Positive solutions of fractional elliptic equation with critical and singular nonlinearity

Jacques Giacomoni; Tuhina Mukherjee; Konijeti Sreenadh

doi:10.1515/anona-2016-0113

Article Open Access

Positive solutions of fractional elliptic equation with critical and singular nonlinearity

Jacques Giacomoni , Tuhina Mukherjee and Konijeti Sreenadh

Published/Copyright: July 30, 2016

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From the journal Advances in Nonlinear Analysis Volume 6 Issue 3

Abstract

In this article, we study the following fractional elliptic equation with critical growth and singular nonlinearity:

(-Δ)s⁢u=u-q+λ⁢u2s*-1,u>0 in ⁢Ω,u=0 in ⁢ℝn∖Ω,

where Ω is a bounded domain in ℝn with smooth boundary ∂⁡Ω, n>2⁢s, s∈(0,1), λ>0, q>0 and 2s*=2⁢nn-2⁢s. We use variational methods to show the existence and multiplicity of positive solutions with respect to the parameter λ.

Keywords: Nonlocal operator; fractional Laplacian; very singular nonlinearities, variational methods in non smooth analysis

MSC 2010: 35R11; 35R09; 35A15

1 Introduction

Let Ω⊂ℝn be a bounded domain with smooth boundary ∂⁡Ω (at least C2), n>2⁢s and s∈(0,1). We consider the following problem with singular nonlinearity:

(Pλ)(-Δ)s⁢u=u-q+λ⁢u2s*-1,u>0 in ⁢Ω,u=0 in ⁢ℝn∖Ω,

where λ>0, 0<q, 2s*=2⁢nn-2⁢s and (-Δ)s is the fractional Laplace operator defined as

(-Δ)su(x)=2Csn(P.V.∫ℝnu⁢(x)-u⁢(y)|x-y|n+2⁢sdy),

where P.V. denotes the Cauchy principal value and Csn=π-n/2⁢22⁢s-1⁢s⁢Γ⁢(n+2⁢s2)/Γ⁢(1-s), with Γ being the Gamma function. The fractional power of Laplacian is the infinitesimal generator of Lévy stable diffusion process and arise in anomalous diffusion in plasma, population dynamics, geophysical fluid dynamics, flames propagation, chemical reactions in liquids and American options in finance, see [3] for instance.

In the local setting (s=1), the paper by Crandal, Rabinowitz and Tartar [10] is the starting point on semilinear problems with a singular nonlinearity. From this pioneering work, a lot of contributions have been made, related to existence, multiplicity, stability and regularity results on problems involving singular nonlinearities. We refer the survey papers [20, 29] for more details and references about the topic. Among the works dealing with elliptic equations with singular nonlinearities and critical growth terms, we cite [1, 27, 28, 30, 31, 19, 17, 18] and references therein, with no attempt to provide an exhaustive list. In [27], Haitao explored existence and multiplicity results for the maximal range of the parameter λ, when 0<q<1, using monotone iterations and the mountain pass lemma in the spirit of [2]. The singular problem for the case 1<q<3 is studied in [1, 12, 25], whereas, using the notion of very weak solutions introduced in [14, 15], Díaz, Hernández and Rakotoso in [13] proved the existence and regularity of weak solutions for any q>0. In the quasilinear case with p-Laplacian, the multiplicity results are proved using Sobolev instead of Hölder minimizers when 0<q<1. These results for q>1 are still open in the non radial case. For related results, we refer to [11, 23, 24, 26, 28] and references therein. For the case q>3, Hirano, Saccon and Shioji in [31] studied the existence of Lloc1 solutions u such that (u-ϵ)+∈H01⁢(Ω) for all ϵ>0, using variational methods and the critical point theory of non-smooth analysis.

Recently, the study of fractional elliptic equations attracted lot of interests by researchers in nonlinear analysis. Subcritical growth problems (without singular nonlinearity) are studied in [8, 36, 34, 35, 42, 44] and Brezis–Nirenberg type critical exponent (and non singular) problems are studied in [6, 38, 37, 45, 43, 46]. We refer also to the survey about variational methods for non local equations [33]. In [5], Barrios et al. considered the problem

(-Δ)s⁢u=λ⁢f⁢(x)uγ+M⁢up,u>0 in ⁢Ω,u=0 in ⁢ℝn∖Ω,

where n>2⁢s, M≥0, 0<s<1, γ>0, λ>0, 1<p<2s*-1 and f∈Lm⁢(Ω), with m≥1, is a nonnegative function. Therein they studied the existence of distributional solutions using the uniform estimates of {un}, which are the unique solutions of regularized problems with the singular term u-γ replaced by (u+1n)-γ. They also discussed multiplicity results when M>0 and for small λ in the subcritical case. The critical exponent problem with singular nonlinearity λ⁢u-q+u2s*-1, 0<q<1, is recently studied in [39]. To the best of our knowledge, there are no works on existence results when q>1.

In this paper we study the existence and multiplicity of positive solutions to a class of problems with a singular type nonlinearity λ⁢u-q+u2s*-1 for all q>0 in the spirit of [31]. Besides, the functional

J(u)=Csn2∥u∥H0s⁢(Ω)2-11-q∫Ω|u|1-qdx-λ2s*∫Ω|u|2s*dx

(taking q≠1 for simplicity), associated to problem (Pλ), is not differentiable, even in the sense of Gâteaux. For the case 0<q<1, the functional I is continuous on X0, but when q≥1, the functional I is neither defined on the whole space nor it is continuous on D⁢(I)≡{u∈H0s⁢(Ω):I⁢(u)<∞}. With these difficulties and taking into account the non local feature of the operator, it is not easy to treat the problem with the usual variational approach. Another difficulty arises in showing that the weak solutions of (Pλ) are classical because the standard bootstrap arguments may not work. Overcoming these difficulties, we prove existence, multiplicity and regularity of solutions for (Pλ). For that we appeal to the critical point theory from non-smooth analysis. Precisely, we use a variant of the linking theorem (see Theorem 2.4) as in [31]. We also use a suitable positive subsolution combined with a weak comparison principle in the non local setting, in order to control the behavior of the singular nonlinearity in the variational setting of (Pλ).

The paper is organized as follows. In Section 2, we recall some results from non-smooth analysis and give the functional setting for the fractional Laplacian.

In Section 3, we prove the existence of the first solution by Perron’s method for non-smooth functionals. Here, we adapt the variational approach in the work of Hirano, Saccon and Shioji [31] to the non local setting. We obtain our results using an approach based on non-smooth analysis, considering solutions of (Pλ) as critical points of I in some suitable non-smooth sense.

In Section 4, we prove the multiplicity result stated in Theorem 2.10. For that we show that the energy functional possesses a linking geometry and apply an appropriate version of the linking theorem. We point out that the multiplicity result obtained here is sharp in the sense that the problem has no solution outside the interval where multiplicity fails.

Finally, in Section 5, we extend the main results obtained in Section 3 and 4 to dimension one. In this case, the critical growth is given by the Orlicz space imbedding, stated in Theorem 5.1. Applying the harmonic extension introduced in [9], we study an equivalent local problem as in [8, 22, 21].

We use the following notations:

For two real valued functions u and v, we define u∨v=max⁡{u,v} and u∧v=min⁡{u,v}.
We say u>v in Ω if ess⁢infK⁡u-v>0 for any compact subset K of Ω.
We denote by |⋅|p the standard norm in Lp⁢(Ω), 1≤p≤∞.
For a Carathéodory function f:Ω×ℝ→ℝ, we denote the partial derivative ∂⁡f∂⁡u⁢(x,u) by f′⁢(x,u).
We set :=⁡d⁢(x)dist⁢(x,∂⁡Ω),x∈Ω.

2 Preliminaries and main results

We recall some definitions for the critical point of a non-smooth function, definitions of function spaces and results that are required in later sections.

2.1 Some definitions and results from non smooth analysis

Definition 2.1.

Let H be a Hilbert space and I:H→(-∞,∞] be a proper (i.e., I≢∞) lower semicontinuous functional.

Let D⁢(I)={u∈H:I⁢(u)<∞} be the domain of I. For every u∈D⁢(I), we define the Fréchet sub-differential of I at u as the set
∂-⁡I⁢(u)={α∈H:lim¯v→u⁡I⁢(v)-I⁢(u)-〈α,v-u〉∥v-u∥H≥0}.
For every u∈H, we define
|||∂-⁡I⁢(u)|||={min⁡{∥α∥H:α∈∂-⁡I⁢(u)}if ⁢∂-⁡I⁢(u)≠∅,∞if ⁢∂-⁡I⁢(u)=∅.

We know that ∂-⁡I⁢(u) is a closed convex set which may be empty. If u∈D⁢(I) is a local minimizer for I, then it can be seen that 0∈∂-⁡I⁢(u).

Remark 2.2.

We remark that if I0:H→(-∞,∞] is a proper, lower semicontinuous, convex functional, I1:H→ℝ is a C1-functional and I=I1+I0, then ∂-⁡I⁢(u)=∇⁡I1⁢(u)+∂⁡I0⁢(u) for every u∈D⁢(I)=D⁢(I0), where ∂⁡I0 denotes the usual subdifferential of the convex functional I0. Thus, u is said to be a critical point of I if u∈D⁢(I0) and for every v∈H, we have

〈∇⁡I1⁢(u),v-u〉+I0⁢(v)-I0⁢(u)≥0.

Definition 2.3.

For a proper, lower semicontinuous functional I:H→(-∞,∞], we say that I satisfies Cerami’s variant of the Palais–Smale condition at level c (in short, I satisfies (CPS)c), if any sequence {un}⊂D⁢(I) such that I⁢(un)→c and (1+∥un∥)⁢|||∂-⁡I⁢(un)|||→0 has a strongly convergent subsequence in H.

Analogous to the mountain pass theorem, we have the following linking theorem for non-smooth functionals.

Theorem 2.4 (see [31, Theorem 2]).

Let H be a Hilbert space. Assume I=I0+I1, where I0:H→(-∞,∞] is a proper, lower semicontinuous, convex functional and I1:H→R is a C1-functional. Let Dn,Sn-1 denote, respectively, the closed unit ball and its boundary in Rn, and let ψ:Sn-1→D⁢(I) be a continuous function such that

Φ:={φ∈C⁢(Dn,D⁢(I)):φ|Sn-1=ψ}≠∅.

Let A be a relatively closed subset of D⁢(I) such that

A∩ψ⁢(Sn-1)=∅,A∩φ⁢(Dn)≠∅ for all ⁢φ∈Φ 𝑎𝑛𝑑 inf⁡I⁢(A)≥sup⁡I⁢(ψ⁢(Sn-1)).

Define

c:=infφ∈Φ⁢supx∈Dn⁡I⁢(φ⁢(x)).

Assume that c is finite and that I satisfies (CPS)c. Then there exists u∈D⁢(I) such that I⁢(u)=c and 0∈∂-⁡I⁢(u). Furthermore, if inf⁡I⁢(A)=c, then there exists u∈A∩D⁢(I) such that I⁢(u)=c and 0∈∂-⁡I⁢(u).

2.2 Functional setting and preliminaries

In [45], Servadei and Valdinoci discussed the Dirichlet boundary value problem for the fractional Laplacian using variational techniques. Due to the nonlocalness of the fractional Laplacian, they introduced the function space (X0,∥⋅∥X0). The space X is defined as

X={u|u:ℝn→ℝ⁢ is measurable,u|Ω∈L2⁢(Ω)⁢ and ⁢(u⁢(x)-u⁢(y))|x-y|n/2+s∈L2⁢(Q)},

where Q=ℝ2⁢n∖(𝒞⁢Ω×𝒞⁢Ω) and 𝒞⁢Ω:=ℝn∖Ω. The space X is endowed with the norm defined as

∥u∥X=∥u∥L2⁢(Ω)+[u]X,

where

[u]X=(∫Q|u⁢(x)-u⁢(y)|2|x-y|n+2⁢s⁢dx⁢dy)1/2=(1Csn⁢∫Ωu⁢(-Δ)s⁢u⁢dx⁢dy)1/2.

Then we define

X0={u∈X:u=0⁢ a.e. in ⁢ℝn∖Ω}.

Also, there exists a constant C>0 such that ∥u∥L2⁢(Ω)≤C⁢[u]X for all u∈X0. 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 ℝn∖Ω. We denote ∥⋅∥=[⋅]X the norm in X0. From the embedding results, we know that X0 is continuously, and compactly embedded in Lr⁢(Ω) when 1≤r<2s* and the embedding is continuous but not compact if r=2s*. We define

Ss=infu∈X0∖{0}⁡∫Q|u⁢(x)-u⁢(y)|2|x-y|n+2⁢s⁢dx⁢dy(∫Ω|u|2s*⁢dx)2/2s*.

Consider the family of functions {Uϵ} defined as

Uϵ⁢(x)=ϵ-(n-2⁢s)/2⁢u*⁢(xϵ),x∈ℝn,

where

u*⁢(x)=u¯⁢(xSs1/(2⁢s)),u¯⁢(x)=u~⁢(x)|u|2s* and u~⁢(x)=α⁢(β2+|x|2)-(n-2⁢s)/2,

with α∈ℝ∖{0} and β>0 being fixed constants. Then, for each ϵ>0, Uϵ satisfies

(-Δ)s⁢u=|u|2s*-2⁢u in ⁢ℝn,

and verifies the equality

∫ℝn∫ℝn|Uϵ⁢(x)-Uϵ⁢(y)|2|x-y|n+2⁢sdxdy=∫ℝn|Uϵ|2s*dx=Ssn/(2⁢s).

For a proof, we refer to [45].

Definition 2.5.

A function u∈Lloc1⁢(Ω) is said to be a weak solution of (Pλ) if the following hold:

infx∈K⁡u⁢(x)>0 for every compact subset K⊂Ω,
u solves the PDE in (Pλ) in the sense of distributions,
(u-ϵ)+∈X0 for every ϵ>0.

In order to prove the existence results for (Pλ), we translate the problem by the solution of the purely singular problem:

(P0)(-Δ)s⁢u=u-q,u>0 in ⁢Ω,u=0 in ⁢ℝn∖Ω.

In [5], it is shown that the problem (P0) has a minimal solution u¯∈L∞⁢(Ω) (by construction). Now we consider the following translated problem:

(P̄λ)(-Δ)s⁢u+u¯-q-(u+u¯)-q=λ⁢(u+u¯)2s*-1,u>0 in ⁢Ω,u=0 in ⁢ℝn∖Ω.

Clearly, we can notice that u+u¯ is a solution of (Pλ) if and only if u∈X0 solves (P̄λ) in the sense of distributions, and hence it is sufficient to show existence and multiplicity results for (P̄λ). We define the function g:Ω×ℝ→ℝ∪{-∞} by

g⁢(x,s)={(u¯⁢(x))-q-(s+u¯⁢(x))-qif ⁢s+u¯⁢(x)>0,-∞otherwise.

We can easily see that g is nonnegative and non-decreasing in s. The required measurability of g⁢(⋅,s) follows from [31, Lemmas 1 and 2]. We now define the notions of subsolution and supersolution for problem (P̄λ).

Definition 2.6.

ϕ∈X is called a subsolution (resp. a supersolution) of (P̄λ) if the following hold:

ϕ+∈X0 (resp. ϕ-∈X0),
g⁢(⋅,ϕ)∈Lloc1⁢(Ω),
For all w∈X0, w≥0, we have
Csn⁢∫Q(ϕ⁢(x)-ϕ⁢(y))⁢(w⁢(x)-w⁢(y))|x-y|n+2⁢s⁢dx⁢dy+∫Ω(g⁢(x,ϕ)-λ⁢(ϕ+u¯)2s*-1)⁢w⁢dx≤0 (resp. ≥0).

Definition 2.7.

A function ϕ is a weak solution of (P̄λ) if it is both a subsolution and a supersolution of (P̄λ). That is, ϕ∈X0, g⁢(⋅,ϕ)∈Lloc1⁢(Ω) and for all ψ∈C0∞⁢(Ω),

Csn⁢∫Q(ϕ⁢(x)-ϕ⁢(y))⁢(ψ⁢(x)-ψ⁢(y))|x-y|n+2⁢s⁢dx⁢dy+∫Ω(g⁢(x,ϕ)⁢ψ-λ⁢(ϕ+u¯)2s*-1⁢ψ)⁢dx=0.

Definition 2.8.

A nonnegative function u∈X0 is called positive weak solution to (P̄λ) if u satisfies Definition 2.7 and ess⁢infK⁡u>0 for any compact set K of Ω.

Definition 2.9.

We say ϕ is a strict subsolution (resp. strict supersolution) of (P̄λ) if ϕ is a subsolution (resp. a supersolution) and

Csn⁢∫Q(ϕ⁢(x)-ϕ⁢(y))⁢(ψ⁢(x)-ψ⁢(y))|x-y|n+2⁢s⁢dx⁢dy+∫Ω(g⁢(x,ϕ)⁢ψ-λ⁢(ϕ+u¯)2s*-1⁢ψ)⁢dx<0 (resp. >0)

for all ψ∈X0∖{0} and ψ≥0.

With this introduction we state our main theorem.

Theorem 2.10.

There exist Λ>0 and α∈(0,1) such that the following hold:

((Pλ)) admits at least two positive solutions in Clocα⁢(Ω)∩L∞⁢(Ω) for every λ∈(0,Λ).
((Pλ)) admits no solution for λ>Λ.
(PΛ) admits at least one positive solution uΛ∈Clocα⁢(Ω)∩L∞⁢(Ω).

3 Regularity of weak solutions of (P¯λ)

In this section, we shall prove some regularity properties of positive weak solutions of (P̄λ). We will need the following important lemma.

Lemma 3.1.

For each w∈X0, w≥0, there exists a sequence {wk} in X0 such that wk→w strongly in X0, where 0≤w1≤w2≤⋯ and wk has compact support in Ω for each k.

Proof.

Let w∈X0,w≥0 and {ψk} be sequence in Cc∞⁢(Ω) such that ψk is nonnegative and converges strongly to w in X0. Define zk=min⁡{ψk,w}. Then zk→w strongly to w in X0. Now we set w1=zr1, where r1>0 is such that ∥zr1-w∥≤1. Then max⁡{w1,zm}→w strongly as m→∞, thus we can find r2>0 such that ∥max{w1,zr2}-w∥≤1/2. We set w2=max⁡{w1,zr2}, and get that 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 wk∈X0 has compact support for each k and ∥max{wk,zrk+1}-w∥≤1/(k+1), which imply that {wk} converges strongly to w in X0 as k→∞. ∎

Lemma 3.2.

Suppose that u is a nonnegative weak solution of (P̄λ). Then, for each w∈X0, g⁢(x,u)⁢w∈L1⁢(Ω) and

Csn⁢∫Q(u⁢(x)-u⁢(y))⁢(w⁢(x)-w⁢(y))|x-y|n+2⁢s⁢dx⁢dy+∫Ω(g⁢(x,u)-λ⁢(u+u¯)2*s-1)⁢w⁢dx=0.

Proof.

Let w∈X0, w≥0. By Lemma 3.1, we obtain a sequence {wk}∈X0 such that {wk}→w strongly in X0, each wk has compact support in Ω and 0≤w1≤w2≤⋯. For each fixed k, we can find a sequence {ψnk}⊂Cc∞⁢(Ω) such that ψnk≥0, ⋃nsupp⁡ψnk is contained in a compact subset of Ω, {∥ψnk∥∞} is bounded and ∥ψnk-wk∥→0 strongly as n→∞. Since u is a weak solution of Pλ¯, we get

Csn⁢∫Q(u⁢(x)-u⁢(y))⁢(ψnk⁢(x)-ψnk⁢(y))|x-y|n+2⁢s⁢dx⁢dy=∫Ωg⁢(x,u)⁢ψnk⁢dx+λ⁢∫Ω(u+u¯)2*s-1⁢ψnk⁢dx.

By Lebesgue’s dominated convergence theorem, as n→∞, we get

∫Ωg⁢(x,u)⁢wk⁢dx=-Csn⁢∫Q(u⁢(x)-u⁢(y))⁢(wk⁢(x)-wk⁢(y))|x-y|n+2⁢s⁢dx⁢dy+λ⁢∫Ω(u+u¯)2*s-1⁢wk⁢dx.

Using the monotone convergence theorem and the nonnegativity of u, we obtain g⁢(x,u)⁢w∈L1⁢(Ω) and

∫Ωg⁢(x,u)⁢w⁢dx=-Csn⁢∫Q(u⁢(x)-u⁢(y))⁢(w⁢(x)-w⁢(y))|x-y|n+2⁢s⁢dx⁢dy+λ⁢∫Ω(u+u¯)2*s-1⁢w⁢dx.

If w∈X0, then w=w+-w- and w+,w-≥0. Since we proved the lemma for each w∈X0, w≥0, we obtain the conclusion. ∎

Theorem 3.3.

Any nonnegative weak solution of (P̄λ) belongs to L∞⁢(Ω).

Proof.

We follow the bootstrap argument used in [4]. We use the following inequality for the fractional Laplacian:

(3.1)(-Δ)s⁢φ⁢(u)≤φ′⁢(u)⁢(-Δ)s⁢u,

where φ is a convex and differentiable function. We define

φ⁢(t)=φT,β⁢(t)⁢{0if ⁢t≤0,tβif ⁢0<t<T,β⁢Tβ-1⁢(t-T)+Tβif ⁢t≥T,

where β>1 and T>0 is large. Then φ is Lipschitz with constant M=β⁢Tβ-1 and φ⁢(u)∈X0. Consequently,

∥φ⁢(u)∥=(∫Q|φ⁢(u⁢(x))-φ⁢(u⁢(y))|2|x-y|n+2⁢s⁢dx⁢dy)1/2≤(∫QM2⁢|u⁢(x)-u⁢(y)|2|x-y|n+2⁢s⁢dx⁢dy)1/2=M2⁢∥u∥.

Using ∥φ⁢(u)∥=(1/Csn)1/2⁢∥(-Δ)s/2⁢φ⁢(u)∥2, we obtain

(3.2)1Csn⁢∫Ωφ⁢(u)⁢(-Δ)s⁢φ⁢(u)=∥φ⁢(u)∥2≥Ss⁢|φ⁢(u)|2s*2,

where Ss is as defined in Section 1. Since φ is convex and φ⁢(u)⁢φ′⁢(u)∈X0, we obtain

∫Ωφ⁢(u)⁢(-Δ)s⁢φ⁢(u)⁢dx≤∫Ωφ⁢(u)⁢φ′⁢(u)⁢(-Δ)s⁢u⁢dx

(3.3)=∫Ωφ⁢(u)⁢φ′⁢(u)⁢(-g⁢(x,u)+λ⁢(u+u¯)2s*-1)⁢dx.

Therefore, using (3.2) and (3.3), we obtain

|φ⁢(u)|2s*2≤C⁢∫Ωφ⁢(u)⁢φ′⁢(u)⁢(-g⁢(x,u)+λ⁢(u+u¯)2s*-1)⁢dx

for some constant C>0. We have u⁢φ′⁢(u)≤β⁢φ⁢(u) and φ′⁢(u)≤β⁢(1+φ⁢(u)), which gives

C⁢∫Ωφ⁢(u)⁢φ′⁢(u)⁢(-g⁢(x,u)+λ⁢(u+u¯)2s*-1)⁢dx≤C⁢λ⁢∫Ωφ⁢(u)⁢φ′⁢(u)⁢(u+u¯)2s*-1⁢dx

≤22s*-2⁢C⁢λ⁢β⁢(∫Ω(ϕ⁢(u))2⁢u2s*-2⁢dx+∫Ω(φ⁢(u)+(φ⁢(u))2)⁢u¯2s*-1⁢dx)

≤22s*-2⁢C⁢λ⁢β⁢(∫Ω(φ⁢(u))2⁢u2s*-2⁢dx+∥u¯∥∞2s*-1⁢∫Ω(φ⁢(u)+(φ⁢(u))2)⁢dx)

≤C1⁢β⁢(∫Ω(φ⁢(u))2⁢u2s*-2⁢dx+∫Ω(φ⁢(u)+(φ⁢(u))2)⁢dx),

where C1=22s*-2⁢λ⁢C⁢max⁡{1,∥u¯∥∞}. Thus, we have

(3.4)|φ⁢(u)|2s*2≤C1⁢β⁢(∫Ω(φ⁢(u))2⁢u2s*-2⁢dx+∫Ω(φ⁢(u)+(φ⁢(u))2)⁢dx).

Next we claim that u∈Lβ1⁢2s*⁢(Ω), where β1=2s*/2. Fixing some K whose appropriate value will be determined later, we can write

∫Ω(φ⁢(u))2⁢u2s*-2⁢dx=∫u≤K(φ⁢(u))2⁢u2s*-2⁢dx+∫u>K(φ⁢(u))2⁢u2s*-2⁢dx

≤K2s*-2⁢∫u≤K(φ⁢(u))2⁢dx+(∫Ω(φ⁢(u))2s*⁢dx)2/2s*⁢(∫u>Ku2s*⁢dx)(2s*-2)/2s*.

Using the monotone convergence theorem, we choose K such that

(∫u>Ku2s*⁢dx)(2s*-2)/2s*≤12⁢C1⁢β,

and this gives

(3.5)(∫Ω(φ⁢(u))2s*⁢dx)2/2s*≤2⁢C1⁢β⁢(∫Ω(φ⁢(u)+(φ⁢(u))2)⁢dx+K2s*-2⁢∫u≤K(φ⁢(u))2⁢dx).

Using φT,β1⁢(u)≤uβ1 in the right-hand side of (3.5) and then letting T→∞ in the left-hand side, we obtain

(∫Ωu2s*⁢β1⁢dx)2/2s*≤2⁢C1⁢β1⁢(∫Ω(u2s*/2+u2s*)⁢dx+K2s*-2⁢∫Ωu2s*⁢dx),

since 2⁢β1=2s*. This proves the claim. Again, from (3.4), using φT,β⁢(u)≤uβ in the right-hand side and then letting T→∞ in the left-hand side, we obtain

(∫Ωu2s*⁢β⁢dx)2/2s*≤2⁢C1⁢β⁢(∫Ω(uβ+u2⁢β)⁢dx+∫Ωu2⁢β+2s*-2⁢dx)

≤2⁢C1⁢β⁢(2⁢|Ω|+2⁢∫u≥1u2⁢β+2s*-2⁢dx+∫Ωu2⁢β+2s*-2⁢dx)

≤2⁢C2⁢β⁢(1+∫Ωu2⁢β+2s*-2⁢dx),

where C2>0 is a constant (independent of β). With further simplifications, we get

(3.6)(1+∫Ωu2s*⁢β⁢dx)1/[2s*⁢(β-1)]≤Cβ1/[2⁢(β-1)]⁢(1+∫Ωu2⁢β+2s*-2⁢dx)1/[2⁢(β-1)],

where Cβ=4⁢C2⁢β⁢(1+|Ω|). For m≥1, let us define βm+1 inductively by

2⁢βm+1+2s*-2=2s*⁢βm,

that is,

(βm+1-1)=2s*2⁢(βm-1)=(2s*2)m⁢(β1-1).

Hence, from (3.6) it follows that

(1+∫Ωu2s*⁢βm+1⁢dx)1/[2s*⁢(βm+1-1)]≤Cβm+11/[2⁢(βm+1-1)]⁢(1+∫Ωu2s*⁢βm⁢dx)1/[2s*⁢(βm-1)],

where Cβm+1=4⁢C2⁢βm+1⁢(1+|Ω|). Setting

Dm+1:=(1+∫Ωu2s*⁢βm)1/[2s*⁢(βm-1)],

we obtain

Dm+1≤{4⁢C2⁢(1+|Ω|)}∑i=2m+11/[2⁢(βi-1)]⁢∏i=2m+1(1+(2s*2)i-1⁢(β1-1))1/[2⁢(2s*/2)i-1⁢(β1-1)]⁢D1.

It is not difficult to show that the following sequence is convergent:

({4⁢C2⁢(1+|Ω|)}∑i=2m+11/[2⁢(βi-1)]⁢∏i=2m+1(1+(2s*2)i-1⁢(β1-1))1/[2⁢(2s*/2)i-1⁢(β1-1)])m∈ℕ

Therefore, there exists a constant C4>0 such that Dm+1≤C4⁢D1, that is,

(3.7)(1+∫Ωu2s*⁢(βm+1)⁢dx)1/[2s*⁢(βm+1-1)]≤C4⁢D1

for all m≥1. Let us assume ∥u∥∞>C4⁢D1. Then there exists η>0 and a measurable subset Ω′⊂Ω such that

u⁢(x)>C4⁢D1+η for all ⁢x∈Ω′.

It follows that

lim infβm→∞(∫Ω′|u|2s*⁢βmdx+1)1/(2s*⁢βm-1)≥lim infβm→∞(C4D1+η)βm/(βm-1)(|Ω′|)1/[2s*⁢(βm-1)]=C4D1+η,

which contradicts (3.7). Hence, ∥u∥∞≤C4⁢D1, that is, u∈L∞⁢(Ω). ∎

Lemma 3.4.

Let r>0 and let v∈L(r+1)/r⁢(Ω) be a positive function and u∈X0∩Lr+1⁢(Ω) a positive weak solution to

(3.8)(-Δ)s⁢u+g⁢(x,u)=v in ⁢Ω,u=0 in ⁢ℝn∖Ω.

Then (u+u¯-ϵ1)+∈X0 for every ϵ1>0. In particular, every positive weak solution u to (P̄λ), belonging to Lr+1⁢(Ω), satisfies (u+u¯-ϵ1)+∈X0 for every ϵ1>0.

Proof.

Let ϵ1,ϵ2>0 and set ψ=min⁡{u,ϵ1-(u¯-ϵ2)+}∈X0. Note that u-ψ=(u+(u¯-ϵ2)+-ϵ1)+∈X0. Since

0≤v⁢(u-ψ)≤v⁢u+v⁢u¯∈L1⁢(Ω),

using the arguments in the proof of Lemma 3.2, we can show that g⁢(⋅,u)⁢(u-ψ)∈L1⁢(Ω) and

Csn⁢∫Q(u⁢(x)-u⁢(y))⁢((u-ψ)⁢(x)-(u-ψ)⁢(y))|x-y|n+2⁢s⁢dx⁢dy+∫Ωg⁢(x,u)⁢(u-ψ)⁢dx-∫Ωv⁢(u-ψ)⁢dx=0.

Let 0≤φ∈Cc∞⁢(Ω). Then, using (3.1), we have

Csn⁢∫Q((u¯-ϵ2)+⁢(x)-(u¯-ϵ2)+⁢(y))⁢(φ⁢(x)-φ⁢(y))|x-y|n+2⁢s⁢dx⁢dy≤Csn⁢∫Q(u¯⁢(x)-u¯⁢(y))⁢(φ⁢(x)-φ⁢(y))|x-y|n+2⁢s⁢dx⁢dy=∫Ωu¯-q⁢φ⁢dx.

So, by arguing as in the proof of Lemma 3.2, we can show that

Csn⁢∫Q((u¯-ϵ2)+⁢(x)-(u¯-ϵ2)+⁢(y))⁢((u-ψ)⁢(x)-(u-ψ)⁢(y))|x-y|n+2⁢s⁢dx⁢dy≤∫Ωu¯-q⁢(u-ψ)⁢dx.

We have u+u¯≥ϵ1 when u≠ψ, (u+u¯)-q⁢(u-ψ)∈L1⁢(Ω) and u¯⁢(u-ψ)∈L1⁢(Ω). Therefore, we have

Csn⁢∫Q|(u+(u¯-ϵ2)+-ϵ1)+⁢(x)-(u+(u¯-ϵ2)+-ϵ1)+⁢(y)|2|x-y|n+2⁢s⁢dx⁢dy

≤∫Ωu¯-q⁢(u-ψ)⁢dx-∫Ωg⁢(x,u)⁢(u-ψ)⁢dx+∫Ωv⁢(u-ψ)⁢dx

=∫Ω(u+u¯)-q⁢(u-ψ)⁢dx+∫Ωv⁢(u-ψ)⁢dx

≤ϵ1-q⁢∫Ω(u-ψ)⁢dx+∫Ωv⁢(u-ψ)⁢dx.

Thus, for any ϵ>0, we have that (u+(u¯-ϵ2)+-ϵ1)+ is bounded in X0 as ϵ2→0+. Hence, we conclude that (u+u¯-ϵ1)+∈X0 for every ϵ1>0. ∎

Lemma 3.5.

Let F∈(X0)* (the dual of X0) and let z,v∈X be such that z,v>0 a.e. in Ω, z-q,v-q∈Lloc1⁢(Ω), (z-ϵ)+∈X0 for all ϵ>0 and

Csn⁢∫Q(z⁢(x)-z⁢(y))⁢(w⁢(x)-w⁢(y))|x-y|n+2⁢s⁢dx⁢dy≤∫Ωz-q⁢w⁢dx+〈F,w〉,

Csn⁢∫Q(v⁢(x)-v⁢(y))⁢(w⁢(x)-w⁢(y))|x-y|n+2⁢s⁢dx⁢dy≥∫Ωv-q⁢w⁢dx+〈F,w〉

for all compactly supported w∈X0∩L∞⁢(Ω) with w≥0. Then z≤v a.e. in Ω.

Proof.

Let us denote Φk:ℝ→ℝ the primitive of the function

s↦{max⁡{-s-q,-k},s>0,-k,s≤0,

such that Φk⁢(1)=0. We define a proper, lower semicontinuous, strictly convex functional f^0,k:L2⁢(Ω)→ℝ as follows:

f^0,k⁢(u)={Csn2⁢∥u∥2+∫ΩΦk⁢(u)⁢dxif ⁢u∈X0,+∞if ⁢u∈L2⁢(Ω)∖X0.

As we know, primitives are usually defined up to an additive constant. To prevent a possible unlikely choice we consider f0,k:L2⁢(Ω)→ℝ defined by

f0,k⁢(u)=f^0,k⁢(u)-min⁡f^0,k=f^0,k⁢(u)-f^0,k⁢(u0,k),

where u0,k∈X0 is the minimum of f^0,k. In general, for every w∈(X0)*, we define

f^w,k⁢(u)={f0,k⁢(u)-〈w,u-u0,k〉if ⁢u∈X0,+∞if ⁢u∈L2⁢(Ω)∖X0.

Let ϵ>0 and k>ϵ-q, and let u be the minimum of the functional fF,k on the convex set

K={u∈X0:0≤u≤v⁢ a.e. in ⁢Ω}.

Then, for all ψ∈K, we can get

(3.9)Csn⁢∫Q(u⁢(x)-u⁢(y))⁢((ψ-u)⁢(x)-(ψ-u)⁢(y))|x-y|n+2⁢s⁢dx⁢dy≥-∫ΩΦk′⁢(u)⁢(ψ-u)⁢dx+〈F,ψ-u〉.

In particular, if 0≤ψ∈Cc∞⁢(Ω) and t>0, we can consider the above inequality with ψt=min⁡{u+t⁢ψ,v} as the test function. Since v is a supersolution of (-Δ)s⁢u=u-q+F, using the definition of Φk, we get v as a supersolution of (-Δ)s⁢u=-Φk′⁢(u)+F. By definition, we have

u≤ψt≤v and ψt-u≤t⁢ψ.

Now using these and (3.9), we get

Csn⁢∫Q((ψt-u)⁢(x)-(ψt-u)⁢(y))2|x-y|n+2⁢s⁢dx⁢dy-∫Ω(-Φk′⁢(ψt)+Φk′⁢(u))⁢(ψt-u)⁢dx

=Csn⁢∫Q(ψt⁢(x)-ψt⁢(y))⁢((ψt-u)⁢(x)-(ψt-u)⁢(y))|x-y|n+2⁢s⁢dx⁢dy

-Csn⁢∫Q(u⁢(x)-u⁢(y))⁢((ψt-u)⁢(x)-(ψt-u)⁢(y))|x-y|n+2⁢s⁢dx⁢dy

+∫ΩΦk′⁢(ψt)⁢(ψt-u)⁢dx-∫ΩΦk′⁢(u)⁢(ψt-u)⁢dx

≤Csn⁢∫Q(ψt⁢(x)-ψt⁢(y))⁢((ψt-u)⁢(x)-(ψt-u)⁢(y))|x-y|n+2⁢s⁢dx⁢dy+∫ΩΦk′⁢(ψt)⁢(ψt-u)⁢dx-〈F,ψt-u〉

=Csn⁢∫Q(ψt⁢(x)-ψt⁢(y))⁢((ψt-u-t⁢ψ)⁢(x)-(ψt-u-t⁢ψ)⁢(y))|x-y|n+2⁢s⁢dx⁢dy+∫ΩΦk′⁢(ψt)⁢(ψt-u-t⁢ψ)⁢dx

-〈F,ψt-u-t⁢ψ〉+t⁢(Csn⁢∫Q(ψt⁢(x)-ψt⁢(y))⁢(ψ⁢(x)-ψ⁢(y))|x-y|n+2⁢s⁢dx⁢dy+∫ΩΦk′⁢(ψt)⁢ψ⁢dx-〈F,ψ〉)

≤Csn⁢∫Q(v⁢(x)-v⁢(y))⁢((ψt-u-t⁢ψ)⁢(x)-(ψt-u-t⁢ψ)⁢(y))|x-y|n+2⁢s⁢dx⁢dy+∫ΩΦk′⁢(v)⁢(ψt-u-t⁢ψ)⁢dx

-〈F,ψt-u-t⁢ψ〉+t⁢(Csn⁢∫Q(ψt⁢(x)-ψt⁢(y))⁢(ψ⁢(x)-ψ⁢(y))|x-y|n+2⁢s⁢dx⁢dy+∫ΩΦk′⁢(ψt)⁢ψ⁢dx-〈F,ψ〉)

≤t⁢(Csn⁢∫Q(ψt⁢(x)-ψt⁢(y))⁢(ψ⁢(x)-ψ⁢(y))|x-y|n+2⁢s⁢dx⁢dy+∫ΩΦk′⁢(ψt)⁢ψ⁢dx-〈F,ψ〉).

This gives

Csn∫Q(ψt⁢(x)-ψt⁢(y))⁢(ψ⁢(x)-ψ⁢(y))|x-y|n+2⁢sdxdy+∫ΩΦk′(ψt)ψdx-〈F,ψ〉≥-∫Ω|Φk′(ψt)-Φk′(u)|(ψt-u)tdx,

which implies

Csn∫Q(ψt⁢(x)-ψt⁢(y))⁢(ψ⁢(x)-ψ⁢(y))|x-y|n+2⁢sdxdy+∫ΩΦk′(ψt)ψdx-〈F,ψ〉≥-∫Ω|Φk′(ψt)-Φk′(u)|ψdx.

Since Φk′⁢(ψt)≤-v-q, using Lebesgue’s dominated convergence theorem and passing to the limit as t→0+, we get

Csn⁢∫Q(u⁢(x)-u⁢(y))⁢(ψ⁢(x)-ψ⁢(y))|x-y|n+2⁢s⁢dx⁢dy≥-∫ΩΦk′⁢(u)⁢ψ⁢dx-〈F,ψ〉.

We can now easily show that the above equation holds for all ψ∈X0 with ψ≥0 a.e. in Ω. In particular, since u≥0, we have (z-u-ϵ)+∈X0 and

(3.10)Csn⁢∫Q(u⁢(x)-u⁢(y))⁢((z-u-ϵ)+⁢(x)-(z-u-ϵ)+⁢(y))|x-y|n+2⁢s⁢dx⁢dy≥-∫ΩΦk′⁢(u)⁢(z-u-ϵ)+⁢dx-〈F,(z-u-ϵ)+〉.

Let us now consider σ∈X0 such that 0≤σ≤z a.e. in Ω. Let {σ^m} be a sequence in Cc∞⁢(Ω) converging to σ in X0 and set σm=min⁡{σ^m,σ}. Then, since z is a subsolution of (-Δ)s⁢u=u-q+F, we have

-Csn⁢∫Q(z⁢(x)-z⁢(y))⁢(σm⁢(x)-σm⁢(y))|x-y|n+2⁢s⁢dx⁢dy≥-∫Ωz-q⁢σm⁢dx-〈F,σm〉.

If z-q⁢σ∈L1⁢(Ω), then passing to the limit as m→∞, we get

-Csn⁢∫Q(z⁢(x)-z⁢(y))⁢(σ⁢(x)-σ⁢(y))|x-y|n+2⁢s⁢dx⁢dy≥-∫Ωz-q⁢σ⁢dx-〈F,σ〉.

If z-q⁢σ∉L1⁢(Ω), then the above inequality is obviously still true. In particular, we have

(3.11)-Csn⁢∫Q(z⁢(x)-z⁢(y))⁢((z-u-ϵ)+⁢(x)-(z-u-ϵ)+⁢(y))|x-y|n+2⁢s⁢dx⁢dy≥-∫Ωz-q⁢(z-u-ϵ)+⁢dx-〈F,(z-u-ϵ)+〉.

Since ϵ-q<k, using (3.1), (3.10) and (3.11), we get

Csn⁢∫Q((z-u-ϵ)+⁢(x)-(z-u-ϵ)+⁢(y))2|x-y|n+2⁢s⁢dx⁢dy

≤Csn⁢∫Q((z-u)⁢(x)-(z-u)⁢(y))⁢((z-u-ϵ)+⁢(x)-(z-u-ϵ)+⁢(y))|x-y|n+2⁢s⁢dx⁢dy

≤∫Ω(z-q+Φk′⁢(u))⁢(z-u-ϵ)+⁢dx

=∫Ω(-Φk′⁢(z)+Φk′⁢(u))⁢(z-u-ϵ)+⁢dx≤0.

Therefore, z≤u+ϵ≤v+ϵ and the assertion follows from the arbitrariness of ϵ. ∎

Lemma 3.6.

Let λ>0 and let z∈X0∩Lr⁢(Ω), r>1, be a weak solution to (Pλ) as it is defined in Definition 2.5. Then z-u¯ is a positive weak solution of (P̄λ) belonging to L∞⁢(Ω).

Proof.

Let us consider problem (3.8) with v=λ⁢z2s*-1. Then 0 is the strict subsolution of (3.8). Let

G⁢(x,s)=∫0sg⁢(x,τ)⁢dτ for ⁢(x,s)∈Ω×ℝ.

We define the corresponding functional I~:X0→(-∞,∞] by

I~⁢(u)={Csn2⁢∥u∥2+∫ΩG⁢(x,u)⁢dx-λ⁢∫Ωz2s*-1⁢u⁢dx if ⁢G⁢(x,u)∈L1⁢(Ω),∞otherwise

for every u∈X0. Also, for every u∈X0, we define the closed convex set K0={u∈X0:u≥0⁢ a.e.} and the functional I~K0 as

I~K0⁢(u)={I~⁢(u)if ⁢u∈K0⁢ and ⁢G⁢(x,u)∈L1⁢(Ω),∞otherwise.

Let {um}∈K0 be the minimizing sequence of I~K0 in K0, i.e., I~K0⁢(um)→infK0⁡I~K0⁢(u). It is easy to check that {um} is bounded in X0 and {G⁢(⋅,um)} is bounded in L1⁢(Ω). Therefore, um⇀u (up to subsequence) weakly for some u∈K0, and by Fatou’s lemma,

∫ΩG⁢(x,u)⁢dx≤lim infm→∞⁡∫ΩG⁢(x,um)⁢dx<∞.

Thus, I~K0⁢(u)=inf⁡I~K0⁢(K0). Hence, 0∈∂-⁡I~K0⁢(u), and by Proposition 4.2 we have that u is a nontrivial, nonnegative, weak solution of (3.8). Also, using Lemma 3.4, we have (u+u¯-ϵ)+∈X0 for every ϵ>0. It can be shown that

Csn⁢∫Q((u+u¯)⁢(x)-(u+u¯)⁢(y))⁢(w⁢(x)-w⁢(y))|x-y|n+2⁢s⁢dx⁢dy-∫Ω((u+u¯)-q-λ⁢z2s*-1)⁢w⁢dx=0

and

Csn⁢∫Q(z⁢(x)-z⁢(y))⁢(w⁢(x)-w⁢(y))|x-y|n+2⁢s⁢dx⁢dy-∫Ω((u+u¯)-q-λ⁢z2s*-1)⁢w⁢dx=0

for w∈X0∩L∞⁢(Ω) with compact support in Ω. Then, using Lemma 3.5, we get z=u+u¯, which implies that u=z-u¯ is a positive weak solution of (P̄λ). Thus, by Lemma 3.3, u∈L∞⁢(Ω). ∎

4 Existence and multiplicity of positive solutions for (Pλ)

4.1 First solution

In this section, we prove the existence of a solution for problem (Pλ). We set the variational framework to problem (P̄λ) in the space X0. For this, recalling that G⁢(x,s)=∫0sg⁢(x,τ)⁢dτ for (x,s)∈Ω×ℝ, we define the functional I:X0→(-∞,∞], corresponding to (P̄λ), by

I⁢(u)={Csn2∫Q|u⁢(x)-u⁢(y)|2|x-y|n+2⁢sdxdy+∫ΩG(x,u)dx-λ2s*∫Ω|u+u¯|2s*dxif ⁢G⁢(⋅,u)∈L1⁢(Ω),∞otherwise.

For a convex subset K⊂X0, we also define the restricted functional IK:X0→(-∞,∞] by

IK⁢(u)={I⁢(u)if ⁢u∈K⁢ and ⁢G⁢(⋅,u)∈L1⁢(Ω),∞otherwise.

We note that u∈D⁢(IK) if and only if u∈K and G⁢(⋅,u)∈L1⁢(Ω). We now state a lemma which characterizes the set ∂-⁡IK⁢(u).

Lemma 4.1.

Let K be a convex subset of X0 and let α∈X0. Let also u∈K with G⁢(⋅,u)∈L1⁢(Ω). Then the following two statements are equivalent:

α∈∂-⁡IK⁢(u).
For every v∈K with G⁢(⋅,v)∈L1⁢(Ω), we have g⁢(⋅,u)⁢(v-u)∈L1⁢(Ω) and
Csn⁢∫Q(u⁢(x)-u⁢(y))⁢((v-u)⁢(x)-(v-u)⁢(y))|x-y|n+2⁢s⁢dx⁢dy+∫Ωg⁢(x,u)⁢(v-u)⁢dx-λ⁢∫Ω(u+u¯)2s*-1⁢(v-u)⁢dx
(4.1)≥〈α,v-u〉.
Moreover, as G⁢(⋅,u) is convex, the last statement implies
Csn⁢∫Q(u⁢(x)-u⁢(y))⁢((v-u)⁢(x)-(v-u)⁢(y))|x-y|n+2⁢s⁢dx⁢dy+∫Ω(G⁢(x,v)-G⁢(x,u))⁢dx-λ⁢∫Ω(u+u¯)2s*-1⁢(v-u)⁢dx
≥〈α,v-u〉.

Proof.

We follow the proof of [31, Lemma 3].

(i) ⇒ (ii) Let v∈K and G⁢(⋅,v)∈L1⁢(Ω), and set w=v-u. Then g⁢(⋅,u)⁢w is measurable and we have G⁢(⋅,u)-G⁢(⋅,v)∈L1⁢(Ω). Since g⁢(x,s) is non decreasing in s, we have g⁢(x,u)⁢w≤G⁢(x,v)-G⁢(x,u), which implies (g⁢(⋅,u)⁢w)∨0∈L1⁢(Ω). The function t↦(G⁢(x,u+t⁢w)-G⁢(x,u))/t, (0,1]→ℝ, is increasing and

IK⁢(u+t⁢w)-IK⁢(u)t=Csn⁢∫Q(u⁢(x)-u⁢(y))⁢(w⁢(x)-w⁢(y))|x-y|n+2⁢s⁢dx⁢dy+t⁢Csn2⁢∥w∥2+∫Ω(G⁢(x,u+t⁢w)-G⁢(x,u))t⁢dx

(4.2)-12s*⁢∫Ω(|u+u¯+t⁢w|2s*-|u+u¯|2s*)t⁢dx.

Letting t→0 on both sides of (4.2) and using the monotone convergence theorem, we get

limt→0⁡IK⁢(u+t⁢w)-IK⁢(u)t=Csn⁢∫Q(u⁢(x)-u⁢(y))⁢((v-u)⁢(x)-(v-u)⁢(y))|x-y|n+2⁢s⁢dx⁢dy

(4.3)+∫Ωg⁢(x,u)⁢(v-u)⁢dx-λ⁢∫Ωu2s*-1⁢(v-u)⁢dx.

Also, α∈∂-⁡IK⁢(u) implies

limt→0⁡IK⁢(u+t⁢w)-IK⁢(u)t≥〈α,v-u〉.

Hence, we get (4.1) from (4.3). From (4.1), we have (g⁢(⋅,u)⁢w)∧0∈L1⁢(Ω), and hence (g⁢(⋅,u)⁢w)∈L1⁢(Ω).

(ii) ⇒ (i) Let v∈K and G⁢(⋅,v)∈L1⁢(Ω). Since G⁢(x,s) is convex in s, (ii) implies

IK⁢(v)-IK⁢(u)=Csn2⁢∥(v-u)∥2+Csn⁢∫Q(u⁢(x)-u⁢(y))⁢((v-u)⁢(x)-(v-u)⁢(y))|x-y|n+2⁢s⁢dx⁢dy

+∫Ω(G⁢(x,v)-G⁢(x,u))⁢dx-λ2s*⁢∫Ω(|v+u¯|2s*-|u+u¯|2s*)⁢dx

≥Csn2⁢∥(v-u)∥2+∫Ω(G⁢(x,v)-G⁢(x,u)-g⁢(x,u)⁢(v-u))⁢dx

-λ⁢∫Ω(|v+u¯|2s*-|u+u¯|2s*2s*-|u+u¯|2s*-1⁢(v-u))⁢dx+〈α,v-u〉

≥Csn2⁢∥(v-u)∥2-λ⁢∫Ω(|v+u¯|2s*-|u+u¯|2s*2s*-|u+u¯|2s*-1⁢(v-u))⁢dx+〈α,v-u〉,

which implies α∈∂-⁡IK⁢(u). ∎

For φ,ψ:Ω→[-∞,+∞], we define

Kφ={u∈X0:φ≤u⁢ a.e.},Kψ={u∈X0:u≤ψ⁢ a.e.} and Kφψ={u∈X0:φ≤u≤ψ⁢ a.e.}.

We state the following proposition which can be thought of as Perron’s method for non-smooth functionals.

Proposition 4.2.

Assume one of the following conditions:

φ1 is a subsolution of ((P̄λ)), G⁢(x,v⁢(x))∈Lloc1⁢(Ω) for all v∈Kφ1, u∈D⁢(IKφ1) and 0∈∂-⁡IKφ1⁢(u).
φ2 is a supersolution of ((P̄λ)), G⁢(x,v⁢(x))∈Lloc1⁢(Ω) for all v∈Kφ2, u∈D⁢(IKφ2) and 0∈∂-⁡IKφ2⁢(u).
φ1 and φ2 are subsolution and supersolution of ((P̄λ)), G⁢(x,φ1⁢(x)),G⁢(x,φ2⁢(x))∈Lloc1⁢(Ω), u∈D⁢(IKφ1φ2) and 0∈∂-⁡IKφ1φ2⁢(u).

Then u is a weak solution of (P̄λ).

Proof.

We follow the proof of [31, Proposition 2]. We have that G⁢(⋅,φ1) and g⁢(⋅,φ1) are measurable and G⁢(x,φ1⁢(x)),G⁢(x,u⁢(x))∈ℝ for a.e. x∈Ω, since G⁢(⋅,u),G⁢(⋅,φ1)∈Lloc1⁢(Ω). So, g⁢(⋅,u) is measurable by [31, Lemma 2 (ii)]. Since

g⁢(x,ϕ1)⁢ψ0≤g⁢(x,u)⁢ψ0≤G⁢(x,u+ψ0)-G⁢(x,u)

for each ψ0∈Cc∞⁢(Ω), we get g⁢(⋅,u)⁢ψ0∈L1⁢(Ω). The arbitrariness of ψ0 implies that g⁢(⋅,u)∈Lloc1⁢(Ω). Let ψ∈Cc∞⁢(Ω) and set vt=(u+t⁢ψ)∨φ1 for 0<t≤1. Then G⁢(⋅,vt)∈Lloc1⁢(Ω) and G⁢(x,vt)=G⁢(x,u) on Ω∖supp⁡ψ, which implies vt∈D⁢(IKφ1). Setting rt=(φ1-(u+t⁢ψ))+, we get vt-u=t⁢ψ+rt. Clearly, rt has a compact support and |rt⁢(x)|≤t⁢|ψ⁢(x)| for each x∈Ω. Using Lemma 4.1, we get g⁢(⋅,u)⁢(vt-u)∈L1⁢(Ω) and

0≤Csn⁢∫Q(u⁢(x)-u⁢(y))⁢((vt-u)⁢(x)-(vt-u)⁢(y))|x-y|n+2⁢s⁢dx⁢dy+∫Ω(g⁢(x,u)-λ⁢(u+u¯)2s*-1)⁢(vt-u)⁢dx

≤t⁢Csn⁢∫Q(u⁢(x)-u⁢(y))⁢(ψ⁢(x)-ψ⁢(y))|x-y|n+2⁢s⁢dx⁢dy+t⁢∫Ω(g⁢(x,u)-λ⁢(u+u¯)2s*-1)⁢ψ⁢dx

(4.4)+Csn⁢∫Q(u⁢(x)-u⁢(y))⁢(rt⁢(x)-rt⁢(y))|x-y|n+2⁢s⁢dx⁢dy+∫Ω(g⁢(x,u)-λ⁢(u+u¯)2s*-1)⁢rt⁢dx.

Fix t∈(0,1] and let {wk} be a non-negative sequence of functions in Cc∞⁢(Ω) such that ⋃ksupp⁡wk is contained in a compact subset of Ω, {∥wk∥∞} is bounded and ∥wk-rt∥→0 as k→∞.

Using the fact that φ1 is a subsolution of (P̄λ), for each k we get

Csn⁢∫Q(φ1⁢(x)-φ1⁢(y))⁢(wk⁢(x)-wk⁢(y))|x-y|n+2⁢s⁢dx⁢dy+∫Ω(g⁢(x,φ1)-λ⁢(φ1+u¯)2s*-1)⁢wk⁢dx≤0.

Taking the limit as k→∞ and using Lebesgue’s dominated convergence theorem, we obtain

(4.5)Csn⁢∫Q(φ1⁢(x)-φ1⁢(y))⁢(rt⁢(x)-rt⁢(y))|x-y|n+2⁢s⁢dx⁢dy+∫Ω(g⁢(x,φ1)-λ⁢(φ1+u¯)2s*-1)⁢rt⁢dx≤0.

From (4.4), (4.5) and since -rt-t⁢ψ≤u-ϕ1 in Ω, we get

0≤t⁢(Csn⁢∫Q(u⁢(x)-u⁢(y))⁢(ψ⁢(x)-ψ⁢(y))|x-y|n+2⁢s⁢dx⁢dy+∫Ω(g⁢(x,u)-λ⁢(u+u¯)2s*-1)⁢ψ⁢dx)

-Csn⁢∥rt∥2-t⁢Csn⁢∫Q(ψ⁢(x)-ψ⁢(y))⁢(rt⁢(x)-rt⁢(y))|x-y|n+2⁢s⁢dx⁢dy

+∫Ω((g⁢(x,u)-g⁢(x,φ1))⁢rt-λ⁢((u+u¯)2s*-1-(φ1+u¯)2s*-1)⁢rt)⁢dx,

which gives

0≤Csn⁢∫Q(u⁢(x)-u⁢(y))⁢(ψ⁢(x)-ψ⁢(y))|x-y|n+2⁢s⁢dx⁢dy+∫Ω(g⁢(x,u)-λ⁢(u+u¯)2s*-1)⁢ψ⁢dx

-Csn⁢∫Q(ψ⁢(x)-ψ⁢(y))⁢(rt⁢(x)-rt⁢(y))|x-y|n+2⁢s⁢dx⁢dy

+∫Ω((g⁢(x,u)-g⁢(x,φ1))⁢rtt-λ⁢((u+u¯)2s*-1-(φ1+u¯)2s*-1)⁢rtt)⁢dx.

Using the inequality |rt⁢(x)|≤t⁢|ψ⁢(x)| for each x∈Ω and 0<t≤1, the limits ∥rt∥→0 as t→0+,

(g⁢(x,u)-g⁢(x,φ1))⁢rtt→0 and λ⁢((u+u¯)2s*-1-(φ1+u¯)2s*-1)⁢rtt→0 a.e. as t→0+,

and the fact that supp⁡ψ is compact and g⁢(⋅,u),g⁢(⋅,φ1)∈L1⁢(Ω), we get

0≤Csn⁢∫Q(u⁢(x)-u⁢(y))⁢(ψ⁢(x)-ψ⁢(y))|x-y|n+2⁢s⁢dx⁢dy+∫Ω(g⁢(x,u)-λ⁢(u+u¯)2s*-1)⁢ψ⁢dx.

Since ψ∈Cc∞⁢(Ω) is arbitrary, u is a weak solution of (P̄λ). The proofs of (ii) and (iii) are similar to those of [31, Proposition 2 (ii) and (iii)]. ∎

Let θ¯∈X0 be the function which satisfies (-Δ)s⁢θ¯=1/2 in Ω in the sense of distributions. From [41, Proposition 1.1], θ¯∈Cs⁢(ℝn). For g and G, we have the following properties.

Proposition 4.3.

Let u∈Lloc1⁢(Ω), satisfying Definition 2.5 (i). Then g⁢(x,u⁢(x)),G⁢(x,u⁢(x))∈Lloc1⁢(Ω).

Proof.

Recall that infK⁡u¯>0 for any K⋐Ω. We have 0≤g⁢(x,u⁢(x))≤u¯-q and 0≤G⁢(⋅,u)≤u¯-q⁢u in Ω. Hence,

∫Kg(x,u(x))dx≤∫K|u¯(x)|-qdx<∞ and ∫K|G(x,u(x))|dx≤(infKu¯)-δ∫K|u|dx<∞.∎

Lemma 4.4.

For each x∈Ω, the following hold:

G⁢(x,r⁢t)≤t2⁢G⁢(x,t) for each r≥1 and t≥0,
G⁢(x,r)-G⁢(x,t)-(g⁢(x,r)+g⁢(x,t))⁢(r-t)/2≥0 for each r, t with r≥t>-θ¯⁢(x),
G⁢(x,r)-g⁢(x,r)⁢r/2≥0 for each r≥0.

Proof.

For a proof we refer to [31, Lemma 4]. ∎

We now proceed to prove some results to obtain the existence of a solution of (P̄λ).

Lemma 4.5.

The following hold:

0 is a strict subsolution of ((P̄λ)) for all λ>0.
θ¯ is a strict supersolution of ((P̄λ)) for all sufficiently small λ>0.
Any positive weak solution z of (P¯μ) is a strict super-solution of ((P̄λ)) for μ>λ>0.

Proof.

(i) Let ψ∈X0∖{0}, ψ≥0. Since g⁢(x,0)=0, we get

Csn∫Q(0⁢(x)-0⁢(y))⁢(ψ⁢(x)-ψ⁢(y))|x-y|n+2⁢sdxdy+∫Ω(g(x,0)ψ-λ(0+u¯)2s*-1ψ)dx=-λ∫Ω|u¯|2s*-1ψdx<0.

(ii) We choose λ>0 such that 1-λ⁢(θ¯+u¯)>0 in Ω. We have g⁢(x,θ¯)∈Lloc1⁢(Ω) and g is nonnegative. So,

Csn⁢∫Q(θ¯⁢(x)-θ¯⁢(y))⁢(ψ⁢(x)-ψ⁢(y))|x-y|n+2⁢s⁢dx⁢dy+∫Ω(g⁢(x,θ¯)⁢ψ-λ⁢(θ¯+u¯)2s*-1⁢ψ)⁢dx≥∫Ω(1-λ⁢|u¯|2s*-1)⁢ψ⁢dx>0.

(iii) Let λ>0 and let z be a positive weak solution of (P¯μ) for some μ>λ. We have g⁢(⋅,z)∈Lloc1⁢(Ω) and g is nonnegative. So,

Csn∫Q(z⁢(x)-z⁢(y))⁢(ψ⁢(x)-ψ⁢(y))|x-y|n+2⁢sdxdy+∫Ω(g(x,z)ψ-λ(z+u¯)2s*-1ψ)dx=(μ-λ)∫Ω|z+u¯|2s*-1ψdx>0,

which gives (iii). ∎

Let

Λ:=sup⁡{λ>0:(P¯λ) admits a weak solution}.

Remark 4.6.

If Λ>0, by Lemma 4.5, we can say that for any λ∈(0,Λ), (P̄λ) has a subsolution (the trivial function 0) and a positive strict supersolution (say z).

Theorem 4.7.

Let φ1,φ2:Ω→[-∞,∞] with φ1≤φ2 such that φ1 is a strict supersolution of (P̄λ). Let also u∈D⁢(IKφ1φ2) be a minimizer for IKφ1φ2. Then u is a local minimizer for IKφ1.

Proof.

For any v∈Kφ1, define

σ⁢(v)=min⁡{v,φ2}=v-(v-φ2)+,

and for any 0≤w∈X0, define

H⁢(w)=Csn⁢∫Q(φ2⁢(x)-φ2⁢(y))⁢(w⁢(x)-w⁢(y))|x-y|n+2⁢s⁢dx⁢dy+∫Ωg⁢(x,φ2)⁢w⁢dx-λ⁢∫Ω(u¯+φ2)2s*-1⁢w⁢dx.

First we see that, there exists 0≤θ≤1 such that

(u¯+u)2s*-1-(u¯+v)2s*-1(u-v)=(2s*-1)⁢((u¯+u)+θ⁢(v-u))2s*-2

≤22s*-3⁢(2s*-1)⁢[u¯2s*-2+((1-θ)⁢u+θ⁢v)2s*-2]

(4.6)≤c1u¯2s*-2+c2max{|u|,|v|}2s*-2,

where c1,c2 are positive constants. For x∈Ω, let us set

mv(x)=(c1u¯2s*-2+c2max{|φ2(x)|,|v(x)|}2s*-2)1{v>φ2}.

We know that G⁢(⋅,σ⁢(v)⁢(⋅)) and g⁢(⋅,σ⁢(v)⁢(⋅))⁢(v⁢(⋅)-σ⁢(v)⁢(⋅)) are measurable by [31, Lemma 2 (i) and (iii)]. Using the fact that σ⁢(v)∈Kφ1φ2, the inequality σ⁢(v)≤v, the convexity of G⁢(x,⋅) and (4.6), we get

IKφ1⁢(v)-IKφ1⁢(u)≥IKφ1⁢(v)-IKφ1⁢(σ⁢(v))

=Csn2⁢∫Q|(v-σ⁢(v))⁢(x)-(v-σ⁢(v))⁢(y)|2|x-y|n+2⁢s⁢dx⁢dy

+Csn⁢∫Q(σ⁢(v)⁢(x)-σ⁢(v)⁢(y))⁢((v-σ⁢(v))⁢(x)-(v-σ⁢(v))⁢(y))|x-y|n+2⁢s⁢dx⁢dy

+∫Ω(G⁢(x,v)-G⁢(x,σ⁢(v)))⁢dx-λ2s*⁢∫Ω((u¯+v)2s*-(u¯+σ⁢(v))2s*)⁢dx

≥Csn2⁢∥v-σ⁢(v)∥2+Csn⁢∫Q(σ⁢(v)⁢(x)-σ⁢(v)⁢(y))⁢((v-σ⁢(v))⁢(x)-(v-σ⁢(v))⁢(y))|x-y|n+2⁢s⁢dx⁢dy

+∫Ωg⁢(x,σ⁢(v))⁢(v-σ⁢(v))⁢dx-λ⁢∫Ω(u¯+σ⁢(v))2s*-1⁢(v-σ⁢(v))⁢dx

-λ2s*⁢∫Ω((u¯+v)2s*-(u¯+σ⁢(v))2s*-2s*⁢(u¯+σ⁢(v))2s*-1⁢(v-σ⁢(v)))⁢dx

=Csn2⁢∥v-σ⁢(v)∥2+Csn⁢∫Q(σ⁢(v)⁢(x)-σ⁢(v)⁢(y))⁢((v-σ⁢(v))⁢(x)-(v-σ⁢(v))⁢(y))|x-y|n+2⁢s⁢dx⁢dy

+∫Ωg⁢(x,σ⁢(v))⁢(v-σ⁢(v))⁢dx-λ⁢∫Ω(u¯+σ⁢(v))2s*-1⁢(v-σ⁢(v))⁢dx

-λ⁢∫Ω∫σ⁢(v)v((u¯+t)2s*-1-(u¯+σ⁢(v))2s*-1)⁢dt⁢dx

(4.7)≥Csn2⁢∥v-σ⁢(v)∥2+H⁢(v-σ⁢(v))-12⁢∫Ωmv⁢(x)⁢(v-σ⁢(v))2⁢dx.

This implies, for any v∈D⁢(IKφ1), that

IKφ1⁢(v)≥IKφ1⁢(u)+Csn2⁢∥v-σ⁢(v)∥2+H⁢((v-φ2)+)-12⁢|mv|2s*/(2s*-2)⁢|(v-φ2)+|2s*2.

Suppose the conclusion of the above theorem does not hold under the considered assumptions. In this case, we can choose a sequence {vk}⊂X0 such that vk∈Kφ1 and

∥vk-u∥≤12k,IKφ1⁢(vk)<IKφ1⁢(u) for all k.

We set l=u+∑k=1∞|vk-u|, which satisfies |vk|≤l a.e. for all k. Also we set

m^v(x)=(c1u¯2s*-2+c2max{|φ2(x)|,|l(x)|}2s*-2)1{v>φ2} for every v∈D⁢(IKφ1).

Then we have

0>IKφ1⁢(vk)-IKφ1⁢(u)

≥IKφ1⁢(vk)-IKφ1⁢(σ⁢(vk))

≥Csn2⁢∥(vk-φ2)+∥2+H⁢((vk-φ2)+)-12⁢∫Ωm^vk⁢(x)⁢((v-φ2)+)2⁢dx

=Csn2⁢∥(vk-φ2)+∥2+H⁢((vk-φ2)+)-12⁢∫{m^vk≤R/Csn}m^vk⁢(x)⁢((v-φ2)+)2⁢dx

-12⁢∫{m^vk>R/Csn}m^vk⁢(x)⁢((v-φ2)+)2⁢dx

≥Csn2∥(vk-φ2)+∥2+H((vk-φ2)+)-R⁢Csn2∫Ω|(v-φ2)+|2dx

-12⁢Ss(∫{m^vk>R/Csn}|m^vk(x)|2s*/(2s*-2)dx)(2s*-2)/2s*∥(vk-φ2)+∥2

for all R>0 and k. As we can choose R>0 such that

12⁢Ss(∫{m^vk>RCsn}|m^vk(x)|2s*/(2s*-2)dx)(2s*-2)/2s*<Csn4 for all k,

we get

(4.8)0>H⁢((vk-φ2)+)+Csn4⁢∥(vk-φ2)+∥2-R⁢Csn2⁢|(vk-φ2)+|22 for all ⁢k.

Let

ν=inf⁡{H⁢(w):w∈A},

where

A={w∈X0:w≥0,|w|2=1,∥w∥≤2⁢R}.

Clearly, A is weakly compact and using Fatou’s lemma, we can show that H is weakly lower semicontinuous on A. So if {wk}⊂A be a minimizing sequence for ν such that wk⇀w weakly as k→∞, then

H⁢(w)≤lim⁡inf⁡H⁢(wk).

Since φ2 is a strict supersolution of (P̄λ), H⁢(w)>0 for all w∈A. This implies ν>0. Since vk→u in X0, there exists k0 such that |(vk0-φ2)+|2≤ν/(R⁢Csn). We consider two cases. If ∥(vk0-φ2)+∥2≥4⁢R⁢|(vk0-φ2)+|22, then from (4.8) we get

0>Csn4⁢∥(vk0-φ2)+∥2-Csn8⁢∥(vk0-φ2)+∥2=Csn8⁢∥(vk0-φ2)+∥2,

which is a contradiction. On the other hand, if ∥(vk0-φ2)+∥2≤4⁢R⁢|(vk0-φ2)+|22, then from (4.8) we get

0>(ν-R⁢Csn2⁢|(vk0-φ2)+|2)⁢|(vk0-φ2)+|2≥ν4⁢R⁢∥(vk0-φ2)+∥,

which is again a contradiction. ∎

Theorem 4.8.

Suppose Λ>0. Let λ∈(0,Λ) and 0,z be the sub and super solutions of (P̄λ), respectively, as in Remark 4.6. Let also K={ϕ∈H01⁢(Ω):0≤ϕ≤z}. Then there exists a weak solution uλ of (P̄λ) with uλ∈K and IK⁢(uλ)=infK⁡IK<0. Furthermore, uλ is a local minimizer for IK0.

Proof.

We have infK⁡IK<0 since IK⁢(0)<0. Let {um} be a minimizing sequence for infK⁡IK in K. Then 0≤um≤z for all m, that is, {um} is bounded in X0. So, there exist uλ∈K such that um⇀uλ weakly in X0 as m→∞. We have that the map v↦∫ΩG⁢(x,v)⁢dx is weakly sequentially lower semicontinuous and, by Lebesgue’s dominated convergence theorem,

limm→∞∫Ω|um+u¯|2s*dx=∫Ω|uλ+u¯|2s*dx.

Thus, IK⁢(uλ)≤lim⁡infm→∞⁡IK⁢(um), which implies IK⁢(uλ)=infK⁡IK. Hence, IK⁢(uλ)<0 and 0∈∂-⁡IK⁢(uλ). Thus, uλ is a weak solution of (P̄λ), by Proposition 4.2. Finally, using Theorem 4.7 with φ1=0 and φ2=z, we conclude that uλ is a local minimizer for IK0. ∎

Lemma 4.9.

We have 0<Λ<∞.

Proof.

First, we prove that Λ>0. From Lemma 4.5, we get 0 as a strict subsolution and θ¯ as a strict supersolution of (P̄λ) for sufficiently small λ>0. We define the convex set K:={ϕ∈X0⁢(Ω):0≤ϕ≤θ¯}. Then, arguing as in the proof of Theorem 4.8, we get that there exist u∈D⁢(IK) such that IK⁢(u)=infK⁡IK. In particular, 0∈∂-⁡IK⁢(u). Thus, u is a weak solution of (P̄λ) for sufficiently small λ>0, by Proposition 4.2. Thus, Λ>0.

Next, we prove that Λ<+∞. Suppose on the contrary that Λ=+∞. So, there exists an increasing sequence {λm}∈ℝ such that λm→+∞, and (P¯λm) admits a weak solution, say uλm as given in Theorem 4.8. Consequently,

(4.9)Csn2∥uλm∥2+∫ΩG(x,uλm)dx-λm2s*∫Ω|uλm+u¯|2s*dx<0.

Also, by the definition of a weak solution, we get

(4.10)Csn∥uλm∥2+∫Ωg(x,uλm)uλmdx-λm∫Ω|uλm+u¯|2s*-1uλmdx=0.

From (4.9) and (4.10), we obtain

∫Ω(G⁢(x,uλm)-12⁢g⁢(x,uλm)⁢uλn)⁢dx+λm⁢∫Ω(12⁢|uλm+u¯|2s*-1⁢uλm-|uλm+u¯|2s*2s*)⁢dx<0.

By Lemma 4.4 (iii) we have G⁢(x,uλm)-g⁢(x,uλm)/2≥0, which implies

(4.11)∫Ω12⁢|uλm+u¯|2s*-1⁢uλm⁢dx<∫Ω|uλm+u¯|2s*2s*⁢dx.

Next since u¯∈L∞⁢(Ω), we note that

limt→∞⁡|t+u¯⁢(x)|2s*|t+u¯⁢(x)|2s*-1⁢t=1

uniformly with respect to x∈Ω. Thus, for any ϵ>0 small enough, there exists M=Mϵ>0 such that

(4.12)12s*∫Ω|uλm+u¯|2s*dx<12+ϵ∫Ω|uλm+u¯|2s*-1uλmdx+M for all m.

From (4.11) and (4.12), we get

supm∫Ω|uλm+u¯|2s*-1uλmdx<∞.

Using (4.10), for each m, we obtain

Csn⁢∥uλm∥2≤λm⁢∫Ω|uλm+u¯|2s*-1⁢uλm⁢dx,

which implies that the sequence {λm-1/2⁢uλm} is bounded in X0. Set vλm:=λm-1/2⁢uλm. Then, up to a subsequence, there exists v∈X0 such that vλm⇀v weakly in X0 as m→∞. Let ψ∈C0∞⁢(Ω) be a non-trivial and non-negative function. Choose m>0 such that u¯≥m on the support of ψ. Then

Csn⁢∫Q(vλm⁢(x)-vλm⁢(y))⁢(ψ⁢(x)-ψ⁢(y))|x-y|n+2⁢s⁢dx⁢dy+∫Ω1mδ⁢λn⁢ψ⁢dx

≥Csn⁢∫Q(vλm⁢(x)-vλm⁢(y))⁢(ψ⁢(x)-ψ⁢(y))|x-y|n+2⁢s⁢dx⁢dy+∫Ωg⁢(x,uλm)λn⁢ψ⁢dx

=λm2s*-1∫Ω|uλm+u¯|2s*-1ψdx

(4.13)≥λm∫Ω|m+u¯|2s*-1ψdx.

Since λm→∞ as m→∞, letting m→∞ in (4.13), we get

Csn⁢∫Q(vλm⁢(x)-vλm⁢(y))⁢(ψ⁢(x)-ψ⁢(y))|x-y|n+2⁢s⁢dx⁢dy=∞,

which is a contradiction. Hence, Λ<∞. ∎

Theorem 4.10.

There exists a positive weak solution of (P¯Λ).

Proof.

Let λm↑Λ and {uλm} be a sequence of positive weak solutions to (P¯λm) such that uλm≤uλm+1 for all m∈ℕ. Then, as in the proof of Lemma 4.9, we have that {uλm} is uniformly bounded in X0. Therefore, up to a subsequence, there exists uΛ∈X0 such that uλm⇀uΛ weakly in X0 as m→∞. Now for any ϕ∈C0∞⁢(Ω) with ϕ≥0, using the monotone convergence theorem, as m→∞, we have

∫Ωg(x,uλn)ϕdx→∫Ωg(x,uΛ)ϕdx and ∫Ω|uλm+u¯|2s*-1ϕdx→∫Ω|uΛ+u¯|2s*-1ϕdx.

Thus,

Csn∫Q(uΛ⁢(x)-uΛ⁢(y))⁢(ϕ⁢(x)-ϕ⁢(y))|x-y|n+2⁢sdxdy+∫Ωg(x,uΛ)ϕdx-Λ∫Ω|uΛ+u¯|2s*-1ϕdx=0.

Now for any ϕ∈C0∞⁢(Ω), taking ϕ=ϕ+-ϕ- and arguing as above, it is easy to check that uΛ is a positive weak solution of (P¯Λ). ∎

4.2 Second solution

Now, we show the existence of at least two distinct positive weak solutions for (P̄λ) with λ∈(0,Λ) . We fix λ∈(0,Λ), and we denote by u the positive weak solution obtained in Theorem 4.8.

Proposition 4.11.

The functional IKu satisfies (CPS)c for each c satisfying

c<IKu⁢(u)+s⁢(Csn⁢Ss)n/(2⁢s)n⁢λ(n-2⁢s)/(2⁢s).

Proof.

Let

c<IKu⁢(u)+Ssn/(2⁢s)n⁢λ(n-2)/(2⁢s)

be fixed and choose a sequence {wm}∈D⁢(IKu) such that

IKu⁢(wm)→c and (1+∥wm∥)⁢|||∂-⁡IKu⁢(wm)|||→0 as m→∞.

There exists βm∈∂-⁡IKu⁢(wm) such that ∥βm∥=|||∂-⁡IKu⁢(wm)||| for each m∈ℕ. Using Lemma 4.1, for every m∈ℕ and v∈D⁢(IKu), we have g⁢(⋅,wm)⁢(v-wm)∈L1⁢(Ω) and

〈βm,v-wm〉≤Csn⁢∫Q(wm⁢(x)-wm⁢(y))⁢((v-wm)⁢(x)-(v-wm)⁢(y))|x-y|n+2⁢s⁢dx⁢dy

(4.14)+∫Ωg⁢(x,wm)⁢(v-wm)⁢dx-λ⁢∫Ω(wm+u¯)2s*-1⁢(v-wm)⁢dx.

By Lemma 4.4 (ii) and since G⁢(⋅,wm)∈L1⁢(Ω), we get G⁢(⋅,2⁢wm)∈L1⁢(Ω), which implies 2⁢wm∈D⁢(IKu) for each m. Substituting v=2⁢wm in (4.14), we get

〈βm,wm〉≤Csn⁢∥wm∥2+∫Ωg⁢(x,wm)⁢wm⁢dx-λ⁢∫Ω(wm+u¯)2s*-1⁢wm⁢dx.

Assuming IKu⁢(wm)≤c+1 for all m and using (4.12), we have

c+1≥Csn2⁢∥wm∥2+∫ΩG⁢(x,wm)⁢dx-λ2s*⁢∫Ω(wm+u¯)2s*⁢dx

≥Csn2⁢∥wm∥2+∫ΩG⁢(x,wm)⁢dx+12+ϵ⁢(〈βm,wm〉-Csn⁢∥wm∥2-∫Ωg⁢(x,wm)⁢wm⁢dx)-λ⁢Mϵ

for ϵ>0 small enough. Using Lemma 4.4 (iii), it can be shown that {wm} is bounded in X0. Thus, up to a subsequence, there exist w∈X0 such that wm⇀w weakly (and almost everywhere) in X0 as m→∞. We assume, again up to a subsequence, that as m→∞,

∥wm-w∥2→a2 and ∫Ω|wm-w|2s*dx→b2s*.

Also, we have

∫ΩG⁢(x,w)⁢dx≥∫ΩG⁢(x,wm)⁢dx+∫Ωg⁢(x,wm)⁢(w-wm)⁢dx

≥∫ΩG⁢(x,wm)⁢dx-λ⁢∫Ω(wm+u¯)2s*-1⁢(wm-w)⁢dx-〈βm,wm-w〉

+Csn⁢∫Q(wm⁢(x)-wm⁢(y))⁢((wm-w)⁢(x)-(wm-w)⁢(y))|x-y|n+2⁢s⁢dx⁢dy,

which gives

∫ΩG⁢(x,w)⁢dx≥∫ΩG⁢(x,w)⁢dx+Csn⁢a2-λ⁢b2s*.

This in turn yields λ⁢b2s*≥Csn⁢a2. Since u is a positive weak solution, we have

(4.15)Csn⁢∫Q(u⁢(x)-u⁢(y))⁢((wm-u)⁢(x)-(wm-u)⁢(y))|x-y|n+2⁢s⁢dx⁢dy+∫Ω(g⁢(x,u)-λ⁢(u+u¯)2s*-1)⁢(wm-u)⁢dx=0.

Since G⁢(⋅,wm),G⁢(⋅,2⁢wm)∈L1⁢(Ω) and u≤2⁢wm-u≤2⁢wm, we have that 2⁢wm-u∈D⁢(IKu). Substituting v=2⁢wm-u in (4.14), we get

〈βm,wm-u〉≤Csn⁢∫Q(wm⁢(x)-wm⁢(y))⁢((wm-u)⁢(x)-(wm-u)⁢(y))|x-y|n+2⁢s⁢dx⁢dy

(4.16)+∫Ω(g⁢(x,wm)-λ⁢(wm+u¯)2s*-1)⁢(wm-u)⁢dx.

By (4.15), (4.16) and Lemma 4.4 (ii), we get

IKu(wm)-IKu(u)=Csn2∥wm∥2+∫ΩG(x,wm)dx-λ2s*∫Ω|wm+u¯|2s*dx

-(Csn2∥u∥2+∫ΩG(x,u)dx-λ2s*∫Ω|u+u¯|2s*dx)

≥∫Ω(G⁢(x,wm)-G⁢(x,u)-12⁢g⁢(x,wm)⁢(wm-u)-12⁢g⁢(x,u)⁢(wm-u))⁢dx

+λ∫Ω(12|wm+u¯|2s*-1(wm-u)-12s*|wm+u¯|2s*

+12|u+u¯|2s*-1(wm-u)+12s*|u+u¯|2s*)dx+12〈βm,wm-u〉

≥λ∫Ω(12|wm+u¯|2s*-1(wm-u)-12s*|wm+u¯|2s*

+12|u+u¯|2s*-1(wm-u)+12s*|u+u¯|2s*)dx+12〈βm,wm-u〉.

Since the map t↦|t+u¯|2s*-1 is convex, using the Brezis–Lieb lemma (see [7]) and letting m→∞ on both sides, we get

c-IKu⁢(u)≥λ⁢s⁢b2s*n+λ⁢∫Ω(12⁢|w+u¯|2s*-1⁢(w-u)-12s*⁢|w+u¯|2s*⁢12⁢|u+u¯|2s*-1⁢(w-u)+12s*⁢|u+u¯|2s*)⁢dx

≥λ⁢s⁢b2s*n+λ∫Ω(12|w+u¯|2s*-1(w-u)+12|u+u¯|2s*-1(w-u)∫uw|t+u¯|2s*-1dt)dx

≥λ⁢s⁢b2s*n.

Suppose a>0. Then λ⁢b2s*≥Csn⁢a2 and a2≥Ss⁢b2 together imply

λ⁢s⁢b2s*n≥s⁢(Csn⁢Ss)n/(2⁢s)n⁢λ(n-2⁢s)/(2⁢s),

which contradicts our hypothesis. Thus, a must be 0, and hence ∥wm∥ strongly converges to w in X0. Therefore, IKu satisfies (CPS)c. ∎

For the sake of simplicity, we assume 0∈Ω. In order to extend Uϵ (defined in Section 1) by zero outside Ω, we fix δ>0 such that B4⁢δ⊂Ω and let ζ∈Cc∞⁢(ℝn) be such that 0≤ζ≤1 in ℝn, ζ≡0 in ℝn∖B2⁢δ and ζ≡1 in Bδ. For each ϵ>0 and x∈ℝn, we define

Φϵ⁢(x):=ζ⁢(x)⁢Uϵ⁢(x).

Moreover, since u is positive and bounded (see Lemma 3.2), we can find m,M>0 such that for each x∈B2⁢δ, m≤u⁢(x)≤M.

Lemma 4.12.

For any sufficiently small ϵ>0,

sup⁡{IKu⁢(u+t⁢Φϵ):t≥0}<IKu⁢(u)+s⁢(Csn⁢Ss)n/(2⁢s)n⁢λ(n-2⁢s)/(2⁢s).

Proof.

We assume ϵ>0 to be sufficiently small. Using [45, Proposition 21], we have

∫Q|Φϵ⁢(x)-Φϵ⁢(y)|2|x-y|n+2⁢s⁢dx⁢dy≤Ssn/(2⁢s)+o⁢(ϵn-2⁢s),

which implies that we can find r1>0 such that

∫Q|Φϵ⁢(x)-Φϵ⁢(y)|2|x-y|n+2⁢s⁢dx⁢dy≤Ssn/(2⁢s)+r1⁢ϵn-2⁢s.

Now, we have

∫Ω|Φϵ|2s*dx=∫ℝn|Uϵ|2s*dx+∫ℝn(ζ(x)2s*-1)|Uϵ(x)|2s*dx

=Ssn/(2⁢s)+∫ℝn∖Bδ(ζ⁢(x)2s*-1)⁢|Uϵ⁢(x)|2s*⁢dx

=Ssn/(2⁢s)+ϵ-n⁢∫ℝn∖Bδ(ζ⁢(x)2s*-1)⁢|u*⁢(xϵ)|2s*⁢dx

≥Ssn/(2⁢s)-ϵn∫ℝn∖Bδ|x|-2⁢ndx

≥Ssn/(2⁢s)-r2⁢ϵn

for some constant r2>0. We now fix 1<ρ<min⁡{2,nn-2⁢s} and have

∫Ω|Φϵ|ρdx=ϵ-(n-2⁢s)⁢ρ/2∫B2⁢δ|ζ(x)u*(xϵ)|ρdx=O(ϵ(n-2⁢s)⁢ρ/2)≤r3ϵ(n-2⁢s)⁢ρ/2

for a constant r3>0. Now we see that

∫Bϵ|Φϵ|2s*-1dx=α2s*-1β-(n+2⁢s)ϵ(n-2⁢s)/2∫|y|<1/(β⁢Ss1/(2⁢s))(1+|y|2)-(n+2⁢s)/2dy≥r4ϵ(n-2⁢s)/2

for some constant r4>0. We also have

G⁢(x,r+t)-G⁢(x,r)-g⁢(x,r)⁢t=∫rr+t(g⁢(x,τ)-g⁢(x,r))⁢dτ=∫rr+t((r+u¯⁢(x))-q-(τ+u¯)-q)⁢dτ≤∫rr+t(r-q-τ-q)⁢dτ.

Thus, we can find γ>0 such that

G⁢(x,r+t)-G⁢(x,r)-g⁢(x,r)⁢t≤γ⁢tρ for each ⁢x∈Ω,r≥m⁢ and ⁢t≥0.

We can find an appropriate constant ρs>0 such that the following inequalities hold:

(c+d)2s*2s*-c2s*2s*-c2s*-1⁢d≥d2s*2s*for all ⁢c,d≥0,

(c+d)2s*2s*-c2s*2s*-c2s*-1⁢d≥d2s*2s*+ρs⁢c⁢d2s*-1r4⁢m⁢(2s*-1)for all ⁢0≤c≤M,d≥1.

Since u is a positive weak solution of (Pλ), using the above inequalities, we obtain

IKu(u+tΦϵ)-IKu(u)=IKu(u+tΦϵ)-IKu(u)-t(Csn∫Q(u⁢(x)-u⁢(y))⁢(Φϵ⁢(x)-Φϵ⁢(y))|x-y|n+2⁢sdxdy

+∫Ω(g(x,u)Φϵ-λ(u+u¯)2s*-1Φϵ)dx)

=t2⁢Csn2⁢∫Q|Φϵ⁢(x)-Φϵ⁢(y)|2|x-y|n+2⁢s⁢dx⁢dy-λ⁢∫Ω12s*⁢(|u+t⁢Φϵ+u¯|2s*-|u+u¯|2s*)⁢dx

+λ⁢t⁢∫Ω(u+u¯)2s*-1⁢Φϵ⁢dx+∫Ω(G⁢(x,u+t⁢Φϵ)-G⁢(x,u)-g⁢(x,u)⁢(t⁢Φϵ))⁢dx

≤t2⁢Csn2(Ssn/(2⁢s)+r1ϵn-2⁢s)-λ⁢t2s*2s*∫Ω|Φϵ|2s*dx+γ∫Ω|tΦϵ|ρdx

≤t2⁢Csn2⁢(Ssn/(2⁢s)+r1⁢ϵn-2⁢s)-λ⁢t2s*2s*⁢(Ssn/(2⁢s)-r2⁢ϵn)+γ⁢r3⁢tρ⁢ϵ(n-2⁢s)⁢ρ/2

for 0≤t<λ-(n-2⁢s)/(4⁢s)/2. Since we can assume t⁢Φϵ≥1 for each t≥λ-(n-2⁢s)/(4⁢s)/2 and |x|≤ϵ, we have

IKu(u+tΦϵ)-IKu(u)≤t2⁢Csn2(Ssn/(2⁢s)+r1ϵn-2⁢s)-λ⁢t2s*2s*∫Ω|Φϵ|2s*dx

-λ⁢ρs⁢t2s*-1r4⁢m⁢(2s*-1)∫|x|≤ϵ(u+u¯)|Φϵ|2s*-1dx+γ∫Ω|tΦϵ|ρdx

≤t2⁢Csn2⁢(Ssn/(2⁢s)+r1⁢ϵn-2⁢s)-λ⁢t2s*2s*⁢(Ssn/(2⁢s)-r2⁢ϵn)-λ⁢ρs⁢t2s*-1(2s*-1)⁢ϵ(n-2⁢s)/2+γ⁢r3⁢tρ⁢ϵ(n-2⁢s)⁢ρ/2.

Now, we define a function hϵ:[0,∞)→ℝ by

hϵ⁢(t)={t2⁢Csn2⁢(Ssn/(2⁢s)+r1⁢ϵn-2⁢s)-λ⁢t2s*2s*⁢(Ssn/(2⁢s)-r2⁢ϵn)+γ⁢r3⁢tρ⁢ϵ(n-2⁢s)⁢ρ/2,t∈[0,λ(n-2⁢s)/4⁢s2),t2⁢Csn2⁢(Ssn/(2⁢s)+r1⁢ϵn-2⁢s)-λ⁢t2s*2s*⁢(Ssn/(2⁢s)-r2⁢ϵn)-λ⁢ρs⁢t2s*-1(2s*-1)⁢ϵ(n-2⁢s)/2+γ⁢r3⁢tρ⁢ϵ(n-2⁢s)⁢ρ/2,t∈[λ(n-2⁢s)/4⁢s2,∞).

With some computations, it can be checked that hϵ attains its maximum at

t=(Csnλ)(n-2⁢s)/(4⁢s)-ρs⁢ϵ(n-2⁢s)/2(2s*-2)⁢Ssn/(2⁢s)+o⁢(ϵ(n-2⁢s)/2),

so we get

sup⁡{I⁢(u+t⁢Φϵ)-I⁢(u):t≥0}≤(Csn)n/(2⁢s)⁢s⁢Ssn/(2⁢s)n⁢λ(n-2⁢s)/(2⁢s)-ρs⁢(Csn)(n+2⁢s)/4⁢s⁢ϵ(n-2⁢s)/2(2s*-1)⁢λ(n-2⁢s)/(4⁢s)+o⁢(ϵ(n-2⁢s)/2)

<s⁢(Csn⁢Ss)n/(2⁢s)n⁢λ(n-2⁢s)/(2⁢s).

This completes the proof. ∎

Proposition 4.13.

For each λ∈(0,Λ), there exist a second positive weak solution of (P̄λ).

Proof.

Let Φ=Φϵ for some sufficiently small ϵ>0, as obtained in the previous lemma. From Theorem 4.8, u is a local minimizer of IKu. So we can choose α>0 small enough such that IKu⁢(v)≥IKu⁢(u) for every v∈Ku with ∥v-u∥≤α. We know that IKu⁢(u+t⁢w)→-∞ as t→∞, which implies that we can choose t>α/∥w∥ such that IKu⁢(u+t⁢w)≤IKu⁢(u). Let us set

Φ={ϕ∈C⁢([0,1],D⁢(IKu)):ϕ⁢(0)=u,ϕ⁢(1)=u+t⁢w},

A={v∈D⁢(IKu):∥v-u∥=α} and c=infϕ∈Φ⁢sup0≤r≤1⁡IKu⁢(ϕ⁢(r)).

The functional IKu satisfies (CPS)c, by Proposition 4.11 and Lemma 4.12. If c=IKu⁢(u), then u∉A, u+t⁢w∉A, inf⁡IKu⁢(A)≥IKu⁢(u)≥IKu⁢(u+t⁢w), and for each ϕ∈Φ, there exist r∈[0,1] such that ∥ϕ⁢(r)-u∥=α. Hence, by Theorem 2.4, we have v∈D⁢(IKu) such that v≠u, IKu⁢(v)=c and 0∈∂-⁡IKu⁢(v). Using Proposition 4.2 (i), we have that v is a positive weak solution of (P̄λ). ∎

Proof of Theorem 2.10.

The proof of Theorem 2.10 follows from Theorems 4.8, 4.10 and 4.13, and Lemmas 3.3–3.6. ∎

5 Fractional problem in the critical dimension n=1

In the critical dimension n=1, the critical growth nonlinearities for the fractional Laplacian is explored in [22]. The analogous critical problem in this case is

(5.1)(-Δ)1/2⁢u=u-q+λ⁢up+1⁢exp⁡(u2),u>0 in ⁢Ω,u=0 on ⁢∂⁡Ω,

where p, q, λ are positive parameters. Fractional problems with exponential growth nonlinearities are motivated by the following version of the Moser–Trudinger inequality [32].

Theorem 5.1.

For u∈H1/2⁢((-1,1)), exp⁡(α⁢u2)∈L1⁢((-1,1)) for any α>0. Moreover, there exists a constant C>0 such that

sup∥(-Δ)1/4⁢u∥L2⁢(-1,1)≤1(∫-11exp(αu2)dx)≤C for all α≤π.

Problem (5.1) can be transformed into a local problem by Dirichlet–Neumann maps introduced by Cafarelli and Silvestre [9]. For any v∈H1/2⁢(ℝ), the unique function w⁢(x,y) that minimizes the weighted integral

ℰ1/2(w)=∫0∞∫ℝ|∇w(x,y)|2dxdy,

over the set

{w⁢(x,y):ℰ1/2⁢(w)<∞,w|y=0=v},

satisfies

∫ℝ|(-Δ)1/2v|2=ℰ1/2(w).

Moreover, w⁢(x,y) solves the boundary value problem

-div⁡(∇⁡w)=0 in ⁢ℝ×ℝ+,w|y=0=v,∂⁡w∂⁡ν=(-Δ)1/2⁢v⁢(x),

where ∂⁡w∂⁡ν=-limy→0+⁡∂⁡w∂⁡y⁢(x,y). So the extension problem corresponding to (5.1) is

(5.2){-div⁡(∇⁡w)=0,w>0in ⁢𝒞:=(-1,1)×(0,∞),∂⁡w∂⁡ν=w-q+λ⁢wp+1⁢exp⁡(w2)on ⁢Ω×{0}.

To solve this, we closely follow the arguments used in [40]. The natural space to look for the solution of this extension problems is the Sobolev space

H0,L1⁢(𝒞)={v∈H1⁢(𝒞):v=0⁢ a.e. in ⁢(-1,1)×(0,∞)},

equipped with the norm ∥w∥=(∫𝒞|∇w|2dxdy)1/2. Now using the relation between the space H1/2⁢((-1,1)) and the square root Laplacian operator (see [16]), we get

∥(-Δ)1/4⁢u∥L2⁢((-1,1))=12⁢π⁢[u]H1/2⁢(ℝ)=∥w∥,

where

[u]H1/2⁢(ℝ)=(∬ℝ2|u⁢(x)-u⁢(y)|2|x-y|2⁢dx⁢dy)1/2.

If w solves the extension problem (5.2), then the trace⁢(w)=w⁢(x,0) solves the given nonlocal problem and vice-versa.

Definition 5.2.

A function w∈Lloc1⁢(𝒞) is said to be a weak solution of (5.2) if the following hold:

inf(x,y)∈K⁡w⁢(x,y)>0 for every compact subset K⊂Ω×[0,∞),
w solves the PDE in (5.2) in the sense of distributions,
(w-ϵ)+∈H0,L1⁢(𝒞) for every ϵ>0.

Let w0 be the minimal weak solution (in the sense of Definition 5.2) of

-Δ⁢w=0 in ⁢(-1,1)×(0,∞),∂⁡w∂⁡y=w-q in ⁢(-1,1)×{0}.

The existence of w0 can be obtained by solving the corresponding equivalent problem (P0) with Ω=(-1,1) and by following the approach used in [5] (see Section 3). Precisely, regularizing the singular nonlinearity in (P0), we introduce for n∈ℕ* the following approximated problem:

(Pn)(-Δ)s⁢u=(u+1n)-q,u>0 in ⁢(-1,1),u=0 in ⁢ℝn∖(-1,1).

This problem admits a unique solution wn in H~1/2⁢(-1,1), the Lions–Magenes space defined by

H~1/2(-1,1):={u∈H1(-1,1):∫-11u2d⁢(x)dx<∞}

={u∈H1/2(ℝ):u≡0 in ℝ∖(-1,1)}

=[H01⁢(-1,1),L2⁢(-1,1)]1/2.

Then, passing wn to the limit as n→∞ in the sense of distributions, we obtain w0⁢(x,0). Using a similar proof to that of Theorem 3.3, we can show that w0∈L∞⁢(𝒞). We can translate the problem, as in Section 3, by w0 as follows:

(P̄λ’){-Δ⁢w=0,w>0in ⁢(-1,1)×(0,∞),∂⁡w∂⁡y+w0-q-(w+w0)-q=λ⁢(w+w0)p+1⁢exp⁡((w+w0)2)in ⁢(-1,1)×{0}.

Note that w+w0 is a solution of (5.2) if w∈H0,L1⁢(𝒞) is a nonnegative distributional solution of (P̄λ’). Hence, it is enough to show existence and multiplicity results for (P̄λ’). It is possible to give a variational framework for problem (P̄λ’) in the space H0,L1⁢(𝒞). Following the arguments used in [40], we define the functions g,f:(-1,1)×ℝ→ℝ by

f⁢(x,s)={(s+w0⁢(x,0))p+1⁢exp⁡((s+w0⁢(x,0))β)if ⁢s+w0⁢(x,0)>0,0 otherwise,

g⁢(x,s)={(w0⁢(x,0))-q-(s+w0⁢(x,0))-q if ⁢s+w0⁢(x)>0,-∞otherwise.

It is easy to see that both g and f are nonnegative and nondecreasing in s. The required measurability of g⁢(⋅,s) and f⁢(⋅,s) follows from [31, Lemmas 1 and 2]. We define the primitives F:(-1,1)×ℝ→ℝ and G:(-1,1)×ℝ→(-∞,∞], respectively, by

F⁢(x,s)=∫0sf⁢(x,τ)⁢dτ and G⁢(x,s)=∫0sg⁢(x,τ)⁢dτ for (x,s)∈(-1,1)×ℝ.

Then we note that there exist M>0, θ>2 such that for all s>0,x∈(-1,1),

F⁢(x,s)≤M⁢(f⁢(x,s)+1) and θ⁢F⁢(x,s)≤f⁢(x,s)⁢s.

Define a functional I:H0,L1⁢(𝒞)→(-∞,∞] corresponding to (5.2) by

I⁢(u)={12∫Ω|∇w|2dxdy+∫-11G(x,w(x,0))dx-λ∫-11F(x,w(x,0))dxif ⁢G⁢(⋅,u)∈L1⁢(Ω),∞otherwise.

Now we can define the weak sub and super solutions and, by following the arguments used in Section 3, we can show the existence of the first solution wλ. Moreover, for :=⁡IKI|K, the following theorem follows from [40, Theorem 3.19].

Theorem 5.3.

Take λ∈(0,Λ). Let z be a strict super-solution of (P̄λ’). Let also wλ∈D⁢(IK) be a minimizer for IK, where K={u∈H0,L1⁢(C):0≤u≤z}. Then wλ is a local minimizer for IH+, where H+={v∈H0,L1(C),v≥0}.

To prove the existence of another solution to problem (5.2), as in [40], we translate this problem about the first solution wλ as follows:

(TPλ){-Δ⁢w=0,w>0in ⁢𝒞,∂⁡w∂⁡y+g⁢(x,w+wλ)-g⁢(x,wλ)=λ⁢(f⁢(x,w+wλ)-f⁢(x,wλ)),w>0in ⁢Ω×{0}.

Clearly, w is a solution of (TPλ) if and only if (w+wλ) solves (5.2). Define

g~⁢(x,s)={g⁢(x,s+wλ)-g⁢(x,wλ)s>0,0s≤0, f~⁢(x,s)={f⁢(x,s+wλ)-f⁢(x,wλ)s>0,0s≤0.

Define the respective primitives:

G~⁢(x,u)=∫0ug~⁢(x,s)⁢ds,F~⁢(x,u)=∫0uf~⁢(x,s)⁢ds.

Thanks to the nondecreasing nature of g and hence g~, we obtain the following inequality:

G~⁢(x,s)≤g~⁢(x,s)⁢s for all ⁢s≥0.

Let us define the energy functional E:H0,L1⁢(𝒞)→(-∞,+∞], associated with (TPλ), as follows:

E⁢(u)={12∫Ω|∇w|2dx+∫-11G~(x,w)dx-λ∫-11F~(x,w)dxif ⁢G~⁢(⋅,u)∈L1⁢(-1,1),∞otherwise.

Recalling the definition of I, we note that

E⁢(u)=I⁢(w++wλ)-I⁢(wλ)+12⁢∥w-∥2 for all ⁢w∈H0,L1⁢(𝒞).

It follows that

D⁢(E)∩H+=D⁢(I)∩H+.

Since wλ is a local minimum of IH+, it follows that 0 is a local minimum of E⁢(u) in H+. Thus, there exists ρ0>0 such that E⁢(u)≥E⁢(0)=0 for all u∈H+ with ∥u∥≤ρ0.

We recall the following version of the Lions compactness lemma (see [21, Lemma 2.3]).

Theorem 5.4.

Let {wk:∥wk∥=1} be a sequence of H0,L1⁢(C) functions converging weakly to a non zero function u. Then, for all p<(1-∥w∥)-1,

supk(∫-11exp(πp|wk|2)dx)<∞.

To show the existence of mountain-pass solution, we need the following sequence of Moser functions concentrating on the boundary, see [22].

Lemma 5.5.

There exists a sequence {ϕk}⊂H0,L1⁢(C) satisfying the following:

ϕk≥0, supp⁢(ϕk)⊂B⁢(0,1)∩ℝ+2 and ∥ϕk∥=1,
ϕk is constant on x∈B⁢(0,1k)∩ℝ+2 and ϕk2=1π⁢log⁡k+O⁢(1) for x∈B⁢(0,1k)∩ℝ+2.

Now we have the following estimate on the level. The proof follows as in [40, Lemma 4.4].

Lemma 5.6.

We have

supt>0⁡E⁢(t⁢ϕn)<π2 for all large n.

Now the proof of the existence of the second solution follows from theorem 5.4 and by closely following the proofs of [40, Lemma 4.9 and Proposition 4.10].

Funding source: Centre National de la Recherche Scientifique

Award Identifier / Grant number: UMI CNRS 3494

Funding statement: The authors were funded by IFCAM (Indo–French Centre for Applied Mathematics) UMI CNRS 3494 under the project “Singular phenomena in reaction diffusion equations and in conservation laws”.

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Received: 2016-5-18

Accepted: 2016-6-1

Published Online: 2016-7-30

Published in Print: 2017-8-1

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Articles in the same Issue

https://doi.org/10.1515/anona-2016-0113

Keywords for this article

Nonlocal operator; fractional Laplacian; very singular nonlinearities, variational methods in non smooth analysis

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