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Advances in Nonlinear Analysis

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


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On Dirichlet problem for fractional p-Laplacian with singular non-linearity

Tuhina Mukherjee / Konijeti Sreenadh
  • Corresponding author
  • Department of Mathematics, Indian Institute of Technology Delhi Hauz Khaz, New Delhi -16, India
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Published Online: 2016-12-02 | DOI: https://doi.org/10.1515/anona-2016-0100

Abstract

In this article, we study the following fractional p-Laplacian equation with critical growth and singular non-linearity:

(-Δp)su=λu-q+uα,u>0in Ω,u=0in nΩ,

where Ω is a bounded domain in n with smooth boundary Ω, n>sp, s(0,1), λ>0, 0<q1 and 1<p<α+1ps*. We use variational methods to show the existence and multiplicity of positive solutions of the above problem with respect to the parameter λ.

Keywords: critical exponent; singular non-linearities

MSC 2010: 35R11; 35R09; 35A15

1 Introduction

Let s(0,1), p(1,) and let Ωn, n2, be a bounded domain with smooth boundary. We consider the following problem with singular non-linearity:

(-Δp)su=λu-q+uα,u>0in Ω,u=0in nΩ,(Pλ)

where λ>0, 0<q1, p<α+1ps*, ps*=npn-sp, n>sp and (-Δp)s is the fractional p-Laplacian operator defined as

(-Δp)su(x)=-2limϵ0nBϵ(x)|u(x)-u(y)|p-2(u(x)-u(y))|x-y|n+sp𝑑yfor all xn.

Recently, a lot of attention is given to the study of fractional and non-local operators of elliptic type due to concrete real-world applications in finance, thin obstacle problem, optimization, quasi-geostrophic flow etc.

The semilinear Dirichlet problem for the fractional Laplacian is recently studied, using variational methods, in [8, 42, 43]. The existence and multiplicity results for non-local operators, like the fractional Laplacian with a combination of convex and concave type non-linearities like uq+λup, p, q>0, is studied in [3, 5, 37, 36, 44, 45]. The eigenvalue problem for the fractional p-Laplacian and properties like the simplicity of the smallest eigenvalue is studied in [16, 32]. The Brezis–Nirenberg type existence result has been studied in [38]. Existence and multiplicity results with convex-concave type regular non-linearities have been studied in [26].

In the local setting (s=1), the paper by Crandal, Rabinowitz and Tartar [12] is the starting point on semilinear problems with singular non-linearity. A lot of work has been done related to the existence and multiplicity results for the Laplacian and the p-Laplacian with singular non-linearity, see [1, 11, 14, 20, 18, 23, 24]. In [11, 14], the singular problems of the type

-Δu=g(x,u)+h(x,λu)in Ω,u=0on Ω,g(x,u)L1(Ω)

was considered with g(x,u)u-α. The existence of solutions under suitable conditions on g and h was studied. In [20, 18], Ghergu and Rădulescu considered singular problems of the type

-Δu+K(x)g(u)+μ|u|a=λf(x,u)in Ω,u=0on Ω,

where λ, μ>0. Here, h,KC0,γ(Ω) for some 0<γ<1 and h>0 in Ω, and f:[0,)[0,) is a Hölder continuous function which is positive on Ω¯×(0,), sublinear at and superlinear at 0. The function gC0,γ(0,), for some 0<γ<1, is non negative and non-increasing function such that lims0+g(s)=+. They proved several results related to existence and non existence of positive solutions of the above problems, by taking into account both the sign of the potential K and the decay rate around the origin of the singular non-linearity g. Several authors considered problems of Lane–Emden–Fowler type with singular non-linearity, see [10, 15, 19]. In addition, some bifurcation results have been proved in [19] for the following problem:

-Δu=g(u)+λ|u|p+μf(x,u),u>0in Ω,u=0on Ω,

where λ,μ>0, 0<p2, f is non-decreasing with respect to the second variable and g(u) behaves like u-α around the origin. The asymptotic behaviour of the solutions is shown by constructing suitable sub- and super-solutions combined with the maximum principle. We also refer to [22, 29] as a part of previous contributions to this field. For detailed study and recent results on singular problems, we refer to [21].

In [23], Giacomoni, Schindler and Taḱaĉ studied the critical growth singular problem

-Δpu=λu-δ+uq,u>0in Ω,u=0on Ω,

where 0<δ<1 and p-1<qp*-1, p*=npn-p and Δpu=div(|u|p-2u). Using variational methods, they proved the existence of multiple solutions with restrictions on p and q in the spirit of [13, 17]. Among the works dealing with elliptic equations with singular and critical growth terms, we cite [1, 34, 2, 9, 25, 40], see also the references therein, with no attempt to provide a complete list.

There are many works on the existence of a solution for fractional elliptic equations with regular non-linearities like uq+λup, p,q>0. Sub-critical growth problems are studied in [8, 42, 43], and critical exponent problems are studied in [5, 37, 36, 38]. Also, the multiplicity of solutions by the method of Nehari manifold and fibering maps has been investigated in [26, 45, 46]. For detailed study and recent results on this subject, we refer to [35].

In [4], Barrios et al. studied the singular problem

(-Δ)su=λf(x)uγ+Mup,u>0in Ω,u=0in nΩ,

where n>2s, M0, 0<s<1, γ,λ>0, 1<p<2s*-1, 2s*=2nn-2s, fLm(Ω) is a non-negative function for m1, and

(-Δ)su(x)=-12nu(x+y)+u(x-y)-2u(x)|y|n+2s𝑑yfor all xn.

In particular, they studied the existence of distributional solutions for small λ using the uniform estimates of {un} which are solutions of the regularised problem with the singular term u-γ replaced by (u+1n)-γ. In [39], the critical growth singular problem, for α=2s*-1 and p=2, is studied where multiplicity results are obtained using the Nehari manifold approach.

There are many works on the study of p-fractional equations with polynomial type non-linearities. In [26], Goyal and the second author studied the subcritical problem using the Nehari manifold and fibering maps. In [38], a Brezis–Nirenberg type critical exponent problem is studied. We also refer readers to [6, 27, 31] and the references therein. To the best of our knowledge, there are no works on the existence of multiplicity results with singular non-linearities involving the fractional p-Laplacian operator.

In this paper, we study existence and multiplicity results with a convex-concave type singular non-linearity. Here we follow the approach of [30]. We obtain our results by studying the existence of minimizers that arise out of the structure of the Nehari manifold. We would like to remark that the results proved here are new even for the case q=1. Also the existence result is sharp in the sense that we show the existence of Λ such that (0,Λ) is the maximal range for λ for which the solution exists. We show the existence of a second solution in the sub-critical case for the maximal range of λ, where the fibering maps have two critical points along with some other condition. We also show some regularity results on the weak solutions of ().

The paper is organised as follows. In Section 2, we present some preliminaries on function spaces required for the variational setting and state the main results. In Section 3, we study the corresponding Nehari manifold and the properties of minimizers. In Sections 4 and 5, we show the existence of minimizers and solutions. In Section 6, we show some regularity results, and Section 7 is devoted to the maximal range of λ for the existence of solutions.

2 Preliminaries and main results

The motivation for defining the function space comes from [42]. In [26], Goyal and the second author discussed the Dirichlet boundary value problem involving the p-fractional Laplace operator using variational techniques. Due to non-localness of the fractional p-Laplace operator, they introduced the function space (X0,X0). The space X is defined as

X={u|u:n is measurable, u|ΩLp(Ω) and (u(x)-u(y))|x-y|n+sppLp(Q)},

where Q=2n(𝒞Ω×𝒞Ω) and 𝒞Ω:=nΩ. The space X is endowed with the norm

uX=uLp(Ω)+[u]X,where [u]X=(Q|u(x)-u(y)|p|x-y|n+sp𝑑x𝑑y)1p.

Then we define X0={uX:u=0 a.e. in nΩ}. There exists a constant C>0 such that uLp(Ω)C[u]X for all uX0. Hence, u=[u]X is a norm on X0, and X0 is a Hilbert space. Note that the norm involves the interaction between Ω and nΩ. We denote Lp(Ω) as ||p and =[]X for the norm in X0. Now, for each β0, we set

Cβ=sup{|u|ββ:uX0,u=1}.(2.1)

Then C0=|Ω| = Lebesgue measure of Ω and Ω|u|βdxCβuβ for all uX0. From the embedding results in [26], we know that X0 is continuously and compactly embedded in Lr(Ω), where 1r<ps* and the embedding is continuous but not compact if r=ps*. We define the best constant S of the embedding as

S=inf{up:uX0,|u|ps*p=1}.

Definition 2.1.

We say that uX0 is a positive weak solution of () if u>0 in Ω and

Q|u(x)-u(y)|p-2(u(x)-u(y))(ψ(x)-ψ(y))|x-y|n+sp𝑑x𝑑y-Ω(λu-q-uα)ψ𝑑x=0

for all ψCc(Ω).

The functional associated to (), Iλ:X0(-,], is defined by

Iλ(u)=1pQ|u(x)-u(y)|p|x-y|n+spdxdy-λΩGq(u)dx-1α+1Ω|u|α+1dx,

where Gq:[-,) is the function defined by

Gq(x)={|x|1-q1-qif 0<q<1,ln|x|if q=1.

For each 0<q1, we set X+={uX0:u0} and X+,q={uX+:u0,Gq(u)L1(Ω)}. Notice that X+,q=X+{0} if 0<q<1 and X+,1 if Ω is, for example, of C2. We will need the following important lemma.

Lemma 2.2.

For each wX+, there exists a sequence {wk} in X0 such that wkw strongly in X0, where 0w1w2 and wk has compact support in Ω for each k.

Proof.

The proof here is adopted from [30]. Let wX+ and let {ψk} be a sequence in Cc(Ω) such that ψk is non negative and converges strongly to w in X0. Define zk=min{ψk,w}. Then zkw strongly in X0. Now, we set w1=zr1 where r1>0 is such that zr1-w1. Then max{w1,zm}w strongly in X0, thus we can find r2>0 such that max{w1,zr2}-w12. We set w2=max{w1,zr2}, and get max{w2,zm}w strongly as m. Consequently, by induction, we set wk+1=max{wk,zrk+1} to obtain the desired sequence, since we can see that wkX0 has compact support for each k and max{wk,zrk+1}-w1k+1, which says that {wk} converges strongly to w in X0. ∎

Let ϕ1>0 be the eigenfunction of (-Δp)s corresponding to the smallest eigenvalue λ1. This is obtained as minimizer of the minimization problem

λ1=min{up:uX0,|u|p=1}.

In [32, 38], it was shown that this minimizer is achieved by a unique positive and bounded function ϕ1. Moreover, (λ1,ϕ1) is the solution of the eigenvalue problem

(-Δp)su=μ|u|p-2u,u>0in Ω,u=0in nΩ.

We assume ϕ1L=1. With these preliminaries, we state our main results.

For each uX+,q, we define the fiber map ϕu:+ by ϕu(t)=Iλ(tu).

Theorem 2.3.

Assume 0<q1. In the case q=1, assume also X+,1. Let Λ1 be a constant defined by

Λ1={sup{λ>0:ϕu(t) has two critical points in (0,) for each uX+,q{0}}if α+1<ps*,sup{λ>0:ϕu(t) has two critical points in (0,) for each uX+,q{0},sup{u:uX+,q,ϕu(1)=0,ϕu′′(1)>0}(pps*)pps*-pSps*ps*-p}if α=ps*.

Then Λ1>0.

Theorem 2.4.

For all λ(0,Λ1), () has at least two distinct solutions in X+,q when α<ps*-1 and at least one solution in the critical case when α=ps*-1.

Definition 2.5.

We say that uX0 is a weak sub-solution of () if u>0 in Ω and

Q|u(x)-u(y)|p-2(u(x)-u(y))(ψ(x)-ψ(y))|x-y|n+sp𝑑x𝑑yΩ(λu-q+uα)ψ𝑑x=0

for all 0ψCc(Ω). Similarly, uX0 is said to be a weak super-solution to () if in the above, the reverse inequalities hold.

Next we study that the existence of a solution with the parameter in the maximal interval. For this, we minimize the functional over the convex set {uX+,q:u¯uu¯}, where u¯ and u¯ are sub- and super-solutions, respectively. Using truncation techniques as in [28], we show that the minimizer is a weak solution.

Theorem 2.6.

Let p<α+1ps* and 0<q1. Then there exists Λ>0 such that () has a solution for all λ(0,Λ) and no solution for λ>Λ.

3 Nehari manifold and fibering maps

We denote Iλ=I for simplicity now. One can easily verify that the energy functional I is not bounded below on the space X0. We will show that it is bounded on the manifold associated to the functional I. In this section, we study the structure of this manifold. We define

𝒩λ={uX+,q:I(u),u=0}.

Theorem 3.1.

The functional I is coercive and bounded below on Nλ.

Proof.

In the case 0<q<1, since u𝒩λ, using the embedding of X0 in L1-q(Ω), we get

I(u)=(1p-1α+1)up-λ(11-q-1α+1)Ω|u|1-qdxc1up-c2u1-q

for some constants c1 and c2. This says that I is coercive and bounded below on 𝒩λ.

In the case q=1, using the inequality ln(|u|)|u| and embedding results for X0, we can similarly get I as bounded below. ∎

From the definition of fiber map ϕu, we have

ϕu(t)={tppup-t1-q1-qΩ|u|1-qdx-tα+1α+1Ω|u|α+1dxif 0<q<1,tppup-λΩln(t|u|)dx-tα+1α+1Ω|u|α+1dxif q=1,

which gives

ϕu(t)=tp-1up-λt-qΩ|u|1-qdx-tαΩ|u|α+1dx,ϕu′′(t)=(p-1)tp-2up+qλt-q-1Ω|u|1-qdx-αtα-1Ω|u|α+1dx.

It is easy to see that the points in 𝒩λ are corresponding to the critical points of ϕu at t=1. So, it is natural to divide 𝒩λ into three sets corresponding to local minima, local maxima and points of inflexion. Therefore, we define

𝒩λ+={u𝒩λ:ϕu(1)=0,ϕu′′(1)>0}={t0u𝒩λ:t0>0,ϕu(t0)=0,ϕu′′(t0)>0},𝒩λ-={u𝒩λ:ϕu(1)=0,ϕu′′(1)<0}={t0u𝒩λ:t0>0,ϕu(t0)=0,ϕu′′(t0)<0}

and 𝒩λ0={u𝒩λ:ϕu(1)=0,ϕu′′(1)=0}.

Lemma 3.2.

There exists λ*>0 such that for each uX+,q{0}, there are unique t1 and t2 with the property that t1<t2, t1uNλ+ and t2uNλ- for all λ(0,λ*).

Proof.

Define A(u)=Ω|u|1-qdx and B(u)=Ω|u|α+1dx. Let uX+,q. Then we have

ddtI(tu)=tp-1up-t-qλA(u)-tαB(u)=t-q(mu(t)-λA(u)),

and we define mu(t):=tp-1+qup-tα+qB(u). Since limtmu(t)=-, we can easily see that mu(t) attains its maximum at

tmax=[(p-1+q)up(α+q)B(u)]1α+1-p

and

mu(tmax)=(α+2-pp-1+q)(p-1+qα+q)α+qα+1-pup(α+q)α+1-pB(u)p-1+qα+1-p.

Now, u𝒩λ if and only if mu(t)=λA(u), and we see that

mu(t)-λA(u)mu(tmax)-λ|u|1-q1-q(α+2-pp-1+q)(p-1+qα+q)α+qα+1-pup(α+q)α+1-pB(u)p-1+qα+1-p-λC1-qu1-q>0

if and only if

λ<=:(α+2-pp-1+q)(p-1+qα+q)α+qα+1-p(Cα+1)-p+1-qα+1-pC1-q-1λ*,

where Cα+1 and C1-q are defined in (2.1).

In the case 0<q<1, we see that mu(t)=λΩ|u|1-qdx if and only if ϕu(t)=0. So for λ(0,λ*), there exist exactly two points 0<t1<tmax<t2 with mu(t1)>0 and mu(t2)<0, that is, t1u𝒩λ+ and t2u𝒩λ-. Thus, ϕu has a local minimum at t=t1 and a local maximum at t=t2. In other words, ϕu is decreasing in (0,t1) and increasing in (t1,t2).

In the case q=1, since limt0ϕu(t)= and limtϕu(t)=-, with similar reasoning as above, we get t1,t2.

In both cases, ϕu has exactly two critical points t1 and t2 such that 0<t1<tmax<t2, ϕu′′(t1)>0 and ϕu′′(t2)<0. Thus, t1u𝒩λ+, t2u𝒩λ-. ∎

Proof of Theorem 2.3.

The proof follows from Lemma 3.2. ∎

Corollary 3.3.

We have Nλ0={0} for all λ(0,Λ1).

Proof.

Let u𝒩λ0 and u0. Then u𝒩λ, that is, t=1 is a critical point of ϕu(t). By Lemma 3.2, ϕu has critical points corresponding to either local minima or local maxima. So, t=1 is the critical point corresponding to either local minima or local maxima of ϕu. Thus, either u𝒩λ+ or u𝒩λ-, which is a contradiction. ∎

We can now show that I is bounded below on 𝒩λ+ and 𝒩λ- as follows.

Lemma 3.4.

The following hold:

  • (i)

    sup{u:u𝒩λ+}<,

  • (ii)

    inf{v:v𝒩λ-}>0 and sup{v:v𝒩λ-,I(v)M}< for each M>0.

Moreover, infI(Nλ+)>- and infI(Nλ-)>-.

Proof.

Let u𝒩λ+ and v𝒩λ-. Then we have

0<ϕu′′(1)(p-1-α)up+λ(α+q)C1-qu1-q,0>ϕv′′(1)(p-1+q)vp-(α+q)Cα+1vα+1.

Thus, we obtain

u(λ(α+q)C1-qα+1-p)1p+q-1andv(p-1+q(α+q)Cα+1)1α+1-p.

This implies that

sup{u:u𝒩λ+}<andinf{v:v𝒩λ-}>0.(3.1)

If I(v)M, using ln(|v|)|v|, we get

M{α+1-pp(α+1)vp-λ(α+q)C1-q(α+1)(1-q)v1-qif 0<q<1,α+1-pp(α+1)vp-λC1v+λα+1if q=1.(3.2)

This implies sup{v:v𝒩λ-,I(v)M}< for each M>0. Using (3.1) and (3.2), it is easy to show that infI(𝒩λ+)>- and infI(𝒩λ-)>-. ∎

Lemma 3.5.

Suppose that uNλ+ and vNλ- are minimizers of I over Nλ+ and Nλ-, respectively. Then, for each wX+, the following hold:

  • (1)

    there exists ϵ0>0 such that I(u+ϵw)I(u) for each ϵ[0,ϵ0],

  • (2)

    tϵ1 as ϵ0+ , where tϵ is the unique positive real number satisfying tϵ(v+ϵw)𝒩λ-.

Proof.

(1) Let wX+, i.e., wX0 and w0. We set

ρ(ϵ)=(p-1)u+ϵwp+λqΩ|u+ϵw|1-qdx-αΩ|u+ϵw|α+1dx

for each ϵ0. Then, using the continuity of ρ, and since u𝒩λ+, ρ(0)=ϕu′′(1)>0, there exists ϵ0>0 such that ρ(ϵ)>0 for ϵ[0,ϵ0]. Since, for each ϵ>0, there exists tϵ>0 such that tϵ(u+ϵw)𝒩λ+, it follows that tϵ1 as ϵ0, and for each ϵ[0,ϵ0], we have

I(u+ϵw)I(tϵ(u+ϵw))infI(𝒩λ+)=I(u).

(2) We define h:(0,)×3 by

h(t,l1,l2,l3)=l1tp-1-λt-ql2-tαl3

for (t,l1,l2,l3)(0,)×3. Then h is a C function. Therefore, we have

dhdt(1,vp,Ω|v|1-qdx,Ω|v|α+1dx)=ϕv′′(1)<0,

and for each ϵ0,

h(tϵ,v+ϵwp,Ω|v+ϵw|1-qdx,Ω|v|α+1dx)=ϕv+ϵw(tϵ)=0.

Also,

h(1,vp,Ω|v|1-qdx,Ω|v|α+1dx)=ϕv(1)=0.

Thus, by the implicit function theorem, there exists an open neighbourhood A(0,) and B3 containing 1 and (vp,Ω|v|1-qdx,Ω|v|α+1), respectively, such that for all yB, h(t,y)=0 has a unique solution t=g(y)A, where g:BA is a continuous function. So,

(v+ϵwp,Ω|v+ϵw|1-qdx,Ω|v+ϵw|α+1dx)B

and

g(v+ϵw)p,Ω|v+ϵw|1-qdx,Ω|v+ϵw|α+1dx)=tϵ,

since

h(tϵ,v+ϵw)p,Ω|v+ϵw|1-qdx,Ω|v+ϵw|α+1dx)=0.

Thus, by the continuity of g, we get tϵ1 as ϵ0+. ∎

Lemma 3.6.

Suppose that uNλ+ and vNλ- are minimizers of I on Nλ+ and Nλ-, respectively. Then, for each wX+, we have u-qw,v-qwL1(Ω),

Q|u(x)-u(y)|p-2(u(x)-u(y))(w(x)-w(y))|x-y|n+sp𝑑x𝑑y-λΩ(u-q+uα)w𝑑x0(3.3)

and

Q|v(x)-v(y)|p-2(v(x)-v(y))(w(x)-w(y))|x-y|n+sp𝑑x𝑑y-λΩ(v-q+vα)w𝑑x0.(3.4)

Proof.

Let wX+. For sufficiently small ϵ>0, by Lemma 3.5,

0I(u+ϵw)-I(u)ϵ=1pϵ(u+ϵwp-up)-λϵΩ(Gq(u+ϵw)-Gq(u))𝑑x-1ϵ(α+1)Ω(|u+ϵw|α+1-|u|α+1)𝑑x.(3.5)

We can easily verify that as ϵ0+,

(u+ϵwp-up)ϵpQ|u(x)-u(y)|p-2(u(x)-u(y))(w(x)-w(y))|x-y|n+sp𝑑x𝑑y,Ω(|u+ϵw|α+1-|u|α+1)ϵdx(α+1)Ω|u|α-1uwdx,

which imply that (Gq(u+ϵw)-Gq(u))ϵL1(Ω). Also, for each xΩ,

Gq(u(x)+ϵw(x))-Gq(u(x))ϵ={1ϵ(|u+ϵw|1-q(x)-|u|1-q(x)1-q)if 0<q<1,1ϵ(ln(|u+ϵw|)-ln(|u|))if q=1,

which increases monotonically as ϵ0 and

limϵ0Gq(u(x)+ϵw(x))-Gq(u(x))ϵ={0if w(x)=0,(u(x))-qw(x)if w(x)>0,u(x)>0,if w(x)>0,u(x)=0.

So, by using the monotone convergence theorem for the sequence {Gq}, we get u-qwL1(Ω). Letting ϵ0 in both sides of (3.5), we get (3.3).

Next we will show these properties for v. For each ϵ>0, there exists tϵ>0 with tϵ(v+ϵw)𝒩λ-. By Lemma 3.5 (2), for sufficiently small ϵ>0, we have

I(tϵ(v+ϵw))I(v)I(tϵv),

which implies I(tϵ(v+ϵw))-I(v)0. Thus, we have

λΩ(Gq(tϵ|v+ϵw|1-q)-Gq(|v|1-q))𝑑xtϵpp(v+ϵwp-vp)-tϵα+1α+1Ω(|v+ϵw|α+1-|v|α+1)𝑑x.

As ϵ0, tϵ1. Thus, using similar arguments as above, we obtain v-qwL1(Ω) and (3.4) follows. ∎

Let η>0 be such that ϕ=ηϕ1 satisfies

Q|ϕ(x)-ϕ(y)|p-2(ϕ(x)-ϕ(y))(ψ(x)-ψ(y))|x-y|n+sp𝑑x𝑑yλQϕ-qψ𝑑x+Qϕαψ𝑑x(3.6)

for all 0ψX0 (i.e., ϕ is a sub-solution of ()) and ϕα+q(x)λ(qα) for each xΩ. Then we have the following lemma.

Lemma 3.7.

Suppose uNλ+ and vNλ- are minimizers of I on Nλ+ and Nλ-, respectively. Then uϕ and vϕ in Ω. In particular, u,v>0.

Proof.

By Lemma 2.2, let {wk} be a sequence in X0 such that supp(wk) is compact, 0wk(ϕ-u)+ for each k and {wk} strongly converges to (ϕ-u)+ in X0. Then

ddt(λt-q+tα)=-qλt-q-1+αtα-10tα+qλ(qα).(3.7)

Using Lemma 3.6 and (3.6), we have

Q(f(u)-f(ϕ))|x-y|n+sp(wk(x)-wk(y))𝑑x𝑑y-Ω(λu-q+uα)wk𝑑x+Ω(λϕ-q+ϕα)wk𝑑x0,

where f(ξ)=|ξ(x)-ξ(y)|p-2(ξ(x)-ξ(y)). Since {wk} converges to (ϕ-u)+ strongly, we get a subsequence of {wk}, still calling it {wk}, such that wk(x)(ϕ-u)+(x) pointwise almost everywhere in Ω, and we write for each xΩ, wk(x)=(ϕ-u)+(x)+o(1) as k. Then

Q(f(u)-f(ϕ))|x-y|n+sp(wk(x)-wk(y))𝑑x𝑑y=Q(f(u)-f(ϕ))|x-y|n+sp((ϕ-u)+(x)-(ϕ-u)+(y))𝑑x𝑑y+o(1)Q(f(u)-f(ϕ))|x-y|n+sp𝑑x𝑑y.

Further, we can see that

Q(f(u)-f(ϕ))|x-y|n+sp((ϕ-u)+(x)-(ϕ-u)+(y))𝑑x𝑑y=(Ω1×Ω1+Ω1×Ω2+Ω2×Ω1+Ω2×Ω2)(f(u)-f(ϕ))|x-y|n+sp((ϕ-u)+(x)-(ϕ-u)+(y))dxdy,(3.8)

where Ω1={x:ϕ(x)u(x)} and Ω2={x:ϕ(x)u(x)}. Now, we separately estimate each integrals and to begin with, first we see that

Ω2×Ω2(f(u)-f(ϕ))|x-y|n+sp((ϕ-u)+(x)-(ϕ-u)+(y))𝑑x𝑑y=0.(3.9)

Next we see that

Ω1×Ω1(f(u)-f(ϕ))|x-y|n+sp((ϕ-u)+(x)-(ϕ-u)+(y))𝑑x𝑑y=-Ω1×Ω1(f(ϕ)-f(u))|x-y|n+sp((ϕ-u)(x)-(ϕ-u)(y))𝑑x𝑑y-12p-2Ω1×Ω1|(ϕ-u)(x)-(ϕ-u)(y)|p|x-y|n+sp𝑑x𝑑y(3.10)

using |a-b|p2p-2(|a|p-2a-|b|p-2b)(a-b), p2 and a,b. Now, consider

Ω1×Ω2(f(u)-f(ϕ))|x-y|n+sp((ϕ-u)+(x)-(ϕ-u)+(y))𝑑x𝑑y=Ω1×Ω2(f(u)-f(ϕ))|x-y|n+sp(ϕ-u)(x)𝑑x𝑑y-12p-2Ω1×Ω2|(ϕ-u)(x)-(ϕ-u)(y)|p|x-y|n+sp𝑑x𝑑y+Ω1×Ω2(f(u)-f(ϕ))|x-y|n+sp(ϕ-u)(y)𝑑x𝑑y,(3.11)

and, similarly, we get

Ω2×Ω1(f(u)-f(ϕ))|x-y|n+sp((ϕ-u)+(x)-(ϕ-u)+(y))𝑑x𝑑y-12p-2Ω2×Ω1|(ϕ-u)(x)-(ϕ-u)(y)|p|x-y|n+sp𝑑x𝑑y-Ω2×Ω1(f(u)-f(ϕ))|x-y|n+sp(ϕ-u)(x)𝑑x𝑑y.(3.12)

Thus, using (3.8)–(3.12), we get

Q(f(u)-f(ϕ))|x-y|n+sp((ϕ-u)+(x)-(ϕ-u)+(y))𝑑x𝑑y-12p-2(ϕ-u)p+Ω1×Ω2(f(u)-f(ϕ))|x-y|n+sp(ϕ-u)(y)𝑑x𝑑y-Ω2×Ω1(f(u)-f(ϕ))|x-y|n+sp(ϕ-u)(x)𝑑x𝑑y=-12p-2(ϕ-u)p.

Since ϕα+q(x)λ(qα), for each xΩ, using (3.7), we get

Ω((λu-q+uα)-(λϕ-q+ϕα))wk𝑑x=Ω{ϕu}((λu-q+uα)-(λϕ-q+ϕα))(ϕ-u)+(x)𝑑x+o(1)0,

which implies

0-12p-2(ϕ-u)+p-Ω(λu-q+uα)wk𝑑x+Ω(λϕ-q+ϕα)wk𝑑x+o(1)-12p-2(ϕ-u)+p+o(1),

and letting k, we get -(ϕ-u)+20. Thus, we showed uϕ. Similarly, we can show vϕ. ∎

4 Existence of minimizer on 𝒩λ+

In this section, we will show that the minimum of I on 𝒩λ+ is achieved in 𝒩λ+. Also, we show that this minimizer is also a solution of ().

Proposition 4.1.

For all λ(0,Λ1), there exist uλNλ+ satisfying I(uλ)=infI(Nλ+).

Proof.

Assume 0<q1 and λ(0,Λ1). We show that there exists uλ𝒩λ+ such that I(uλ)=infI(𝒩λ+). Let {uk}𝒩λ+ be a sequence such that I(uk)infI(𝒩λ+) as k. Now, by (3.1), we can assume that there exists uλX0 such that ukuλ weakly in X0 (up to a subsequence). First we will show that infI(𝒩λ+)<0. Let u0𝒩λ+. We have ϕu0′′(1)>0, which gives

(p-1+qα+q)u0p>Ω|u0|α+1dx.

Therefore, using α+1>p, we obtain

I(u0)=(1p-11-q)u0p+(11-q-1α+1)Ω|u0|α+1dx-(p+q-1)p(1-q)u0p+(p+q-1)(α+1)(1-q)u0p=(1α+1-1p)(p+q-11-q)u0p<0.

This proves that infI(𝒩λ+)<0. Case (I): α+1<ps* and 0<q1. First, we claim that uλX+,q. When 0<q<1, if uλ=0, then 0=I(uλ)lim¯kI(uk)<0, which is a contradiction. In the case q=1, the sequence {Ωln(|uk|)} is bounded, since the sequence {I(uk)} and {uk} is bounded. So, using Fatou’s lemma and the fact that ln(|uk|)uk for each k, we get

-<lim¯kΩln(|uk|)𝑑xΩlim¯kln(|uk|)dx=Ωln(|uλ|)𝑑x,

which implies uλ0 and thus, in both cases we have shown uλX+,q. We claim that ukuλ strongly in X0. Suppose not. Then, we may assume uk-uλc>0. Using the Brezis–Lieb lemma and the embedding results for X0 in the subcritical case, we have

limkϕuk(1)=ϕuλ(1)+cp,(4.1)

which implies ϕuλ(1)+cp=0, using the fact that ϕuk(1)=0 for each k. Since λ(0,Λ1), there exist 0<t1<t2 (by fibering map analysis) such that ϕuλ(t1)=ϕuλ(t2)=0 and t1uλ𝒩λ+. By (4.1), we have ϕuλ(1)<0, which gives two cases: 1<t1 or t2<1. When t1>1, we have

infI(𝒩λ+)=limI(uk)=I(uλ)+cpp=ϕuλ(1)+cpp>ϕuλ(1)>ϕuλ(t1)infI(𝒩λ+),

which is a contradiction. Thus, we have t2<1. We set f(t)=ϕuλ(t)+cptp2 for t>0. From (4.1), we get f(1)=0, and since 0<t2<1, f(t2)=t2p-1cp>0. So, f is increasing in [t2,1], and we obtain

infI(𝒩λ+)=I(uλ)+cpp=ϕuλ(1)+cpp=f(1)>f(t2)>ϕuλ(t2)>ϕuλ(t1)infI(𝒩λ+),

which gives a contradiction. Hence, c=0, and thus ukuλ strongly in X0. Since λ(0,Λ1), we have ϕuλ′′(1)>0, so we obtain uλ𝒩λ+ and I(uλ)=infI(𝒩λ+). Case (II): α=ps*-1 and 0<q<1. We set wk:=uk-uλ and claim that ukuλ strongly in X0. Suppose wkpcp0 and Ω|wk|ps*dxdps* as k. Since uk𝒩λ+, using the Brezis–Lieb lemma, we get

0=limkϕuk(1)=ϕuλ(1)+cp-dps*,(4.2)

which implies

uλp+cp=λΩ|uλ|1-qdx+Ω|uk|ps*dx+dps*.

We claim that uλX+,q. Suppose uλ0. If 0<q<1 and c=0, then 0>infI(𝒩λ+)=I(0)=0, which is a contradiction, and if c0, then

infI(𝒩λ+)=I(0)+cpp-dps*ps*=cpp-dps*ps*.(4.3)

But we have Sukps*pukp, which gives cpSdp. Also from (4.2) we have cp=dps*. Then (4.3) implies

0>infI(𝒩λ+)=(1p-1ps*)cpsnSnsp,

which is again a contradiction. In the case q=1, the sequence {Ωln(|uk|)} is bounded, since the sequences {I(uk)} and {uk} are bounded. So using Fatou’s lemma and ln(|uk|)uk, for each k, we get

-<lim¯kΩln(|uk|)𝑑xΩlim¯kln(|uk|)dx=Ωln(|uλ|)𝑑x.

which implies uλ0. Thus, in both cases we have shown that uλX+,q. So, there exist 0<t1<t2 such that ϕuλ(t1)=ϕuλ(t2)=0 and t1uλ𝒩λ+. Then, the following three cases arise:

  • (i)

    t2<1,

  • (ii)

    t21 and cpp-dps*ps*<0,

  • (iii)

    t21 and cpp-dps*ps*0.

(i) Let h(t)=ϕuλ(t)+cptpp-dps*tps*ps* for t>0. By (4.2), we get h(1)=ϕuλ(1)+cp-dps*=0 and

h(t2)=ϕuλ(t2)+t2pcp-t2ps*dps*=t2p(cp-t2ps*-pdps*)>t2p(cp-dps*)>0,

which implies that h increases in [t2,1]. Then we get

infI(𝒩λ+)=limkI(uk)ϕu(1)+cpp-dps*ps*=h(1)>h(t2)=ϕu(t2)+cpt2pp-dps*t2ps*ps*ϕu(t2)+t2pp(cp-dps*)>ϕu(t2)>ϕu(t1)infI(𝒩λ+),

which is a contradiction.

(ii) In this case, we have (cpp-dps*ps*)<0 and Sdpcp. Since λ(0,Λ1), we have

sup{up:u𝒩λ+}(pps*)pps*-pSps*ps*-p<cpsup{up:u𝒩λ+},

which gives a contradiction.

Consequently, only (iii) holds true, where we have

infI(𝒩λ+)=I(uλ)+cpp-dps*ps*I(uλ)=ϕuλ(1)ϕuλ(t1)infI(𝒩λ+).

Clearly, this holds when t1=1 and (cpp-dps*ps*)=0, which yields c=0 and uλ𝒩λ+. Thus, ukuλ strongly in X0 as k and I(uλ)=infI(𝒩λ+). ∎

Proposition 4.2.

uλ is a positive weak solution of ().

Proof.

Let ψCc(Ω). By Lemma 3.7, since ϕ>0, we can find β>0 such that uλβ on the support of ψ. Then uλ+ϵψ0 for small ϵ. With similar reasoning as in the proof of Lemma 3.5, I(uλ+ϵψ)I(uλ) for sufficiently small ϵ>0. Then we have

0limϵ0I(uλ+ϵψ)-I(uλ)ϵ=Q|uλ(x)-uλ(y)|p-2(uλ(x)-uλ(y))(ψ(x)-ψ(y))|x-y|n+ps𝑑x𝑑y-λΩuλ-qψ𝑑x-Ωuλαψ𝑑x.

Since ψCc(Ω) is arbitrary, we conclude that uλ is a positive weak solution of (). ∎

We recall the following comparison principle from [32].

Lemma 4.3.

Let u,vX0 be such that uv in RnΩ and

Q(|u(x)-u(y)|p-2(u(x)-u(y))-|v(x)-v(y)|p-2(v(x)-v(y)))(ψ(x)-ψ(y))|x-y|n+ps𝑑x𝑑y0

for all 0ψX0. Then uv in Ω.

Proof.

The proof follows by taking ψ=(v-u)+ and using the equality

|b|p-2b-|a|p-2a=(p-1)(b-a)01|a+t(b-a)|p-2dt.

As a consequence, we have the following.

Lemma 4.4.

We have Λ1<.

Proof.

Suppose Λ1=. Then, from Proposition 4.2, () has a solution for all λ. Now, we choose λ large enough such that

λt-q+tps*-1>(λ1+ϵ)tp-1for all t(0,).

Then u¯:=uλ is a super solution of the eigenvalue problem

uX0,(-Δp)su=(λ1+ϵ)|u|p-2uin Ω.(Qϵ)

Also we can choose r small enough such that u¯:=rϕ1 is a subsolution of ().

Then, using the boundedness of uλ (see Theorem 6.4) and ϕ1, we can further choose r small enough such that u¯u¯.

Now, we consider the following monotone iterations:

u0=rϕ1,unX0 such that (-Δp)suk=(λ1+ϵ)|uk-1|p-2un-1 in Ω.

Then, by the weak comparison Lemma 4.3, we get

rϕ1(x)u1(x)u2(x)uk-1(x)uk(x)uλ(x)for all xΩ.

Therefore, the sequence {uk} is bounded in X0, and hence has a weakly convergent subsequence, still calling it {uk}, that converges to u0 in X0. Thus, u0 is a weak solution of (). Since ϵ>0 is arbitrary, we get a contradiction to the simplicity and isolatedness of λ1. ∎

5 Existence of minimizer on 𝒩λ-

In this section we show the existence of second solution for () in the subcritical case. We assume α+1<ps* throughout this section.

Proposition 5.1.

For all λ(0,Λ1), there exist vλNλ- satisfying I(vλ)=infI(Nλ-).

Proof.

Assume 0<q1 and λ(0,Λ1). We will show that there exists vλ𝒩λ- with I(vλ)=infI(𝒩λ-). Let {vk}𝒩λ- be a sequence such that limkI(vk)=infI(𝒩λ-). Using Lemma 3.4, we can assume that vkvλ weakly as k in X0. We claim that vλX+,q. When 0<q<1, if vλ=0, then {vk} converges strongly to 0, which contradicts Lemma 3.4. If q=1, we similarly have -<Ωln(|vk|)𝑑x as in the earlier section. So, in both cases we get vλX+,q. Next we claim that {vk} converges strongly to vλ in X0. Suppose not. Then, we may assume vk-vλd>0 and we have the following:

  • (1)

    infI(𝒩λ-)=limI(vk)I(vλ)+dpp,

  • (2)

    ϕvk(1)=0 and ϕvk′′(1)<0 for each k ϕvλ(1)+dp=0 and ϕvλ′′(1)+dp0.

By (2), we have ϕvλ(1)<0 and ϕvλ′′(1)<0. So, there exists t2(0,1) such that ϕvλ(t2)=0 and ϕvλ′′(t2)<0. Thus, t2vλ𝒩λ-. Define g:+ as g(t)=ϕvλ(t)+dptp2 for t>0. From (2), we get g(1)=0 and, since 0<t2<1, g(t2)=dpt2p-1>0. So, g is increasing on [t2,1]. Now, we obtain

infI(𝒩λ-)I(vλ)+dpp=ϕvλ(1)+dpp=g(1)g(t2)>ϕvλ(t2)=I(t2vλ)infI(𝒩λ-),

which gives a contradiction. Hence, d=0, and thus {vk} converges strongly to vλ in X0. Since λ(0,Λ1), we have ϕvλ′′(1)<0. Therefore, we obtain vλ𝒩λ- and I(vλ)=infI(𝒩λ-). This completes the proof of this proposition for the subcritical case. ∎

Proposition 5.2.

For λ(0,Λ1), vλ is a positive weak solution of ().

Proof.

Let ψCc(Ω). Using Lemma 3.7, since ϕ>0 in Ω, we can find β>0 such that vλβ on the support of ψ. Also, tϵ1 as ϵ0+, where tϵ is the unique positive real number corresponding to (vλ+ϵψ) such that tϵ(vλ+ϵψ)𝒩λ-. Then, by Lemma 3.5, we have

0limϵ0I(tϵ(vλ+ϵψ))-I(vλ)ϵlimϵ0I(tϵ(vλ+ϵψ))-I(tϵvλ)ϵ=Q|vλ(x)-vλ(y)|p-2(vλ(x)-vλ(y))(ψ(x)-ψ(y))|x-y|n+sp𝑑x𝑑y-Ω(λvλ-q+vλα)ψ𝑑x.

Since ψCc(Ω) is arbitrary, we conclude that vλ is a positive weak solution of (). ∎

Proof of Theorem 2.4.

The proof follows from Propositions 4.2 and 5.2. ∎

Remark 5.3.

To prove the existence of second positive solution in the critical case, one requires to know the classification of exact solutions of the problem

(-Δp)su=|u|ps*-2uin n.

These are the minimizers of S, the best constant of the embedding X0 into Lps*. In [6, 38], several estimates on these minimizers were obtained, and it was conjectured that the solutions are dilations and translations of the radial function

U(x)=1(1+|x|p)N-spp,xn,

where p=pp-1. In the case p=2, these classifications were proved in [41], where, in addition, it was proved that all solutions are classified by dilations and translations of U(x). Using these classifications, in [39], it was shown that

sup{I(uλ+tUϵ):t0}<I(uλ)+snSn2s,

where Uϵ=ϵ-n-2s2U(xϵ), xn, ϵ>0 and uλ is the minimizer on 𝒩λ+. Then, by carefully analysing the related fiber maps, it was shown that uλ+tUϵ𝒩λ- for large t. From this it follows

infI(𝒩λ-)<I(uλ)+snSn2s.

Then the existence of a minimizer of Nλ- was shown using the analysis of fibering maps in Lemma 3.2.

6 Regularity of weak solutions

In this section, we shall prove some regularity properties of positive weak solutions of (). We begin with the following lemma.

Lemma 6.1.

Suppose u is a weak solution of (). Then, for each wX0, u-qwL1(Ω) and

Q|u(x)-u(y)|p-2(u(x)-u(y))(w(x)-w(y))|x-y|n+sp𝑑x𝑑y-Ω(λu-q+uα)w𝑑x=0.(6.1)

Proof.

Let u be a weak solution of () and wX+. By Lemma 2.2, we get a sequence {wk}X0 such that {wk}w strongly in X0. Each wk has compact support in Ω and 0w1w2. Therefore, u is a positive weak solution of () and, for each k, we get

λΩu-qwk𝑑x=Q|u(x)-u(y)|p-2(u(x)-u(y))(wk(x)-wk(y))|x-y|n+sp𝑑x𝑑y-Ωuαwk𝑑x.

Using the monotone convergence theorem, we get u-qwL1(Ω) and

λΩu-qw𝑑x=Q|u(x)-u(y)|p-2(u(x)-u(y))(w(x)-w(y))|x-y|n+sp𝑑x𝑑y-Ωuαw𝑑x.

If wX0, then w=w+-w- and w+,w-X+. Since we proved the lemma for each wX+, we obtain the conclusion. ∎

Before proving our next result, let us recall some estimates or inequalities from [7].

Lemma 6.2.

Let 1<p< and f:RR be a 𝐶1 convex function. If τ0, t,a,bR and A,B>0, then

|f(a)-f(b)|p-2(f(a)-f(b))(A-B)|a-b|p-2(a-b)(A|f(a)|p-2f(a)-B|f(b)|p-2f(b)).

Lemma 6.3.

Let 1<p< and g:RR be an increasing function. Then we have

|G(a)-G(b)|p|a-b|p-2(a-b)(g(a)-g(b)),

where G(t)=0tg(τ)1p𝑑τ for tR.

Theorem 6.4.

Let u be a positive solution of (). Then uL(Ω).

Proof.

The proof here is adopted from Brasco and Parini [7]. Let ϵ>0 be very small and define

fϵ(t)=(ϵ2+t2)12,

which is smooth, convex and Lipschitz. Let 0<ψCc(Ω) and take φ=ψ|fϵ(u)|p-2fϵ(u) as the test function in (6.1). By taking the choices

a=u(x),b=u(y),A=ψ(x),B=ψ(y)

in Lemma 6.2, we get

Q|fϵ(u(x))-fϵ(u(y))|p-2(fϵ(u(x))-fϵ(u(y)))(ψ(x)-ψ(y))|x-y|n+sp𝑑x𝑑yΩ(|λu-q+uα|)|fϵ(u)|p-1ψ𝑑x.

As t0, fϵ(t)|t| and we have |fϵ(t)|1. So using Fatou’s lemma, we let ϵ0 in the above inequality to get

Q||u(x)|-|u(y)||p-2(|u(x)|-|u(y)|)(ψ(x)-ψ(y))|x-y|n+sp𝑑x𝑑yΩ(|λu-q+uα|)ψ𝑑x(6.2)

for every 0<ψCc(Ω). The above inequality still holds for 0ψX0 ( similar proof as of Lemma 6.1). Now, define uK=min{(u-1)+,K}X0 for K>0. For β>0 and ρ>0, we take ψ=(uK+ρ)β-ρβ as test function in (6.2) and get

Q|u(x)|-|u(y)|p-2(|u(x)|-|u(y)|)((uK(x)+ρ)β-(uK(y)+ρ)β)|x-y|n+sp𝑑x𝑑yΩ(|λu-q+uα|)((uK+ρ)β-ρβ)𝑑x.

Then, using Lemma 6.3 with the function g(u)=(uK+ρ)β, we get

Q|(uK(x)+ρ)β+p-1p-(uK(y)+ρ)β+p-1p|n+spx-y𝑑x𝑑y(β+p-1)pβppQ||u(x)|-|u(y)||p-2(|u(x)|-|u(y)|)((uK(x)+ρ)β-(uK(y)+ρ)β)|x-y|n+sp𝑑x𝑑y(β+p-1)pβppΩλ|u-q|((uK+ρ)β-ρβ)dx+Ω|uα|((uK+ρ)β-ρβ)dx.

Now, from the support of uK, we have

Ωλ|u-q|((uK+ρ)β-ρβ)dx+Ω|uα|((uK+ρ)β-ρβ)dx={u1}λ|u-q|((uK+ρ)β-ρβ)dx+{u1}|uα|((uK+ρ)β-ρβ)dxC1{u1}(1+|u|α)((uK+ρ)β-ρβ)𝑑x2C1{u1}|u|α((uK+ρ)β-ρβ)dx2C1|u|ps*α|(uK+ρ)β|r,

where C1=max{λ,1} and r=ps*ps*-α. Using the Sobolev inequality, given in [33, Theorem 1], we get

Q|(uK(x)+ρ)β+p-1p-(uK(y)+ρ)β+p-1p|x-yn+sp𝑑x𝑑y1Tp,s|(uK+ρ)β+p-1p-ρβ+p-1p|ps*p1Tp,s((ρ2)p-1|(uK+ρ)βp|ps*p-ρβ+p-1|Ω|pps*),

where Tp,s is a nonnegative constant and the last inequality follows from the triangle inequality and (uK+ρ)β+p-1ρp-1(uK+ρ)β. Using all these estimates, we now have

|(uK+ρ)βp|ps*pC(Tp,s(2ρ)p-1((β+p-1)pβpp)|u|ps*α|(uK+ρ)β|r+ρβ|Ω|pps*),

where C=C(p)>0 is a constant. By the convexity of the map ttp, we can show that

1β(β+p-1p)p1.

Using this, we can also check that

ρβ|Ω|pps*1β(β+p-1p)p|Ω|1-1r-spn|(uK+ρ)β|r.

Hence, we have

|(uK+ρ)βp|ps*pC1β(β+p-1p)p|(uK+ρ)β|r(Tp,s|u|ps*αρp-1+|Ω|1-1r-spn)

for C=C(p)>0 constant. We now choose

ρ=(Tp,s|u|ps*α)1p-1|Ω|-1p-1(1-1r-spn),

and let β1 be such that

1β(β+p-1p)pβp-1.

In addition, if we let τ=βr and ν=ps*pr>1, then the above inequality reduces to

|(uK+ρ)|ντ(C|Ω|1-1r-spn)rτ(τr)(p-1)rτ|(uK+ρ)|τ.(6.3)

At this stage, if we take K, then we can say that (u-1)+Lm(Ω) for all m. This will imply that uLm(Ω) for all m. Now, we iterate (6.3) using τ0=r, and let τm+1=ντm=νm+1r, which gives

|(uK+ρ)|τm+1(C|Ω|1-1r-spn)i=0mrτi(i=0m(τir)rτi)p-1|(uK+ρ)|r.(6.4)

Since ν>1,

i=0rτi=i=0m1νi=νν-1andi=0((τir)rτi)p-1=νν(ν-1)2.

Taking limit as m0 in (6.4), we finally get

|uK|(Cνν(ν-1)2)p-1(|Ω|1-1r-spn)νν-1|(uK+ρ)|r.

Since uK(u-1)+, using the triangle inequality in the above inequality, we get

|uK|C(νν(ν-1)2)p-1(|Ω|1-1r-spn)νν-1(|(u-1)+|r+ρ|Ω|1r)

for some constant C=C(p)>0. If we now let K, we get

|(u-1)+|C(νν(ν-1)2)p-1(|Ω|1-1r-spn)νν-1(|(u-1)+|r+ρ|Ω|1r).

Hence, in particular, we have that uL(Ω). ∎

Theorem 6.5.

Let u be a positive solution of (). Then there exists γ(0,s] such that uClocγ(Ω).

Proof.

Let ΩΩ. Then, using Lemma 3.7 and the above regularity result, for any ψCc(Ω), we get

λΩu-qψ𝑑x+Ωuαψ𝑑xλΩϕ1-qψ𝑑x+uαΩψ𝑑xCΩψ𝑑x

for some constant C>0, since we can find k>0 such that ϕ1>k on Ω. Thus, we have |(-Δp)su|C weakly in Ω. So, using [31, Theorem 4.4] and applying a covering argument on the above inequality as in [31, Corollary 5.5], we can prove that there exists γ(0,s] such that uCγ(Ω) for all ΩΩ. Therefore, we get uClocγ(Ω). ∎

7 Global existence of solution

In this section, we show the existence of solution for maximal range of λ. Let us define

Λ=sup{λ>0:(Pλ) has a solution}.

Lemma 7.1.

We have Λ<+.

Proof.

The proof follows similarly to the proof of Lemma 4.4. ∎

In the following lemmas, we will show the existence of a solution of () when λ(0,Λ).

Lemma 7.2.

If u¯X0 is a weak sub-solution and u¯X0 is a weak super-solution of () such that u¯u¯ a.e. in Ω, then there exists a weak solution uX0 satisfying u¯uu¯.

Proof.

We follow [28]. We know that the functional I is non- differentiable in X0. Let M:={uX0:u¯uu¯}. Then M is closed and convex, and I is weakly lower semicontinuous on M. We can see that if {uk}M and uku weakly in X0 as k, then we may assume uku pointwise a.e. in Ω (along a subsequence). Since uM, Ω|u¯|α+1dx<+ and ΩGq(u¯)𝑑x<+, so, by Lebesgue’s dominated convergence theorem,

Ω|uk|α+1dxΩ|u|α+1dx and ΩGq(uk)dxΩGq(u)dx.

So, lim¯kI(uk)I(u). Thus, there exists uM such that I(u)=infI(M). We claim that u is a weak solution of (). For ϵ>0 and φX0, define vϵ=u+ϵφ-φϵ+φϵM, where φϵ=(u+ϵφ-u¯)+0 and φϵ=(u+ϵφ-u¯)-0. For t(0,1), u+t(vϵ-u)M and we have

0I(u+t(vϵ-u))-I(u)t=limt0(1pt(u+t(vϵ-u)p-up)+λΩ(Gq(u+t(vϵ-u))-Gq(u))tdx-1α+1Ω|u+t(vϵ-u)|α+1-|u|α+1tdx)=Q|u(x)-u(y)|p-2(u(x)-u(y))((vϵ-u)(x)-(vϵ-u)(y))|x-y|n+sp𝑑x𝑑y-λΩu-q(vϵ-u)𝑑x-Ωuα(vϵ-u)𝑑x,

which gives

Q|u(x)-u(y)|p-2(u(x)-u(y))(φ(x)-φ(y))|x-y|n+sp𝑑x𝑑y-Ω(λu-q+uα)φ𝑑x1ϵ(Hϵ-Hϵ),(7.1)

where

Hϵ=Q|u(x)-u(y)|p-2(u(x)-u(y))(φϵ(x)-φϵ(y))|x-y|n+sp𝑑x𝑑y-Ω(λu-q+uα)φϵ𝑑x,Hϵ=Q|u(x)-u(y)|p-2(u(x)-u(y))(φϵ(x)-φϵ(y))|x-y|n+sp𝑑x𝑑y-Ω(λu-q+uα)φϵ𝑑x.

Now, we consider

1ϵHϵ=1ϵ(Q|u(x)-u(y)|p-2(u(x)-u(y))(φϵ(x)-φϵ(y))|x-y|n+sp𝑑x𝑑y-Ω(λu-q+uα)φϵ𝑑x).

Let Ω1={u+ϵφu¯>u} and Ω2={u+ϵφ<u¯}. Then, using the technique of Lemma 3.7, we get

1ϵQ|u(x)-u(y)|p-2(u(x)-u(y))(φϵ(x)-φϵ(y))|x-y|n+sp𝑑x𝑑y=1ϵ(Ω1×Ω1+Ω1×Ω2+Ω2×Ω1)|u(x)-u(y)|p-2(u(x)-u(y))(φϵ(x)-φϵ(y))|x-y|n+spdxdy=1ϵΩ1×Ω1|u(x)-u(y)|p-2(u(x)-u(y))((u-u¯)(x)-(u-u¯)(y))|x-y|n+sp𝑑x𝑑y   +Ω1×Ω1|u(x)-u(y)|p-2(u(x)-u(y))(φ(x)-φ(y))|x-y|n+sp𝑑x𝑑y   +1ϵΩ1×Ω2|u(x)-u(y)|p-2(u(x)-u(y))|x-y|n+sp(u-u¯)(x)𝑑x𝑑y   +Ω1×Ω2|u(x)-u(y)|p-2(u(x)-u(y))|x-y|n+spφ(x)𝑑x𝑑y   -1ϵΩ2×Ω1|u(x)-u(y)|p-2(u(x)-u(y))|x-y|n+sp(u-u¯)(y)𝑑x𝑑y   -Ω2×Ω1|u(x)-u(y)|p-2(u(x)-u(y))|x-y|n+spφ(y)𝑑x𝑑y3ϵ2p-2Ω1×Ω1|(u-u¯)(x)-(u-u¯)(y)|p|x-y|n+sp𝑑x𝑑y   +Ω1×Ω1|u(x)-u(y)|p-2(u(x)-u(y))(φ(x)-φ(y))|x-y|n+sp𝑑x𝑑yΩ1×Ω1|u(x)-u(y)|p-2(u(x)-u(y))(φ(x)-φ(y))|x-y|n+sp𝑑x𝑑y,

where we used the inequality |a-b|p2p-2(|a|p-2a-|b|p-2b)(a-b) for p2 and a,b. Thus,

1ϵHϵΩ1×Ω1|u(x)-u(y)|p-2(u(x)-u(y))(φ(x)-φ(y))|x-y|n+sp𝑑x𝑑y-Ω1(λu-q+uα)φϵ𝑑xΩ1×Ω1|u(x)-u(y)|p-2(u(x)-u(y))(φ(x)-φ(y))|x-y|n+spdxdy-Ω1|λu¯-q-u-q||φ|dx=o(1)as ϵ0,

since |Ω1|0 as ϵ0. Similarly, as ϵ0, we can show that

1ϵHϵo(1).

Therefore, taking ϵ0 in (7.1), we get

Q|u(x)-u(y)|p-2(u(x)-u(y))(φ(x)-φ(y))|x-y|n+sp𝑑x𝑑y-Ω(λu-q+uα)φ𝑑xo(1).

Since φX0 is arbitrary, for all φX0, we get

Q|u(x)-u(y)|p-2(u(x)-u(y))(φ(x)-φ(y))|x-y|n+sp𝑑x𝑑y-Ω(λu-q+uα)φ𝑑x=0.

Proposition 7.3.

For λ(0,Λ), () has a weak solution uλX0.

Proof.

We fix λ(0,Λ). By the definition of Λ, there exists λ0(λ,Λ) such that (Pλ0) has a solution uλ0, say. Then u¯=uλ0 becomes a super-solution of (). Now, consider the function ϕ1 as the eigenfunction of (-Δp)s corresponding to the smallest eigenvalue λ1. Then ϕ1L(Ω) and

(-Δp)sϕ1=λ1|ϕ1|p-2ϕ1,ϕ1>0in Ω,ϕ1=0on nΩ.

Let us choose t>0 such that tϕ1u¯ and tp+q-1ϕ1p+q-1λλ1. If we define u¯=tϕ1, then

(-Δp)su¯=λ1tp-1ϕ1p-1λt-qϕ1-qλt-qϕ1-q+tαϕ1α=λu¯-q+u¯α,

that is, u¯ is a sub-solution of () and u¯u¯. Applying Lemma 7.2, we conclude that () has a solution for all λ(0,Λ). This completes the proof. ∎

Proof of Theorem 2.6.

The proof follows from Proposition 7.3 and Lemma 7.1. ∎

Remark 7.4.

We remark that using the method in Lemma 7.2, we can show the existence of solution for the following pure singular problem:

(-Δp)su=λu-qin Ω,u=0in nΩ,(7.2)

where 0<q<1. We define u to be a positive weak solution of (7.2) if u>0 in Ω, uX0 and

Q|u(x)-u(y)|p-2(u(x)-u(y))(ψ(x)-ψ(y))|x-y|n+sp𝑑x𝑑y-λΩu-q𝑑x=0for all 0ψX0.

Also, we say that uX0 is a positive weak sub-solution of (7.2) if u>0 and

Q|u(x)-u(y)|p-2(u(x)-u(y))(ψ(x)-ψ(y))|x-y|n+sp𝑑x𝑑yλΩu-q𝑑xfor all 0ψX0.

We define the functional Jλ:X0(-,] by

Jλ(u)=1pQ|u(x)-u(y)|p|x-y|n+sp𝑑x𝑑y-λΩGq(u)𝑑x,

where Gq is as defined in Section 2. One can easily see that Jλ is coercive, bounded below and weakly lower semicontinuous in X0. Thus, there exist u0X0 such that infI(X0)=I(u0). We claim that u0 is a positive weak solution of (7.2). We choose t>0 such that tϕ1u0 in Ω and tϕ1 is a sub-solution of (7.2) (ϕ1 is defined in Proposition 7.3). Let us define M:={uX0:u¯u}, where u¯ is a weak sub-solution of (7.2). Then u0M, and following the proof of Lemma 7.2 with vϵ=u0+ϵφ+φϵ, where ϵ>0, φϵ=(u0+ϵφ-u¯)- and φX0, we can show that u0 is a positive weak solution of (7.2).

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About the article

Received: 2016-04-26

Revised: 2016-07-26

Accepted: 2016-09-18

Published Online: 2016-12-02


Citation Information: Advances in Nonlinear Analysis, Volume 8, Issue 1, Pages 52–72, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2016-0100.

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