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Volume 9, Issue 1

# Continuity results for parametric nonlinear singular Dirichlet problems

Yunru Bai
• Jagiellonian University in Krakow, Faculty of Mathematics and Computer Science, ul. Lojasiewicza 6, 30348, Krakow, Poland
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/ Dumitru Motreanu
/ Shengda Zeng
• Corresponding author
• Jagiellonian University in Krakow, Faculty of Mathematics and Computer Science, ul. Lojasiewicza 6, 30348, Krakow, Poland
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Published Online: 2019-06-01 | DOI: https://doi.org/10.1515/anona-2020-0005

## Abstract

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 Sλ the set of positive solutions of the problem corresponding to the parameter λ, we establish the following essential properties of Sλ:

1. there exists a smallest element $\begin{array}{}{u}_{\lambda }^{\ast }\end{array}$ in Sλ, and the mapping λ$\begin{array}{}{u}_{\lambda }^{\ast }\end{array}$ is (strictly) increasing and left continuous;

2. the set-valued mapping λSλ is sequentially continuous.

MSC 2010: 35J92; 35J25; 35P30

## 1 Introduction

Elliptic equations with singular terms represent a class of hot-point problems because they are mathematically significant and appear in applications to chemical catalysts processes, non-Newtonian fluids, and in models for the temperature of electrical conductors (see [3, 9]). An extensive literature is devoted to such problems, especially focusing on their theoretical analysis. For instance, Ghergu-Rădulescu [18] established several existence and nonexistence results for boundary value problems with singular terms and parameters; Gasínski-Papageorgiou [15] studied a nonlinear Dirichlet problem with a singular term, a (p − 1)-sublinear term, and a Carathéodory perturbation; Hirano-Saccon-Shioji [21] proved Brezis-Nirenberg type theorems for a singular elliptic problem. Related topics and results can be found in Crandall-Rabinowitz-Tartar [7], Cîrstea-Ghergu-Rădulescu [6], Dupaigne-Ghergu-Rădulescu [10], Gasiński-Papageorgiou [17], Averna-Motreanu-Tornatore [2], Papageorgiou-Winkert [33], Carl [4], Faria-Miyagaki-Motreanu [11], Carl-Costa-Tehrani [5], Liu-Motreanu-Zeng [26] Papageorgiou-Rădulescu-Repovš [30], and the references therein.

Let Ω ⊂ ℝN be a bounded domain with a C2-boundary ∂ Ω and let y ∈ (0, 1) and 1 < p < + ∞. Recently, Papageorgiou-Vetro-Vetro [32] have considered the following parametric nonlinear singular Dirichlet problem

$−△pu(x)=λu(x)−y+f(x,u(x)) in Ωu(x)>0 in Ωu=0 on ∂Ω,$(1)

where the operator Δp stands for the p-Laplace differential operator

$Δpu=div(|∇u|p−2∇u) for all u∈W01,p(Ω).$

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. xΩ, f(x, 0) = 0, f(x, s) ≥ 0 for all s ≥ 0, and

1. for every ρ > 0, there exists aρL(Ω) such that

$|f(x,s)|≤aρ(x) for a.e. x∈Ω and for all |s|≤ρ;$

2. there exists an integer m ≥ 2 such that

$lims→+∞f(x,s)sp−1=λ^m uniformly for a.e. x∈Ω,$

where λ̂m is the m-th eigenvalue of (-Δp, $\begin{array}{}{W}_{0}^{1,p}\end{array}$(Ω)), and denoting

$F(x,t)=∫0sf(x,t)dt,$

then

$pF(x,s)−f(x,s)s→+∞ as s→+∞, uniformly for a.e. x∈Ω;$

3. for some r > p, there exists c0 ≥ 0 such that

$0≤lim infs→0+f(x,s)sr−1≤lim sups→0+f(x,s)sr−1≤c0 uniformly for a.e. x∈Ω;$

4. for every ρ > 0, there exists ξ̂ρ > 0 such that for a.e. xΩ the function

$s↦f(x,s)+ξ^ρsp−1$

is nondecreasing on [0, ρ].

The following bifurcation type result is proved in [32, Theorem 2].

#### Theorem 1

If hypotheses H(f) hold, then there exists a critical parameter value λ* > 0 such that

1. for all λ ∈ (0, λ*) problem (1) has at least two positive solutions u0, u1$\begin{array}{}\mathrm{i}\mathrm{n}\mathrm{t}\left({C}_{0}^{1}\left(\overline{\mathit{\Omega }}{\right)}_{+}\right)\end{array}$;

2. for λ = λ* problem (1) has at least one positive solution u*$\begin{array}{}\mathrm{i}\mathrm{n}\mathrm{t}\left({C}_{0}^{1}\left(\overline{\mathit{\Omega }}{\right)}_{+}\right)\end{array}$;

3. for all λ > λ* problem (1) has no positive solutions.

In what follows, we denote

$L:={λ>0:problem (1) admits a (positive) solution}=(0,λ∗],Sλ={u∈W01,p(Ω):u is a (positive) solution of problem (1)}$

for λ ∈ 𝓛. In this respect, Theorem 1 asserts that the above hypotheses, in conjunction with the nonlinear regularity theory (see Liebermann [24, 25]) and the nonlinear strong maximum principle (see Pucci-Serrin [34]), ensure that there holds

$Sλ⊂int(C01(Ω¯)+).$

Also, we introduce the set-valued mapping $\begin{array}{}\mathit{\Lambda }:\left(0,{\lambda }^{\ast }\right]\to {2}^{{C}_{0}^{1}\left(\overline{\mathit{\Omega }}\right)}\end{array}$ by

$Λ(λ)=Sλ for all λ∈(0,λ∗].$

The following open questions need to be answered:

1. Is there a smallest positive solution to problem (1) for each λ ∈ (0, λ*] ?

2. If for each λ ∈ (0, λ*] problem (1) has a smallest positive solution $\begin{array}{}{u}_{\lambda }^{\ast }\end{array}$, then the function Γ : (0, λ*] → $\begin{array}{}{C}_{0}^{1}\left(\overline{\mathit{\Omega }}\right)\end{array}$ with Γ(λ) = $\begin{array}{}{u}_{\lambda }^{\ast }\end{array}$ is it monotone ?

3. If for each λ ∈ (0, λ*] problem (1) has a smallest positive solution $\begin{array}{}{u}_{\lambda }^{\ast }\end{array}$, then is the function Γ continuous ?

4. Is the solution mapping Λ upper semicontinuous ?

5. Is the solution mapping Λ lower semicontinuous ?

In this paper we answer in the affirmative the above open questions.

#### Theorem 2

Assume that hypotheses H(f) hold. Then there hold:

1. the set-valued mapping $\begin{array}{}\mathit{\Lambda }:\mathcal{L}\to {2}^{{C}_{0}^{1}\left(\overline{\mathit{\Omega }}\right)}\end{array}$ is sequentially continuous;

2. for each λ ∈ 𝓛, problem (1) has a smallest positive solution $\begin{array}{}{u}_{\lambda }^{\ast }\in \mathrm{i}\mathrm{n}\mathrm{t}\left({C}_{0}^{1}\left(\overline{\mathit{\Omega }}{\right)}_{+}\right)\end{array}$, and the map Γ from 𝓛 to $\begin{array}{}{C}_{0}^{1}\left(\overline{\mathit{\Omega }}\right)\end{array}$ given by Γ(λ) = $\begin{array}{}{u}_{\lambda }^{\ast }\end{array}$ is

1. (strictly) increasing, that is, if 0 < μ < λλ*, then

$uλ∗−uμ∗∈int(C01(Ω¯)+);$

2. left continuous.

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.

## 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 1 < p < ∞ and p′ be its Hölder conjugate defined by $\begin{array}{}\frac{1}{p}+\frac{1}{{p}^{\prime }}=1\end{array}$. In what follows, the Lebesgue space Lp(Ω) is endowed with the standard norm

$∥u∥p=(∫Ω|u(x)|pdx)1p for all u∈Lp(Ω).$

The Sobolev space $\begin{array}{}{W}_{0}^{1,p}\end{array}$(Ω) is equipped with the usual norm

$∥u∥=(∫Ω|∇u(x)|pdx)1p for all u∈W01,p(Ω).$

In addition, we shall use the Banach space

$C01(Ω¯)={u∈C1(Ω¯):u=0 on ∂Ω}.$

Its cone of nonnegative functions

$C01(Ω¯)+={u∈C01(Ω¯):u≥0 in Ω}$

has a nonempty interior given by

$int(C01(Ω¯)+)={u∈C01(Ω¯):u>0 in Ω with ∂u∂n|∂Ω<0},$

where $\begin{array}{}\frac{\mathrm{\partial }u}{\mathrm{\partial }n}\end{array}$ 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 (W1,p(Ω)*, W1,p(Ω)). Also, we define the nonlinear operator A : W1,p(Ω) → W1,p(Ω)* by

$〈A(u),v〉=∫Ω|∇u(x)|p−2(∇u(x),∇v(x))RNdx for all u,v∈W1,p(Ω).$(2)

The following statement is a special case of more general results (see Gasiński-Papageorgiou [14], Motreanu-Motreanu-Papageorgiou [29]).

#### Proposition 3

The map A : W1,p(Ω) → W1,p(Ω)* introduced in (2) is continuous, bounded (that is, it maps bounded sets to bounded sets), monotone (hence maximal monotone) and of type(S+), i.e., if unu in W1,p(Ω) and

$lim supn→∞〈A(un),un−u〉⩽0,$

then unu in W1,p(Ω).

For the sake of clarity we recall the following notion regarding order.

#### Definition 4

Let (P, ≤ ) be a partially ordered set. A subset EP is called downward directed if for each pair u, vE there exists wE such that wu and wv.

For any u, v$\begin{array}{}{W}_{0}^{1,p}\end{array}$(Ω) with u(x) ≤ v(x) for a.e. xΩ, we set the ordered interval

$[u,v]:={w∈W01,p(Ω):u(x)≤w(x)≤v(x) for a.e. x∈Ω}.$

For s ∈ ℝ, we denote s± = max{± s, 0}. It is clear that if u$\begin{array}{}{W}_{0}^{1,p}\end{array}$(Ω) then it holds

$u±∈W01,p(Ω),u=u+−u−,|u|=u++u−.$

We recall a few things regarding upper and lower semicontinuous set-valued mappings.

#### Definition 5

Let X and Y be topological spaces. A set-valued mapping F : X → 2Y is called

1. upper semicontinuous (u.s.c., for short) at xX if for every open set OY with F(x) ⊂ O there exists a neighborhood N(x) of x such that

$F(N(x)):=⋃y∈N(x)F(y)⊂O;$

if this holds for every xX, F is called upper semicontinuous;

2. lower semicontinuous (l.s.c., for short) at xX if for every open set OY with F(x) ∩ O ≠ ∅ there exists a neighborhood N(x) of x such that

$F(y)∩O≠∅ for all y∈N(x);$

if this holds for every xX, F is called lower semicontinuous;

3. continuous at xX if F is both upper semicontinuous and lower semicontinuous at xX; if this holds for every xX, F is called continuous.

The propositions below provide criteria of upper and lower semicontinuity.

#### Proposition 6

The following properties are equivalent:

1. F : X → 2Y is u.s.c.;

2. for every closed subset CY, the set

$F−(C):={x∈X∣F(x)∩C≠∅}$

is closed in X.

#### Proposition 7

The following properties are equivalent:

1. F : X → 2Y is l.s.c.;

2. if uX, {uλ}λJX is a net such that uλu, and u*F(u), then for each λJ there is $\begin{array}{}{u}_{\lambda }^{\ast }\end{array}$F(uλ) with $\begin{array}{}{u}_{\lambda }^{\ast }\end{array}$u* in Y.

## 3 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 Sλ is downward directed (see Definition 4).

#### Lemma 8

For each λ ∈ 𝓛 = (0, λ*], the solution set Sλ of problem (1) is downward directed, i.e., if u1, u2Sλ, then there exists uSλ such that

$u≤u1andu≤u2.$

#### Proof

Fix λ ∈ (0, λ*] and u1, u2Sλ. Corresponding to any ε > 0 we introduce the truncation ηε : ℝ → ℝ as follows

$ηε(t)=0if t≤0tεif 0

which is Lipschitz continuous. It results from Marcus-Mizel [27] that

$ηε(u2−u1)∈W01,p(Ω)$

and

$∇(ηε(u2−u1))=ηε′(u2−u1)∇(u2−u1).$

Then for any function v$\begin{array}{}{C}_{0}^{\mathrm{\infty }}\left(\mathit{\Omega }\right)\end{array}$(Ω) with v(x) ≥ 0 for a.e. xΩ, we have

$ηε(u2−u1)v∈W01,p(Ω)$

and

$∇(ηε(u2−u1)v)=v∇(ηε(u2−u1))+ηε(u2−u1)∇v.$

Since u1, u2Sλ, there hold

$∫Ω|∇ui(x)|p−2(∇ui(x),∇φ(x))RNdx=λ∫Ωui(x)−yφ(x)dx+∫Ωf(x,ui(x))φ(x)dx for all φ∈W01,p(Ω),i=1,2.$

Inserting φ = ηε(u2u1)v for i = 1 and φ = (1 − ηε(u2u1))v for i = 2, and summing the resulting inequalities yield

$∫Ω|∇u1(x)|p−2(∇u1(x),∇(ηε(u2−u1)v)(x))RNdx+∫Ω|∇u2(x)|p−2(∇u2(x),∇((1−ηε(u2−u1))v)(x))RNdx=∫Ω[λu1(x)−y+f(x,u1(x))](ηε(u2−u1)v)(x)dx+∫Ω[λu2(x)−y+f(x,u2(x))](1−ηε(u2−u1))v)(x)dx.$

We note that

$∫Ω|∇u1(x)|p−2(∇u1(x),∇(ηε(u2−u1)v)(x))RNdx=1ε∫{0

and

$∫Ω|∇u2(x)|p−2(∇u2(x),∇((1−ηε(u2−u1))v)(x))RNdx=−1ε∫{0

Altogether, we obtain

$∫Ω|∇u1(x)|p−2(∇u1(x),∇v(x))RNηε(u2(x)−u1(x))dx+∫Ω|∇u2(x)|p−2(∇u2(x),∇v(x))RN(1−ηε(u2(x)−u1(x)))dx≥∫Ω[λu1(x)−y+f(x,u1(x))](ηε(u2−u1)v)(x)dx+∫Ω[λu2(x)−y+f(x,u2(x))](1−ηε(u2−u1))v)(x)dx.$

Now we pass to the limit as ε → 0+. Using Lebesgue’s Dominated Convergence Theorem and the fact that

$ηε((u2−u1)(x))→χ{u1

we find

$∫{u1(3)

Here the notation χD stands for the characteristic function of a set D, that is,

$χD(t)=1 if t∈D0 otherwise.$

The gradient of u := min{u1, u2} ∈ $\begin{array}{}{W}_{0}^{1,p}\end{array}$(Ω) is equal to

$∇u(x)=∇u1(x) for a.e. x∈{u1

Consequently, we can express (3) in the form

$∫Ω|∇u(x)|p−2(∇u(x),∇v(x))RNdx≥∫Ω[λu(x)−y+f(x,u(x))]v(x)dx$(4)

for all v$\begin{array}{}{C}_{0}^{\mathrm{\infty }}\left(\mathit{\Omega }\right)\end{array}$ with v(x) ≥ 0 for a.e. xΩ. Actually, the density of $\begin{array}{}{C}_{0}^{\mathrm{\infty }}\left(\mathit{\Omega }{\right)}_{+}\end{array}$ in $\begin{array}{}{W}_{0}^{1,p}\end{array}$(Ω)+ ensures that (4) is valid for all v$\begin{array}{}{W}_{0}^{1,p}\end{array}$(Ω)+.

Let λ be the unique solution of the purely singular elliptic problem

$−Δpu(x)=λu(x)−y in Ωu>0 in Ωu=0 on ∂Ω.$

Proposition 5 of Papageorgiou-Smyrlis [31] guarantees that λ$\begin{array}{}\text{int}\left({C}_{0}^{1}\left(\overline{\mathit{\Omega }}{\right)}_{+}\right)\end{array}$. We claim that

$u~λ≤u for all u∈Sλ.$(5)

For every uSλ, there holds

$∫Ω|∇u(x)|p−2(∇u(x),∇v(x))RNdx=∫Ω[λu(x)−y+f(x,u(x))]v(x)dx$(6)

whenever v$\begin{array}{}{W}_{0}^{1,p}\end{array}$(Ω). Inserting v = (λu)+$\begin{array}{}{W}_{0}^{1,p}\end{array}$(Ω) in (6) and using the fact that f(x, u(x)) ≥ q 0, we derive

$∫Ω|∇u(x)|p−2(∇u(x),∇(u~λ−u)+(x))RNdx=∫Ω[λu(x)−y+f(x,u(x))](u~λ−u)+(x)dx≥∫Ωλu(x)−y(u~λ−u)+(x)dx≥∫Ωλu~λ(x)−y(u~λ−u)+(x)dx=∫Ω|∇u~λ|p−2(∇u~λ(x),∇(u~λ−u)+(x))RNdx.$

Then the monotonicity of −Δp leads to (5).

Since u1, u2Sλ and u := min{u1, u2} ∈ $\begin{array}{}{W}_{0}^{1,p}\end{array}$(Ω), we conclude that uλ. Corresponding to the truncation

$g~(x,s)=λu~λ(x)−y+f(x,u~λ(x)) if s(7)

we consider the intermediate Dirichlet problem

$−Δpw(x)=g~(x,w(x)) in Ωw>0 in Ωw(x)=0 on ∂Ω.$(8)

By [32, Proposition 7] there exists u$\begin{array}{}{W}_{0}^{1,p}\end{array}$(Ω) such that

$〈A(u_),h〉=∫Ωg~(x,u_(x))h(x)dx$

for all h$\begin{array}{}{W}_{0}^{1,p}\end{array}$(Ω). Inserting h = (uu)+, through (4) and (7), we infer that

$〈A(u_),(u_−u)+〉=∫Ω[λu(x)−y+f(x,u(x))](u_−u)+(x)dx≤〈A(u),(u_−u)+〉.$

It turns out that uu. Through the same argument, we also imply uλ. So by virtue of (7) and (8) we arrive at uSλ and u ≤ min{u1, u2}. □

We are in a position to prove that problem (1) admits a smallest solution for every λ ∈ 𝓛.

#### Lemma 9

If hypotheses H(f) hold and λ ∈ 𝓛 = (0, λ*], then problem (1) has a smallest (positive) solution $\begin{array}{}{u}_{\lambda }^{\ast }\end{array}$Sλ, that is,

$uλ∗≤ufor all u∈Sλ.$

#### Proof

Fix λ ∈ (0, λ*]. Invoking Hu-Papageorgiou [22, Lemma 3.10], we can find a decreasing sequence {un} ⊂ Sλ such that

$infSλ=infnun.$

On the basis of (5) we note that

$u~λ≤un for all n.$(9)

Next we verify that the sequence {un} is bounded in $\begin{array}{}{W}_{0}^{1,p}\end{array}$(Ω). Arguing by contradiction, suppose that a relabeled subsequence of {un} satisfies ∥un∥ → ∞. Set $\begin{array}{}{y}_{n}=\frac{{u}_{n}}{\parallel {u}_{n}\parallel }\end{array}$. This ensures

$yn→y weakly in W01,p(Ω) and yn→y strongly in Lp(Ω) with y≥0.$(10)

From (6) and {un} ⊂ Sλ we have

$〈A(yn),v〉=∫Ω|∇yn(x)|p−2(∇yn(x),∇v(x))RNdx=∫Ω[λun(x)−y∥un∥p−1+f(x,un(x))∥un∥p−1]v(x)dx$(11)

for all v$\begin{array}{}{W}_{0}^{1,p}\end{array}$(Ω). On the other hand, hypotheses H(f)(i) and (ii) entail

$0≤f(x,s)≤c1(1+|s|p−1) for a.e. x∈Ω and all s≥0,$(12)

with some c1 > 0. By (10) and (12) we see that the sequence

${f(⋅,un(⋅))∥un∥p−1} is bounded in Lp′(Ω).$

Due to hypothesis H(f)(ii) and Aizicovici-Papageorgiou-Staicu [1, Proposition 16], we find that

${f(⋅,un(⋅))∥un∥p−1}→λ^myp−1 weakly in Lp′(Ω).$

Then inserting v = yny in (11) and using (9) lead to

$limn→∞〈A(yn),yn−y〉=0.$

We can apply Proposition 3 to obtain yny in $\begin{array}{}{W}_{0}^{1,p}\end{array}$(Ω). Letting n → ∞ in (11) gives

$〈A(y),v〉=λ^m∫Ωyp−1vdx for all v∈W01,p(Ω),$

so y is a nontrivial nonnegative solution of the eigenvalue problem

$−Δpy(x)=λ^my(x)p−1 in Ωy=0 on ∂Ω.$

Consequently, y must be nodal because m ≥ 2 and y ≠ 0, which contradicts that y ≥ 0 in Ω. This contradiction proves that the sequence {un} is bounded in $\begin{array}{}{W}_{0}^{1,p}\end{array}$(Ω).

Along a relabeled subsequence, we may assume that

$un→uλ∗ weakly in W01,p(Ω) and un→uλ∗ in Lp(Ω),$(13)

for some $\begin{array}{}{u}_{\lambda }^{\ast }\in {W}_{0}^{1,p}\left(\mathit{\Omega }\right)\end{array}$. In addition, we may suppose that

$un(x)−y→uλ∗(x)−y for a.e. x∈Ω.$(14)

From λ$\begin{array}{}\text{int}\left({C}_{0}^{1}\left(\overline{\mathit{\Omega }}{\right)}_{+}\right)\end{array}$ and (5), through the Lemma in Lazer-Mckenna [23], we obtain

$0≤un−y≤u~λ−y∈Lp′(Ω).$(15)

On account of (13)-(15) we have

$un−y→(uλ∗)−y weakly in Lp′(Ω)$(16)

Setting u = unSλ and v = un$\begin{array}{}{u}_{\lambda }^{\ast }\in {W}^{1,p}\left(\mathit{\Omega }\right)\end{array}$ in (6), in the limit as n → ∞ we get

$limn→∞〈Aun,un−uλ∗〉=0.$

The property of A to be of type (S+) (according to Proposition 3) implies

$un→uλ∗ in W01,p(Ω).$

The above convergence and Sobolev embedding theorem enable us to deduce

$∫Ω|∇uλ∗(x)|p−2(∇uλ∗(x),∇v(x))RNdx=∫Ω[λuλ∗(x)−y+f(x,uλ∗(x))]v(x)dx$

for all v$\begin{array}{}{W}_{0}^{1,p}\end{array}$(Ω). Consequently, we have

$uλ∗∈Sλ⊂int(C01(Ω¯)+) and uλ∗=infSλ,$

which completes the proof. □

In the next lemma we examine monotonicity and continuity properties of the map λ$\begin{array}{}{u}_{\lambda }^{\ast }\end{array}$ from 𝓛 = (0, λ*] to $\begin{array}{}{C}_{0}^{1}\left(\overline{\mathit{\Omega }}\right)\end{array}$.

#### Lemma 10

Suppose that hypotheses H(f) hold. Then the map Γ : 𝓛 = (0, λ*] → $\begin{array}{}{C}_{0}^{1}\left(\overline{\mathit{\Omega }}\right)\end{array}$ given by Γ(λ) = $\begin{array}{}{u}_{\lambda }^{\ast }\end{array}$ fulfills:

1. Γ is strictly increasing, in the sense that

$0<μ<λ≤λ∗impliesuλ∗−uμ∗∈int(C01(Ω¯)+);$

2. Γ is left continuous.

#### Proof

1. It follows from [32, Proposition 5] that there exists a solution uμSμ$\begin{array}{}\text{int}\left({C}_{0}^{1}\left(\overline{\mathit{\Omega }}{\right)}_{+}\right)\end{array}$ such that

$uλ∗−uμ∈int(C01(Ω¯)+).$

The desired conclusion is the direct consequence of the inequality $\begin{array}{}{u}_{\mu }^{\ast }\le {u}_{\mu }\end{array}$.

2. Let {λn} ⊂ (0, λ*] and λ ∈ (0, λ*] satisfy λnλ. Denote for simplicity un = $\begin{array}{}{u}_{{\lambda }_{n}}^{\ast }\end{array}$ = Γ(λn) ∈ Sλn$\begin{array}{}\text{int}\left({C}_{0}^{1}\left(\overline{\mathit{\Omega }}{\right)}_{+}\right)\end{array}$. It holds

$〈A(un),v〉=∫Ω[λnun(x)−y+f(x,un(x))]v(x)dx$(17)

for all v$\begin{array}{}{W}_{0}^{1,p}\end{array}$(Ω). By assertion (i) we know that

$0≤u1≤un≤uλ∗∗.$(18)

Choosing v = un in (17) and proceeding as in the proof of Lemma 9, we verify that the sequence {un} is bounded in $\begin{array}{}{W}_{0}^{1,p}\end{array}$(Ω). Given r > N, it is true that $\begin{array}{}\left({u}_{{\lambda }_{1}}^{\ast }{\right)}^{r}\in \text{int}\left({C}_{0}^{1}\left(\overline{\mathit{\Omega }}{\right)}_{+}\right)\end{array}$, so there is a constant c2 > 0 such that

$u~1≤c2(uλ1∗)r=c2u1r,$

or

$u~1−yr≥c2−yru1−y.$

We can make use of the Lemma in Lazer-Mckenna [23] for having

$0≤un−y≤u1−y∈Lr(Ω) for all n.$

Moreover, hypothesis H(f)(i) and (18) render that

$the sequence {f(⋅,un(⋅))} is bounded in Lr(Ω).$

Therefore, utilizing Guedda-Véron [20, Proposition 1.3] we obtain the uniform bound

$∥un∥L∞(Ω)≤c3 for all n,$(19)

with some c3 > 0. Besides, the linear elliptic problem

$−Δv(x)=gλn(x) in Ωv=0 on ∂Ω,$

where gλn(⋅) = λnun(⋅)y + f(⋅, un(⋅)) ∈ Lr(Ω), has a unique solution vλn$\begin{array}{}{W}_{0}^{2,r}\left(\mathit{\Omega }\right)\end{array}$ (see, e.g., [19, Theorem 9.15]). Owning to r > N, the Sobolev embedding theorem provides

$vλn∈C01,α(Ω¯),$

with $\begin{array}{}\alpha =1-\frac{N}{r}\end{array}$. For wn := ∇ vλn, we have wnC0,α(Ω, ℝN) and

$−div(|∇un(x)|p−2∇un(x)−wn(x))=0 in Ωun=0 on ∂Ω.$

This allows us to apply the nonlinear regularity up to the boundary in Liebermann [24, 25] finding that un$\begin{array}{}{C}_{0}^{1,\beta }\left(\overline{\mathit{\Omega }}\right)\end{array}$ with some β ∈ (0, 1) for all n. Here the uniform estimate in (19) is essential. The compactness of the embedding of $\begin{array}{}{C}_{0}^{1,\beta }\left(\overline{\mathit{\Omega }}\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{in}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{C}_{0}^{1}\left(\overline{\mathit{\Omega }}\right)\end{array}$ and the monotonicity of the sequence {un} guarantee

$un→u¯λ in C01(Ω¯)$

for some uλ$\begin{array}{}{C}_{0}^{1}\left(\overline{\mathit{\Omega }}\right)\end{array}$.

We claim that uλ = $\begin{array}{}{u}_{\lambda }^{\ast }\end{array}$. Arguing by contradiction, suppose that there exists x*Ω satisfying

$u¯λ(x∗)

The known monotonicity property of {un} entails

$uλ∗(x∗)

which contradicts assertion (i). It results that uλ = $\begin{array}{}{u}_{\lambda }^{\ast }\end{array}$ = Γ(λ), thereby

$Γ(λn)=un→u¯λ=Γ(λ) as n→∞,$

completing the proof. □

Next we turn to the semicontinuity properties of the set-valued mapping Λ.

#### Lemma 11

Assume that hypotheses H(f) hold. Then the set-valued mapping Λ : 𝓛 → $\begin{array}{}{2}^{{C}_{0}^{1}\left(\overline{\mathit{\Omega }}\right)}\end{array}$ is sequentially upper semicontinuous.

#### Proof

According to Proposition 6 we are going to show that for any closed set D$\begin{array}{}{C}_{0}^{1}\left(\overline{\mathit{\Omega }}\right)\end{array}$, one has that

$Λ−(D):={λ∈R:Λ(λ)∩D≠∅}$

is closed in ℝ. Let {λn} ⊂ Λ(D) verify λnλ as n → ∞. So,

$Λ(λn)∩D≠∅,$

hence there exists a sequence {un} ⊂ $\begin{array}{}\text{int}\left({C}_{0}^{1}\left(\overline{\mathit{\Omega }}{\right)}_{+}\right)\end{array}$ satisfying

$un∈Λ(λn)∩D for all n∈N,$

in particular

$∫Ω|∇un(x)|p−2(∇un(x),∇v(x))RNdx=∫Ω[λnun(x)−y+f(x,un(x))]v(x)dx$(20)

for all v$\begin{array}{}{W}_{0}^{1,p}\end{array}$(Ω). As in the proof of Lemma 9, we can show that the sequence {un} is bounded in $\begin{array}{}{W}_{0}^{1,p}\end{array}$(Ω). Therefore we may assume that

$un→u weakly in W01,p(Ω) and un→u in Lp(Ω).$(21)

for some u$\begin{array}{}{W}_{0}^{1,p}\end{array}$(Ω). Furthermore, the sequences {f(⋅, un(⋅))} and $\begin{array}{}\left\{{u}_{n}^{-y}\right\}\end{array}$ are bounded in Lp′(Ω) as already demonstrated in the proofs of Lemmas 9 and 10. In (20), we choose v = unu$\begin{array}{}{W}_{0}^{1,p}\end{array}$(Ω) and then pass to the limit as n → ∞. By means of (21) we are led to

$limn→∞〈A(un),un−u〉=0.$

Since A is of type (S+), we can conclude

$un→u in W01,p(Ω).$(22)

On account of (20), the strong convergence in (22) and Sobolev embedding theorem imply

$∫Ω|∇u(x)|p−2(∇u(x),∇v(x))RNdx=∫Ω[λu(x)−y+f(x,u(s))]v(x)dx$

for all v$\begin{array}{}{W}_{0}^{1,p}\end{array}$(Ω). This reads as uSλ = Λ(λ).

It remains to check that uD. Fix λ ∈ 𝓛 such that

$λ_<λn≤λ∗ for all n.$

By Lemma 10 (i) we know that

$uλ_∗

The same argument as in the proof of Lemma 10 confirms that, for r > N fixed, the function xλnun(x)y + f(x, un(x)) is bounded in Lr(Ω). Let gλn(x) = λnun(x)y + f(x, un(x)) ∈ Lr(Ω) and consider the linear Dirichlet problem

$−Δv(x)=gλn(x) in Ωv=0 on ∂Ω.$(23)

The standard existence and regularity theory (see, e.g., Gilbarg-Trudinger [19, Theorem 9.15]) ensure that problem (23) has a unique solution

$vλn∈W2,r(Ω)⊂C01,α(Ω¯) with ∥vλn∥C01,α(Ω¯)≤c4,$

with a constant c4 > 0 and $\begin{array}{}\alpha =1-\frac{N}{r}\end{array}$. Denote wn(x) = ∇ vλn(x) for all xΩ. It holds wnC0,α(Ω) thanks to $\begin{array}{}{v}_{{\lambda }_{n}}\in {C}_{0}^{1,\alpha }\left(\overline{\mathit{\Omega }}\right)\end{array}$. Notice that

$−div(|∇un(x)|p−2∇un(x)−wn(x))=0 in Ωun=0 on ∂Ω.$

The nonlinear regularity up to the boundary in Liebermann [24, 25] reveals that un$\begin{array}{}{C}_{0}^{1,\beta }\left(\overline{\mathit{\Omega }}\right)\end{array}$ for all n ∈ ℕ with some β ∈ (0, 1). The compactness of the embedding of $\begin{array}{}{C}_{0}^{1,\beta }\left(\overline{\mathit{\Omega }}\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{in}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{C}_{0}^{1}\left(\overline{\mathit{\Omega }}\right)\end{array}$ and (22) yield the strong convergence

$un→u in C01(Ω¯).$

Recalling that D is closed in $\begin{array}{}{C}_{0}^{1}\left(\overline{\mathit{\Omega }}\right)\end{array}$ it results that uΛ(λ) ∩ D, i.e., λΛ(D). □

#### Lemma 12

Suppose that hypotheses H(f) hold. Then the set-valued mapping Λ : 𝓛 → $\begin{array}{}{2}^{{C}_{0}^{1}\left(\overline{\mathit{\Omega }}\right)}\end{array}$ is sequentially lower semicontinuous.

#### Proof

In order to refer to Proposition 7, let {λn} ⊂ 𝓛 satisfy λnλ ≠ 0 as n → ∞ and let wSλ$\begin{array}{}\text{int}\left({C}_{0}^{1}\left(\overline{\mathit{\Omega }}{\right)}_{+}\right)\end{array}$. For each n ∈ ℕ, we formulate the Dirichlet problem

$−Δpu(x)=λnw(x)−y+f(x,w(x)) in Ωu>0 in Ωu=0 on ∂Ω.$(24)

In view of wλ$\begin{array}{}\text{int}\left({C}_{0}^{1}\left(\overline{\mathit{\Omega }}{\right)}_{+}\right)\end{array}$ (see (5)) and

$λnw(x)−y+f(x,w(x))≥0 for all x∈Ωλnw(x)−y+f(x,w(x))≢0,$

it is obvious that problem (24) has a unique solution $\begin{array}{}{u}_{n}^{0}\in \text{int}\left({C}_{0}^{1}\left(\overline{\mathit{\Omega }}{\right)}_{+}\right)\end{array}$. 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 $\begin{array}{}\left\{{u}_{n}^{0}\right\}\end{array}$ is bounded in $\begin{array}{}{W}_{0}^{1,p}\end{array}$(Ω). Then Proposition 1.3 of Guedda-Véron [20] implies the uniform boundedness

$un0∈L∞(Ω) and ∥un0∥L∞(Ω)≤c5 for all n∈N,$

with a constant c5 > 0. As in the proof of Lemma 11, we set gλn(x) = λnw(x)y + f(x, w(x)) and consider the Dirichlet problem (23) to obtain that $\begin{array}{}\left\{{u}_{n}^{0}\right\}\end{array}$ is contained in $\begin{array}{}{C}_{0}^{1,\beta }\left(\overline{\mathit{\Omega }}\right)\end{array}$ for some β ∈ (0, 1). Due to the compactness of the embedding of $\begin{array}{}{C}_{0}^{1,\beta }\left(\overline{\mathit{\Omega }}\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{in}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{C}_{0}^{1}\left(\overline{\mathit{\Omega }}\right)\end{array}$, we may assume

$un0→u in C01(Ω¯) as n→∞,$

with some u$\begin{array}{}{C}_{0}^{1}\left(\overline{\mathit{\Omega }}\right)\end{array}$. Then (24) yields

$−Δpu(x)=λw(x)−y+f(x,w(x)) in Ωu>0 in Ωu=0 on ∂Ω.$

Thanks to wΛ(λ), a simple comparison justifies u = w. Since every convergent subsequence of {un} converges to the same limit w, it is true that

$limn→∞un0=w.$

Next, for each n ∈ ℕ, we consider the Dirichlet problem

$−Δpu(x)=λnun0(x)−y+f(x,un0(x)) in Ωu>0 in Ωu=0 on ∂Ω.$

Carrying on the same reasoning, we can show that this problem has a unique solution $\begin{array}{}{u}_{n}^{1}\end{array}$ belonging to $\begin{array}{}\text{int}\left({C}_{0}^{1}\left(\overline{\mathit{\Omega }}{\right)}_{+}\right)\end{array}$ and that

$limn→∞un1=w.$

Continuing the process, we generate a sequence $\begin{array}{}\left\{{u}_{n}^{k}{\right\}}_{n,k\ge 1}\end{array}$ such that

$−Δpunk(x)=λnunk−1(x)−y+f(x,unk−1(x)) in Ωunk>0 in Ωunk=0 on ∂Ω,$

and

$limn→∞unk=w for all k∈N.$(25)

Fix n ≥ 1. As before, based on the nonlinear regularity [24, 25], we notice that the sequence $\begin{array}{}\left\{{u}_{n}^{k}{\right\}}_{k\ge 1}\end{array}$ is relatively compact in $\begin{array}{}{C}_{0}^{1}\left(\overline{\mathit{\Omega }}\right)\end{array}$, so we may suppose

$unk→un in C01(Ω¯) as k→∞,$

for some un$\begin{array}{}{C}_{0}^{1}\left(\overline{\mathit{\Omega }}\right)\end{array}$. Then it appears that

$−Δpun(x)=λnun(x)−y+f(x,un(x)) in Ωun>0 in Ωun=0 on ∂Ω,$

which means that unΛ(λn).

The convergence in (23) and the double limit lemma (see, e.g., [13, Proposition A.2.35]) result in

$un→w in C01(Ω¯) as n→∞.$

By Proposition 7 we conclude that Λ is lower semicontinuous. □

#### Proof of Theorem 2

1. It suffices to apply Lemmas 11 and 12.

2. The stated conclusion is a direct consequence of Lemmas 9 and 10. □

## Acknowledgement

Project supported by the European Union’s Horizon 2020 Research and Innovation Programme under the Marie Skłodowska-Curie grant agreement No. 823731 – CONMECH, the National Science Center of Poland under Maestro Project No. UMO-2012/06/A/ST1/00262, and National Science Center of Poland under Preludium Project No. 2017/25/N/ST1/00611. It is also supported by the International Project co-financed by the Ministry of Science and Higher Education of Republic of Poland under Grant No. 3792/GGPJ/H2020/2017/0. D. Motreanu received Visiting Professor fellowship from CNPQ/Brazil PV- 400633/2017-5.

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Tel.: +86-18059034172

Accepted: 2018-12-08

Published Online: 2019-06-01

Published in Print: 2019-03-01

Citation Information: Advances in Nonlinear Analysis, Volume 9, Issue 1, Pages 372–387, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496,

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