Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Advances in Nonlinear Analysis

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

IMPACT FACTOR 2018: 6.636

CiteScore 2018: 5.03

SCImago Journal Rank (SJR) 2018: 3.215
Source Normalized Impact per Paper (SNIP) 2018: 3.225

Mathematical Citation Quotient (MCQ) 2017: 1.89

Open Access
See all formats and pricing
More options …

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
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Dumitru Motreanu / Shengda Zeng
  • Corresponding author
  • Jagiellonian University in Krakow, Faculty of Mathematics and Computer Science, ul. Lojasiewicza 6, 30348, Krakow, Poland
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2019-06-01 | DOI: https://doi.org/10.1515/anona-2020-0005


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



    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


    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


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


    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


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


    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


    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


    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



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




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




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


We note that




Altogether, we obtain


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


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


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


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


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


for all hW01,p(Ω). Inserting h = (uu)+, through (4) and (7), we infer that


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,



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


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


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


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


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


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


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


  2. Γ is left continuous.


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


    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


    for all vW01,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 W01,p(Ω). Given r > N, it is true that (uλ1)rint(C01(Ω¯)+), so there is a constant c2 > 0 such that




    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


    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


    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.


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


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


hence there exists a sequence {un} ⊂ int(C01(Ω¯)+) satisfying

unΛ(λn)D for all nN,

in particular


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


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


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.


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


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


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,


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. □


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.


  • [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

Tel.: +86-18059034172

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.

Export Citation

© 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

Comments (0)

Please log in or register to comment.
Log in