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Introduction This article concerns the existence of solutions and estimates of the intervals of parameters for which the problem −∆pu=a(x)f(u) +λb(x)g(u) +µV(x,∇u) in Ω, u >0 in Ω, u= 0 on∂Ω, (1.1) has a solution

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ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu

QUASILINEAR PROBLEMS WITH TWO PARAMETERS INCLUDING SUPERLINEAR AND GRADIENT TERMS

MANUELA C. REZENDE, CARLOS A. SANTOS

Abstract. In this article, we establish conditions for the existence of solu- tions for a quasilinear elliptic two-parameter problem with dependence on the gradient term in smooth bounded domains or in the whole space RN. We consider superlinear and asymptotically linear terms. Estimates on the values of two parameters for which the problem have solutions are provided.

1. Introduction

This article concerns the existence of solutions and estimates of the intervals of parameters for which the problem

−∆pu=a(x)f(u) +λb(x)g(u) +µV(x,∇u) in Ω,

u >0 in Ω, u= 0 on∂Ω, (1.1)

has a solution. Here, ∆pu = div(|∇u|p−2∇u), 1 < p < ∞, denotes the usual p-Laplacian operator; λ > 0; µ ≥ 0 are real parameters; f, g : (0,∞) → [0,∞);

a, b : Ω → [0,∞) with a, b 6= 0; V : Ω×RN → [0,∞) are continuous functions satisfying appropriate hypotheses and either Ω⊂RN is a smooth bounded domain or Ω =RN. When Ω =RN, the conditionu= 0 on∂Ω means thatu(x)→0 when

|x| → ∞.

By a solution of (1.1) we mean a functionu=uλ,µ∈C1(Ω)∩C(Ω), withu >0 in Ω,u= 0 on∂Ω and

Z

|∇u|p−2∇u∇φ dx= Z

[a(x)f(u) +λb(x)g(u) +µV(x,∇u)]φ dx, for allφ∈C0(Ω).

In this article we say that a functionh: (0,∞)→[0,∞) is (p−1)-sublinear at 0 or at +∞, if lims→0h(s)/sp−1=∞or lims→∞h(s)/sp−1= 0 respectively; (p−1)- superlinear at 0 or at +∞, if lims→0h(s)/sp−1 = 0 or lims→∞h(s)/sp−1 = ∞ respectively and (p−1)-asymptotically linear, if there are positive and finite num- bers that correspond to the values of these limits. To abbreviate, we say sublinear, superlinear and asymptotically linear nonlinearities, respectively. In particular, a sublinear termhat 0 is called singular at 0, if lims→0+h(s) =∞.

2000Mathematics Subject Classification. 35J92, 35J75, 35B08, 35B09.

Key words and phrases. p-Laplacian; entire solutions; superlinear; sublinear; gradient term.

c

2014 Texas State University - San Marcos.

Submitted April 10, 2013. Published October 21, 2014.

1

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For µ = 0, problems like (1.1) have been intensively studied in recent years, including sublinear and superlinear nonlinearities terms at zero and/or infinity and singular terms at zero. In [5, 15, 13, 18] and references therein studies on the bounded domain case are found. For Ω =RN, we refer the reader to [4, 14, 19, 21]

and their references.

However, there are not many results in the case where the nonlinearities depend on the gradient of the solution, that is,µ6= 0, withp6= 2. In general, variational techniques are not suitable to handle (1.1). In the case p = 2, an interesting exception can be seen in [6].

One of the novelties in this article is that we improve a regularization-mono- tonicity technique (see Section 2). This allows us to treat (1.1) with superlinear nonlinearities both in bounded domain and RN. This improvement also makes possible for us to study (1.1) in RN by creating a sequence of solutions of (1.1) in bounded domains, locally bounded below by a positive function and bounded above by a carefully constructed function.

We emphasize that our results do not require any monotonicity condition and (or) singularity of the functionsf andg. We are particularly interested in the cases wheref andgmay have singularity at 0. Problems including singular nonlinearities arise in electrical conductivity, the theory of pseudoplastic fluids, singular minimal surfaces, reaction-diffusion processes, the obtaining of various geophysical indexes and industrial processes, among others; see [3, 10] for a detailed discussion.

Considering the problem (1.1) in smooth bounded domains, we quote Zhang and Yu [27] who, in 2000, studied the problem

−∆u=u−α+λ+µ|∇u|q in Ω,

u >0 in Ω, u= 0 on∂Ω, (1.2)

whereµ, λ≥0,α >0 andq∈(0,2]. Using a change of variables, the authors proved that the problem (1.2) has classical solutions for µλ < λ1, if q= 2 or µ∈[0, µ), if 0< q <2, with µ(q, λ), whereλ1 >0 denotes the first eigenvalue of the Dirichlet problem inW01,2(Ω).

Ghergu and Radulescu [11] considered

−∆u=h(u) +λf(x, u) +µ|∇u|q in Ω,

u >0 in Ω, u= 0 on∂Ω, (1.3)

under the conditions f >0 in Ω×(0,∞), ∂f /∂s(x, s)≥0,s > 0,f(x, s)/s non- increasing in s > 0, lims→∞f(x, s)/s = 0, lims→0h(s) = +∞, h∈ C0,α((0,∞)), h >0 non-increasing andλ= 1. They proved that

(i) if 0< q <1, then (1.3) has solution for eachµ≥0,

(ii) if 1≤q≤2, there exists µ>0 such that (1.3) has a solution for 0≤µ <

µ. Moreover, if 1< q≤2, thenµ<∞.

In 2010, Alves, Carri˜ao and Faria [1] used the Galerkin method to study

−∆u=g(x, u) +µV(x,∇u) in Ω,

u >0 in Ω, u= 0 on∂Ω, (1.4)

whereg andV are locally H¨older continuous functions such that b|s|r1 ≤g(x, s)≤a1(x) +a2(x)|s|r2+a3(x)

|s|r3 , 0≤V(x, ξ)≤a5(x) +a4(x)|ξ|r4,

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with b > 0, ri ∈(0,1), i= 1, . . . ,4 are constants and ai, i = 1, . . . ,5 are positive continuous functions. Under these conditions, it was shown that (1.4) has a solution for eachµ≥0.

Liu, Shi and Wei [16], still withp= 2, recently showed, by using Morse theory and an iterative method, existence of solution for a problem like (1.1) with terms that have asymptotically linear growth at zero and infinity. Considering singular terms at 0 and permitting p6= 2, Loc and Schmitt [17] used the lower and upper solution method to show existence of solution for (1.1) with the nonlinearity of the gradient term bounded above by the natural growth.

For Ω =RN in the problem (1.1). In 2007, Ghergu and Radulescu [12] showed existence of solution for the problem

−∆u=a(x)[f(u) +g(u) +|∇u|q] inRN, u >0 in RN, and u(x)|x|→∞−→ 0,

(1.5) whereq∈(0,1),f ∈C1((0,∞)) is positive and decreasing, lims→0+f(s) =∞and the functiong: [0,∞)→[0,∞) satisfies

g0≥0, g(s)

s is non-increasing ins >0, lim

s→0+

g(s) s = +∞

and lim

s→∞

g(s) s = 0.

Concerning the functiona, they assumed that 0< a∈C0,α(RN) and Z

0

rφ(r)dr <∞, where φ(r) = max

|x|=ra(x). (1.6) In the same year, Xue and Zhang in [24] assumed (1.6) and studied the problem (1.5) without requiring any monotonicity condition overf andg. They just assumed

lim

s→0+

g(s)

s = +∞, lim

s→∞

g(s)

s = 0, lim

s→0+

f(s)

s = +∞, lim

s→∞

f(s) s = 0.

For the rest of this article, givenσ: (0,∞)→(0,∞), we denote byσi, σi ∈[0,∞]

the following limits σi:= lim

s→iσ(s) and σi:= lim

s→i

σ(s)

sp−1, fori= 0 ori=∞

and we assume that there exists a ρ ∈ C(Ω)∩L(Ω), ρ ≥ 0, ρ 6= 0 such that ρ≤a, b. We denote byλ1,Ω(ρ)>0 the first eigenvalue and byϕ1,Ω>0 the first eigenfunction of the problem

−∆pϕ=λρ(x)|ϕ|p−2ϕ in Ω,

ϕ >0 in Ω, ϕ= 0 on∂Ω, (1.7)

where Ω ⊂ RN is a smooth bounded domain. Moreover, we denote by λ1(ρ) = limR→∞λ1,BR(0)(ρ)≥0, whereBR(0) is the ball centered at the origin ofRN with radiusR >0.

Also, we let us assume:

(V1) V(x, ξ)≤α(x)|ξ|q+β(x) in Ω×RN for some 0 ≤α, β ∈C(Ω)∩L(Ω) andq≥0,

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(M1) there existsωM ∈C1(Ω) (ωM ∈C1(Ω)∩W1,∞(Ω) if Ω =RN) satisfying

−∆pωM =M(x) in Ω,

ωM >0 in Ω, ωM = 0 on∂Ω, (1.8)

whereM(x) := max{2a(x),2b(x), α(x), β(x)},x∈Ω, (F1)

(F0) f0<1/kωMkp−1L(Ω), or (F) f<1/kωMkp−1L(Ω). Remark 1.1. With respect to the hypotheses (M1) and (F1), we note that:

(1) If Ω⊂RN is a smooth bounded domain, then (M1) occurs if, for example, M ∈ Lq(Ω) for some q > N > 1. See, for instance, Perera and Zhang [20]. This allows we take singular potentials of the type a(x) = b(x) = 1/(1− |x|)γ, withγ <1 and Ω =B1(0)⊂RN in (1.1).

(2) If Ω =RN, it is known that (1.8) has a solution ifM is a bounded contin- uous function and satisfies

M:=

Z 0

h s1−N

Z s 0

tN−1Mˆ(t)dtip−11

ds <∞,

where ˆM(t) = max|x|=tM(x),t ≥0. The existence and L-boundedness of a solution of (1.8) imply its regularity (see [8]). In addition, if we assume thatN ≥3 and

Z 1

rp−11p−11 (r)dr <∞ or Z

1

r(p−2)N+1p−1 Mˆ(r)dr <∞,

if 1 < p ≤2 or p ≥2, respectively, then M <∞. In [25], we have an example that shows that the converse of this fact is not true.

(3) Condition (F0) holds iff is superlinear at 0 (f0= 0) and (F) occurs iff is sublinear at∞(f= 0).

To state our results, we assume 0< g0+f0≤ ∞and denote by λ(g0) :=





0, ifg0= 0 andf0> λ(ρ), max{0,λ(ρ)−fg 0

0 }, if 0< g0<∞,

0, ifg0=∞,

where λ1(ρ) =λ(ρ), if Ω =RN. We have thatλ = 0 in all the previous works, becauseg0=∞there.

Regarding problem (1.1) in bounded domains we have the following result.

Theorem 1.2. Assume that (F1), (M1), (V1) with q ∈ [0, p] hold. Then there existsλ ∈(0,∞] such that for eachλ < λ < λ, there exist µλ >0 and a u=uλ,µ∈C1(Ω)∩C(Ω) solution of (1.1)for each 0≤µ < µ. Additionally:

(i) u≥cϕfor somec >0,

(ii) if(Fi)holds, for i∈ {0,∞}, then λ≥ 1

gi

1

Mkp−1L(Ω)

−fi

:=λi, (iii) there exists a constant d >0 such that

µλ≥dmin

[fi+λgi]p−1−qp−1 , fi+λgi , if q∈[0, p−1].

For Ω =RN and 1< p < N our main result is the following.

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Theorem 1.3. Assume that (F1), (M1), (V1)with q∈[0, p−1]hold. Then there existsλ ∈(0,∞] such that for eachλ < λ < λ, there exist µλ >0 and a u=uλ,µ∈C1(RN)solution of (1.1)for each0≤µ < µ. Moreover, if(Fi)holds, fori∈ {0,∞}, then there is a constantd >0 such that

(i) λ≥λi (ii) µλ≥dmin

[fi+λgi]p−1−qp−1 , fi+λgi for0< λ < λi.

Remark 1.4. In the definition of λ, the possibility f0 > λ1(ρ) does not permit (F0) to occur, becauseλ(ρ)≥λ(M)≥ kwMk1−pL(Ω)and as a consequence of this, we have λ1(ρ) ≥λ1(M)≥ kwMk1−pL(RN) also (see Santos [21]). In this situation, (F) should occur, as in [12] and [24].

Theorem 1.3 improves previous results principally because it addresses the p- Laplacian operator, obtains estimates for λ and µ, no monotonicity or growth restriction on the nonlinearities are required, the cases q = 0 and q = p−1 are included and we assume the hypothesis (M1) that is weaker than (1.6). We point out that problem (1.1) has no solution forp≥N (see Serrin and Zou [22]).

This paper is organized as follows: In section 2 we construct several auxiliary functions for the terms f and g and we study their properties. Because of the singularities allowed onf andg, we regularize the problem (1.1) and we obtain an upper solution for it in bounded domain and inRN, in sections 3 and 5, respectively.

After that, we use section 4 to prove Theorem 1.2. In section 6, we generalize this result forRN.

2. Auxiliary functions

To prove Theorems 1.2 and 1.3 we refine a regularization-motonicity technique used, among others, by Feng and Liu [9], Zhang [26] and Mohammed [19].

Observing that we do not assume monotonicity on the nonlinearities, we intro- duce a truncation of the terms f and g through a parameter γ > 0 and build auxiliary functions which allow us to obtain not only the monotonicity, but also the necessary regularity for the proof of our results. Parallel to this, the inclusion of a parameter θ < 1, in this construction, makes it possible solving the problem (1.1) for the caseq > p−1.

Analyzing the behavior of these auxiliary functions, the parametersλ, γ, θ and the fact that the problem (1.8) has a solution, we determine a Λ-curve whose behavior allow us to find region of variation for the parameterλ, and consequently, obtain an estimate from below for that region.

With these purposes, let us define the continuous functions, depending on real parameterγ >0, as

fγ(s) :=

(f(s), if 0< s≤γ

f(γ)

γp−1sp−1, ifs≥γ

and gγ(s) :=

(g(s), if 0< s≤γ

g(γ)

γp−1sp−1, ifs≥γ.

Now, for eachs >0, defining the function ζλ,γ(s) =sp−1supfγ(t)

tp−1, t > s +λsp−1supgγ(t)

tp−1, t > s , λ≥0 (2.1) we obtain, from the above definitions, that

(i) ζλ,γsp−1(s) is non-increasing ins >0;

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(ii) ζλ,γ(s)≥fγ(s) +λgγ(s),s >0;

(iii) lims→∞ζλ,γsp−1(s)= γf(γ)p−1γg(γ)p−1. Now, defining

Hλ,γ(s) = s2 Rs

0 t ζλ,γ(t)p−11

dt, s >0, and using (i) and (iii) above, we have the following lemma.

Lemma 2.1. The function H satisfies:

(i) Hλ,γ ∈C1((0,∞),(0,∞));

(ii) ζλ,γ(s)≤[Hλ,γ(s)]p−1,s >0;

(iii) Hλ,γs(s) is non-increasing in s >0;

(iv)

s→∞lim

Hλ,γ(s)

s =hf(γ)

γp−1 +λg(γ) γp−1

ip−11 .

After these, introducing a parameterθ∈(0,1] and defining the function Γλ(γ) = Γλ,θ(γ) = θ

γ Z γ

0

tθ

Hλ,γ(tθ)dt, γ >0 (2.2) we obtain, from the previously defined functions and their properties, the following result.

Lemma 2.2. Suppose(M1)and(F1)hold. Then for eachθ∈(kwMkfi1/(p−1),1], for eitheri= 0or i=∞, we have:

(i) limγ→∞Γλ,θ(γ) = θ

(f+λg)p−11

, for each λ≥0;

(ii) limγ→0Γλ,θ(γ) = θ

(f0+λg0)p−11

, for each λ≥0;

(iii) Γλ,θ is decreasing in λ >0, for eachγ >0;

(iv) there exists aγ˜= ˜γ(Ω, θ)>0such that Γ0,θ(˜γ)>kωMkL(Ω). By Lemma 2.2, we can define the nonempty set

A=AΩ,θ:={γ∈(0,∞) : Γ0,θ(γ)>kωMkL(Ω)}.

Now, as a consequence of limλ→∞Γλ,θ(γ) = 0, limλ→0Γλ,θ(γ) = Γ0,θ(γ) and of the above lemma, we have that the function Λ= ΛΩ,θ :A →(0,∞) that associate for eachγ∈ Athe unique number Λ(γ) satisfying

ΓΛ(γ),θ(γ) =kωMkL(Ω), (2.3) is well defined.

Thus, we can define the positive number

λθ(Ω) := sup{Λ(γ) :γ∈ A}. (2.4) After these, we infer the following lemma.

Lemma 2.3. Suppose(M1)and(F1)hold. Then for eachθ∈(kwMkfi1/(p−1),1], we have

λθ(Ω)≥ 1 gi

θ

Mkp−1L(Ω)

−fi :=λiθ.

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Proof. If (F0) occurs andg0 <∞, then for each 0< δ < λ0θ, from Lemma 2.2 (ii) it follows that

lim inf

γ→0δ,θ(γ)− kωMk) = θ (f0+δg0)p−11

− kωMk

> θ

(f00θg0)p−11

− kωMk= 0.

Now, if (F) occurs andg<∞, using Lemma 2.2 (i), we have lim inf

γ→∞δ,θ(γ)− kωMk) = θ (f+δg)p−11

− kωMk

> θ

(fθ g)p−11

− kωMk= 0, for each 0< δ < λθ .

So, in both cases, there exists a γ00(δ) >0 such that Γδ,θ0)> kωMk. As a consequence of this and Lemma 2.3(iii), we have that γ0 ∈ A, because Γ0,θ0)>Γδ,θ0) >kωMk. So, from (2.3) there is a unique Λ0) such that ΓΛ0),θ0) =kωMk. Now, using ΓΛ0),θ0)<Γδ,θ0) and Lemma 2.3(iii), we obtain Λ0) > δ. So, by the arbitrariness of δ, it follows the proof of the

Lemma.

Now, defining

ηλ(s) =ηλ,θ(s) = θ γ

Z s 0

tθ

Hλ,γ(tθ)dt, s >0, γ∈ A, λ >0, (2.5) it follows that

ηλ,θ(γ) = Γλ,θ(γ)>kωMk+ ¯σ, (2.6) for each 0< λ <Λ(γ), where ¯σ= ¯σ(λ, θ, γ) = Γλ,θ(γ)− kωMk

/2>0.

Besides this, the following lemma follows from the previous results.

Lemma 2.4. Suppose(M1)and(F1)hold. Then, for each 0< λ < λθ(Ω) given:

(i) [¯σ,kωMk+ ¯σ]⊂Im(ηλ);

(ii) ηλ∈C2((0,∞),Im(ηλ))is increasing in s >0;

(iii) ηλ−1:=ψλ∈ C2((Im(ηλ)\{0},(0,∞))is increasing in s >0;

(iv) ψ0λ(s) = γHλ,γ0λ(s)

θ)

θψλ(s)θ ,s >0;

(v) ψ00λ(s)≤0,s >0;

(vi) ηλ is decreasing inλ.

3. An auxiliary problem

To solve the problem (1.1) with the gradient term in the presence of nonlinearities f and g already described, we will explore the behavior of the auxiliary λ, γ, θ- functions given in the previous section considering different intervals of variation forq∈[0, p] and an appropriate division of the domain Ω⊂RN. All this together with the behavior of the Λ-curve will allow us to determine a µ-curve whose behavior will define the region of variation of the parameterµ≥0.

As a consequence of the hypotheses (M1), (F1) and of the behavior of Λ, µ- curves, we obtain aγ0which allow us to show the existence of solution (-uniformly

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limited in L(Ω)) of the -family of problems (3.1) below, for appropriateλ > 0 andµ≥0.

In this sense, we will construct a positive bounded upper solution for the-family of problems

−∆pu=a(x)f(u+) +λb(x)g(u+) +µV(x,∇u) in Ω

u >0 in Ω, u= 0 on∂Ω, (3.1)

for sufficiently small >0.

Proposition 3.1. Assume (F1), (M1), (V1)withq∈[0, p]hold. Then there exists aλ>0 such that for each 0< λ < λ, there exist real numbersσ=σ(λ)>0and µλ >0, both independent of , such that if0 < σ≤σ and0≤µ < µ, then there exists avσ =vσ,λ∈C1(Ω) upper solution of(3.1). Additionally:

(i) ψλ(σ)θ0 ≤vσ ≤γ0θ0 for some θ00(λ) ∈(kwMkfi1/(p−1),1] and γ0 = γ0(λ)>0;

(ii) if(Fi)holds, for i∈ {0,∞}, then λ≥ 1

gi

1

Mkp−1L(Ω)

−fi :=λi;

(iii) there exists a constant d >0 such that for 0< λ < λi, we have µλ≥dmin

[fi+λgi]p−1−qp−1 , fi+λgi ifq∈[0, p−1].

Proof. Because the possible singular behavior of the nonlinearities, we divide this proof into two parts, depending on the value of the exponentqof the gradient term in the hypothesis (V1).

Case one: q ∈ [0, p−1]. In this case, we pick θ0 = 1 and take θ = θ0 in the functions Γλ,θ andηλ,θ. So, given 0< λ < λ :=λ1(Ω) we define, for eachγ >0, the positive number

µλ(γ) =µλ,Ω(γ) := minn[f(γ) +λg(γ)]p−1−qp−1 4k∇ωMkqL(Ω)

,f(γ) +λg(γ) 4

o

. (3.2) Now, we can define

µλλ,Ω:= sup{µλ(γ) :γ∈ Aand λ <Λ(γ)} ∈(0,∞]. (3.3) So, from (2.4), there existsγ∈ Asuch thatλ <Λ(γ). That is,µλ≥µλ(γ)>0.

Thus, given 0 ≤µ < µλ there is a γ0 = γ0(λ) ∈ Asuch that λ < Λ0) and µ < µλ0). Now, we fix thisγ0.

From the hypothesis (M1) and Lemma 2.4 (ii), we define vσ = vσ,λ ∈ C1(Ω), increasing inσ, by

vσ(x) :=ψλM(x) +σ), x∈Ω (3.4) for each 0< σ≤σ, where ¯¯ σ= ¯σ(λ) is given in (2.6). So, vσ(x)> ψλ(σ) in Ω and vσ(x) =ψλ(σ) on∂Ω, becauseωM(x)>0 in Ω andωM(x) = 0 on∂Ω.

Besides this, from (2.6), Lemma 2.4 (iii) and 0 < λ < Λ0) we have that vσ(x)< γ0, x∈Ω. So, there exists an >0, which is sufficiently small, such that

kvσkL(Ω)< γ0−, 0< σ≤σ.¯ (3.5)

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Now, it follows from (3.4), Lemmas 2.1, 2.4 and the assumption (M1), that Z

|∇vσ|p−2∇vσ∇φ dx

= Z

0λM) +σ]p−1|∇ωM|p−2∇ωM∇φ dx

= Z

|∇ωM|p−2∇ωM∇([ψλ0M +σ)]p−1φ)dx

−(p−1) Z

|∇ωM|pλ0M +σ)]p−2ψ00λM +σ)φ dx

≥ Z

M(x)[ψ0λM+σ)]p−1φ dx

= Z

M(x)γ0p−1hHλ,γ0λM+σ)) ψλM+σ)

ip−1 φ dx

(3.6)

for eachφ∈C0(Ω),φ≥0.

The study of this inequality will be divided in two parts. One of them will produce an estimate foraf+λbg while the other will result in an estimate forµV. We note that from the definitions and properties of the functions defined in the Section 2 and (3.5) that

1 2

Z

M(x)γ0p−1hHλ,γ0λM +σ)) ψλM +σ)

ip−1

φ

≥ 1 2

Z

M(x)γ0p−1ζλ,γ0(vσ+) (vσ+)p−1 φ

≥ 1 2

Z

M(x)γ0p−1ζλ,γ0(vσ+) (γ0)p−1 φ

≥ Z

[a(x)f(vσ+) +λb(x)g(vσ+)]φ

(3.7)

for each >0 and 0< σ <σ.¯

On the other hand, from Lemma 2.1 (iii)-(iv) and 0≤q≤p−1, it follows that 1

2 Z

M(x)γ0p−1hHλ,γ0λM +σ)) ψλM +σ)

ip−1

φ dx

≥ 1 4

Z

M(x)γp−10 hf(γ0)

γ0p−1 +λg(γ0) γ0p−1

iφ dx

+1 4

Z

M(x)γ0p−1−qhHλ,γ0λM +σ)) ψλM +σ)

ip−1−q0Hλ,γ0λM+σ)) ψλM +σ)

iq

φ dx

≥ [f(γ0) +λg(γ0)]

4

Z

M(x)φ dx +{[f(γ0) +λg(γ0)]p−11 }p−1−q

4

Z

M(x)[ψ0λM+σ)]qφ dx

≥ [f(γ0) +λg(γ0)]

4

Z

β(x)φ dx

+[f(γ0) +λg(γ0)]p−1−qp−1 4k∇ωMkqL(Ω)

Z

M(x)[ψ0λM+σ)]q|∇ωM|qφ dx.

(10)

Now, using (V1) and (3.2) we can write 1

2 Z

M(x)γ0p−1hHλ,γ0λM+σ)) ψλM+σ)

ip−1

φ dx

≥µλ0) Z

β(x)φ dx+µλ0) Z

α(x)|∇vσ|qφ dx≥µ Z

V(x,∇vσ)φ dx.

(3.8)

So, replacing (3.7) and (3.8) in (3.6), we conclude the proof of Proposition 3.1.

Case two: q ∈ (p−1, p]. If gi < ∞, we define λ := lim infθ%1λθ(Ω), where λθ(Ω) was defined in (2.4). Note that, by Lemma 2.3, we have λ ≥ λiθ with θ= 1. So, given 0< λ < λ, there is aθ00(λ)∈(kwMkfi1/(p−1),1) such that 0< λ < λθ

0(Ω).Now, we fix thisθ0 in the functions Γλ,θ andηλ,θ defined in (2.2) and (2.5), respectively. So, if gi =∞, we choose a θ0 ∈(kwMkfi1/(p−1),1) and we setλθ

0(Ω). In this case, we haveλ≥λiθ

0.

In both cases, given 0< λ < λ, we set the positive number µλ(γ) :=µλ,Ω(γ) by

minn γp−1−q 4C2k∇ωMkqL(Ω)

γ(p−1)(θ0−1)[f(γ) +λg(γ)]

4 ,

(1−θ0)(p−1)[γHλ,γ(1)]p−q 4kαkL(Ω)

o

(3.9)

for each γ > 0 and for some constant C2 =C2(γ) > 0 to be chosen posteriorly.

Now, we define

µλλ,Ω:= sup

µλ(γ) :γ∈ Aandλ <Λ(γ) .

As in Case one, we claim thatµλ>0 and given 0≤µ < µλ, there is aγ00(λ)∈ Asuch thatλ <Λ0) andµ < µλ0). From now on, we fix thisγ0.

SinceωM ∈C1(Ω) and ∂ωM/∂ν <0 on∂Ω, there are δ0 >0 sufficiently small andk0=k00)>0 such that

|∇ωM|p> k00) forx∈Ωδ0, (3.10) where Ωδ0 ={x∈Ω : dist(x, ∂Ω)< δ0} andν is the exterior normal to the∂Ω.

In a similar way to that done in (3.4), we obtain that

vσ(x) := [ψλM(x) +σ)]θ0, x∈Ω (3.11) is well-defined,ψλ(σ)θ0 ≤vσ ∈C1(Ω) andkvσkL(Ω)< γ0θ0, for each 0< σ ≤σ.¯ In the last conclusion, we used Lemma 2.4 and the inequality (2.6).

That is, there is a sufficiently small >0 such that

kvσkL(Ω)< γ0θ0−. (3.12) Since lims→0ψλ(s) = 0, we can take 0<σ <˜ σ¯ sufficiently small such that

ψλ(˜σ)θ0< 1

2 and k00)

λ(˜σ)θ0 >k∇ωMkqL(Ω). (3.13) So, from Lemma 2.4 (iii), it follows thatvσ(x)< v˜σ(x) in Ω, for each 0< σ <σ.˜ Moreover, sincev˜σ(x) =ψλ(˜σ)θ0 on∂Ω, it follows from Lemma 2.4 (iii) again, that there exists aδ11(˜σ)>0 sufficiently small such that

vσ(x)< v˜σ(x)<2ψλ(˜σ)θ0, forx∈Ωδ1, σ∈(0,σ).˜ (3.14)

(11)

Then, from (3.10), (3.13) and (3.14), we have

|∇ωM(x)|p

vσ(x) > k00)

λ(˜σ)θ0 >|∇ωM(x)|q, (3.15) for eachx∈Ωδ, whereδ= min{δ0, δ1}>0.

Now, given φ∈C0(Ω) withφ≥0 and 0< σ <σ, we take˜ τ ∈C0(Ω) defined byτ = 1 in Ω\Ωδ andτ= 0 in Ωδ/2with 0≤τ≤1. So, writingφ=τ φ+ (1−τ)φ, we have that

Z

|∇vσ|p−2∇vσ∇φ= Z

Ω\Ωδ/2

|∇vσ|p−2∇vσ∇(τ φ) + Z

δ

|∇vσ|p−2∇vσ∇(1−τ)φ.

(3.16) In Ω\Ωδ/2, it follows from the definition ofvσ, that

Z

Ω\Ωδ/2

|∇vσ|p−2∇vσ∇(τ φ)

= Z

Ω\Ωδ/2

|∇ωM|p−2∇ωM

θ0p−1λM +σ)]0−1)(p−1)λ0M +σ)]p−1τ φ

−(θ0−1)(p−1) Z

Ω\Ωδ/2

|∇ωM|pθ0p−1

×[ψλM +σ)]0−1)(p−1)−10λM+σ)]pτ φ

−(p−1) Z

Ω\Ωδ/2

|∇ωM|pθ0p−1λM +σ)]0−1)(p−1)

×[ψλ0M +σ)]p−2ψ00λM +σ)τ φ

Now, recalling thatθ0∈(kwMkfi1/(p−1),1),ψ0λ≥0,ψ00λ≤0 (see Lemma 2.4) and noting that

θ0p−1λM +σ)]0−1)(p−1)λ0M +σ)]p−1τ φ∈W01,p(Ω), it follows from (M1) and Lemma 2.4 (iv) that

Z

Ω\Ωδ/2

|∇vσ|p−2∇vσ∇(τ φ)dx

≥ Z

Ω\Ωδ/2

|∇ωM|p−2∇ωM∇ θ0p−1

λM +σ)]0−1)(p−1)

×[ψλ0M +σ)]p−1τ φ

≥ Z

Ω\Ωδ/2

M(x)θ0p−1λM +σ)]0−1)(p−1)λ0M +σ)]p−1τ φ

= Z

Ω\Ωδ/2

M(x)θ0p−1v

0−1)(p−1) θ0

σ

γ0p−1 θp−10

hHλ,γ0((ψλM+σ))θ0) (ψλM+σ))θ0

ip−1

τ φ.

(3.17)

(12)

As in Case one, the analysis of this inequality will be divided in two parts. So, from the properties of auxiliary functions, Lemma 2.1 (ii) and (3.12), we have

1 2

Z

Ω\Ωδ/2

|∇vσ|p−2∇vσ∇τ φ dx

≥ 1 2

Z

Ω\Ωδ/2

M(x)γ00−1)(p−1)γ0p−1ζλ,γ0(vσ+) (vσ+)p−1 τ φ

≥ 1 2

Z

Ω\Ωδ/2

M(x)γ0(p−1)θ0ζλ,γ0(vσ+) γ0θ0(p−1)

τ φ

≥ 1 2

Z

Ω\Ωδ/2

M(x)[fγ0(vσ+) +λgγ0(vσ+)]τ φ

≥ Z

Ω\Ωδ/2

[a(x)f(vσ+) +λb(x)g(vσ+)]τ φ,

(3.18)

for eachλ∈(0, λ),σ∈(0,σ),˜ >0.

Now, denoting by

v(x) := lim

σ→0vσ(x) = [ψλM(x))]θ0, x∈Ω, (3.19) it follows from Lemma 2.1 (iii),vσ> v >0 in Ω\Ωδ/2and q∈(p−1, p] that

hHλ,γ0(vσ) vσ

iq−(p−1)

≤hHλ,γ0(v) v

iq−(p−1)

Hλ,γ0(v) v

q−(p−1) L(Ω\Ωδ/2)

=C2h min

Ω\Ωδ/2

vi0

−1)(p−1−q) θ0

≤C2v

0−1)(p−1−q) θ0

< C2v

0−1)(p−1−q) θ0

σ , for allx∈Ω\Ωδ/2,

(3.20)

where

C2=

Hλ,γ0(v) v

q−(p−1) L(Ω\Ωδ/2)

h min

Ω\Ωδ/2

vi0

−1)(p−1−q) θ0

>0 is independent ofσ.

Now we show that 1

2 Z

Ω\Ωδ/2

|∇vσ|p−2∇vσ∇τ φ dx≥ γ(p−1)(θ0 0−1)[f(γ0) +λg(γ0)]

4

Z

Ω\Ωδ/2

β(x)τ φ dx

+ γ0p−1−q 4C2k∇ωMkqL(Ω)

Z

Ω\Ωδ/2

α(x)|∇vσ|q]τ φ dx and as a consequence of this, using (3.9), we obtain

1 2 Z

Ω\Ωδ/2

|∇vσ|p−2∇vσ∇τ φ dx≥µ Z

Ω\Ωδ/2

V(x,∇vσ)τ φ dx for each 0≤µ < µλ.

(13)

By (3.20) and Lemma 2.1 (iii)-(iv) in (3.17), we have 1

2 Z

Ω\Ωδ/2

|∇vσ|p−2∇vσ∇τ φ dx

≥1 2

Z

Ω\Ωδ/2

M(x)v

0−1)(p−1) θ0

σ γ0p−1hHλ,γ0(vσ) vσ

ip−1

τ φ

≥1 4

Z

Ω\Ωδ/2

M(x)γ00−1)(p−1)γp−10 hf(γ0)

γ0p−1 +λg(γ0) γ0p−1 i

τ φ

+1 4 Z

Ω\Ωδ/2

M(x)v

0−1)(p−1−q) θ0

σ γ0p−1−qhHλ,γ0(vσ) vσ

ip−1θq0 θq0v

0−1)q θ0

σ γ0qτ φ

γ0(p−1)θ0[f(γ0)

γ0p−1g(γ0)

γ0p−1] 4

Z

Ω\Ωδ/2

M(x)τ φ +γ0p−1−q

4C2

Z

Ω\Ωδ/2

M(x)θ0qv

0−1)q θ0

σ

γ0q θ0q

hHλ,γ0(vσ) vσ

iq τ φ.

Using (3.9), Lemma 2.4 (iv), the definition ofM and (V1), we obtain 1

2 Z

Ω\Ωδ/2

|∇vσ|p−2∇vσ∇τ φ dx

≥ γ(p−1)(θ0 0−1)[f(γ0) +λg(γ0)]

4

Z

Ω\Ωδ/2

M(x)τ φ dx +µλ0)k∇ωMkqL(Ω)

Z

Ω\Ωδ/2

M(x)[θ0ψλM+σ)θ0−1ψλ0M +σ)]qτ φ dx

≥ γ(p−1)(θ0 0−1)[f(γ0) +λg(γ0)]

4

Z

Ω\Ωδ/2

β(x)τ φ dx

λ0) Z

Ω\Ωδ/2

α(x)[θ0ψλM +σ)θ0−1ψλ0M +σ)|∇ωM|]qτ φ dx

≥µλ0) Z

Ω\Ωδ/2

[β(x) +α(x)|∇vσ|q]τ φ dx

≥µ Z

Ω\Ωδ/2

V(x,∇vσ)τ φ dx.

(3.21) Going back to (3.17) and using (3.18) and (3.21), we obtain

Z

Ω\Ωδ/2

|∇vσ|p−2∇vσ∇τ φ dx

≥ Z

Ω\Ωδ/2

[a(x)f(vσ+) +λb(x)g(vσ+) +µV(x,∇vσ)]τ φ,

(3.22)

for each 0< λ < λ, 0≤µ < µλ, >0.

(14)

Below we work on the ring Ωδ. As before, using the definition of vσ, it follows that

Z

δ

|∇vσ|p−2∇vσ∇(1−τ)φ dx

= Z

δ

|∇ωM|p−2∇ωM

θ0p−1λM +σ)]0−1)(p−1)

×[ψλ0M +σ)]p−1(1−τ)φ

−(θ0−1)(p−1) Z

δ

|∇ωM|pθ0p−1λM+σ)]0−1)(p−1)−1

×[ψλ0M +σ)]p(1−τ)φ

−(p−1) Z

δ

|∇ωM|pθ0p−1λM+σ)]0−1)(p−1)0λM+σ)]p−2ψλ00M+σ)(1−τ)φ.

(3.23)

In a way similar to the one for (3.18), we have 1

2 Z

δ

|∇vσ|p−2∇vσ∇(1−τ)φ dx

≥ θ0p−1 2

Z

δ

M(x)[ψλM +σ)]0−1)(p−1)λ0M +σ)]p−1(1−τ)φ dx

≥ Z

δ

[a(x)f(vσ+) +λb(x)g(vσ+)](1−τ)φ dx,

(3.24)

for eachλ∈(0, λ),σ∈(0,σ),˜ >0. Besides this, we will show that 1

2 Z

δ

|∇vσ|p−2∇vσ∇(1−τ)φ dx

≥ (1−θ0)(p−1)[γ0Hλ,γ0(1)]p−q 4kαk

Z

δ

α(x)|∇vσ|q(1−τ)φ dx +γ0(p−1)(θ0−1)[f(γ0) +λg(γ0)]

4

Z

δ

β(x)(1−τ)φ dx and as a consequence of this, using (3.9), we obtain

1 2 Z

δ

|∇vσ|p−2∇vσ∇(1−τ)φ dx≥µ Z

δ

V(x,∇vσ)(1−τ)φ dx for each 0≤µ < µλ.

In fact, from the properties of the auxiliary functions and (M1), we have 1

2 Z

δ

|∇vσ|p−2∇vσ∇(1−τ)φ dx

≥ −(θ0−1)(p−1) 4

Z

δ

|∇ωM|pθp−10λM +σ)]0−1)(p−1)−1

×[ψ0λM+σ)]p(1−τ)φ +1

4 Z

δ

|∇ωM|p−2∇ωM∇{θ0p−1λM+σ)]0−1)(p−1)

×[ψ0λM+σ)]p−1(1−τ)φ}

(15)

= (1−θ0)(p−1) 4

Z

δ

|∇ωM|pθp−10 v

p(θ0−1)−θ0 θ0

σ

γ0p θp0

hHλ,γ0(vσ) vσ

ip

(1−τ)φ +1

4 Z

δ

M(x)θ0p−1λM+σ)]0−1)(p−1)0λM+σ)]p−1(1−τ)φ.

That is, from (3.15) and Lemma 2.1, we have 1

2 Z

δ

|∇vσ|p−2∇vσ∇(1−τ)φ dx

=(1−θ0)(p−1) 4

Z

δ

|∇ωM|p vσ

θ0p−1 θ0p−1v

p(θ0−1) θ0

σ γ0phHλ,γ0(vσ) vσ

ip

(1−τ)φ dx +1

4 Z

δ

M(x)θ0p−1v

0−1)(p−1) θ0

σ

γ0p−1 θp−10

hHλ,γ0(vσ) vσ

ip

(1−τ)φ dx

≥(1−θ0)(p−1) 4

Z

δ

|∇ωM|qv

p(θ0−1) θ0

σ γ0phHλ,γ0(vσ) vσ

ip

(1−τ)φ dx +1

4 Z

δ

M(x)γ00−1)(p−1)γ0p−1hf(γ0)

γ0p−1 +λg(γ0) γ0p−1 i

(1−τ)φ dx.

(3.25)

Using thatvσ<1 in Ωδ (see (3.13)),q < pandθ0<1, we obtain

[vσ(x)]

0−1)p

θ0 >[vσ(x)]

0−1)q

θ0 , for eachx∈Ωδ (3.26) and from Lemma 2.1, we have

[Hλ,γ0(1)]p−qhHλ,γ0(vσ) vσ

iq

≤hHλ,γ0(vσ) vσ

ip

, x∈Ωδ. (3.27)

From (3.26) and (3.27), we rewrite (3.25) as 1

2 Z

δ

|∇vσ|p−2∇vσ∇(1−τ)φ dx

≥ (1−θ0)(p−1) 4

Z

δ

|∇ωM|qv

0−1)q θ0

σ γ0p−qγ0qθ0q

θ0q[Hλ,γ0(1)]p−q

×hHλ,γ0(vσ) vσ

iq

(1−τ)φ dx+

γ(p−1)θ0 0[f(γ0)

γp−10g(γ0)

γ0p−1] 4

Z

δ

M(x)(1−τ)φ dx

= (1−θ0)(p−1)[γ0Hλ,γ0(1)]p−q 4kαk

Z

δ

kαkθ0qv

0−1)q θ0

σ

0Hλ,γ0(vσ) θ0vσ

iq

× |∇ωM|q(1−τ)φ+γ(p−1)(θ0 0−1)[f(γ0) +λg(γ0)]

4

Z

δ

M(x)(1−τ)φ dx.

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