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Advances in Nonlinear Analysis

Editor-in-Chief: Radulescu, Vicentiu / Squassina, Marco


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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
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  • 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 uλ in Sλ, and the mapping λuλ is (strictly) increasing and left continuous;

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

Keywords: Parametric singular elliptic equation; p-Laplacian; smallest solution; sequential continuity; monotonicity

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|p2u) for all uW01,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. xΩ and for all |s|ρ;

  2. there exists an integer m ≥ 2 such that

    lims+f(x,s)sp1=λ^m uniformly for a.e. xΩ,

    where λ̂m is the m-th eigenvalue of (-Δp, W01,p(Ω)), and denoting

    F(x,t)=0sf(x,t)dt,

    then

    pF(x,s)f(x,s)s+ as s+, uniformly for a.e. xΩ;

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

    0lim infs0+f(x,s)sr1lim sups0+f(x,s)sr1c0 uniformly for a.e. xΩ;

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

    sf(x,s)+ξ^ρsp1

    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, u1int(C01(Ω¯)+);

  2. for λ = λ* problem (1) has at least one positive solution u*int(C01(Ω¯)+);

  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={uW01,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

Sλint(C01(Ω¯)+).

Also, we introduce the set-valued mapping Λ:(0,λ]2C01(Ω¯) 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 uλ, then the function Γ : (0, λ*] → C01(Ω¯) with Γ(λ) = uλ is it monotone ?

  3. If for each λ ∈ (0, λ*] problem (1) has a smallest positive solution uλ, 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 Λ:L2C01(Ω¯) is sequentially continuous;

  2. for each λ ∈ 𝓛, problem (1) has a smallest positive solution uλint(C01(Ω¯)+), and the map Γ from 𝓛 to C01(Ω¯) given by Γ(λ) = uλ is

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

      uλuμint(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 1p+1p=1. In what follows, the Lebesgue space Lp(Ω) is endowed with the standard norm

up=(Ω|u(x)|pdx)1p for all uLp(Ω).

The Sobolev space W01,p(Ω) is equipped with the usual norm

u=(Ω|u(x)|pdx)1p for all uW01,p(Ω).

In addition, we shall use the Banach space

C01(Ω¯)={uC1(Ω¯):u=0 on Ω}.

Its cone of nonnegative functions

C01(Ω¯)+={uC01(Ω¯):u0 in Ω}

has a nonempty interior given by

int(C01(Ω¯)+)={uC01(Ω¯):u>0 in Ω with un|Ω<0},

where un 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)|p2(u(x),v(x))RNdx for all u,vW1,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 supnA(un),unu0,

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, vW01,p(Ω) with u(x) ≤ v(x) for a.e. xΩ, we set the ordered interval

[u,v]:={wW01,p(Ω):u(x)w(x)v(x) for a.e. xΩ}.

For s ∈ ℝ, we denote s± = max{± s, 0}. It is clear that if uW01,p(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)):=yN(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)OforallyN(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):={xXF(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 uλF(uλ) with uλu* 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

uu1anduu2.

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 t0tεif 0<t<ε1otherwise,

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

ηε(u2u1)W01,p(Ω)

and

(ηε(u2u1))=ηε(u2u1)(u2u1).

Then for any function vC0(Ω)(Ω) with v(x) ≥ 0 for a.e. xΩ, we have

ηε(u2u1)vW01,p(Ω)

and

(ηε(u2u1)v)=v(ηε(u2u1))+ηε(u2u1)v.

Since u1, u2Sλ, there hold

Ω|ui(x)|p2(ui(x),φ(x))RNdx=λΩui(x)yφ(x)dx+Ωf(x,ui(x))φ(x)dx for all φW01,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)|p2(u1(x),(ηε(u2u1)v)(x))RNdx+Ω|u2(x)|p2(u2(x),((1ηε(u2u1))v)(x))RNdx=Ω[λu1(x)y+f(x,u1(x))](ηε(u2u1)v)(x)dx+Ω[λu2(x)y+f(x,u2(x))](1ηε(u2u1))v)(x)dx.

We note that

Ω|u1(x)|p2(u1(x),(ηε(u2u1)v)(x))RNdx=1ε{0<u2u1<ε}|u1(x)|p2(u1(x),(u2u1)(x))RNv(x)dx+Ω|u1(x)|p2(u1(x),v(x))RNηε(u2(x)u1(x))dx

and

Ω|u2(x)|p2(u2(x),((1ηε(u2u1))v)(x))RNdx=1ε{0<u2u1<ε}|u2(x)|p2(u2(x),(u2u1)(x))RNv(x)dx+Ω|u2(x)|p2(u2(x),v(x))RN(1ηε(u2(x)u1(x)))dx.

Altogether, we obtain

Ω|u1(x)|p2(u1(x),v(x))RNηε(u2(x)u1(x))dx+Ω|u2(x)|p2(u2(x),v(x))RN(1ηε(u2(x)u1(x)))dxΩ[λu1(x)y+f(x,u1(x))](ηε(u2u1)v)(x)dx+Ω[λu2(x)y+f(x,u2(x))](1ηε(u2u1))v)(x)dx.

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

ηε((u2u1)(x))χ{u1<u2}(x) for a.e. xΩ as ε0+,

we find

{u1<u2}|u1(x)|p2(u1(x),v(x))RNdx+{u1u2}|u2(x)|p2(u2(x),v(x))RNdx{u1<u2}[λu1(x)y+f(x,u1(x))]v(x)dx+{u1u2}[λu2(x)y+f(x,u2(x))]v(x)dx.(3)

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

χD(t)=1 if tD0 otherwise.

The gradient of u := min{u1, u2} ∈ W01,p(Ω) is equal to

u(x)=u1(x) for a.e. x{u1<u2}u2(x) for a.e. x{u1u2}.

Consequently, we can express (3) in the form

Ω|u(x)|p2(u(x),v(x))RNdxΩ[λu(x)y+f(x,u(x))]v(x)dx(4)

for all vC0(Ω) with v(x) ≥ 0 for a.e. xΩ. Actually, the density of C0(Ω)+ in W01,p(Ω)+ ensures that (4) is valid for all vW01,p(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 λint(C01(Ω¯)+). We claim that

u~λu for all uSλ.(5)

For every uSλ, there holds

Ω|u(x)|p2(u(x),v(x))RNdx=Ω[λu(x)y+f(x,u(x))]v(x)dx(6)

whenever vW01,p(Ω). Inserting v = (λu)+W01,p(Ω) in (6) and using the fact that f(x, u(x)) ≥ q 0, we derive

Ω|u(x)|p2(u(x),(u~λu)+(x))RNdx=Ω[λu(x)y+f(x,u(x))](u~λu)+(x)dxΩλu(x)y(u~λu)+(x)dxΩλu~λ(x)y(u~λu)+(x)dx=Ω|u~λ|p2(u~λ(x),(u~λu)+(x))RNdx.

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

Since u1, u2Sλ and u := min{u1, u2} ∈ W01,p(Ω), we conclude that uλ. Corresponding to the truncation

g~(x,s)=λu~λ(x)y+f(x,u~λ(x)) if s<u~λ(x)λsy+f(x,s) if u~λ(x)su(x)λu(x)y+f(x,u(x)) if u(x)<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 uW01,p(Ω) such that

A(u_),h=Ωg~(x,u_(x))h(x)dx

for all hW01,p(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)dxA(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 uλSλ, that is,

uλuforalluSλ.

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~λun for all n.(9)

Next we verify that the sequence {un} is bounded in W01,p(Ω). Arguing by contradiction, suppose that a relabeled subsequence of {un} satisfies ∥un∥ → ∞. Set yn=unun. This ensures

yny weakly in W01,p(Ω) and yny strongly in Lp(Ω) with y0.(10)

From (6) and {un} ⊂ Sλ we have

A(yn),v=Ω|yn(x)|p2(yn(x),v(x))RNdx=Ω[λun(x)yunp1+f(x,un(x))unp1]v(x)dx(11)

for all vW01,p(Ω). On the other hand, hypotheses H(f)(i) and (ii) entail

0f(x,s)c1(1+|s|p1) for a.e. xΩ and all s0,(12)

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

{f(,un())unp1} is bounded in Lp(Ω).

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

{f(,un())unp1}λ^myp1 weakly in Lp(Ω).

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

limnA(yn),yny=0.

We can apply Proposition 3 to obtain yny in W01,p(Ω). Letting n → ∞ in (11) gives

A(y),v=λ^mΩyp1vdx for all vW01,p(Ω),

so y is a nontrivial nonnegative solution of the eigenvalue problem

Δpy(x)=λ^my(x)p1 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 W01,p(Ω).

Along a relabeled subsequence, we may assume that

unuλ weakly in W01,p(Ω) and unuλ in Lp(Ω),(13)

for some uλW01,p(Ω). In addition, we may suppose that

un(x)yuλ(x)y for a.e. xΩ.(14)

From λint(C01(Ω¯)+) and (5), through the Lemma in Lazer-Mckenna [23], we obtain

0unyu~λyLp(Ω).(15)

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

uny(uλ)y weakly in Lp(Ω)(16)

(see also Gasiński-Papageorgiou [16, p. 38]).

Setting u = unSλ and v = unuλW1,p(Ω) in (6), in the limit as n → ∞ we get

limnAun,unuλ=0.

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

unuλ in W01,p(Ω).

The above convergence and Sobolev embedding theorem enable us to deduce

Ω|uλ(x)|p2(uλ(x),v(x))RNdx=Ω[λuλ(x)y+f(x,uλ(x))]v(x)dx

for all vW01,p(Ω). Consequently, we have

uλSλint(C01(Ω¯)+) and uλ=infSλ,

which completes the proof. □

In the next lemma we examine monotonicity and continuity properties of the map λuλ from 𝓛 = (0, λ*] to C01(Ω¯).

Lemma 10

Suppose that hypotheses H(f) hold. Then the map Γ : 𝓛 = (0, λ*] → C01(Ω¯) given by Γ(λ) = uλ fulfills:

  1. Γ is strictly increasing, in the sense that

    0<μ<λλimpliesuλuμint(C01(Ω¯)+);

  2. Γ is left continuous.

Proof

  1. It follows from [32, Proposition 5] that there exists a solution uμSμint(C01(Ω¯)+) such that

    uλuμint(C01(Ω¯)+).

    The desired conclusion is the direct consequence of the inequality uμuμ.

  2. Let {λn} ⊂ (0, λ*] and λ ∈ (0, λ*] satisfy λnλ. Denote for simplicity un = uλn = Γ(λn) ∈ Sλnint(C01(Ω¯)+). It holds

    A(un),v=Ω[λnun(x)y+f(x,un(x))]v(x)dx(17)

    for all vW01,p(Ω). By assertion (i) we know that

    0u1unuλ.(18)

    Choosing v = un in (17) and proceeding as in the proof of Lemma 9, we verify that the sequence {un} is bounded in W01,p(Ω). Given r > N, it is true that (uλ1)rint(C01(Ω¯)+), so there is a constant c2 > 0 such that

    u~1c2(uλ1)r=c2u1r,

    or

    u~1yrc2yru1y.

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

    0unyu1yLr(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

    unL(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 vλnW02,r(Ω) (see, e.g., [19, Theorem 9.15]). Owning to r > N, the Sobolev embedding theorem provides

    vλnC01,α(Ω¯),

    with α=1Nr. For wn := ∇ vλn, we have wnC0,α(Ω, ℝN) and

    div(|un(x)|p2un(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 unC01,β(Ω¯) with some β ∈ (0, 1) for all n. Here the uniform estimate in (19) is essential. The compactness of the embedding of C01,β(Ω¯)inC01(Ω¯) and the monotonicity of the sequence {un} guarantee

    unu¯λ in C01(Ω¯)

    for some uλC01(Ω¯).

    We claim that uλ = uλ. Arguing by contradiction, suppose that there exists x*Ω satisfying

    u¯λ(x)<uλ(x).

    The known monotonicity property of {un} entails

    uλ(x)<un(x)=uλn(x) for all n,

    which contradicts assertion (i). It results that uλ = uλ = Γ(λ), thereby

    Γ(λn)=unu¯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 → 2C01(Ω¯) is sequentially upper semicontinuous.

Proof

According to Proposition 6 we are going to show that for any closed set DC01(Ω¯), one has that

Λ(D):={λR: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} ⊂ int(C01(Ω¯)+) satisfying

unΛ(λn)D for all nN,

in particular

Ω|un(x)|p2(un(x),v(x))RNdx=Ω[λnun(x)y+f(x,un(x))]v(x)dx(20)

for all vW01,p(Ω). As in the proof of Lemma 9, we can show that the sequence {un} is bounded in W01,p(Ω). Therefore we may assume that

unu weakly in W01,p(Ω) and unu in Lp(Ω).(21)

for some uW01,p(Ω). Furthermore, the sequences {f(⋅, un(⋅))} and {uny} are bounded in Lp′(Ω) as already demonstrated in the proofs of Lemmas 9 and 10. In (20), we choose v = unuW01,p(Ω) and then pass to the limit as n → ∞. By means of (21) we are led to

limnA(un),unu=0.

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

unu in W01,p(Ω).(22)

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

Ω|u(x)|p2(u(x),v(x))RNdx=Ω[λu(x)y+f(x,u(s))]v(x)dx

for all vW01,p(Ω). This reads as uSλ = Λ(λ).

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

λ_<λnλ for all n.

By Lemma 10 (i) we know that

uλ_<uλnun for all n.

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

vλnW2,r(Ω)C01,α(Ω¯) with vλnC01,α(Ω¯)c4,

with a constant c4 > 0 and α=1Nr. Denote wn(x) = ∇ vλn(x) for all xΩ. It holds wnC0,α(Ω) thanks to vλnC01,α(Ω¯). Notice that

div(|un(x)|p2un(x)wn(x))=0 in Ωun=0 on Ω.

The nonlinear regularity up to the boundary in Liebermann [24, 25] reveals that unC01,β(Ω¯) for all n ∈ ℕ with some β ∈ (0, 1). The compactness of the embedding of C01,β(Ω¯)inC01(Ω¯) and (22) yield the strong convergence

unu in C01(Ω¯).

Recalling that D is closed in C01(Ω¯) 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 → 2C01(Ω¯) is sequentially lower semicontinuous.

Proof

In order to refer to Proposition 7, let {λn} ⊂ 𝓛 satisfy λnλ ≠ 0 as n → ∞ and let wSλint(C01(Ω¯)+). 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 wλint(C01(Ω¯)+) (see (5)) and

λnw(x)y+f(x,w(x))0 for all xΩλnw(x)y+f(x,w(x))0,

it is obvious that problem (24) has a unique solution un0int(C01(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 {un0} is bounded in W01,p(Ω). Then Proposition 1.3 of Guedda-Véron [20] implies the uniform boundedness

un0L(Ω) and un0L(Ω)c5 for all nN,

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 {un0} is contained in C01,β(Ω¯) for some β ∈ (0, 1). Due to the compactness of the embedding of C01,β(Ω¯)inC01(Ω¯), we may assume

un0u in C01(Ω¯) as n,

with some uC01(Ω¯). 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

limnun0=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 un1 belonging to int(C01(Ω¯)+) and that

limnun1=w.

Continuing the process, we generate a sequence {unk}n,k1 such that

Δpunk(x)=λnunk1(x)y+f(x,unk1(x)) in Ωunk>0 in Ωunk=0 on Ω,

and

limnunk=w for all kN.(25)

Fix n ≥ 1. As before, based on the nonlinear regularity [24, 25], we notice that the sequence {unk}k1 is relatively compact in C01(Ω¯), so we may suppose

unkun in C01(Ω¯) as k,

for some unC01(Ω¯). 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

unw 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.

References

  • [1]

    S. Aizicovici, N.S. Papageorgiou, V. Staicu, Degree theory for operators of monotone type and nonlinear elliptic equations with inequality constraints, Mem. Amer. Math. Soc. 196 (2008), no. 915, vi+70 pp. Google Scholar

  • [2]

    D. Averna, D. Motreanu, E. Tornatore, Existence and asymptotic properties for quasilinear elliptic equations with gradient dependence, Appl. Math. Lett. 61 (2016), 102-107. CrossrefGoogle Scholar

  • [3]

    A. Callegari, A. Nachman, A nolinear singular boundary value problem in the theory of pseudoplastic fluids, SIAM J. Appl. Math. 38 (1980), 275–281. CrossrefGoogle Scholar

  • [4]

    S. Carl, Extremal solutions of p-Laplacian problems in 𝓓1,p(ℝN) via Wolff potential estimates, J. Differential Equations 263 (2017), 3370–3395. Web of ScienceCrossrefGoogle Scholar

  • [5]

    S. Carl, D.G. Costa, H. Tehrani, 𝓓1,2(ℝN) versus 𝓒(ℝN) local minimizer and a Hopf-type maximum principle, J. Differential Equations 261 (2016), 2006–2025. Web of ScienceCrossrefGoogle Scholar

  • [6]

    F. Cîrstea, M. Ghergu, V.D. Rădulescu, Combined effects of asymptotically linear and singular nonlinearities in bifurcation problems of Lane-Emden-Fowler type, J. Math. Pures Appl. 84 (2005), 493–508. CrossrefGoogle Scholar

  • [7]

    M.G. Crandall, P.H. Rabinowitz, L. Tartar, On a Dirichlet problem with a singular nonlinearity, Comm. Partial Differential Equations 2 (1977), 193–222. CrossrefGoogle Scholar

  • [8]

    Z. Denkowski, S. Migórski, N.S. Papageorgiou, An Introduction to Nonlinear Analysis: Applications, Kluwer Academic/Plenum Publishers, Boston, Dordrecht, London, New York, 2003. Google Scholar

  • [9]

    J. Díaz, M. Morel, L. Oswald, An elliptic equation with singular nonlinearity, Comm. Partial Differential Equations 12 (1987), 1333–1344. CrossrefGoogle Scholar

  • [10]

    L. Dupaigne, M. Ghergu, V.D. Rădulescu, Lane-Emden-Fowler equations with convection and singular potential, J. Math. Pures Appl. 87 (2007), 563–581. Web of ScienceCrossrefGoogle Scholar

  • [11]

    L.F.O. Faria, O.H. Miyagaki, D. Motreanu, Comparison and positive solutions for problems with the (p, q)-Laplacian and a convection term, Proc. Edinb. Math. Soc. 57 (2014), 687–698. Web of ScienceCrossrefGoogle Scholar

  • [12]

    M. Filippakis, N.S. Papageorgiou, Multiple constant sign and nodal solutions for nonlinear elliptic equations with the p-Laplacian, J. Differential Equations 245 (2008), 1883–1922. CrossrefWeb of ScienceGoogle Scholar

  • [13]

    L. Gasiński, N.S. Papageorgiou, Nonlinear Analysis, Chapman & Hall/CRC, Boca Raton, FL, 2006. Google Scholar

  • [14]

    L. Gasiński, N.S. Papageorgiou, Existence and multiplicity of solutions for Neumann p-Laplacian-type equations, Adv. Nonlinear Stud. 8 (2008), 843–870. Google Scholar

  • [15]

    L. Gasiński, N.S. Papageorgiou, Nonlinear elliptic equations with singular terms and combined nonlinearities, Ann. Henri Poincaré 13 (2012), 481–512. CrossrefGoogle Scholar

  • [16]

    L. Gasiński, N.S. Papageorgiou, Exercises in Analysis. Part 2: Nonlinear Analysis, Springer, Heidelberg, 2016. Google Scholar

  • [17]

    L. Gasiński, N.S. Papageorgiou, Asymmetric (p, 2)-equations with double resonance, Calc. Var. Partial Differential Equations 56:3 (2017), Art. 88, 23 pp. Web of ScienceGoogle Scholar

  • [18]

    M. Ghergu, V.D. Rădulescu, Sublinear singular elliptic problems with two parameters, J. Differential Equations 195 (2003), 520–536. CrossrefGoogle Scholar

  • [19]

    D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 1998. Google Scholar

  • [20]

    M. Guedda, L. Véron, Quasilinear elliptic equations involving critical Sobolev exponents, Nonlinear Anal. TMA 13 (1989), 879–902. CrossrefGoogle Scholar

  • [21]

    N. Hirano, C. Saccon, N. Shioji, Brezis-Nirenberg type theorems and multiplicity of positive solutions for a singular elliptic problem, J. Differential Equations 245 (2008), 1997–2037. CrossrefWeb of ScienceGoogle Scholar

  • [22]

    S. Hu, N.S. Papageorgiou, Handbook of Multivalued Analysis. Vol. I: Theory, Kluwer Academic Publishers, Dordrecht, 1997. Google Scholar

  • [23]

    A.C. Lazer, P.J. McKenna, On a singular nonlinear elliptic boundary-value problem, Proc. Amer. Math. Soc. 111 (1991), 721–730. CrossrefGoogle Scholar

  • [24]

    G.M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal. TMA 12 (1988), 1203–1219. CrossrefGoogle Scholar

  • [25]

    G.M. Lieberman, The natural generalization of the natural conditions of Ladyzhenskaya and Uraľtseva for elliptic equations, Comm. Partial Differential Equations 16 (1991), 311–361. CrossrefGoogle Scholar

  • [26]

    Z.H. Liu, D. Motreanu, S.D. Zeng, Positive solutions for nonlinear singular elliptic equations of p-Laplacian type with dependence on the gradient, Calc. Var. Partial Di erential Equations, 98 (2019), 22 pp,  CrossrefGoogle Scholar

  • [27]

    M. Marcus, V. Mizel, Absolute continuity on tracks and mappings of Sobolev spaces, Arch. Rational Mech. Anal. 45 (1972), 294–320. CrossrefGoogle Scholar

  • [28]

    S. Migórski, A. Ochal, M. Sofonea, Nonlinear Inclusions and Hemivariational Inequalities. Models and Analysis of Contact Problems, Advances in Mechanics and Mathematics 26, Springer, New York, 2013. Google Scholar

  • [29]

    D. Motreanu, V.V. Motreanu, N.S. Papageorgiou, Multiple constant sign and nodal solutions for nonlinear Neumann eigenvalue problems, Ann. Sc. Norm. Super. Pisa Cl. Sci. 10 (2011), 729–755. Google Scholar

  • [30]

    N.S. Papageorgiou, V.D. Rădulescu, D.D. Repovš, Positive solutions for nonlinear parametric singular Dirichlet problems, Bulletin of Mathematical Sciences, (2018), . CrossrefGoogle Scholar

  • [31]

    N.S. Papageorgiou, G. Smyrlis, A bifurcation-type theorem for singular nonlinear elliptic equations, Methods Appl. Anal. 22 (2015), 147–170. Web of ScienceGoogle Scholar

  • [32]

    N.S. Papageorgiou, C. Vetro, F. Vetro, Parametric nonlinear singular Dirichlet problems, Nonlinear Anal. RWA 45 (2019), 239–254. CrossrefGoogle Scholar

  • [33]

    N.S. Papageorgiou, P. Winkert, Singular p-Laplacian equations with superlinear perturbation, J. Differential Equations, 266 (2019), 1462–1487. CrossrefWeb of ScienceGoogle Scholar

  • [34]

    P. Pucci, J. Serrin, The Maximum Principle, Progress in Nonlinear Differential Equations and their Applications, 73, Birkhäuser Verlag, Basel, 2007. Google Scholar

  • [35]

    E. Zeidler, Nonlinear Functional Analysis and Applications II A/B, Springer, New York, 1990. Web of ScienceGoogle Scholar

About the article

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Received: 2018-09-05

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, DOI: https://doi.org/10.1515/anona-2020-0005.

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© 2020 Y. Bai et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0

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