Multiple positive solutions for singular anisotropic Dirichlet problems

Multiple positive solutions for singular anisotropic Dirichlet problems

Zhenhai Liu

and Nikolaos S. Papageorgiou

1Guangxi Colleges and Universities Key Laboratory of Complex System Optimization and Big Data Processing, Yulin Normal University, Yulin 537000, P.R. China.

2Guangxi Key Laboratory of Hybrid Computation and IC Design Analysis, Guangxi University for Nationalities, Nanning, Guangxi, 530006, P.R. China

3Department of Mathematics, National Technical University, Zografou Campus, 15780 Athens, Greece

Received 22 April 2021, appeared 10 July 2021 Communicated by Gabriele Bonanno

Abstract. We consider a nonlinear Dirichlet problem driven by the variable exponent (anisotropic) p-Laplacian and a reaction that has the competing effects of a singular term and of a superlinear perturbation. There is no parameter in the equation (non- parametric problem). Using variational tools together with truncation and comparison techniques, we show that the problem has at least two positive smooth solutions.

Keywords: variable exponent, anisotropic regularity, anisotropic maximum principle, positive solutions, critical point theory.

2020 Mathematics Subject Classification: 35J75, 35J60.

1 Introduction

Let Ω ⊆ _R^N be a bounded domain with a C²-boundary ∂Ω. In this paper we study the following anisotropic singular Dirichlet problem

−_∆p(z)u(z) =u(z)^−η(z)+ f(z,u(z)) inΩ, u|_∂Ω=0, u>0. (1.1) In this problem the exponentp:Ω→_Rin the differential operator, is Lipschitz continuous (that is p ∈ C^0,1(_Ω)) _{and 1} < _p− = _{minΩp. By ∆p(z)} we denote the anisotropic p-Laplace operator defined by

∆_p(z)u=div(|Du|^p(z)−2Du) ∀u∈W₀^1,p(z)(_Ω).

In problem (1.1) we have the competing effects of a singular termx^−η(z)withη∈C(_Ω), 0<

η(z) < _{1 for all} z ∈ _Ω and a Carathéodory perturbation f(z,x) (that is, for all x ∈ _R,z → f(z,x) is measurable and for a.a. z ∈ _Ω,x → f(z,x) is continuous), which is (p+−1)- superlinear as x → +_∞ (here p+ = max_Ωp), but need not satisfy the usual for superlinear

BCorresponding author. Email: zhhliu@hotmail.com

problems Ambrosetti–Rabinowitz condition (the AR-condition for short). We are looking for positive solutions. Using a combination of variational tools based on the critical point theory, together with truncation and comparison techniques, we show that the problem has at least two positive smooth solutions.

While anisotropic boundary value problems have been studied extensively in the last few years (see the books of Diening–Harjulehto–Hästö–R ˚užiˇcka [2] and of R˘adulescu–Repovš [12]

and the references therein), the study of singular anisotropic problems is lagging behind.

Only a very limited number of works exist on this subject and they all concern parametric problems (see the works of Byun-Ko [1] and Saoudi–Ghanmi [13]). The presence of parameter in the equation is very helpful, since by varying the parameter, we achieve certain desirable geometric configurations which in turn permit the use of the minimax theorems of critical point theory. In problem (1.1) there is no parameter to facilitate the analysis.

2 Mathematical background – hypotheses

The study of problem (1.1) requires the use of Lebesgue and Sobolev spaces with variable ex- ponents. A comprehensive presentation of these spaces can be found in the book of Diening–

Harjulehto–Hästö–R ˚užiˇcka [2].

For everyr∈ C(_Ω)we set

r− =_min

Ω r andr+=_max

Ω r.

Let E₁ = {r ∈ C(_Ω) _{: 1} < r−} _and M(_Ω) = {u : Ω → _Rmeasurable}. As usual, we identify two such functions which differ only on a Lebesgue-null set. Forr ∈ E₁, the variable exponent Lebesgue spaceL^r(z)(_Ω)is defined by

L^r(z)(_Ω) =

u∈ M(_Ω): Z

Ω|u|^r(z)dz<_∞

. We equip this space with the so-called “Luxemburg norm” defined by

kuk_r(z) =inf

"

λ>0 : Z

Ω

|u(z)|

λ r(z)

dz≤1

#

, u∈ L^r(z)(_Ω).

With this norm the space L^r(z)(_Ω) is a Banach space which is separable and reflexive (in fact uniformly convex). Let r⁰ ∈ E₁ be defined by r⁰(z) = _r(^r_z⁽₎₋^z)

1 for all z ∈ _Ω (that is,

1

r(z)+ _r0(¹z) =1 for allz∈Ω). Then we have

L^r(z)(_Ω)^∗ = L^r0(z)(_Ω) and the following version of Hölder’s inequality is true

Z

Ω|uv|dz≤ 1

r−

+ ¹ r⁰₋

kuk_r(z)kvk_r0(z), ∀u∈ L^r(z)(_Ω),∀v∈ L^r0(z)(_Ω). Note that ifr₁,r₂ ∈E₁ andr₁(z)≤r₂(z)for allz∈ Ω, then we have

L^r2(z)(_Ω),→L^r1(z)(_Ω) continuously.

Using the variable exponent Lebesgue spaces, we can introduce variable exponent Sobolev spaces. Givenr ∈E₁, the anisotropic Sobolev spaceW^1,r(z)(_Ω)is defined by

W^1,r(z)(_Ω) ={u∈ L^r(z)(_Ω)_:|Du| ∈L^r(z)(_Ω)}_,

where Dudenotes the gradient ofuin the weak sense. This space is equipped with the norm kuk_1,r(z)=kuk_r(z)+kDuk_r(z), u∈W^1,r(z)(_Ω) (herekDuk_r(z)= k|Du|k_r(z)). Ifr ∈ E₁∩C^0,1(_Ω), then we define

W₀^1,r(z)(_Ω) =C^∞_c (_Ω)^k·k1,r(z).

The spaces W^1,r(z)(_Ω) and W₀^1,r(z)(_Ω)are separable, reflexive (in fact uniformly convex).

For the spaceW₀^1,r(z)(_Ω)the Poincaré inequality holds, that is, there exists ˆc>0 such that kuk_r(z)≤_bckDuk_{r(z) for all}u∈W₀^1,r(z)(_Ω)_.

This implies that onW₀^1,r(z)(_Ω)we can use the equivalent norm

|u|_1,r(z)=kDuk_r(z), u∈W₀^1,r(z)(_Ω). Forr∈ E₁, we set

r^∗(z) =







Nr(z)

N−r(z)^{, ifr}(z)< N +_∞, if N≤r(z)

∀z∈_Ω.

Let r,q ∈ E₁∩C^0,1(_Ω) and suppose that q(z) ≤ r^∗(z) (resp. q(z) < r^∗(z)) for all z ∈ _Ω.

Then we have the anisotropic Sobolev embedding theorem W₀^1,r(z)(_Ω),→L^q(z)(_Ω) continuously (resp.W₀^1,r(z)(_Ω),→ L^q(z)(_Ω) compactly).

In the study of these spaces, central role plays the following modular function ρ_r(u) =

Z

Ω|u|^r(z)dz for all u ∈L^r(z)(_Ω). Ifu∈W₀^1,r(z)(_Ω)or u∈W^1,r(z)(_Ω), thenρ_r(Du) =ρ_r(|Du|). This modular function is closely related to the Luxemburg norm.

Proposition 2.1. If r∈ E₁and{un,u}_n∈N⊆ L^r(z)(_Ω), then we have (a) For allλ>0,

kuk_r(z) =λ⇔ρ_r(^u

λ) =_1;

(b) kuk_r(z) <₁⇔ kuk^r+

r(z)≤ ρ_r(u)≤ kuk^r−

r(z), kuk_r(z) >1⇔ kuk^r−

r(z)≤ ρ_r(u)≤ kuk^r+

r(z);

(c) ku_nk_r(z) →0⇔ρ_r(u_n)→0;

(d) ku_nk_r(z) →_∞⇔ρ_r(u_n)→+_∞.

Also forr∈ E₁∩C^0,1(_Ω), we have

W₀^1,r(z)(_Ω)^∗ =W^−1,r0(z)(_Ω).

Consider the operator Ar :W₀^1,r(z)(_Ω)→W^−1,r0(z)(_Ω)defined by hA_r(z)(u),hi=

Z

Ω|Du|^r(z)−2(Du,Dh)_RNdz, for allu,h∈W₀^1,r(z)(_Ω).

This operator has the following properties (see Gasi ´nski–Papageorgiou [5], Proposition 2.5 and R˘adulescu–Repovš [12], p.40).

Proposition 2.2. If r ∈E₁∩C^0,1(_Ω)and A_r:W₀^1,r(z)(_Ω)→W^−1,r0(z)(_Ω)is defined as above, then A_r(·)is bounded (maps bounded sets to bounded sets), continuous, strictly monotone (hence maximal monotone too) and is of type(S)+, that is, it has the following property:

“if u_n −→^w u in W₀^1,r(z)(_Ω)and lim sup

n→_∞

hA_r(z)(u_n),u_n−ui ≤0, then u_n →u in W₀^1,r(z)(_Ω).”

For everyu∈W₀^1,r(z)(_Ω), we defineu^±=_max{±u, 0}_{. Then}

u^± ∈W₀^1,r(z)(_Ω), u=u⁺−u⁻, |u|=u⁺+u⁻.

Suppose u,v : Ω → _R are measurable functions such thatu(z)≤ v(z)for a.a z ∈ _{Ω. We} define

[u,v] =ⁿh∈W₀^1,r(z)(_Ω):u(z)≤ h(z)≤v(z) for a.a. z∈_Ω^o, [u) =ⁿh∈W₀^1,r(z)(_Ω):u(z)≤ h(z) for a.a. z ∈_Ω^o.

Another space that we will need isC¹₀(_Ω) = {u ∈ C¹(_Ω) : u|_∂Ω = 0}. This is an ordered Banach space with positive (order) cone C+ = {u ∈ C¹₀(_Ω) : u(z) ≥ 0 for allz ∈ _Ω}. This cone has a nonempty interior given by

intC+ =

u∈C+:u(z)>0 for all z∈_Ω, ∂u

∂n ∂Ω

<0

. withn(·)being the outward unit normal on∂Ω.

LetX be a Banach space andϕ∈ C¹(X). We introduce the set K_ϕ ={u∈X :ϕ⁰(u) =0} (the critical set of ϕ).

We say that ϕ(·)satisfies the “C-condition”, if it has the following property:

“Every sequence{u_n}_n∈N⊆ X such that{ϕ(u_n)}_n∈N⊆_Ris bounded

and(1+kunk_X)ϕ⁰(un)→0in X^∗ as n →_∞,admits a strongly convergent subsequence.”

Now we are ready to introduce our hypotheses on the data of problem (1.1).

H₀: p∈ C^0,1(_Ω), 1< p−=min_Ωp,η∈ C(_Ω), 0<η(z)<1 for allz∈_Ω.

H₁: f :Ω×_R→_Ris a Carathéodory function such that f(z, 0) =0 for a.a.z∈_Ωand (i) |f(z,x)| ≤ a(z)[1+x^r(z)−1] for a.a. z ∈ _{Ω, all} x ≥ 0, with r ∈ C(_Ω) and p(z) <

r(z)< p^∗(z)for allz∈ _Ω;

(ii) if F(z,x) = Rx

0 f(z,s)ds, then limx→+_∞ ^F(z,x)

x^p+ = +_∞ uniformly for a.a. z ∈ _Ωand there existsτ∈C(_Ω)such that

τ(z)∈

(r+−p−)max N

p−

, 1

,p^∗(z)

for allz∈_Ω, 0<η_b0 ≤lim inf

x→+_∞

f(z,x)x−p+F(z,x)

x^τ(z) uniformly for a.a. z∈ _Ω;

(iii) there existsθ >0 such that

θ^−η(z)+ f(z,θ)≤ −_bc<0 for a.a. z∈ _Ω;

(iv) there existδ >_{0 and}q∈E₁ such thatq+ < p− such that

c₁x^q(x)−1 ≤ f(z,x) for a.a. z ∈_{Ω, all 0}≤x ≤δ, withc₁ >0;

(v) there existsξb_θ >0 such that for a.a.z ∈Ω, the function

x → f(z,x) +ξb_θx^p(z)−1 is nondecreasing on [0,θ].

Remark 2.3. Since we look for positive solutions and all the above hypotheses concern the positive semiaxis R+ = [0,∞), we can always assume without any loss of generality that f(z,x) = 0 for a.a. z ∈ _{Ω, all} x ≤ 0. Hypotheses H₁(ii)implies that for a.a. z ∈ _Ω f(z,·) is (p+−1)-superlinear. However, it need not satisfy the AR-condition which is common in the literature when dealing with superlinear problems (see, for example, Saoudi–Ghanmi [13], hypothesis (H4) and Byun–Ko [1, p. 76]). ConditionH₁(ii)is less restrictive and incorporates in our framework also superlinear nonlinearities with “slower” growth asx→ +∞, which fail to satisfy the AR-condition. For example, the following function f(z,x)satisfies hypotheses H₁but fails to satisfy the AR-condition:

f(z,x) =

((x⁺)^q(z)−1−2(x⁺)^k(z)−1 if x≤1 x^p+−1lnx−x^p(z)−1 if 1< x,

with q ∈ E₁ as in hypothesis H₁(iv), k ∈ C(_Ω), τ(z) < k(z) for az ∈ Ω. Evidently for this f(z,x)we can chooseθ =1. HypothesesH₁(iii),(iv)dictate an oscillatory behavior for f(z,·) near 0⁺ since it starts positive near zero (see hypothesisH₁(v)) and drops to negative values as we approachθ >0 (see hypothesisH₁(iii)). Also, hypothesisH₁(v)implies the presence of a concave term near zero.

3 An auxiliary problem

When dealing with singular problems, a major difficulty that we encounter, is that the pres- ence of the singularity leads to an energy functional which is not C¹. This fact prevents us

from using the results of critical point theory. So, we need to find a way to bypass the singu- larity and deal withC¹-functions in order to use the minimax theorems of critical point theory.

This is done by using the solution of an auxiliary problem which we introduce and solve in this section. The auxiliary problem is suggested by a unilateral growth condition satisfied by f(z,·). More precisely note that on account of hypothesesH₁(i),(iv), we can findc2>0 such that

f(z,x)≥c₁x^q(z)−1−c₂x^r(z)−1 for a.az∈_{Ω, all} x≥0. (3.1) Motivated by this unilateral growth condition on f(z,·)and using hypothesis H₁(iii), we introduce the Carathéodory functiong:Ω×_R→_Rdefined by

g(z,x) =

(c₁(x⁺)^q(z)−1−c₂(x⁺)^r(z)−1 ifx≤ θ

c₁θ^q(z)−1−c2θ^r(z)−1 ifθ <x. (3.2) Then we consider the following Dirichlet problem

−_∆p(z)u(z) = g(z,u(z)) inΩ, u|_∂Ω =0,u>0. (3.3) Proposition 3.1. If hypotheses H0 hold, then problem(3.3)has a unique positive solution u∈intC+

and0≤u(z)≤θfor all z∈ _Ω.

Proof. First we show the existence of a positive solution for problem (3.3). To this end, let ψ₀ :W₀^1,p(z)(_Ω)→_Rbe theC¹-functional defined by ψ₀(u) =R

Ω 1

p(z)|Du|^p(z)dz−R

ΩG(z,u)dz for allu∈W₀^1,p(z)(_Ω), whereG(z,x) =Rx

0 g(z,s)ds. From (3.2), we see that ψ₀(u)≥ ¹

pρ_p(Du)−c₃ for somec₃>0,

⇒ψ₀(·) is coercive (see Proposition2.1).

Also, from the anisotropic Sobolev embedding theorem, we see that ψ0(·) is sequentially weakly lower semicontinuous.

So, by the Weierstrass–Tonelli theorem, we can findu∈W₀^1,p(z)(_Ω)such that

ψ₀(u) =min[ψ₀(u):u∈W₀^1,p(z)(_Ω)]. (3.4) Letu ∈intC+and choose t ∈ (0, 1)small so that 0≤ tu(z)≤ θ for allz∈ Ω. Then using (3.2), we have

ψ₀(tu)≤ ^tp− p−

ρ_p(Du) + ^t

r−

r−

ρ_τ(u)− ^tq+ q+

ρ_q(u)

≤c₄t^p−−c₅t^q+ for somec₄,c₅ >0. (3.5) (since 1<q+< p−< r− andt ∈(0, 1)).

From (3.5) we see that by takingt ∈(0, 1)even smaller if necessary, we have ψ₀(tu)<_0,

⇒ψ₀(u)<0=ψ₀(0) (see (3.4)),

⇒u6=0.

From (3.4) we have that ψ⁰₀(u) =0,

⇒ hA_p(z)(u),hi=

Z

Ωg(z,u)hdz, for all h∈W₀^1,p(z)(_Ω). (3.6) In (3.6) first we chooseh=−u⁻∈W₀^1,p(z)(_Ω)_{and obtain}

ρ_p(Du⁻) =0 (see (3.2)),

⇒u ≥0, u6=0.

Next in (3.6) first we chooseh= [u−θ]⁺∈W₀^1,p(z)(_Ω). We obtain hA_p(z)(u),(u−θ)⁺i=

Z

Ω[c₁θ^q(z)−1−c₂θ^r(z)−1](u−θ)⁺dz (see (3.2))

≤

Z

Ωf(z,θ)(u−θ)⁺dz (see (3.1))

≤0=hA_p(z)(θ),(u−θ)⁺i (seeH₁(iii)),

⇒u≤θ.

So, we have proved that

u∈[0,θ], u6=0. (3.7)

From (3.7),(3.2) and (3.6), we infer thatu6= 0 is a positive solution of problem (3.3). From Fan [3] (Theorem 1.3), we have that u∈C+\{0}. Moreover, we have

∆_p(z)(u)≤c₂θ^r(z)−p(z)u(z)^p(z)−1≤c₆u(z)^p(z)−1 inΩfor somec₆ >0.

Then the anisotropic maximum principle of Zhang [15, Theorem 1.2] implies that

u ∈intC+. (3.8)

Next we show that this positive solution of (3.3) is in fact unique. Letv ∈ W₀^1,p(z)(_Ω) be another positive solution of (3.3). Again we have

v∈intC+. (3.9)

From (3.8) and (3.9) and using Proposition 4.1.22, p. 274, of Papageorgiou–R˘adulescu- Repovš [9], we have that

u

v ∈ L^∞(_Ω) and v

u ∈ L^∞(_Ω). (3.10)

Letj:L¹(_Ω)→_R=_R∪ {+_∞}be the integral functional defined by j(u) =

(R

Ω 1

p(z)|Du^1/p−|^p(z)dz if u≥0,u^1/p− ∈W₀^1,p(z)(_Ω), +_∞ otherwise.

Let domj = {u ∈ L¹(_Ω) : j(u) < _∞}(the effective domain of j(·)). From Theorem 2.2 of Takaˇc–Giacomoni [14], we know that j(·) is convex. Let h = u^p−−v^p− ∈ W₀^1,p(z)(_Ω). On account of (3.10), for|t|<1 small, we have

u^p−+th∈_domj and v^p−+th∈_domj.

Then the convexity ofj(·)implies the Gateaux differentiability ofj(·)atu^p− and atv^p− in the directionh. Moreover, using Green’s theorem, we obtain

j⁰(u^p−)(h) = ¹ p−

Z

Ω

−_∆p(z)u

u^p−−1 hdz = ¹ p−

Z

Ω

c₁

u^p−−q(z) −c₂u^r(z)−p−

hdz,

j⁰(v^p−)(h) = ¹ p−

Z

Ω

−_∆p(z)v

v^p−−1 hdz = ¹ p−

Z

Ω

c₁

v^p−−q(z) −c₂v^r(z)−p−

hdz.

The convexity of j(·)implies the monotonicity ofj⁰(·). So, we have

0≤

Z

Ω

c₁

1

u^p−−q(z) − ¹ v^p−−q(z)

−c₂(u^r(z)−p−−v^r(z)−p−)

(u^p−−v^p−)dz≤0 (sinceq+ < p− <r−)

⇒u= v.

This proves the uniqueness of the positive solutionu∈intC+.

In what follows, letdb(·) =d(·,∂Ω)andub₁is the positive,L^p+-normalized (that is,ku_b1k_p+ = 1) eigenfunction corresponding to the principal eigenvalue of (−_∆p+,W^1,p+(_Ω)). We know thatub₁ ∈intC+ (see, for example, Gasi ´nski–Papageorgiou [4, p. 739]).

Proposition 3.2. If Hypotheses H0hold and u∈ intC+ is the unique solution of problem(3.3), then u(·)^−η(·) ∈L¹(_Ω)and for every h∈W₀^1,p(z)(_Ω),u(·)^−η(·)h(·)∈ L¹(_Ω).

Proof. From Lemma 14.16, p. 355 of Gilbarg–Trudinger [6], we can find δ0 > 0 such that, if Ωδ0 = {z ∈ _Ω : db(z) < δ₀}, then db ∈ C²(_Ωδ0). If follows that db∈ intC+ and so by Proposition 4.1.22, p. 274, of Papageorgiou–R˘adulescu–Repovš [9], we can find c₇ > 0 such that

c₇ub₁ ≤db and c₇db≤u (recallu∈intC+). (3.11) From (3.11) we infer that

u^−η(·)≤ c₈ub⁻₁^η(·) for somec₈ >0.

Then the Lemma (in fact its proof to be precise) of Lazer–McKenna [8], implies thatub⁻₁^η(·)∈ L¹(_Ω). Therefore we have

u^−η(·)∈ L¹(_Ω). On the other hand, for everyh∈W₀^1,p(z)(_Ω), we have

Z

Ω|u^−η(z)h|dz=

Z

Ωu^1−η(z)|h| u dz

≤c9

Z

Ω

|h|

u dz for somec9>0

(recall thatu∈ intC+and see hypothesesH₀)

≤c₁₀ Z

Ω

|h|

dbdz for somec₁₀>0 (see (3.11))

≤c₁₁k^h

dbk_p(z) for somec₁₁ >0

≤c₁₂kDhk_{p(z) for some}c₁₂>_0.

This last inequality is a consequence of the anisotropic Hardy inequality due to Harjulehto–

Hästö–Koskenoja [7]. So, finally we have

u(·)^−η(·)h(·)∈L¹(_Ω) _{for all}h∈W₀^1,p(z)(_Ω)_.

4 Multiple positive solutions

In this section usingu ∈intC+, the unique positive solution of (3.3), we are able to bypass the singularity and have C¹-functionals. Working with them, we show that problem (1.1) has at least two positive smooth solutions.

Theorem 4.1. If hypotheses H₀,H₁hold, then problem(1.1)has at least two positive solutions u₀,ub∈ intC+,u06= u,_b u0(_z)< θfor all z∈ _Ω.

Proof. Let u ∈ intC⁺ be the unique positive solution of problem (3.3) produced in Proposi- tion3.1. We introduce the Carathéodory functiong:Ω×_R→_Rdefined by

g(z,x) =

(u^−η(z)+ f(z,u(z)) if x≤u(z)

x^−η(z)+ f(z,x) if u(z)< x. (4.1) From Proposition3.1we know that 0≤u(z)≤θ for allz∈ Ω. Hence we can consider the truncation of g(z,·)atθ, that is, the Carathéodory function bg:Ω×_R→_Rdefined by

bg(z,x) =

(g(z,x) if x≤θ

g(z,θ) ifθ <x. (4.2)

We set G(z,x) = Rx

0 g(z,s)ds andGb(z,x) = Rx

0 gb(z,s)ds and consider the functionsψ,ψb: W₀^1,p(z)(_Ω)→_Rdefined by

ψ(u) =

Z

Ω

1

p(z)|Du|^p(z)dz−

Z

ΩG(z,u)dz, ψb(u) =

Z

Ω

1

p(z)|Du|^p(z)dz−

Z

ΩGb(z,u)dz, for allu∈W₀^1,p(z)(_Ω).

On account of Proposition3.2, these functionals are well-defined and in fact Proposition3.1 of Papageorgiou–Smyrlis [11] implies thatψ,ψb∈ C¹(W₀^1,p(z)(_Ω)).

For everyu∈W₀^1,p(z)(_Ω), we have ψb(u)≥ ¹

p+

ρ_p(Du)−c₁₃ for somec₁₃ >0, (see (4.1),(4.2) and Proposition3.2)

⇒ψb(·)is coercive.

(see Proposition2.1and use Poincaré’s inequality).

The anisotropic Sobolev embedding theorem implies thatψb(·)is sequentially weakly lower semicontinuous.

So, by the Weierstrass–Tonelli theorem, we can findu₀∈W₀^1,p(z)(_Ω)such that

ψb(u₀) =_min[ψb(u)_: u∈W₀^1,p(z)(_Ω)]_{, (4.3)}

From (4.3) we have

hψb⁰(u₀),hi=0 for allh∈W₀^1,p(z)(_Ω),

⇒ hA_p(z)(u₀),hi=

Z

Ωbg(z,u₀)hdz for allh ∈W₀^1,p(z)(_Ω). (4.4) In (4.4) first we chooseh = [u−u₀]⁺∈W₀^1,p(z)(_Ω). We have

hA_p(z)(u₀)_,(u−u₀)⁺i=

Z

Ω[u^−η(z)+ f(z,u)](u−u₀)⁺dz (see (4.1),(4.2))

≥

Z

Ω f(z,u)(u−u₀)⁺dz (since u∈intC+)

= hA_p(z)(u),(u−u0)⁺i (see Proposition3.1),

⇒u ≤u0 (see Proposition2.2). Next in (4.4) we chooseh= [u₀−θ]⁺ ∈W₀^1,p(z)(_Ω). We have

hA_p(z)(u0),(u0−θ)⁺i=

Z

Ω[θ^−η+ f(z,θ)](u0−θ)⁺dz (see (4.1),(4.2))

≤0= hA_p(z)(θ)_,(_u0−θ)⁺i (see hypothesisH₁(_iii))_,

⇒u₀ ≤_θ.

So, we have proved that

u₀ ∈[u,θ]_{. (4.5)}

From (4.5), (4.1), (4.2) and (4.4), we have that u₀ is a positive solution of (1.1). Invoking Theorem 13.1 of Saaudi–Ghanmi [13] (see also Theorem 3.2 of Byun–Ko [1]), we have that u₀ ∈intC+(recallu∈intC+).

Now letξb_θ >0 be as postulated by hypothesis H₁(v)_{. We have}

−_∆p(z)u₀+ξb_θu₀^p(z)−1−u⁻₀^η(z)

= f(z,u₀) +ξb_θu₀^p(z)−1

≤ f(z,θ) +ξb_θθ^p(z)−1 (see (4.5) and hypothesis H₁(v))

≤ −_∆p(z)θ+ξb_θθ^p(z)−1−θ^−η(z) (see hypothesis H₁(iii))_,

⇒u₀(z)<θ for allz∈_Ω (4.6)

(from Proposition A4 of Papageorgiou–R˘adulescu–Zhang [10]). It is clear from (4.1) and (4.2) that

ψ|_[0,θ] =ψb|_[0,θ]. Sinceu₀ ∈intC+, we infer that

u₀ is a localC¹₀(_Ω)minimizer ofψ(·) (see (4.6)),

⇒u₀ is a localW₀^1,p(z)(_Ω)minimizer ofψ(·) (_{see [10,13]})_{. (4.7)} Using (4.1) and the anisotropic regularity theory, we can see that K_ψ ⊆ [u)∩intC+. So, we may assume thatK_ψ is finite or otherwise on account of (4.1) we see that we already have

a whole sequence of distinct positive smooth solutions and so we are done. Then from (4.7) and Theorem 5.7.6, p. 449, of Papageorgiou–R˘adulescu–Repovš [9], we know that we can find ρ∈(0, 1)small such that

ψ(u0)<inf[ψ(u):ku−u0k=ρ] =mρ. (4.8) Moreover, hypothesis H₁(ii)implies that ifu∈intC+, then

ψ(tu)→ −_∞ ast→+_∞. (4.9)

Finally from Proposition 4.1 of Gasi ´nski–Papageorgiou [5] (see hypothesisH₁(ii)), we have that

ψ(·) satisfies the C-condition. (4.10)

Then (4.8), (4.9) and (4.10) permit the use of the mountain pass theorem. Therefore we can find ub∈W₀^1,p(z)(_Ω)such that

ub∈ Kψ⊆ [u)∩intC+,mρ ≤ψ(u_b), (4.11)

⇒u_b∈intC+is a positive solution of (1.1) (see (4.1)), ub6=u₀ (see (4.8) and (4.11))_,u₀(z)<θ for allz ∈_Ω.

Acknowledgements

The work was supported by NNSF of China Grant No. 12071413, NSF of Guangxi Grant No.

2018GXNSFDA138002.

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