Continuity results for parametric nonlinear singular Dirichlet problems

: In this paper we study from a qualitative point of view the nonlinear singular Dirichlet problem depending on a parameter λ > 0 that was considered in [32]. Denoting by S λ the set of positive solutions of the problem corresponding to the parameter λ , we establish the following essential properties of S λ : (i) there exists a smallest element u * λ in S λ , and the mapping λ (cid:55)→ u * λ is (strictly) increasing and left continuous; (ii) the set-valued mapping λ (cid:55)→ S λ is sequentially continuous.

Let Ω ⊂ R N be a bounded domain with a C -boundary ∂Ω and let γ ∈ ( , ) and < p < +∞. Recently, Papageorgiou-Vetro-Vetro [32] have considered the following parametric nonlinear singular Dirichlet problem where the operator ∆p stands for the p-Laplace di erential operator ∆p u = div |∇u| p− ∇u for all u ∈ W ,p (Ω).
The nonlinear function f is assumed to satisfy the following conditions: H(f ): f : Ω × R → R is a Carathéodory function such that for a.e. x ∈ Ω, f (x, ) = , f (x, s) ≥ for all s ≥ , and (i) for every ρ > , there exists aρ ∈ L ∞ (Ω) such that |f (x, s)| ≤ aρ(x) for a.e. x ∈ Ω and for all |s| ≤ ρ; (ii) there exists an integer m ≥ such that where λm is the m-th eigenvalue of (−∆p , W ,p (Ω)), and denoting (iv) for every ρ > , there exists ξρ > such that for a.e. x ∈ Ω the function The following bifurcation type result is proved in [32, Theorem 2].
Is the solution mapping Λ lower semicontinuous ?
In this paper we answer in the a rmative the above open questions.

Theorem 2.
Assume that hypotheses H(f ) hold. Then there hold: for each λ ∈ L, problem (1) has a smallest positive solution u * λ ∈ int(C (Ω)+), and the map Γ from L to C (Ω) given by The rest of the paper is organized as follows. In Section 2 we set forth the preliminary material needed in the sequel. In Section 3 we prove our main results formulated as Theorem 2.

Preliminaries
In this section we gather the preliminary material that will be used to prove the main result in the paper. For more details we refer to [8,13,16,19,22,28,29,35]. Let < p < ∞ and p be its Hölder conjugate de ned by p + p = . In what follows, the Lebesgue space L p (Ω) is endowed with the standard norm The Sobolev space W ,p (Ω) is equipped with the usual norm In addition, we shall use the Banach space Its cone of nonnegative functions C (Ω)+ = u ∈ C (Ω) : u ≥ in Ω has a nonempty interior given by where ∂u ∂n is the normal derivative of u and n(·) is the outward unit normal to the boundary ∂Ω. Hereafter by ·, · we denote the duality brackets for (W ,p (Ω) * , W ,p (Ω)). Also, we de ne the nonlinear operator A : W ,p (Ω) → W ,p (Ω) * by (2) The following statement is a special case of more general results (see , Motreanu-Motreanu-Papageorgiou [29]).
For the sake of clarity we recall the following notion regarding order.

De nition 4. Let (P, ≤) be a partially ordered set. A subset E
We recall a few things regarding upper and lower semicontinuous set-valued mappings.

De nition 5. Let X and Y be topological spaces. A set-valued mapping F
if this holds for every x ∈ X, F is called upper semicontinuous; if this holds for every x ∈ X, F is called lower semicontinuous; continuous at x ∈ X if F is both upper semicontinuous and lower semicontinuous at x ∈ X; if this holds for every x ∈ X, F is called continuous.
The propositions below provide criteria of upper and lower semicontinuity.
The following properties are equivalent: Proposition 7. The following properties are equivalent:

Proof of the main result
In this section we prove Theorem 2. We start with the fact that, for each λ ∈ L, problem (1) has a smallest solution. To this end, we will use the similar technique employed in [12,Lemma 4.1] to show that the solution set S λ is downward directed (see De nition 4).
Proof. Fix λ ∈ ( , λ * ] and u , u ∈ S λ . Corresponding to any ε > we introduce the truncation ηε : R → R as follows which is Lipschitz continuous. It results from Marcus-Mizel [27] that and . )v for i = , and summing the resulting inequalities yield We note that Altogether, we obtain Now we pass to the limit as ε → + . Using Lebesgue's Dominated Convergence Theorem and the fact that Here the notation χ D stands for the characteristic function of a set D, that is, The gradient of u := min{u , u } ∈ W ,p (Ω) is equal to Consequently, we can express (3) in the form for all v ∈ C ∞ (Ω) with v(x) ≥ for a.e. x ∈ Ω. Actually, the density of C ∞ (Ω)+ in W ,p (Ω)+ ensures that (4) is valid for all v ∈ W ,p (Ω)+. Let u λ be the unique solution of the purely singular elliptic problem Proposition 5 of Papageorgiou-Smyrlis [31] guarantees that u λ ∈ int C (Ω)+ . We claim that For every u ∈ S λ , there holds whenever v ∈ W ,p (Ω). Inserting v = ( u λ − u) + ∈ W ,p (Ω) in (6) and using the fact that f (x, u(x)) ≥ , we derive Then the monotonicity of −∆p leads to (5).
Since u , u ∈ S λ and u := min{u , u } ∈ W ,p (Ω), we conclude that u ≥ u λ . Corresponding to the truncation we consider the intermediate Dirichlet problem By [32,Proposition 7] there exists u ∈ W ,p (Ω) such that for all h ∈ W ,p (Ω). Inserting h = (u − u) + , through (4) and (7), we infer that It turns out that u ≤ u. Through the same argument, we also imply u ≥ u λ . So by virtue of (7) and (8) we arrive at u ∈ S λ and u ≤ min{u , u }.

2
We are in a position to prove that problem (1) admits a smallest solution for every λ ∈ L.
Next we verify that the sequence {un} is bounded in W ,p (Ω). Arguing by contradiction, suppose that a relabeled subsequence of {un} satis es un → ∞. Set yn = un un . This ensures yn → y weakly in W ,p (Ω) and yn → y strongly in L p (Ω) with y ≥ .
From (6) and {un} ⊂ S λ we have for all v ∈ W ,p (Ω). On the other hand, hypotheses H(f )(i) and (ii) entail x ∈ Ω and all s ≥ , with some c > . By (10) and (12) we see that the sequence f (·, un(·)) un p− is bounded in L p (Ω).
We can apply Proposition 3 to obtain yn → y in W ,p (Ω). Letting n → ∞ in (11) gives so y is a nontrivial nonnegative solution of the eigenvalue problem Consequently, y must be nodal because m ≥ and y ≠ , which contradicts that y ≥ in Ω. This contradiction proves that the sequence {un} is bounded in W ,p (Ω). Along a relabeled subsequence, we may assume that un → u * λ weakly in W ,p (Ω) and un → u * λ in L p (Ω), for some u * λ ∈ W ,p (Ω). In addition, we may suppose that From u λ ∈ int(C (Ω)+) and (5), through the Lemma in Lazer-Mckenna [23], we obtain On account of (13)- (15) we have u −γ n → (u * λ ) −γ weakly in L p (Ω) (16) (see also p. 38]). Setting u = un ∈ S λ and v = un − u * λ ∈ W ,p (Ω) in (6), in the limit as n → ∞ we get The property of A to be of type (S+) (according to Proposition 3) implies The above convergence and Sobolev embedding theorem enable us to deduce which completes the proof.

2
In the next lemma we examine monotonicity and continuity properties of the map λ → u * λ from L = ( , λ * ] to C (Ω).
Γ is strictly increasing, in the sense that (ii) Γ is left continuous.

It holds
for all v ∈ W ,p (Ω). By assertion (i) we know that Choosing v = un in (17) and proceeding as in the proof of Lemma 9, we verify that the sequence {un} is bounded in W ,p (Ω). Given r > N, it is true that (u * λ ) r ∈ int(C (Ω)+), so there is a constant c > such that We can make use of the Lemma in Lazer-Mckenna [23] for having ≤ u −γ n ≤ u −γ ∈ L r (Ω) for all n.
for all v ∈ W ,p (Ω). As in the proof of Lemma 9, we can show that the sequence {un} is bounded in W ,p (Ω). Therefore we may assume that un → u weakly in W ,p (Ω) and un → u in L p (Ω).
for some u ∈ W ,p (Ω). Furthermore, the sequences {f (·, un(·))} and {u −γ n } are bounded in L p (Ω) as already demonstrated in the proofs of Lemmas 9 and 10. In (20), we choose v = un − u ∈ W ,p (Ω) and then pass to the limit as n → ∞. By means of (21) we are led to lim n→∞ A(un), un − u = .
Since A is of type (S+), we can conclude un → u in W ,p (Ω).
On account of (20), the strong convergence in (22) and Sobolev embedding theorem imply for all v ∈ W ,p (Ω). This reads as u ∈ S λ = Λ(λ). It remains to check that u ∈ D. Fix λ ∈ L such that λ < λn ≤ λ * for all n.
By Lemma 10 (i) we know that The same argument as in the proof of Lemma 10 con rms that, for r > N xed, the function x → λn un(x) −γ + f (x, un(x)) is bounded in L r (Ω). Let g λn (x) = λn un(x) −γ + f (x, un(x)) ∈ L r (Ω) and consider the linear Dirichlet problem The standard existence and regularity theory (see, e.g., Gilbarg-Trudinger [19,Theorem 9.15]) ensure that problem (23) has a unique solution with a constant c > and α = − N r . Denote wn(x) = ∇v λn (x) for all x ∈ Ω. It holds wn ∈ C ,α (Ω) thanks to v λn ∈ C ,α (Ω). Notice that The nonlinear regularity up to the boundary in Liebermann [24,25] reveals that un ∈ C ,β (Ω) for all n ∈ N with some β ∈ ( , ). The compactness of the embedding of C ,β (Ω) in C (Ω) and (22) yield the strong convergence Recalling that D is closed in C (Ω) it results that u ∈ Λ(λ) ∩ D, i.e., λ ∈ Λ − (D). Proof. In order to refer to Proposition 7, let {λn} ⊂ L satisfy λn → λ ≠ as n → ∞ and let w ∈ S λ ⊂ int(C (Ω) +). For each n ∈ N, we formulate the Dirichlet problem In view of w ≥ u λ ∈ int(C (Ω)+) (see (5)) and it is obvious that problem (24) has a unique solution u n ∈ int(C (Ω)+). Relying on the growth condition for f (see hypotheses H(f )(i) and (ii)), through the same argument as in the proof of Lemma 9 we show that the sequence {u n } is bounded in W ,p (Ω). Then Proposition 1.3 of Guedda-Véron [20] implies the uniform boundedness u n ∈ L ∞ (Ω) and u n L ∞ (Ω) ≤ c for all n ∈ N, with a constant c > . As in the proof of Lemma 11, we set g λn (x) = λn w(x) −γ + f (x, w(x)) and consider the Dirichlet problem (23) to obtain that {u n } is contained in C ,β (Ω) for some β ∈ ( , ). Due to the compactness of the embedding of C ,β (Ω) in C (Ω), we may assume u n → u in C (Ω) as n → ∞, with some u ∈ C (Ω). Then (24) yields Thanks to w ∈ Λ(λ), a simple comparison justi es u = w. Since every convergent subsequence of {un} converges to the same limit w, it is true that lim n→∞ u n = w.
Next, for each n ∈ N, we consider the Dirichlet problem Carrying on the same reasoning, we can show that this problem has a unique solution u n belonging to int(C (Ω)+) and that lim n→∞ u n = w.
Continuing the process, we generate a sequence {u k n } n,k≥ such that Fix n ≥ . As before, based on the nonlinear regularity [24,25], we notice that the sequence {u k n } k≥ is relatively compact in C (Ω), so we may suppose which means that un ∈ Λ(λn). The convergence in (25) and the double limit lemma (see, e.g., [13,Proposition A.2.35]) result in un → w in C (Ω) as n → ∞.
By Proposition 7 we conclude that Λ is lower semicontinuous. (ii) The stated conclusion is a direct consequence of Lemmas 9 and 10.