q = , but also for < q < .
In the present work we contribute to these topics by proving that, if there is a solution to (P ), then there are at least two di erent solutions to (P λ ) for all λ > small enough provided q and α satisfy certain relations involving also the dimension N. We prove also that the branch of positive solutions bifurcates from in nity to the right of the axis λ = .

(H1)
Observe that µ is bounded away from zero but not necessarily constant. We introduce here the main result of this paper: Theorem 1.1. Assume that (H1) holds and that (P ) admits a solution u . If q > N N− , suppose also that Then, there existsλ ∈ ( , λ ) such that problem (P λ ) admits at least two di erent solutions for all λ ∈ ( ,λ]. Moreover, zero is the unique bifurcation point from in nity to problem (P λ ).
Even though this result deals only with the range λ > , in order to make a more complete picture we will gather and prove in Section 3 some existence, nonexistence and uniqueness results about problem (P λ ) for λ ≤ . We stress that the uniqueness result for λ ≤ , apart from being new in the literature, shows that λ = is a critical point beyond which the nature of the problem changes drastically, as in the well-known case q = and α = .
Concerning the proof of Theorem 1.1, the idea is to derive a priori estimates of the solutions to (P λ ) for all λ > λ which are independent of λ > . This idea rst appeared in [15] for q = and α = , but the approach for deriving the estimates does not work in our framework. For our purposes, it is more convenient to use the arguments developed in [16], which allow us to nd L p estimates of supersolutions. After that, we establish a bootstrap argument, which works thanks to some results in [9], that yields an L ∞ estimate. Actually, these results are valid only in the nonsingular case α = , so we will extend some parts of them to our singular framework. After writing the present work, it came to the author's knowledge that similar results extending [9] to a more general setting have been recently obtained in [36].
Hypothesis (1.1) in Theorem 1.1 deserves some comments. It appears in the proof as a result of the combination of the mentioned techniques from [16] and the bootstrap from [9]. However, we presume that this is a technical assumption forced by the tools we employed, so the theorem might admit some improvements. In order to clarify the meaning of this condition, we derive two corollaries below in which simpler conditions assuring (1.1) are imposed. For instance, if we consider the sequence then q ∈ ( , Q N ] \ { } implies (1.1), with no extra hypotheses on α apart from ≤ α < q − (see Corollary 3.17). Observe that Qn > but limn→∞ Qn = . This means that, if N is large, then q has to be chosen close to . However, one would expect a multiplicity result for any q ∈ ( , ) and any N. This still remains as an open problem. In any case, Corollary 3.17 represents a remarkable advance, in particular, about the nonsingular problem (R λ ). Changing the point of view, we give in Corollary 3.18 below a condition on α that is su cient for applying Theorem 1.1 even for q close to and for N large.
With the aim of having a deeper insight into problem (P λ ), we also consider in this work the case q − < α ≤ . In contrast to the previous situation ( ≤ α < q − ), we will prove that existence and uniqueness hold for λ > small enough. For this purpose, we will need the following assumption on Ω: There exist r , θ > such that, if x ∈ ∂Ω and < r < r , then |Ωr| ≤ ( − θ )|Br(x)| for every connected component Ωr of Ω ∩ Br(x). (A) Note that, if ∂Ω is Lipschitz, then Ω satis es (A) (see [7]), so this represents only a mild restriction. The precise hypotheses that we need are gathered here: We emphasize that µ is allowed to vanish in subsets of Ω with nonzero measure.
The statement of the main result in the q − < α ≤ case is the following: Theorem 1.2. Assume that (H2) holds. Then there exists a solution to (P λ ) for all λ < λ , and there exists no solution to (P λ ) for all λ ≥ λ . Moreover, the solution is unique for all λ ≤ and, if f satis es that then the solution is unique for all λ < λ . Finally, λ is the unique bifurcation point from in nity to problem (P λ ).
Even though we are specially interested in the uniqueness part, the existence statement in Theorem 1.2 deserves also attention. Observe that one has existence of solution if and only if λ < λ . This suggests that the nonlinear term does not play an essential role in this case, since the situation is analogous to the linear problem (µ ≡ ). Recall that this is not the case when α = q − , for which one has existence if and only if λ < λ * , where λ * < λ provided µ > (see [1,Remark 6.3]).
The proof of the existence of solution in Theorem 1.2 is performed by passing to the limit in certain family of approximate nonsingular problems. We will derive Hölder continuous a priori estimates on the solutions to such a family, which will allow us to pass to the limit. For proving such estimates, the assumption α ≤ is essential (see Remark 3.3 below). Moreover, the continuity of the solutions is also essential to prove their uniqueness. Indeed, we state and prove in Section 2 two comparison principles valid for continuous lower and upper solutions to singular equations. As far as we know, these two results are new, and they are interesting by themselves as only few uniqueness results for singular equations are known (see [1,33,[37][38][39]). We follow in their proofs the arguments in [7] and [1].
As a summary, our results contribute to the theory of equations with subquadatic growth in the gradient, extending what it is known about the multiplicity of solutions in the quadratic case. On the other hand, they can be seen as a link between the singular and nonsingular theory, in the sense that they show that the presence or not of a singularity is determining only if it is strong enough. Finally, new existence and uniqueness results are given for strong singularities, where the uniqueness part is specially remarkable.
We organize the paper as follows: in Section 2 we deal with the mentioned comparison principles; we devote Section 3 to prove Theorem 1.1 as well as some auxiliary results and some consequences of the mentioned theorem; Section 4 contains the proof of Theorem 1.2, and Section 5 is an appendix where we prove a continuation result needed in the proof of Theorem 1.1.

Acknowledgments
The author wants to thank warmly T. Leonori and J. Carmona for their helpful contributions to this work. He wants to thank also A.J. Fernández and M. Magliocca for useful conversations and suggestions.

Notation
• For every x ∈ R N , the distance from x to ∂Ω will be denoted as δ(x). Furthermore, for p ≥ we will denote as L p (Ω, δ) the space of measurable functions u : Ω → R such that identifying functions equal up to a set of zero measure. • For p ≥ , we will denote the usual Marcinkiewicz space as M p (Ω), i.e., the space of measurable functions u : Ω → R for which there exists c > such that |{|u| > k}|k p ≤ c for all k > . In this case, we denote • For k ≥ , the usual truncation functions will be written as T k (s) = max{−k, min{s, k}} and G k (s) = s − T k (s) for all s ∈ R. • The principal eigenvalue of the −∆ operator in Ω under zero Dirichlet boundary conditions will be denoted as λ . In other words, λ is the unique real number satisfying that the equation −∆φ = λ φ has a solution < φ ∈ H (Ω). We will write φ for the positive eigenfunction associated with λ such that φ L ∞ (Ω) = .

Comparison principles
We start with a comparison principle valid for singular equations. The proof basically follows the steps of a similar result in [7]. However, up to our knowledge this is the rst time that a comparison result has been proved including a general positive singular lower order term on the right hand side of the equation (see the comparison results in [1], where a speci c -homogeneous singular term is considered).
for every ≤ ϕ ∈ H (Ω) ∩ L ∞ (Ω) with compact support. Suppose also that the following boundary condition holds: Then, u ≤ v in Ω.
Remark 2.2. Theorem 2.1 is valid for a wide class of lower order terms. For instance, the model example is e. x ∈ Ω, ∀s > , for any α > and ≤ µ ∈ L ∞ loc (Ω). In particular, the growth of the singularity is irrelevant in the proof. Nonetheless, the comparison principle does not work for λ > . Indeed, as we pointed out in the Introduction, if the singularity is mild enough in some sense, then a multiplicity phenomenon appears for λ > . Thus, for the model case, the comparison result is sharp in terms of the sign of λ. Remark 2.3. In Theorem 2.1, u, v ∈ C(Ω) are not assumed to be continuous up to ∂Ω, so a suitable ordering condition on the boundary is given by (2.3). However, if u, v ∈ C(Ω), then (2.3) is equivalent to the usual and more natural condition Proof of Theorem 2.1. Let us denote w = u − v. For k > , we consider the function ϕ = (w − k) + , and we also denote Notice that supp(ϕ) ⊂ A k . Moreover, condition (2.3) implies that A k ⊂⊂ Ω, so ϕ has compact support. In particular, ϕ ∈ H (Ω) ∩ L ∞ (Ω), so it can be taken as test function in (2.1) and (2.2), obtaining that and Since λ ≤ , we deduce that Assume in order to achieve a contradiction that w + ≢ , and let k ∈ ( , w + L ∞ (Ω) ). Let also ω ⊂⊂ Ω be an open set such that A k ⊂ ω. Observe that A k ⊂ A k for all k ≥ k . Then, using the properties of g, it is clear that . For every j ∈ R, let us denote Ω j = {x ∈ Ω : |w(x)| = j}, and consider also the set J = {j ∈ R : |Ω j | ≠ }.
Since |Ω| < ∞, then J is at most countable, which implies that the set j∈J Ω j is measurable, and we also have that Hence, if we de ne the set Z = Ω \ j∈J Ω j , we deduce from (2.7) that Taking into account that u, v ∈ W ,N loc (Ω) and A k ⊂⊂ Ω, we have that Hence, from (2.8) we derive that Let us now de ne the function F : and F( w + L ∞ (Ω) ) = . It is clear that F is nonincreasing and continuous. Thus, choosing k close enough to w + L ∞ (Ω) , we deduce from (2.9) that (w − k) + ≡ . That is to say, w ≤ k in Ω. But this is not possible since k < w + L ∞ (Ω) = sup Ω (w). In conclusion, we have proved that w + ≡ , i.e., w ≤ in Ω.
Next theorem is another comparison principle which works for λ > . In turn, one has to impose stronger hypotheses on g and h. The proof is similar to the one above combined with some ideas in [1]. Theorem 2.4. Let < q ≤ , λ ∈ R, ≤ h ∈ L loc (Ω) and g : Ω × ( , +∞) → [ , +∞) satisfying s → s q− g(x, s) is nonincreasing for a.e. x ∈ Ω, is locally essentially bounded for all s > .
We claim that w + ε ≡ for any ε > . Suppose by contradiction that there exists ε > such that w + ε ≢ . Let us x k ∈ , w + ε L ∞ (Ω) and ε ∈ ( , ε ), the latter to be chosen small enough later. It is clear that wε ≤ wε in Ω, so w + ε ≢ . For k ∈ [k , w + ε L ∞ (Ω) ], let us denote Notice that supp(wε − k) + ⊂ A k . By (2.11), we also have that lim sup x→x wε(x) ≤ for all x ∈ ∂Ω, which implies that A k ⊂⊂ Ω. Then, the function (wε −k) + has compact support, and in particular, (wε −k) + ∈ H (Ω)∩L ∞ (Ω). Therefore, we may take (wε−k) + u as test function in (2.1), and (wε−k) + v+ε in (2.2), obtaining and, using that g ≥ , Let ω ⊂⊂ Ω be an open set such that A k ⊂ ω. Observe that A k ⊂ A k for all k ≥ k . Then, it is clear that Moreover, we have that whenever λ ≤ . On the other hand, if λ > , let us take where cω is the constant given by (2.10). With this choice, it is straightforward to deduce that (2.14) holds again.

Multiplicity for ≤ α < q −
In this section we will study problem (P λ ) under condition (H1). In this case observe that, if < u ∈ W , loc (Ω) and t > , then Since α < q − , then q − α > . That is to say, the lower order term has superlinear homogeneity.
The concept of solution we will adopt is gathered in the following de nition. and Reciprocally, a supersolution to (P λ ) is a function u ∈ H (Ω)∩L ∞ (Ω) such that u > a.e. in Ω, µ |∇u| q u α ∈ L loc (Ω) and satis es the reverse inequality. Finally, a solution to (P λ ) is a function u ∈ H (Ω) ∩ L ∞ (Ω) which is both a subsolution and a supersolution to (P λ ).

Remark 3.2. Arguing as in
This fact allows us, in particular, to take u itself as test function.

Remark 3.3.
Assume that (H1) holds. By taking φ as test function in the weak formulation of (P λ ) one easily deduces that, if u is a solution to (P λ ), then λ < λ . Furthermore, since α ∈ [ , ], it can be proved as in [1,Appendix], which follows the ideas in [40], that every solution u to (P λ ), for any λ < λ , satis es that u ∈ C ,η (Ω) for some η ∈ ( , ). Finally, since the solutions to (P λ ) are positive in compact subsets of Ω, then it can be seen again as in the mentioned appendix that u ∈ W ,N loc (Ω) for every solution to (P λ ) for any λ < λ .
Our rst result is concerned with the existence and uniqueness of solution to (P λ ) for λ ≤ . The existence is well-known from the works that are quoted in the proof below. However, a precise statement for unbounded datum f is required for our purposes. In any case, the uniqueness is new up to our knowledge.

Proposition 3.4.
Assume that (H1) holds. Then, problem (P λ ) has a unique solution for all λ < . Moreover, assume additionally that either α > or the following smallness condition holds: where B ( ) denotes the unit ball in R N , and a, b > are such that Then (P ) has a unique solution.
Observe now that, by Young's inequality, there exist C , C > such that Then, the hypotheses of [31, Proposition 4.1] are ful lled, so there exists a solution u ∈ H (Ω)∩ L ∞ (Ω) to (P ) in some weaker sense than De nition 3.1. Nonetheless, since f in Ω, then the strong maximum principle implies that u > in Ω, so u is in fact a solution to (P λ ) in the sense of De nition 3.1.
Concerning the existence for λ < , we argue by approximation as follows. For all n ∈ N, let us consider the problem Since (3.1) and (3.2) hold, we know from [29] that there exists a solution un Hence, Theorem 2.1 applies (see Remark 3.3) and yields In other words, {un} is bounded in L ∞ (Ω). By taking un as test function in the weak formulation of (3.3), we immediately deduce that {un} is also bounded in H (Ω). Hence, there exists u ∈ H (Ω) ∩ L ∞ (Ω) such that, passing to a subseqence, un u weakly in H (Ω) and un → u strongly in L p (Ω) for any p ∈ [ , ∞).
Observe also that, again by comparison, un ≥ z for all n, Now, the strong maximum principle applied on z implies that ∀ω ⊂⊂ Ω ∃cω > : un ≥ cω a.e. in ω, ∀n. The uniqueness of u is a direct consequence of Theorem 2.1 and Remark 3.3.

Remark 3.5.
For the sake of simplicity, we have assumed (H1) in Proposition 3.4. Nevertheless, as it has been shown in the proof, the condition µ ≥ µ is not needed, only µ ≥ is su cient.
Next result shows that, if α = , then the existence of solution to (P ) may fail if f or µ are too large in some sense, in contrast to the case α > . Thus, the smallness assumption in Proposition 3.4 is justi ed. This result is basically contained in [10, Theorem 2.1]. We include the statement and proof in our context for completeness.
Proposition 3.6. Assume that (H1) holds with α = , and suppose that (P λ ) admits a solution for some λ ≥ . Then, Proof. Let u be a solution to (P λ ), and let ≤ ϕ ∈ W ,q (Ω) ∩ L ∞ (Ω). Since q > , then ϕ q ∈ H (Ω) ∩ L ∞ (Ω), so it can be taken as test function in the weak formulation of (P λ ) to obtain, after using Young's inequality, that Hence, it is now clear that the result follows.
Our aim in the next two subsections is to prove, for a xed λ > , an L ∞ estimate for the solutions to (P λ ) for all λ > λ . Such an estimate implies that zero is the only possible bifurcation point from in nity to problem (P λ ). This fact will be the key to prove multiplicity of solutions to (P λ ) for λ > small enough.

. A priori L p estimates
This subsection is devoted to proving an L p estimate on the supersolutions to (P λ ) for λ > . The techniques employed here have been taken from [16].
The rst result of the subsection provides an apparently weak local estimate on the solutions to (P λ ). Notwithstanding, this is the starting point for proving the L ∞ estimate we are aiming at. Concerning the proof, we will argue similarly as in Proposition 3.6. Lemma 3.7. Assume that (H1) holds. Then, for every λ > and ω ⊂⊂ Ω there exists C > such that ω u ≤ C. (3.4) for every supersolution u to (P λ ) with λ > λ .
Proof. Let ϕ ∈ C c (Ω) be such that ω ⊂⊂ supp(ϕ), ≤ ϕ ≤ in Ω and ϕ = in ω. Taking ϕ β ∈ C c (Ω) for some β > as test function in (P λ ) and using Young's inequality twice we obtain that Taking β = q q− −α , the last term in the previous inequality is bounded. Therefore, so (3.4) follows by taking into account that ϕ = in ω.
The following is a slightly more general version of [ Proof. Let us consider the following problem for all n ∈ N: It is well-known that it has a unique solution vn ∈ C ,ν (Ω) for all ν ∈ ( , We conclude the proof by letting n tend to in nity.
Next lemma is an immediate consequence of Lemma 3.8.
Combining Lemmas 3.7 and 3.9 we obtain in the following result some estimates in weighted Lebesgue spaces.
Proof. Integrating both sides of inequality (3.5) over any open set ω ⊂⊂ Ω and using the estimate (3.4) we deduce that In particular, and this is equivalent to item (2). Regarding item (1), observe that Hence, by [42, Proposition 2.2] we obtain directly item (1).
We nish the subsection with the best L p estimate for supersolutions that we obtain with these techniques. for every supersolution u to Proof. Let us denote v = u − α q . Since − α q > , we can argue as in [1, Lemma 2.6] to prove that v ∈ H (Ω). Then, [16,Proposition 2] implies that Hence, by Lemma 3.10 we derive that Now, [16,Lemma 3] implies that It is easy to check that, in fact, γ = . Therefore, recalling that m = b − α q , by (3.8) and (3.7) we conclude that and the result holds true.

. A priori L ∞ estimates
In this subsection we will show how to obtain L ∞ estimates on the solutions to (P λ ) for λ > by combining the L p estimate given by Lemma 3.11 and a bootstrapp argument. We will make use of several results in [9]. In fact, the ideas in such a paper will be used also to derive some new results which provide analogous estimates in our singular framework.
We start the subsection with the easier case α = , which is interesting itself; we will deal with the singular case α ∈ ( , q − ) later. Thus we state and prove the following for every solution u to (P λ ) with λ > λ .
Proof. In this proof, C denotes a positive constant independent of u and λ whose value may vary from line to line.
We start by assuming that < q < N N− . Observe that N N− < Q N , so q ≤ Q N is not a restriction in this case. Let us denote h(x) = (λ + )u + f (x). Then, u satis es We know from Lemma 3.11 that u L m (Ω) ≤ C, where m = (q−α)N N−q+ , so h L p (Ω) ≤ C, where p = min{m, p }. If m > N , and taking into account that p > N , then [9, Theorem 5.8, item (i)] implies that u L ∞ (Ω) ≤ C.
Let us assume now that m = N . Then, [9, Theorem 5.8, item (ii)] implies that u L p (Ω) ≤ C for all p < ∞. In particular, h L p (Ω) ≤ C. Since p > N , then again item (i) of the same mentioned theorem yields the L ∞ estimate.
Suppose now that ( * ) < m < N . Let us de ne the sequence {mn} inductively as where m = m. This is clearly an increasing sequence. Moreover, using one more time [9, Theorem 5.8, item (iii)], it is easy to see that u L mn (Ω) ≤ C for n ∈ N as long as mn < N . In particular, the same holds for h.
Assume by contradiction that mn < N for all n ∈ N. Since {mn} is increasing and bounded from above, there exists l ≤ N such that, passing to a not relabeled subsequence, mn → l. Consequently, From this equality we deduce that l = . But this is a contradiction because m > and the sequence is increasing. Therefore, mn ≥ N for some n ∈ N, so the previous cases imply that u L ∞ (Ω) ≤ C.
It only remains to consider the case < m ≤ ( * ) . Now, item (iv) of the same theorem implies that On the other hand, it is straightforward to prove that, for any a ∈ ( , ), there exists a constant b > such that Then, with mn = m ** n− and m = m, as before, In particular, h L m (Ω) ≤ C. It can be proved inductively that u L mn (Ω) ≤ C as long as mn ≤ ( * ) . Arguing as above, we deduce that {mn} is increasing and divergent. Hence, mn > ( * ) for some n ∈ N, and the proof concludes using the previous cases.
We now turn to the range N N− < q < . The procedure is the same as above, but in this case, instead of Theorem 5.8, one has to apply (a nite number of times) either [9,Theorem 4.9] or [9, Theorem 3.8], depending on the value of q. In both cases, one has to verify in the rst step of the bootstrap that h ∈ L (q− )N q (Ω) so that the hypotheses of both theorems are satis ed. We know by virtue of Lemma 3.11 that h ∈ L m (Ω), so we have to impose that One can easily check that the previous inequality is satis ed if and only if q ≤ Q N .
It is left to consider the case q = N N− . Since N N− < Q N , we can take ε > small enough so that Moreover, we have by Young's inequality that where h(x) = (λ + )u + f (x) and hε(x) = h(x) + Cε for some Cε > . Therefore, the previous case can be applied and the proof concludes.
We deal now with the singular case. For this purpose, it is necessary to derive results similar to the ones from [9] mentioned in the previous proof, but valid for singular equations. Even though our results are not proper extensions in the whole generality (as in [9] the solutions are weaker than ours and the terms in their equation are not explicit and only satisfy growth restrictions), they are new in considering singular terms.
The mentioned results will be concerned with the following auxiliary problem: where the parameters satisfy < q < , α ∈ [ , q − ), β > , µ ∈ L ∞ (Ω). The following result provides estimates on solutions to (3.10) when q is large and h has enough summability. Observe also that, if

so the condition in item (1) may be ful lled only if
We will assume consequently that q belongs to such an interval. In fact, we will divide the proof of this item into several steps, considering di erent ranges for p and q. It can be easily checked that each of these ranges is nonempty.
Then, it is clear that the following relation is satis ed: Let us now consider the following functions de ned for every t ≥ : where ζ > will be xed later. First of all observe that for any v ∈ H (Ω). Moreover, using (3.13) and also that σ − = * σ p , it can be proved respectively that For k > , let us take Φ (G k (u)) ∈ H (Ω) ∩ L ∞ (Ω) as test function in the weak formulation of (3.10), so that we obtain Let us now estimate the nonlinear term. Thanks to (3.14) we derive that We now focus on the last term in (3.16). Using (3.15) we deduce that If we denote Y k = Φ (G k (u)) H (Ω) , we have proved so far that Hence, using Young's inequality we obtain that or equivalently, for some C , C > independent of k and ζ . Let us de ne the function F : [ , +∞) → R by Since q < , it easy to see that < q + * − q .
This means that F is positive near zero, negative far from zero, and has a unique maximum F * > with a corresponding unique maximizer Z * > .
We now choose ζ = min , F * * σ . Thus, Let us now consider Hence, for any ρ > , the equation F(Y) = ζ * σ + hχ {h(x)≥β(k * +ρ)} σ L p (Ω) has two roots Z and Z such that Z < Z * < Z . By virtue of inequality (3.17), it holds that for every k ≥ k * + ρ, either Y k ≤ Z or Y k ≥ Z . But the function k → Y k is continuous and tends to zero as k tends to in nity. Therefore, If we let now ρ tend to zero, we obtain that Hence, we have that We claim now that Indeed, let us de ne the real functions for all t ≥ : It is easy to see that and also that y(t) ≤ Cz(t) ∀t ≥ , for some C > .
Now we take T k (u)y(u) as test function in the weak formulation of (3.10) and get where, by virtue of (3.18), Gathering (3.20), (3.21) and (3.22) together we deduce that We will show now that there exists k > independent of h L p (Ω) such that k * ≤ k . Indeed, the absolute continuity of the integral implies that there exists ρ > such that, if |{|h(x)| ≥ βk }| < ρ for some k > , where h L p (Ω) ≤ M, then |{|h(x)| ≥ βk }| < ρ and k does not depend on h L p (Ω) , as we wanted to show. Therefore, we can estimate k * in (3.24) and, by virtue of (3.18), we obtain that We nally arrive at (3.19) by using Young's inequality and by the fact that z is a bounded function. This concludes the proof of Case 1. Recalling the de nitions of ϕ, Φ and Φ in the previous case, for some k > we take Φ (G k (u)) as test function in the weak formulation of (3.10), so that we obtain It can be easily proved that for some C > . Thus, using this inequality in the singular term of (3.25), we deduce that

Now we claim that
for some k > large enough. Indeed, since q < + N , we can apply Hölder's inequality with exponent N(q− ) > and obtain that, for any k > , Therefore, by Case 1 and Sobolev's inequality, Hence, the fact that σr < implies that and the proof of the claim is done. As a consequence, it can be shown, again by virtue of the absolute continuity of the integral, that the limit is uniform in u. Hence, from (3.26) we deduce that there exists k > independent of u such that Then, we derive from (3.25) that By virtue of (3.15) we immediately obtain the estimate We conclude this case similarly as Case 1.

Case 3:
Thus, the proof of Case 1 can be reproduced here. We conclude this way the proof of item (1). (2).

Proof of
In this case, there exists θ ∈ , q − − N (3.13) holds. Now, for k > , let us take G k (u) σ− as test function in the weak formulation of (3.10). Notice that this choice is valid since σ > . Then, following the arguments of the proof of Case 1 of item (1) we obtain that where k * = inf k > : hχ {|h(x)|≥βk} L p (Ω) < F * and F * > is the unique maximum of the function Observe that Therefore, Now we take T k * + (u) as test function in the weak formulation of (3.10) so we get Again, the absolute continuity of the integral implies that k * ≤ k for some k > independent of h L p (Ω) . Thus we can estimate k * in the last inequality and, using Young's inequality, deduce that Ω |∇T k * + (u)| ≤ C.
Summarizing, Ω |∇u| ≤ C, which proves the rst part of item (2). Moreover, Thus, the proof of Case 1 is concluded.
Let us denote, as above, r = N(q− ) , so Case 1 of item (2) can be applied. For some k > , we take G k (u) σ− as test function in the weak formulation of (3.10), so we obtain In order to estimate the nonlinear term, notice that Hence, we can use Hölder inequality with those three exponents, and we deduce that Now, thanks to Case 1 of item (2) and the absolute continuity of the integral, there exists k > independent of u such that Then, from (3.27) we derive that Since ( σ − )p = * σ, we conclude that Clearly, Finally, using that u is bounded in H (Ω) (from Case 1), we obtain that This proves Case 2.
Case 3: N N− < q < + N and N N+ ≤ p < N . Here one can argue as in Case 2 of the proof of item (1), but considering this time ϕ(s) = s σ− for all s ≥ .
Let us take G k (u) as test function in the weak formulation of (3.10) for some k > , so we obtain this time, removing the term with β, (3.28) We consider now two di erent cases.
In this case, we have that r = N(q− ) q ∈ N N+ , N , so σr = (N− )r (N− r) ≥ . On the other hand, it can be checked that Then, we can use Hölder's inequality in such a way that Next, by item (2) we can take k ≥ k , with k independent of u, so that G k (u) q− is small enough. Then, from (3.28) we deduce that We conclude by using the Stampacchia's method in a direct way.
Case 2: N N− < q < + N . In this case, Hölder's inequality yields By Case 2 of item (2), we can take k ≥ k , with k independent of u, such that Ω |∇G k (u)| N(q− ) is small enough. Then, from (3.28) we deduce that and we can apply again Stampacchia's method.
The proof is now concluded.
We prove now a result analogous to Proposition 3.13 for q small.
Proposition 3.14. Assume that q, α, β, µ satisfy (3.11), and assume in addition that Then, for all M > and p ≥ , there exists C > such that, for any h ∈ L p (Ω) with h L p (Ω) ≤ M and for any solution u to problem (3.10), the following holds: 1. If p = , then u Proof. We will prove rst item (1). Thus, for j, k > , let us take T j (G k (u)) as test function in the weak formulation of (3.10), so we obtain On the one hand, it is clear that On the other hand, concerning the right hand side of (3.29), we obtain that In sum, we deduce that Then, we apply [43,Lemma 4.2], so that we deduce that Since q < N N− , we have the immersions Therefore, We now consider the function F : [ , ∞) → R de ned as and we denote Y k = ∇G k (u) L q (Ω) .
Thus we have proved that The proof of this part concludes as in the previous proposition. The proofs of the rest of the items follow the same arguments of Proposition 3.13. We only stress that the estimate Therefore, the estimate holds by virtue of item (1).
The same arguments of the proof of Proposition 3.12 (but using Propositions 3.13 and 3.14 instead of the results in [9]) are valid also for proving the main result of this subsection.

. Proof of the main result and consequences
We prove now the main result of the paper.
We conclude the section by stating and proving two corollaries of Theorem 1.1. The rst one provides multiplicity of solutions for q small, but for any α ∈ [ , q − ).
It can be proved that z is increasing. Indeed, Using that N ≥ and q < , it is straightforward to deduce that which means that z (s) > for all s ∈ [ , q − ). Moreover, since q ≤ Q N , then z( ) ≥ (see Proposition 3.12). Thus, z(α) ≥ , or equivalently, condition (1.1) holds and Theorem 1.1 can be applied.
The second corollary gives multiplicity of solutions for all q ∈ ( , ) at the expense of taking α close to q − .

Uniqueness for q − < α ≤
We will consider in this section problem (P λ ) under condition (H2). Observe that if < u ∈ W , loc (Ω) and t > , then In this case, α > q − , so q − α < . That is to say, the lower order term has sublinear homogeneity. We will prove the existence of solution to (P λ ) after deriving certain a priori estimates on an approximate problem and passing eventually to the limit, in a way that such a limit will be the solution we look for. Thus, consider the following approximate problem: (4.1) In the next lemma we show that problem (4.1) admits a solution. Proof. Fix n ∈ N and λ < λ . Then, the following linear problem has a solution < ψ ∈ H (Ω) ∩ L ∞ (Ω): Clearly, ψ is a supersolution to (4.1). Moreover, ψ = is a subsolution to (4.1). Since ψ ≤ ψ, then there exists a solution un ∈ H (Ω) ∩ L ∞ (Ω) to (4.1) (see [4]).
We prove now the key estimates for proving the existence of solution to problem (P λ ).
Step 1: H estimate. Let us take un as test function in the weak formulation of (4.1). Then we obtain by using Poincaré's and Hölder's inequalities that Now, since α > q − , then ( −α) −q < < * . Hence, we can apply Sobolev's inequality to get that Observe now that q+ −α < . Therefore, we deduce that un H (Ω) ≤ C.
Step 2: L ∞ estimate. Assume now, in order to achieve a contradiction, that { un L ∞ (Ω) } n∈N is unbounded, and choose a not relabeled divergent subsequence. Then, the function vn = un un L ∞ (Ω) satis es in Ω, vn > in Ω, vn = on ∂Ω. (4.2) Notice that vn L ∞ (Ω) = for all n, and also that ≤ µ(x)Tn(|∇un| q ) Then, it is standard to prove that vn C ,η (Ω) ≤ C for all n and for some η ∈ ( , ) independent of n following the arguments in [40] (see [1,Appendix] Using now that {vn} n∈N is bounded in H (Ω), we conclude that Ω µ(x)Tn(|∇un| q )ϕ un L ∞ (Ω) un + n q− +α → as n → ∞. Finally, we pass to the limit in (4.2) and obtain that This contradicts the fact that λ < λ .
We are ready now to prove the main theorem of this section. Finally, similar arguments as in the proof of Step 2 in Proposition 4.3 can be used to prove that λ is the only possible bifurcation point from in nity. Actually, reasoning by contradiction and using that there is no solution to (P λ ), it is also standard to prove that λ is, indeed, a bifurcation point from in nity.

Appendix: Existence of an unbounded continuum
For every w ∈ L ∞ (Ω) and λ ∈ R, let us consider the following problem: in Ω, u = on ∂Ω. We will prove next that K is a completely continuous operator, i.e., it is continuous and maps bounded sets to relatively compact sets. Proposition 5.1. Assume that (H1) holds. Then, the operator K is completely continuous.
Now we can argue as in [1,Appendix] to prove that {un} is, in fact, bounded in C ,η (Ω) for some η ∈ ( , ). Therefore, Arzelà-Ascoli theorem implies that {un} admits a uniformly convergent subsequence. Say, up to a not relabeled subsequence, un → u uniformly in Ω for some u ∈ C(Ω).
Using that {un} and {(λn , wn)} are bounded in L ∞ (Ω) and in R×L ∞ (Ω), and also that α < q− < , the previous equality clearly implies that {un} is bounded in H (Ω). Then, u ∈ H (Ω) and, up to a new subsequence, un u in H (Ω). Moreover, by [44], ∇un → ∇u strongly in L q (Ω) N . Furthermore, a lower local estimate on {un} can be derived by comparison in the usual way. With all these estimates and convergences, the passing to the limit in (5.1) is standard. Therefore, u ∈ H (Ω) ∩ L ∞ (Ω) is the unique solution to (5.1). This means that K(λ, w) = u. Thus, we have proved that, up to a subsequence, K(λn , wn) → K(λ, w) strongly in L ∞ (Ω). Actually, since (λ, w) was xed from the beginning, the whole sequence, and not just a subseqence, converges to (λ, w). That is to say, K is continuous.
It is left to prove that K maps bounded sets to relatively compact sets. In other words, that for every sequence {(λn , wn)} bounded in R × L ∞ (Ω), there exists (λ, w) ∈ R × L ∞ (Ω) such that, up to a subsequence, K(λn , wn) → K(λ, w) strongly in L ∞ (Ω). Indeed, it is well-known that, up to a subsequence, λn → λ in R and wn → w weakly* in L ∞ (Ω) for some (λ, w) ∈ R × L ∞ (Ω). This convergence is enough to pass to the limit in the term with wn. In the rest of the terms, we pass to limit arguing as above. Thus, up to a subsequence, K(λn , wn) → K(λ, w), and the proof is nished.
Proof of Proposition 5.2. By virtue of Proposition 5.1, K is completely continuous. Moreover, since (P ) admits at most one solution (by virtue of [7]), then u is the unique solution to Φ( , u) = (see Remark 5.3). In particular, it is isolated. We will prove now that i(Φ( , ·), u ) ≠ by using the properties of the Leray-Schauder degree. In conclusion, we can now apply [15,Theorem 2.2], which is essentially [45,Theorem 3.2], and the proof is nished.