# Gravity at cosmological distances:
Explaining the accelerating expansion without dark energy

Junpei Harada<EMAIL_ADDRESS>Health Sciences University of
Hokkaido, 1757 Kanazawa, Tobetsu-cho, Ishikari-gun, Hokkaido 061-0293, Japan

(August 15, 2023)

###### Abstract

Three theoretical criteria for gravitational theories beyond general
relativity are considered: obtaining the cosmological constant as an
integration constant, deriving the energy conservation law as a consequence of
the field equations, rather than assuming it, and not necessarily considering
conformally flat metrics as vacuum solutions. Existing theories, including
general relativity, do not simultaneously fulfill all three criteria. To
address this, a new gravitational field equation is proposed that satisfies
these criteria. From this equation, a spherically symmetric exact solution is
derived, which is a generalization of the Schwarzschild solution. It
incorporates three terms: the Schwarzschild term, the de Sitter term, and a
newly discovered term, which is proportional to $\displaystyle r^{4}$ in a
radial coordinate, that becomes significant only at large distances. The
equation is further applied to cosmology, deriving an equation for the scale
factor. It then presents a solution that describes the transition from
decelerating to accelerating expansion in a matter-dominated universe. This is
achieved without the need for negative pressure as dark energy or the positive
cosmological constant. This provides a novel explanation for the current
accelerating expansion of the universe.

##  I Introduction

In certain gravitational theories beyond general relativity, the cosmological
constant $\displaystyle\Lambda$ is derived as a constant of integration. This
feature provides a notable theoretical advantage over the Einstein equations
in general relativity. Therefore, it is pertinent to establish the following
theoretical criteria for gravitational theories:

  1. 1.

The cosmological constant $\displaystyle\Lambda$ is obtained as a constant of
integration.

In the Einstein equations, the presence or absence of the cosmological
constant $\displaystyle\Lambda$ is fixed from the beginning. Therefore,
general relativity does not meet this criterion. However, the trace-free
Einstein equations, denoted by $\displaystyle R_{\mu\nu}-Rg_{\mu\nu}/4=8\pi
G(T_{\mu\nu}-Tg_{\mu\nu}/4)$, which were initially investigated by Einstein
himself, do satisfy this criterion only if the conservation law,
$\displaystyle\nabla_{\mu}T^{\mu}{}_{\nu}=0$, is assumed as an additional
assumption [1, 2]. The need for this assumption is theoretically a
disadvantage. Hence, it is appropriate to require the second theoretical
criterion:

  1. 2.

The conservation law, $\displaystyle\nabla_{\mu}T^{\mu}{}_{\nu}=0$, is derived
as a consequence of the gravitational field equations, rather than being
assumed.

General relativity fulfills the second criterion due to the Bianchi identity
but fails to satisfy the first criterion. On the other hand, the trace-free
Einstein equations fulfill the first criterion but not the second one.

Conformal gravity [3] and Cotton gravity [4, 5] satisfy both the first and the
second criteria. In these theories, the gravitational field equation does not
include the cosmological constant; it arises as a constant of integration.
Additionally, the conservation law,
$\displaystyle\nabla_{\mu}T^{\mu}{}_{\nu}=0$, is derived from the field
equations, as in general relativity, due to the Bianchi identity.
Unfortunately, in these theories, any conformal flat metric serves as a vacuum
solution. This may be a potential disadvantage, as it allows for unphysical
solutions. For example, in cosmology, the conformally flat Friedmann-Lemaître-
Robertson-Walker metric is a vacuum solution even if the scale factor
$\displaystyle a(t)$ is an arbitrary function of $\displaystyle t$, and in
this case, it conflicts with observations. Therefore, it is reasonable to
consider the third theoretical criterion:

  1. 3.

A conformally flat metric is not necessarily a solution in vacuum.

To date, no known theory simultaneously satisfies all three criteria. It
remains uncertain whether such a theory is even possible. Hence, the following
questions arise: Does a gravitational field equation satisfying the three
criteria exist? If so, what is its form? What are the physical implications of
such an equation?

This paper provides answers to these questions. First, a new gravitational
field equation is proposed, which satisfies all three criteria. Subsequently,
from this equation, a spherically symmetric solution is derived, which is a
generalization of the Schwarzschild solution. The solution contains three
terms: the Schwarzschild term $\displaystyle(\propto 1/r)$, the de Sitter term
$\displaystyle(\propto r^{2})$, and a newly discovered term
$\displaystyle(\propto r^{4})$ that only becomes significant at large
distances, being negligible at small distances.

The equation is further applied to cosmology. By assuming isotropy and spatial
homogeneity of the universe, an equation of motion for the scale factor is
derived. The solution to this equation exhibits a significant property: even
in the absence of dark energy or the cosmological constant, with only matter
present, the universe undergoes a transition from decelerating to accelerating
expansion. In fact, in this theory, the accelerating expansion naturally and
inevitably emerges as a consequence of the gravitational field equation,
rather than being attributed to negative pressure. This offers a novel
explanation for the current accelerating expansion of the universe.

This paper is organized as follows. In Sec. II, we present the gravitational
field equation that satisfies the three criteria mentioned earlier. Section
III explores a generalized solution of the Schwarzschild solution. In Sec.
IV.1, we derive the equation of motion for the scale factor, which serves as a
generalization of the Friedmann equation. In Sec. IV.2, we present a solution
that describes a transition from decelerating to accelerating expansion in a
matter-dominated universe. Finally, Sec. V provides a summary and conclusions.

Throughout this paper, we set $\displaystyle c=8\pi G=1$, although
$\displaystyle 8\pi G$ is explicitly stated in some cases. The covariant
derivative uses the Levi-Civita connection, and the metric signature is
$\displaystyle(-,+,+,+)$.

##  II Gravitational field equation

Two different approaches satisfying the first criterion mentioned in the
Introduction are known.

The first approach involves demanding that the gravitational field equation be
traceless, as originally proposed by Einstein. However, while this approach
satisfies the first criterion, it fails to meet the second criterion, thus
requiring us to consider an alternative approach.

The second approach employs derivatives of the curvature tensors instead of
the curvature tensor itself. This approach includes conformal gravity, Cotton
gravity, and Yang’s gravitational field equation [6]. In a previous study, the
author explored a scenario in which the gravitational field equation possesses
the same symmetry as $\displaystyle\nabla_{\mu}R^{\mu}{}_{\nu\rho\sigma}$.
While this approach satisfies the first and the second criteria, it was found
to fail to fulfill the third criterion. Therefore, alternative symmetries need
to be considered in place of
$\displaystyle\nabla_{\mu}R^{\mu}{}_{\nu\rho\sigma}$.

Based on these observations, we consider the following scenario. We examine
two possible totally symmetric derivatives of the curvature:

$\displaystyle\displaystyle\nabla_{\rho}R_{\mu\nu}+\nabla_{\mu}R_{\nu\rho}+\nabla_{\nu}R_{\rho\mu},$
(1a)
$\displaystyle\displaystyle(g_{\mu\nu}\partial_{\rho}+g_{\nu\rho}\partial_{\mu}+g_{\rho\mu}\partial_{\nu})R.$
(1b) Here, $\displaystyle\nabla_{\mu}$ represents the covariant derivative,
$\displaystyle R_{\mu\nu}$ is the Ricci tensor, and $\displaystyle R$ is the
Ricci scalar. These two terms, Eqs. (1a) and (1b), are linearly independent,
allowing for a linear combination of (1a) and (1b) to serve as the left-hand
side of the gravitational field equation.

A similar representation can be employed for the right-hand side of the
gravitational field equation, which comprises two potential terms:

$\displaystyle\displaystyle\nabla_{\rho}T_{\mu\nu}+\nabla_{\mu}T_{\nu\rho}+\nabla_{\nu}T_{\rho\mu},$
(1c)
$\displaystyle\displaystyle(g_{\mu\nu}\partial_{\rho}+g_{\nu\rho}\partial_{\mu}+g_{\rho\mu}\partial_{\nu})T.$
(1d)

Here, $\displaystyle T_{\mu\nu}$ is the energy-momentum tensor, and
$\displaystyle T$ denotes its trace.

Hence, the gravitational field equation can be expressed as follows:

$\displaystyle\displaystyle
a(\nabla_{\rho}R_{\mu\nu}+\nabla_{\mu}R_{\nu\rho}+\nabla_{\nu}R_{\rho\mu})$
$\displaystyle\displaystyle\qquad+b(g_{\mu\nu}\partial_{\rho}+g_{\nu\rho}\partial_{\mu}+g_{\rho\mu}\partial_{\nu})R$
$\displaystyle\displaystyle=c(\nabla_{\rho}T_{\mu\nu}+\nabla_{\mu}T_{\nu\rho}+\nabla_{\nu}T_{\rho\mu})$
$\displaystyle\displaystyle\qquad+d(g_{\mu\nu}\partial_{\rho}+g_{\nu\rho}\partial_{\mu}+g_{\rho\mu}\partial_{\nu})T,$
(2)

where $\displaystyle a$, $\displaystyle b$, $\displaystyle c$, and
$\displaystyle d$ are coefficients.

The coefficients $\displaystyle a$, $\displaystyle b$, $\displaystyle c$, and
$\displaystyle d$ can be determined as follows. By multiplying Eq. (2) by
$\displaystyle g^{\nu\rho}$, we obtain

$2(a+3b)\partial_{\mu}R=2c\nabla_{\lambda}T^{\lambda}{}_{\mu}+(c+6d)\partial_{\mu}T,$
(3)

where we have used the identity $\displaystyle
2\nabla_{\mu}R^{\mu}{}_{\nu}=\partial_{\nu}R$.

To ensure that the conservation law
$\displaystyle\nabla_{\mu}T^{\mu}{}_{\nu}=0$ is satisfied identically, we can
derive the following conditions from Eq. (3):

$\displaystyle\displaystyle a+3b$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle 0,$ (4a) $\displaystyle\displaystyle c+6d$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle 0.$ (4b)

We also impose the condition that every solution of the Einstein equations
satisfies Eq. (2). By substituting $\displaystyle
T_{\mu\nu}=R_{\mu\nu}-Rg_{\mu\nu}/2$ (with $\displaystyle 8\pi G=1$) and
$\displaystyle T=-R$ into the right-hand side of Eq. (2), we obtain

$\displaystyle\displaystyle(a-c)(\nabla_{\rho}R_{\mu\nu}+\nabla_{\mu}R_{\nu\rho}+\nabla_{\nu}R_{\rho\mu})$
$\displaystyle\displaystyle+\left(b+\frac{c}{2}+d\right)(g_{\mu\nu}\partial_{\rho}+g_{\nu\rho}\partial_{\mu}+g_{\rho\mu}\partial_{\nu})R=0.$
(5)

From this equation, we can obtain the following conditions:

$\displaystyle\displaystyle a-c$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle 0,$ (6a) $\displaystyle\displaystyle
b+\frac{c}{2}+d$ $\displaystyle\displaystyle=$ $\displaystyle\displaystyle 0.$
(6b)

Using Eqs. (4a), (4b), (6a), and (6b) (three of them are linearly
independent), we can determine the coefficients $\displaystyle a$,
$\displaystyle b$, $\displaystyle c$, and $\displaystyle d$ as follows:

$\displaystyle\displaystyle a=1,$ $\displaystyle\displaystyle b=-\frac{1}{3},$
$\displaystyle\displaystyle c=1,$ $\displaystyle\displaystyle d=-\frac{1}{6},$
(7)

where we have set $\displaystyle a=1$ as a normalization.

Here, it is convenient to define the tensor $\displaystyle H_{\mu\nu\rho}$ as

$\displaystyle\displaystyle H_{\mu\nu\rho}$ $\displaystyle\displaystyle\equiv$
$\displaystyle\displaystyle\nabla_{\rho}R_{\mu\nu}+\nabla_{\mu}R_{\nu\rho}+\nabla_{\nu}R_{\rho\mu}$
(8)
$\displaystyle\displaystyle-\frac{1}{3}(g_{\mu\nu}\partial_{\rho}+g_{\nu\rho}\partial_{\mu}+g_{\rho\mu}\partial_{\nu})R,$

which is totally symmetric in $\displaystyle\mu$, $\displaystyle\nu$, and
$\displaystyle\rho$. It satisfies

$\displaystyle\displaystyle g^{\nu\rho}H_{\mu\nu\rho}=0.$ (9)

Another convenient definition is

$\displaystyle\displaystyle T_{\mu\nu\rho}$ $\displaystyle\displaystyle\equiv$
$\displaystyle\displaystyle\nabla_{\rho}T_{\mu\nu}+\nabla_{\mu}T_{\nu\rho}+\nabla_{\nu}T_{\rho\mu}$
(10)
$\displaystyle\displaystyle-\frac{1}{6}(g_{\mu\nu}\partial_{\rho}+g_{\nu\rho}\partial_{\mu}+g_{\rho\mu}\partial_{\nu})T,$

which is also totally symmetric in $\displaystyle\mu$, $\displaystyle\nu$, and
$\displaystyle\rho$. It satisfies

$\displaystyle\displaystyle
g^{\nu\rho}T_{\mu\nu\rho}=2\nabla_{\nu}T^{\nu}{}_{\mu}.$ (11)

Consequently, we obtain the gravitational field equation expressed by third-
order totally symmetric tensors,

$H_{\mu\nu\rho}=8\pi GT_{\mu\nu\rho},$ (12)

where we explicitly show $\displaystyle 8\pi G=1$.

Multiplying Eq. (12) by $\displaystyle g^{\nu\rho}$ and using Eqs. (9) and
(11), we can confirm that the conservation law,
$\displaystyle\nabla_{\mu}T^{\mu}{}_{\nu}=0$, is satisfied as

$g^{\nu\rho}H_{\mu\nu\rho}=16\pi G\nabla_{\nu}T^{\nu}{}_{\mu}=0.$ (13)

Here, we provide three remarks on Eq. (12). First, every solution of the
Einstein equations satisfies Eq. (12). This means the following: by
substituting $\displaystyle 8\pi GT_{\mu\nu}=G_{\mu\nu}$ into the right-hand
side of Eq. (12), we can confirm that Eq. (12) is satisfied. Furthermore, by
substituting $\displaystyle 8\pi GT_{\mu\nu}=G_{\mu\nu}+\Lambda g_{\mu\nu}$,
where $\displaystyle\Lambda$ is nonvanishing, into the right-hand side of Eq.
(12), we can confirm that Eq. (12) is still satisfied. Thus, Eq. (12) does not
distinguish between $\displaystyle 8\pi GT_{\mu\nu}=G_{\mu\nu}$ and
$\displaystyle 8\pi GT_{\mu\nu}=G_{\mu\nu}+\Lambda g_{\mu\nu}$. This implies
that the cosmological constant $\displaystyle\Lambda$ arises as an integration
constant. Second, as shown in Eq. (13), the conservation law,
$\displaystyle\nabla_{\mu}T^{\mu}{}_{\nu}=0$, is satisfied without being
assumed. Third, it should be noted that the vanishing of the Weyl tensor
$\displaystyle C_{\mu\nu\rho\sigma}$ does not mean the vanishing of
$\displaystyle H_{\mu\nu\rho}$. Therefore, a conformally flat spacetime is not
necessarily a vacuum solution of $\displaystyle H_{\mu\nu\rho}=0$.
Consequently, the gravitational field equation (12) simultaneously satisfies
all three criteria stated in the Introduction.

##  III Spherically symmetric static vacuum solution

We consider the Schwarzschild-like metric given by

$ds^{2}=-e^{\nu(r)}dt^{2}+e^{-\nu(r)}dr^{2}+r^{2}d\Omega^{2},$ (14)

where $\displaystyle d\Omega^{2}\equiv d\theta^{2}+\sin^{2}\theta d\phi^{2}$.
Substituting this into Eq. (8), we find that the component of $\displaystyle
H_{\mu\nu\rho}$ is expressed as follows:

$\displaystyle\displaystyle-2H^{1}{}_{11}=-\frac{8}{r^{3}}$
$\displaystyle\displaystyle+e^{\nu}\left(\nu^{\prime\prime\prime}+3\nu^{\prime}\nu^{\prime\prime}+(\nu^{\prime})^{3}-\frac{2\nu^{\prime\prime}}{r}-\frac{2(\nu^{\prime})^{2}}{r}-\frac{2\nu^{\prime}}{r^{2}}+\frac{8}{r^{3}}\right),$
(15)

where a prime denotes the derivative with respect to $\displaystyle r$. The
other components vanish except for $\displaystyle H^{0}{}_{01}$,
$\displaystyle H^{2}{}_{12}$, and $\displaystyle H^{3}{}_{13}$, which are
proportional to $\displaystyle H^{1}{}_{11}$. By making the substitution

$y(r)=\left(\nu^{\prime}-\frac{2}{r}\right)e^{\nu},$ (16)

Eq. (15) can be simplified to

$-2H^{1}{}_{11}=y^{\prime\prime}-\frac{6y}{r^{2}}-\frac{8}{r^{3}}.$ (17)

The solution to the equation $\displaystyle H^{1}{}_{11}=0$ is given by

$y(r)=-\frac{2}{r}+\frac{c_{1}}{r^{2}}+c_{2}r^{3},$ (18)

where $\displaystyle c_{1}$ and $\displaystyle c_{2}$ are constants of
integration. Therefore, Eq. (16) can be rewritten as

$\left(\nu^{\prime}-\frac{2}{r}\right)e^{\nu}=-\frac{2}{r}+\frac{c_{1}}{r^{2}}+c_{2}r^{3}.$
(19)

This equation can be solved as

$e^{\nu}=1-\frac{c_{1}}{3r}+c_{3}r^{2}+\frac{c_{2}}{2}r^{4},$ (20)

where $\displaystyle c_{3}$ is a constant of integration.

If we rename the constants as $\displaystyle c_{1}=6M$, $\displaystyle
c_{3}=-\Lambda/3$, and $\displaystyle c_{2}=-2\lambda/5$, then the solution is
given by

$-g_{00}=1/g_{11}=e^{\nu}=1-\frac{2M}{r}-\frac{\Lambda}{3}r^{2}-\frac{\lambda}{5}r^{4}.$
(21)

This solution is exact. The term $\displaystyle 2M/r$ represents the
Schwarzschild term. The term $\displaystyle\Lambda r^{2}/3$ corresponds to the
de Sitter term, indicating that the cosmological constant
$\displaystyle\Lambda$ arises as a constant of integration, as expected. The
last term, $\displaystyle\lambda r^{4}/5$, is a newly discovered term that
does not emerge from the Einstein equations. When $\displaystyle\lambda$
vanishes (or when $\displaystyle r$ is sufficiently small to ignore the term
$\displaystyle\lambda r^{4}/5$), Eq. (21) is the Schwarzschild–de Sitter
metric, thus remaining consistent with observations. The term
$\displaystyle\lambda r^{4}/5$ only becomes significant at large distances,
such as cosmological distances, and can be ignored at small distances.

We present the curvature invariants as follows:

$\displaystyle\displaystyle R^{\mu\nu\rho\sigma}R_{\mu\nu\rho\sigma}$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\frac{48M^{2}}{r^{6}}+\frac{8\Lambda^{2}}{3}+\frac{48M\lambda}{5r}$
(22a) $\displaystyle\displaystyle\quad+8\Lambda\lambda
r^{2}+\frac{212\lambda^{2}r^{4}}{25},$ $\displaystyle\displaystyle
R^{\mu\nu}R_{\mu\nu}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle 4\Lambda^{2}+12\Lambda\lambda
r^{2}+10\lambda^{2}r^{4},$ (22b) $\displaystyle\displaystyle R$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle 4\Lambda+6\lambda
r^{2},$ (22c) $\displaystyle\displaystyle
C^{\mu\nu\rho\sigma}C_{\mu\nu\rho\sigma}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\frac{48M^{2}}{r^{6}}+\frac{48M\lambda}{5r}+\frac{12\lambda^{2}r^{4}}{25}.$
(22d)

Hence, we observe that $\displaystyle\lambda$ contributes to both the Ricci
tensor and the Weyl tensor. On the other hand, the $\displaystyle M$ only
contributes to the Weyl tensor, while the cosmological constant
$\displaystyle\Lambda$ contributes solely to the Ricci tensor.

##  IV Accelerating universe

In this section, we apply our gravitational field equation to cosmology.
First, we derive the equation of motion for the scale factor. Then, we find a
solution that describes the accelerating expansion of the universe.

###  IV.1 Equation of motion for the scale factor

We assume that the universe is isotropic and spatially homogeneous. This
assumption leads us to choose a spacetime coordinate system where the metric
takes the Friedmann-Lemaître-Robertson-Walker metric [7, 8, 9, 10, 11, 12],

$ds^{2}=-dt^{2}+a^{2}(t)\left(\frac{dr^{2}}{1-kr^{2}}+r^{2}d\Omega^{2}\right).$
(23)

Here, $\displaystyle a(t)$ represents the scale factor, and $\displaystyle k$
is a constant that represents the curvature of three-dimensional space. The
requirements of isotropy and spatial homogeneity dictate that the components
of the energy-momentum tensor take the form

$T^{\mu}{}_{\nu}={\rm diag}(-\rho(t),p(t),p(t),p(t)),$ (24)

and its trace is

$T\equiv T^{\mu}{}_{\mu}=-\rho(t)+3p(t).$ (25)

The energy conservation law gives

$0=-\nabla_{\mu}T^{\mu}{}_{0}=\dot{\rho}+3\frac{\dot{a}}{a}(\rho+p),$ (26)

where $\displaystyle\dot{\rho}\equiv d\rho/dt$ and $\displaystyle\dot{a}\equiv
da/dt$.

We now focus on the gravitational field equation (12). For the Friedmann-
Lemaître-Robertson-Walker metric, the components of $\displaystyle
H_{\mu\nu\rho}$, defined by Eq. (8), are given by

$\displaystyle\displaystyle H^{1}{}_{01}$
$\displaystyle\displaystyle=H^{2}{}_{02}=H^{3}{}_{03}=-\frac{1}{3}H^{0}{}_{00}$
$\displaystyle\displaystyle=-4\left(\frac{\dot{a}}{a}\right)^{3}-\frac{\dddot{a}}{a}+5\frac{\dot{a}\ddot{a}}{a^{2}}-4k\frac{\dot{a}}{a^{3}}$
$\displaystyle\displaystyle=\frac{d}{dt}\left[2\left(\frac{\dot{a}}{a}\right)^{2}-\frac{\ddot{a}}{a}+\frac{2k}{a^{2}}\right],$
(27)

where dots denote time derivatives. The remaining components of $\displaystyle
H_{\mu\nu\rho}$ vanish.

We also require the components of the tensor $\displaystyle T_{\mu\nu\rho}$ as
defined by Eq. (10). From Eqs. (23)–(25), we obtain the following expressions:

$T^{0}{}_{00}=-\frac{1}{2}(5\dot{\rho}+3\dot{p})$ (28)

and

$\displaystyle\displaystyle T^{1}{}_{01}$
$\displaystyle\displaystyle=T^{2}{}_{02}=T^{3}{}_{03}$
$\displaystyle\displaystyle=-2\frac{\dot{a}}{a}(\rho+p)+\frac{1}{6}(\dot{\rho}+3\dot{p}).$
(29)

Using Eq. (26), we find that

$\displaystyle\displaystyle
T^{1}{}_{01}=T^{2}{}_{02}=T^{3}{}_{03}=-T^{0}{}_{00}/3.$ (30)

The remaining components of $\displaystyle T_{\mu\nu\rho}$ vanish.

The gravitational field equation (12) is therefore

$\frac{d}{dt}\left[2\left(\frac{\dot{a}}{a}\right)^{2}-\frac{\ddot{a}}{a}+\frac{2k}{a^{2}}\right]=8\pi
G\frac{d}{dt}\left[\frac{1}{6}(5\rho+3p)\right].$ (31)

By integrating this equation, we obtain

$2\left(\frac{\dot{a}}{a}\right)^{2}-\frac{\ddot{a}}{a}+\frac{2k}{a^{2}}=\frac{4\pi
G}{3}(5\rho+3p)+{\rm const},$ (32)

where $\displaystyle{\rm const}$ is a constant of integration. If we rename it
as $\displaystyle\Lambda/3$, then we can find that the Friedmann equations

$\displaystyle\displaystyle\left(\frac{\dot{a}}{a}\right)^{2}$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle\frac{8\pi
G}{3}\rho-\frac{k}{a^{2}}+\frac{\Lambda}{3},$ (33)
$\displaystyle\displaystyle\frac{\ddot{a}}{a}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle-\frac{4\pi G}{3}(\rho+3p)+\frac{\Lambda}{3},$ (34)

satisfy Eq. (32). Thus, in our gravitational theory, the cosmological constant
$\displaystyle\Lambda$ is indeed a constant of integration, as expected.

By substituting $\displaystyle\Lambda/3$ for const in Eq. (32), we obtain the
following equation for the scale factor:

$2\left(\frac{\dot{a}}{a}\right)^{2}-\frac{\ddot{a}}{a}=\frac{4\pi
G}{3}(5\rho+3p)-\frac{2k}{a^{2}}+\frac{\Lambda}{3}.$ (35)

This is the equation of motion for the scale factor in our gravity theory,
which is a generalization of the Friedmann equation in general relativity. It
should be noted that the Friedmann equations, Eqs. (33) and (34), with certain
$\displaystyle\rho$ and $\displaystyle p$, satisfy Eq. (35) with the same
$\displaystyle\rho$ and $\displaystyle p$. However, the inverse is not
necessarily true; Eq. (35), with certain $\displaystyle\rho$ and
$\displaystyle p$, does not necessarily satisfy Eqs. (33) and (34) with the
same $\displaystyle\rho$ and $\displaystyle p$. Given $\displaystyle p$ as a
function of $\displaystyle\rho$, we can solve Eq. (26) to find
$\displaystyle\rho$ as a function of $\displaystyle a$. Then, using the
obtained $\displaystyle\rho$ as a function of $\displaystyle a$, we can solve
Eq. (35) to determine $\displaystyle a$ as a function of $\displaystyle t$.

###  IV.2 Accelerating expansion

####  IV.2.1 Preliminary

In general relativity, Eq. (34) indicates that the accelerating expansion of
the universe $\displaystyle(\ddot{a}>0)$ requires either a positive
cosmological constant $\displaystyle\Lambda$ or a negative
$\displaystyle\rho+3p$ (representing dark energy). Therefore, in a matter-
dominated universe ($\displaystyle p=0$) with $\displaystyle\Lambda=0$, Eq.
(34) implies that $\displaystyle-\ddot{a}/a\propto\rho$, indicating
decelerating expansion $\displaystyle(\ddot{a}<0)$.

However, in our gravitational theory, the result differs significantly from
general relativity. For a matter-dominated universe ($\displaystyle p=0$),
with spatial flatness ($\displaystyle k=0$) and $\displaystyle\Lambda=0$, Eq.
(35) yields $\displaystyle 2(\dot{a}/a)^{2}-\ddot{a}/a\propto\rho$. This does
not necessarily imply that $\displaystyle\ddot{a}<0$, because $\displaystyle
2(\dot{a}/a)^{2}$ is positive. In the following, by solving the equation of
motion for the scale factor, we will demonstrate the existence of a solution
that describes the accelerating expansion in a matter-dominated universe.

Figure 1: The ratio of the scale factors $\displaystyle a(t)/a_{0}$ is shown
as a function of time $\displaystyle t$ (present is $\displaystyle t=0$) in
Gyr. The Hubble constant is assumed to be $\displaystyle H_{0}=68$ km s-1
Mpc-1. The four dashed lines represent Eq. (46) for $\displaystyle\Omega_{\rm
m}=(0.1,0.3,0.5,0.7)$. These lines clearly demonstrate that even in the
absence of dark energy or the cosmological constant, and when only matter is
present, the universe undergoes a transition from decelerating to accelerating
expansion. The time of this transition, for each case of
$\displaystyle\Omega_{\rm m}=(0.1,0.3,0.5,0.7)$, is $\displaystyle-11.4$,
$\displaystyle-6.2$, $\displaystyle-3.7$, and $\displaystyle-1.5$ Gyr,
respectively. The negative sign of the transition time indicates that the
transition occurred in the past $\displaystyle(t<0)$. The age of the universe
$\displaystyle t_{0}$, for each case of $\displaystyle\Omega_{\rm
m}=(0.1,0.3,0.5,0.7)$, is $\displaystyle 21.4,14.9,12.4$, and $\displaystyle
11.0$ Gyr, respectively. If we impose the constraint of $\displaystyle
t_{0}>13.0$ Gyr, which is motivated by observations, the range for
$\displaystyle\Omega_{\rm m}$ is determined to be $\displaystyle\Omega_{\rm
m}<0.44$. The solid blue line represents the Einstein–de Sitter universe with
$\displaystyle\Omega_{\rm m}=1$, while the solid orange line represents the
$\displaystyle\Lambda$CDM model following the Friedmann equation with
$\displaystyle\Omega_{\rm m}=0.3$ and $\displaystyle\Omega_{\Lambda}=0.7$. The
age of the universe $\displaystyle t_{0}$ is 9.6 Gyr for the Einstein–de
Sitter universe, and the age is 13.9 Gyr for the $\displaystyle\Lambda$CDM
with $\displaystyle\Omega_{\rm m}=0.3$ and
$\displaystyle\Omega_{\Lambda}=0.7$.

####  IV.2.2 An accelerating solution

For simplicity, we assume that the universe is spatially flat
$\displaystyle(k=0)$ and $\displaystyle\Lambda=0$. For a matter-dominated
universe ($\displaystyle p=0$), Eq. (26) yields

$\rho(t)=\rho_{0}\left(\frac{a(t)}{a_{0}}\right)^{-3},$ (36)

where $\displaystyle a_{0}$ represents the scale factor at the present time,
and $\displaystyle\rho_{0}$ is the matter density at the present time.
Substituting Eq. (36), along with $\displaystyle p=k=\Lambda=0$, into Eq.
(35), we obtain

$2\left(\frac{\dot{a}}{a}\right)^{2}-\frac{\ddot{a}}{a}=\frac{5}{2}H_{0}^{2}\Omega_{\rm
m}\left(\frac{a}{a_{0}}\right)^{-3},$ (37)

where $\displaystyle H_{0}$ is the Hubble constant. The density parameter
$\displaystyle\Omega_{\rm m}$, defined as

$\Omega_{\rm m}\equiv\frac{\rho_{0}}{\rho_{\rm
c}},\quad\rho_{c}\equiv\frac{3H_{0}^{2}}{8\pi G},$ (38)

represents the ratio of matter density to the critical density
$\displaystyle\rho_{\rm c}$. It should be noted that in our gravity theory,
$\displaystyle\Omega_{\rm m}$ does not necessarily satisfy
$\displaystyle\Omega_{\rm m}+\Omega_{\rm others}=1$, where
$\displaystyle\Omega_{\rm others}$ represents the density parameter for other
components, if they exist. Even if $\displaystyle\Omega_{\rm others}=0$,
$\displaystyle\Omega_{\rm m}$ is not necessarily equal to 1, because the
Friedmann equations are not necessarily satisfied. Therefore, we assume
$\displaystyle\Omega_{\rm m}\leq 1$.

The solution to Eq. (37) is given by

$\displaystyle\displaystyle t+t_{0}=\frac{2}{3H_{0}\sqrt{\Omega_{\rm
m}}}\left(\frac{a}{a_{0}}\right)^{3/2}{}_{2}F_{1}\left(\frac{3}{10},\frac{1}{2};\frac{13}{10};A\left(\frac{a}{a_{0}}\right)^{5}\right),$
(39)

where $\displaystyle t_{0}$ and $\displaystyle A$ are two constants of
integration, and $\displaystyle{}_{2}F_{1}\left(a,b;c;x\right)$ is the
hypergeometric function. The scale factor $\displaystyle a(t)$ reaches zero at
$\displaystyle t=-t_{0}$. By substituting $\displaystyle t=0$ (representing
present time) in Eq. (39), we obtain

$\displaystyle\displaystyle t_{0}=\frac{2}{3H_{0}\sqrt{\Omega_{\rm
m}}}{}_{2}F_{1}\left(\frac{3}{10},\frac{1}{2};\frac{13}{10};A\right),$ (40)

where we have used $\displaystyle a_{0}=a(t=0)$.

We also need to determine the constant $\displaystyle A$. This can be done as
follows. Differentiating Eq. (39) with respect to $\displaystyle t$, we have

$\displaystyle\displaystyle 1=\frac{1}{H_{0}\sqrt{\Omega_{\rm
m}}\sqrt{1-A(a/a_{0})^{5}}}\left(\frac{a}{a_{0}}\right)^{1/2}\frac{\dot{a}}{a_{0}}.$
(41)

Using Eqs. (39) and (41) to eliminate $\displaystyle 1/(H_{0}\sqrt{\Omega_{\rm
m}})$, we find that the Hubble parameter $\displaystyle H(t)\equiv\dot{a}/a$
is given by

$\displaystyle\displaystyle
H(t)=\frac{2\sqrt{1-A(a/a_{0})^{5}}}{3(t+t_{0})}{}_{2}F_{1}\left(\frac{3}{10},\frac{1}{2};\frac{13}{10};A\left(\frac{a}{a_{0}}\right)^{5}\right).$
(42)

Therefore, the Hubble constant $\displaystyle H_{0}=H(t=0)$ is

$\displaystyle\displaystyle
H_{0}=\frac{2\sqrt{1-A}}{3t_{0}}{}_{2}F_{1}\left(\frac{3}{10},\frac{1}{2};\frac{13}{10};A\right).$
(43)

By substituting Eq. (40) into Eq. (43), we obtain

$\sqrt{(1-A)\Omega_{\rm m}}=1.$ (44)

This yields

$A=1-\frac{1}{\Omega_{\rm m}}=\frac{\Omega_{\rm m}-1}{\Omega_{\rm m}}.$ (45)

Taking the ratio between Eqs. (39) and (40) and using Eq. (45), we find that
the scale factor $\displaystyle a(t)$ satisfies

$\displaystyle\displaystyle\frac{t+t_{0}}{t_{0}}=\frac{{}_{2}F_{1}\left(\frac{3}{10},\frac{1}{2};\frac{13}{10};\frac{\Omega_{\rm
m}-1}{\Omega_{\rm
m}}\left(\frac{a}{a_{0}}\right)^{5}\right)}{{}_{2}F_{1}\left(\frac{3}{10},\frac{1}{2};\frac{13}{10};\frac{\Omega_{\rm
m}-1}{\Omega_{\rm m}}\right)}\left(\frac{a}{a_{0}}\right)^{3/2}.$ (46)

Here, the age of the universe $\displaystyle t_{0}$ is given by

$\displaystyle\displaystyle t_{0}=\frac{2}{3H_{0}\sqrt{\Omega_{\rm
m}}}{}_{2}F_{1}\left(\frac{3}{10},\frac{1}{2};\frac{13}{10};\frac{\Omega_{\rm
m}-1}{\Omega_{\rm m}}\right).$ (47)

These equations, Eqs. (46) and (47), are fundamental equations. Using the
observed value of the Hubble constant $\displaystyle H_{0}$, Eq. (47)
determines $\displaystyle t_{0}$ as a function of $\displaystyle\Omega_{\rm
m}$. Subsequently, using $\displaystyle t_{0}$ as a function of
$\displaystyle\Omega_{\rm m}$, Eq. (46) determines $\displaystyle a(t)$ as a
function of $\displaystyle t$ and $\displaystyle\Omega_{\rm m}$.

In the special case where $\displaystyle\Omega_{\rm m}=1$, Eqs. (46) and (47)
simplify to

$\displaystyle\displaystyle\frac{a}{a_{0}}=\left(\frac{t+t_{0}}{t_{0}}\right)^{2/3},$
$\displaystyle\displaystyle t_{0}=\frac{2}{3H_{0}}.$ (48)

These equations represent the Einstein–de Sitter universe, and thus Eqs. (46)
and (47) include the result derived from general relativity as a special case.

In general cases where $\displaystyle\Omega_{\rm m}\not=1$, Fig. 1 illustrates
the behavior of $\displaystyle a(t)/a_{0}$ as a function of time. The figure
demonstrates that even in the absence of dark energy or the cosmological
constant, and with only matter present, the universe undergoes a transition
from decelerating to accelerating expansion.

This transition, from deceleration to acceleration, occurs at $\displaystyle
t=-t_{\star}$, which is the time when the acceleration $\displaystyle\ddot{a}$
reaches zero. By performing a straightforward calculation, we can determine
the time $\displaystyle t_{\star}$ as follows:

$t_{\star}=t_{0}\left[1-\left(\frac{\Omega_{\rm m}}{4(1-\Omega_{\rm
m})}\right)^{3/10}\frac{{}_{2}F_{1}\left(\frac{3}{10},\frac{1}{2};\frac{13}{10};-\frac{1}{4}\right)}{{}_{2}F_{1}\left(\frac{3}{10},\frac{1}{2};\frac{13}{10};\frac{\Omega_{\rm
m}-1}{\Omega_{\rm m}}\right)}\right],$ (49)

where $\displaystyle{}_{2}F_{1}(3/10,1/2;13/10;-1/4)\approx 0.97383$. This
equation indicates that $\displaystyle t_{\star}$ is positive if
$\displaystyle\Omega_{\rm m}$ is less than 0.8. A positive $\displaystyle
t_{\star}$ (or equivalently negative $\displaystyle-t_{\star}$) implies that
the transition from deceleration to acceleration occurred in the past
($\displaystyle t<0$). Therefore, we can conclude that in our gravitational
theory, even in the absence of dark energy, the current universe is in an
accelerating phase if $\displaystyle\Omega_{\rm m}<0.8$. The transition time
($\displaystyle t=-t_{\star}$) for typical values of $\displaystyle\Omega_{\rm
m}$ is provided in the caption of Fig. 1.

##  V Summary and conclusions

In this study, we have set three theoretical criteria for gravitational
theories, as outlined in the Introduction:

  1. 1.

The gravitational field equations should not explicitly contain the
cosmological constant $\displaystyle\Lambda$, but it can emerge as a constant
of integration.

  2. 2.

The conservation law $\displaystyle\nabla_{\mu}T^{\mu}{}_{\nu}=0$ should be
derived as a consequence of the field equations, rather than being introduced
as an additional assumption.

  3. 3.

A conformally flat metric should not necessarily be a vacuum solution.

These criteria impose stringent restrictions on gravitational theories, and so
far, no theory has been known to fulfill all three criteria. In this paper, we
have presented the gravitational field equation, Eq. (12), which satisfies all
three criteria. Our construction provides an explicit model, and while it is a
unique model that the author could find, it may not be the only one. These
criteria and their fulfillment are summarized in Table. 1.

Table 1: A summary of typical gravitational theories and their fulfillment against the three criteria.  Criterion | GR111General relativity. | TFE222Trace-free Einstein equations. | CG333Conformal gravity. | Cotton444Cotton gravity. | This work
---|---|---|---|---|---
1 | No | Yes | Yes | Yes | Yes
2 | Yes | No | Yes | Yes | Yes
3 | Yes | Yes | No | No | Yes

Additionally, we have derived a spherically symmetric solution that
generalizes the Schwarzschild solution. This solution consists of three terms:
the Schwarzschild term $\displaystyle(\propto 1/r)$, the de Sitter term
$\displaystyle(\propto r^{2})$, and a newly discovered term
$\displaystyle(\propto r^{4})$. The $\displaystyle r^{4}$ term only becomes
significant at large distances while being negligible at short distances. This
indicates that gravity described by Eq. (12) differs from general relativity
primarily at large distances, such as cosmological distances.

Motivated by this observation, we have applied our gravitational equations to
cosmology. Assuming the isotropy and spatial homogeneity of the universe, we
have derived an equation for the scale factor, Eq. (35), which serves as a
generalization of the Friedmann equation. Through our analysis, we have
demonstrated that even in the absence of dark energy or the cosmological
constant, the universe undergoes a transition from a decelerating phase to an
accelerating phase. Thus, in our gravitational theory, the current
accelerating expansion is a natural and inevitable consequence in a matter-
dominated universe.

###### Acknowledgements.

This work was supported by JSPS KAKENHI Grant No. JP22K03599.

## References

  * Ellis  _et al._ [2011] G. F. R. Ellis, H. van Elst, J. Murugan, and J.-P. Uzan, On the trace-free Einstein equations as a viable alternative to general relativity, Class. Quant. Grav. 28, 225007 (2011), arXiv:1008.1196 [gr-qc] .
  * Ellis [2014] G. F. R. Ellis, The trace-free Einstein equations and inflation, Gen. Rel. Grav. 46, 1619 (2014), arXiv:1306.3021 [gr-qc] .
  * Mannheim and Kazanas [1989] P. D. Mannheim and D. Kazanas, Exact vacuum solution to conformal Weyl gravity and galactic rotation curves, Astrophys. J. 342, 635 (1989).
  * Harada [2021] J. Harada, Emergence of the Cotton tensor for describing gravity, Phys. Rev. D 103, L121502 (2021), arXiv:2105.09304 [gr-qc] .
  * Mantica and Molinari [2023] C. A. Mantica and L. G. Molinari, Codazzi tensors and their space-times and Cotton gravity, Gen. Rel. Grav. 55, 62 (2023), arXiv:2210.06173 [gr-qc] .
  * Yang [1974] C. N. Yang, Integral Formalism for Gauge Fields, Phys. Rev. Lett. 33, 445 (1974).
  * Friedmann [1922] A. Friedmann, On the curvature of space, Z. Phys. 10, 377 (1922).
  * Friedmann [1924] A. Friedmann, On the possibility of a world with constant negative curvature of space, Z. Phys. 21, 326 (1924).
  * Lemaître [1927] G. Lemaître, A homogeneous universe of constant mass and growing radius accounting for the radial velocity of extragalactic nebulae, Annales Soc. Sci. Bruxelles A 47, 49 (1927).
  * Lemaître [1931a] G. Lemaître, A homogeneous universe of constant mass and increasing radius accounting for the radial velocity of extra-galactic nebulae, Mon. Not. Roy. Astron. Soc. 91, 483 (1931a).
  * Lemaître [1931b] G. Lemaître, The expanding universe, Mon. Not. Roy. Astron. Soc. 91, 490 (1931b).
  * Robertson [1935] H. P. Robertson, Kinematics and world-structure, Astrophys. J. 82, 284 (1935).


# A note on the global regularity results for strongly nonhomogeneous
$p,q$-fractional problems and applications

Jacques Giacomoni 1 111e-mail<EMAIL_ADDRESS>Deepak Kumar
2222e-mail<EMAIL_ADDRESS>and Konijeti Sreenadh 2333 e-mail:
<EMAIL_ADDRESS>

${}^{1}\,$Université de Pau et des Pays de l’Adour, LMAP (UMR E2S-UPPA CNRS
5142)
Bat. IPRA, Avenue de l’Université F-64013 Pau, France
${}^{2}\,$Department of Mathematics, Indian Institute of Technology Delhi,
Hauz Khaz, New Delhi-110016, India

###### Abstract

In this article, we communicate with the glimpse of the proofs of new global
regularity results for weak solutions to a class of problems involving
fractional $(p,q)$-Laplacian, denoted by
$(-\Delta)^{s_{1}}_{p}+(-\Delta)^{s_{2}}_{q}$, for $s_{2},s_{1}\in(0,1)$ and
$1<p,q<\infty$. We also obtain the boundary Hölder continuity results for the
weak solutions to the corresponding problems involving at most critical growth
nonlinearities. These results are almost optimal. Moreover, we establish Hopf
type maximum principle and strong comparison principle. As an application to
these new results, we prove the Sobolev versus Hölder minimizer type result,
which provides the multiplicity of solutions in the spirit of seminal work
[2].

###### Résumé

Dans cette note, nous présentons de nouveaux résultats de régularité
Höldérienne des solutions faibles d’une classe de problèmes faisant intervenir
des opérateurs de diffusion fractionnaire non linéaires et non homogènes de la
forme $(-\Delta)^{s_{1}}_{p}+(-\Delta)^{s_{2}}_{q}$ avec $s_{2},s_{1}\in(0,1)$
et $1<p,q<\infty$. Précisément, nous obtenons des résultats de régularité
intérieure et près du bord pour les solutions faibles de ces problèmes alors
que la nonlinéarité du membre de droite est de croissance critique au sens de
l’injection de Sobolev. Ce résultat étend les principaux résultats de
régularité intérieure de [1] où le cas de l’opérateur homogène
$(-\Delta)^{s_{1}}_{p}$ est investi, améliore de façon optimale et complète
ceux de [8].

Nous établissons par ailleurs un lemme de Hopf et un principe de comparaison
fort pour cette classe de problèmes. Nous appliquons ensuite ces résultats
pour démontrer la propriété que les minimiseurs locaux de l’énergie associée
dans $C^{\alpha}(\overline{\Omega})$ avec $\alpha\in(0,s_{1})$ sont aussi
minimiseurs locaux dans $W^{s_{1},p}_{0}(\Omega)$ dans l’esprit de l’article
pionnier [2]. Ceci conduit à des nouveaux résultats de muliplicité de
solutions pour ces problèmes non locaux et fortement non homogènes.

Keywords: Fractional $(p,q)$-Laplacian, non-homogeneous nonlocal operator,
Hölder continuity up to the boundary, maximum principle, strong comparison
principle.
2010 Mathematics Subject Classification: 35J60, 35R11, 35B45, 35D30.

##  1 Introduction

In this note we study the Hölder continuity results and maximum principle for
weak solutions to the following problem:

$\begin{array}[]{rllll}(-\Delta)^{s_{1}}_{p}u+(-\Delta)^{s_{2}}_{q}u=f\quad\text{in
}\Omega,\end{array}$ ($\mathcal{P}$)

where $\Omega\subset\mathbb{R}^{N}$ is a bounded domain with $C^{1,1}$
boundary, $2\leq q,p<\infty$, $0<s_{2}\leq s_{1}<1$ and $f\in
L^{\infty}_{\mathrm{loc}}(\Omega)$. The fractional $p$-Laplace operator
$(-\Delta)^{s}_{p}$ is defined as

${(-\Delta)^{s}_{p}u(x)}=2\lim_{\varepsilon\to 0}\int_{\mathbb{R}^{N}\setminus
B_{\varepsilon}(x)}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{N+ps}}dy.$

The leading differential operator,
$(-\Delta)^{s_{1}}_{p}+(-\Delta)^{s_{2}}_{q}$, in problem ($\mathcal{P}$) is
known as the fractional $(p,q)$-Laplacian. The operator is non-homogeneous in
the sense that for any $t>0$, there does not exist any $\sigma\in\mathbb{R}$
such that
$((-\Delta)^{s_{1}}_{p}+(-\Delta)^{s_{2}}_{q})(tu)=t^{\sigma}((-\Delta)^{s_{1}}_{p}u+(-\Delta)^{s_{2}}_{q}u)$
holds for all $u\in W^{s_{1},p}(\Omega)\cap W^{s_{2},q}(\Omega)$.

The regularity results and maximum principles for the equations involving the
homogeneous nonlocal operators are well known, see [1, 4, 9, 10, 12], whereas
the regularity issues for the problems involving the fractional
$(p,q)$-Laplacian is still developing and only few continuity results are
available, see for instance, [7, 6]. The strong non-homogeneity, in this case,
feature creates an additional difficulty while handling the distance function
in order to prove the boundary behavior of the weak solutions.

In the present work, we obtain interior regularity results for local weak
solutions, which improves the regularity results of [7] for larger class of
exponents. Our proof of the improved local Hölder continuity result (see
Theorem 2.1) relies on Moser’s iteration technique to obtain suitable
embedding for Besov spaces into the Hölder spaces. Here, we stress that we do
not assume any order relation on the exponents $p$ and $q$. Subsequently, we
establish the asymptotic behavior of the fractional $q$-Laplacian
($(-\Delta)_{q}^{s_{2}}$) of the distance function $d^{s_{1}}$ near the
boundary, which in turn gives us almost optimal (and optimal in some cases,
see Remark 3) boundary behavior of the weak solution. This coupled with the
interior Hölder regularity result of Theorem 2.1 provides the almost optimal
Hölder continuity result. As a consequence of this, we obtain the Hopf type
maximum principle for non-negative solutions. Additionally, under the
restriction that the fractional $q$-Laplacian of the subsolution is bounded
from below, we prove a strong comparison principle. In the last section, as an
application to these results, we obtain the multiplicity results for problem
involving the subcritical nonlinearity by establishing Sobolev versus Hölder
type minimization result for nonlinearities with atmost critical growth.
Complete proofs of the regularity main results and other applications (in
particular to singular problems) can be found in [7].

##  2 Preliminaries and main results

We denote $[t]^{p-1}:=|t|^{p-2}t$, for all $p>1$ and $t\in\mathbb{R}$. For
$(\ell,s)\in\\{(p,s_{1}),(q,s_{2})\\}$ and for $S_{1}\times
S_{2}\subset\mathbb{R}^{N}\times\mathbb{R}^{N}$, we set

$\displaystyle A_{\ell}(u,v,S_{1}\times S_{2})=\int_{S_{1}\times
S_{2}}\frac{[u(x)-u(y)]^{\ell-1}(v(x)-v(y))}{|x-y|^{N+\ell
s}}\leavevmode\nobreak\ dxdy.$

We define the distance function as
$d(x):=\mathrm{dist}(x,\mathbb{R}^{N}\setminus\Omega)$ and a neighborhood of
the boundary as $\Omega_{\varrho}:=\\{x\in\Omega\ :\ d(x)<\varrho\\}$, for
$\varrho>0$.
We will follow the notation $p^{*}_{s_{1}}:=Np/(N-ps_{1})$ if $N>ps_{1}$,
otherwise an arbitrarily large number.

###  2.1 Function Spaces

For $E\subset\mathbb{R}^{N}$, $p\in[1,\infty)$ and $s\in(0,1)$, the fractional
Sobolev space $W^{s,p}(E)$ is defined as

$\displaystyle W^{s,p}(E):=\left\\{u\in
L^{p}(E):[u]_{W^{s,p}(E)}<\infty\right\\}$

endowed with the norm $\|u\|_{W^{s,p}(E)}:=\|u\|_{L^{p}(E)}+[u]_{W^{s,p}(E)}$,
where

$\displaystyle[u]_{W^{s,p}(E)}:=\left(\int_{E}\int_{E}\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}\leavevmode\nobreak\
dxdy\right)^{1/p}.$

For any (proper) subset $E$ of $\mathbb{R}^{N}$, we have

$\displaystyle W^{s,p}_{0}(E):=\\{u\in W^{s,p}(\mathbb{R}^{N})\ :\
u=0\quad\mbox{in }\mathbb{R}^{N}\setminus E\\}$

which is a uniformly convex Banach space when equipped with the norm
$[\cdot]_{W^{s,p}(\mathbb{R}^{N})}$ (hereafter, it will be denoted by
$\|\cdot\|_{W^{s,p}_{0}(E)}$). Next, we define

$\mathcal{W}(E):=W^{s_{1},p}(E)\cap W^{s_{2},q}(E)$

equipped with the norm
$\|\cdot\|_{\mathcal{W}(E)}:=\|\cdot\|_{W^{s_{1},p}(E)}+\|\cdot\|_{W^{s_{2},q}(E)}$.
The space $\mathcal{W}_{0}(E)$ is defined analogously. We say that
$u\in\mathcal{W}_{\rm loc}(E)$ if $u\in\mathcal{W}(E^{\prime})$, for all
$E^{\prime}\Subset E$. Note that for $1<q\leq p<\infty$, $0<s_{2}<s_{1}<1$ and
the domain $E$ with Lipschitz boundary, $W^{s_{1},p}_{0}(E)$ coincides with
the space $X_{p,s_{1}}$, as defined in [8]. Indeed, from [8, Lemma 2.1], we
have

$\displaystyle\|u\|_{W^{s_{2},q}_{0}(E)}\leq
C\|u\|_{W^{s_{1},p}_{0}(E)},\quad\text{for all }\;u\in W^{s_{1},p}_{0}(E),$

for some $C=C(|E|,\;N,\;p,\;q,\;s_{1},\;s_{2})>0$. Additionally, we define

$\displaystyle\widetilde{W}^{s,p}(\Omega):=\bigg{\\{}u\in L^{p}_{\rm
loc}(\mathbb{R}^{N})\ :\ \exists\Omega^{\prime}\Supset\Omega\mbox{ s.t. }u\in
W^{s,p}(\Omega^{\prime}),\
\int_{\mathbb{R}^{N}}\frac{|u(x)|^{p-1}}{(1+|x|)^{N+ps}}dx<\infty\bigg{\\}}.$

######  Definition 1.

Let $u:\mathbb{R}^{N}\to\mathbb{R}$ be a measurable function and
$0<m,\alpha<\infty$. We define the tail space and the nonlocal tail,
respectively, as below:

$\displaystyle L^{m}_{\alpha}(\mathbb{R}^{N})=\bigg{\\{}u\in L^{m}_{\rm
loc}(\mathbb{R}^{N}):\int_{\mathbb{R}^{N}}\frac{|u(x)|^{m}dx}{(1+|x|)^{N+\alpha}}<\infty\bigg{\\}},\
\
T_{m,\alpha}(u;x_{0},R)=\left(R^{\alpha}\int_{B_{R}(x_{0})^{c}}\frac{|u(y)|^{m}dy}{|x_{0}-y|^{N+\alpha}}\right)^{\frac{1}{m}}.$

Set $T_{m,\alpha}(u;R)=T_{m,\alpha}(u;0,R)$. We will follow the notation
$T_{p-1}(u;x,R):=T_{p-1,s_{1}p}(u;x,R)$ and
$T_{q-1}(u;x,R):=T_{q-1,s_{2}q}(u;x,R)$, unless otherwise stated.

###  2.2 Statements of main results

In this subsection, we state our main results. We start with the definition of
local weak solution.

######  Definition 2 (Local weak solution).

A function $u\in\mathcal{W}_{\rm loc}(\Omega)\cap
L^{p-1}_{s_{1}p}(\mathbb{R}^{N})\cap L^{q-1}_{s_{2}q}(\mathbb{R}^{N})$ is said
to be a local weak solution of problem ($\mathcal{P}$) if

$\displaystyle
A_{p}(u,\phi,\mathbb{R}^{N}\times\mathbb{R}^{N})+A_{q}(u,\phi,\mathbb{R}^{N}\times\mathbb{R}^{N})=\int_{\Omega}f\phi
dx,$

for all $\phi\in\mathcal{W}(\Omega)$ with compact support contained in
$\Omega$.

Our first main theorem is the following higher local Hölder continuity result.

######  Theorem 2.1.

Suppose that $(q-p+2)s_{2}<2$. Let $u\in\mathcal{W}_{\rm loc}(\Omega)\cap
L^{p-1}_{s_{1}p}(\mathbb{R}^{N})\cap L^{q-1}_{s_{2}q}(\mathbb{R}^{N})$ be a
locally bounded local weak solution to problem ($\mathcal{P}$). Then, for
every $\sigma\in(0,\Theta)$, $u\in C^{0,\sigma}_{\mathrm{loc}}(\Omega)$, where

$\displaystyle\Theta\equiv\Theta(p,s_{1},q,s_{2})=\begin{cases}\min\\{1,ps_{1}/(p-1)\\}\mbox{
if }qs_{2}<ps_{1}+2(1-s_{1}),\\\ \min\\{1,qs_{2}/(q-1)\\}\mbox{ if
}ps_{1}<qs_{2}.\end{cases}$

Moreover, for $B_{2\bar{R}_{0}}(x_{0})\Subset\Omega$ with
$\bar{R}_{0}\in(0,1)$, there holds

$\displaystyle[u]_{C^{\sigma}(B_{\bar{R}_{0}/2}(x_{0}))}\leq
C\big{(}K_{2}(u)(\|u\|_{W^{s_{1},p}(B_{\bar{R}_{0}}(x_{0}))}+1)\big{)}^{j_{\infty}}$

where $C=C(N,s_{1},p,s_{2},q,\sigma)>0$ is a constant and $K_{2}$ is given by

$\displaystyle
K_{2}=1+T_{p-1}(u;x_{0},\bar{R}_{0})^{p-1}+T_{q-1}(u;x_{0},\bar{R}_{0})^{q-1}+\|u\|^{\frac{(\ell_{1}+j_{\infty})(\ell_{1}-1)}{\ell_{1}-2}}_{L^{\infty}(B_{\bar{R}_{0}}(x_{0}))}+\|u\|^{q-1}_{L^{\infty}(B_{R_{0}})}+\|f\|_{L^{\infty}(B_{\bar{R}_{0}}(x_{0}))}$

with $\ell_{1}=\max\\{p,q\\}$ and $j_{\infty}\in\mathbb{N}$ depends only on
$N,\sigma$ and $(p,s_{1})$ or $(q,s_{2})$.

######  Remark 1.

We remark that for $1<q\leq p<\infty$, the conclusion of Theorem 2.1 holds for
some $\sigma<\min\\{\frac{ps_{1}}{p-1},\frac{qs_{2}}{q-1}\\}$. See Theorem 2.1
and Corollary 2.1 of [6] for details.

Next, we have the following global Hölder continuity result.

######  Theorem 2.2.

Suppose that $(q-p+2)s_{2}<2$. Let $u\in\mathcal{W}_{0}(\Omega)$ be a solution
to problem ($\mathcal{P}$) with $f\in L^{\infty}(\Omega)$. Then, for every
$\sigma\in(0,s_{1})$, $u\in C^{0,\sigma}(\overline{\Omega})$. Moreover,

$\displaystyle\|u\|_{C^{\sigma}(\overline{\Omega})}\leq C,$ (2.1)

where $C=C(\Omega,N,p,s_{1},q,s_{2},\sigma,\|f\|_{L^{\infty}(\Omega)})>0$ is a
constant (which depends as a non-decreasing function of
$\|f\|_{L^{\infty}(\Omega)}$).

######  Corollary 2.3.

Suppose that $2\leq q\leq p<\infty$. Let $u\in W^{s_{1},p}_{0}(\Omega)$ be a
solution to problem ($\mathcal{P}$) with $f(x):=f(x,u)$, a Carathéodory
function satisfying $|f(x,t)|\leq C_{0}(1+|t|^{p^{*}_{s_{1}}-1})$, where
$C_{0}>0$ is a constant. Then, $u\in C^{0,\sigma}(\overline{\Omega})$, for all
$\sigma\in(0,s_{1})$, and (2.1) holds.

Now, we mention our strong comparison theorem.

######  Theorem 2.4 (Strong Comparison principle).

Suppose that $1<q\leq p<\infty$. Let $u,v\in W^{s_{1},p}_{0}(\Omega)\cap
C(\overline{\Omega})$ be such that $0<v\leq u$ in $\Omega$ with $u\not\equiv
v$, and for some $K,K_{1}>0$, the following holds:

$(-\Delta)_{p}^{s_{1}}v+(-\Delta)_{q}^{s_{2}}v\leq(-\Delta)_{p}^{s_{1}}u+(-\Delta)_{q}^{s_{2}}u\leq
K\quad\mbox{and }(-\Delta)_{q}^{s_{2}}v\geq-K_{1},\ \ \ \mbox{weakly in
}\Omega.$

Then $u>v$ in $\Omega$. Moreover, for $s_{1}\neq q^{\prime}s_{2}$,
$\frac{u-v}{d^{s_{1}}}\geq C>0$ in $\Omega$.

##  3 Hölder regularity results

We first recall some boundedness results.

######  Proposition 3.1 (Local boundedness).

Suppose $1<q\leq p<\infty$. Let $u\in W^{s_{1},p}_{\rm loc}(\Omega)\cap
L^{p-1}_{s_{1}p}(\mathbb{R}^{N})\cap L^{q-1}_{s_{2}q}(\mathbb{R}^{N})$ be a
local weak solution to the problem ($\mathcal{P}$). Then, $u\in
L^{\infty}_{\mathrm{loc}}(\Omega)$, and the following holds

$\displaystyle\|u\|_{L^{\infty}(B_{r/2}(x_{0}))}\leq
C\left(\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{r}}|u|^{p}dx\right)^{1/p}+T_{p-1}(u;x_{0},\frac{r}{2})+T_{q-1}(u;x_{0},\frac{r}{2})^{\frac{q-1}{p-1}}+\|f\|_{L^{\infty}(B_{r})}^{1/(p-1)}+1,$

where $C(N,p,q,s_{1})>0$ is a constant.

Proceeding similar to [3, Theorem 3.3] and noticing that the terms
corresponding to the fractional $q$-Laplacian will be non-negative (using the
similar inequality, as in the fractional $p$-Laplacian, for different
$G_{\beta}$ in there), we can prove the following boundedness property.

######  Theorem 3.2.

Let $1<q\leq p<\infty$. Let $u\in W^{s_{1},p}_{0}(\Omega)$ be a weak solution
to problem ($\mathcal{P}$) with $f(x):=f(x,u)$ satisfying $|f(x,t)|\leq
C_{0}(1+|t|^{p^{*}_{s_{1}}-1})$, for all $t\in\mathbb{R}$ and a.e.
$x\in\Omega$, where $C_{0}>0$ is a constant. Then, $u\in L^{\infty}(\Omega)$.

######  Remark 2.

We remark that, as in [3, Remark 3.4], the quantity
$\|u\|_{L^{\infty}(\Omega)}$ depends only on the constants $C_{0}$, $N$, $p$,
$s_{1}$, $\|u\|_{W^{s_{1},p}_{0}(\Omega)}$ and the constant $M>0$ satisfying
$\int_{\\{|u|\geq M\\}}|u|^{p^{*}_{s_{1}}}<\epsilon$, for given
$\epsilon\in(0,1)$.

######  Corollary 3.3.

Suppose that $1<q\leq p<\infty$. Let $u_{\varepsilon}\in
W^{s_{1},p}_{0}(\Omega)$, for $\varepsilon\in(0,1)$, be the family of weak
solution to problem ($\mathcal{P}$) with
$f_{\epsilon}(x):=f_{\epsilon}(x,u_{\epsilon})$ satisfying
$|f_{\epsilon}(x,t)|\leq C_{0}(1+|t|^{p^{*}_{s_{1}}-1})$, for all
$t\in\mathbb{R}$ and a.e. $x\in\Omega$, where $C_{0}$ is independent of
$\epsilon$. Assume that the sequence
$\\{\|u_{\varepsilon}\|_{W^{s_{1},p}_{0}(\Omega)}\\}_{\varepsilon}$ is bounded
and $u_{\varepsilon}\to u_{0}$ in $L^{p^{*}_{s_{1}}}(\Omega)$, as
$\varepsilon\to 0$. Then the sequence
$\\{\|u_{\varepsilon}\|_{L^{\infty}(\Omega)}\\}_{\varepsilon}$ is also
bounded.

###  3.1 Interior regularity

Let $u:\mathbb{R}^{N}\to\mathbb{R}$ be a measurable function and
$h\in\mathbb{R}^{N}$, then we define

$\displaystyle
u_{h}(x)=u(x+h),\quad\delta_{h}u(x)=u_{h}(x)-u(x),\quad\delta^{2}_{h}u(x)=\delta_{h}(\delta_{h}u(x))=u_{2h}(x)+u(x)-2u_{h}(x).$

For $1\leq m<\infty$ and $u\in L^{m}(\mathbb{R}^{N})$, we set

$\displaystyle[u]_{\mathcal{B}^{\beta,m}_{\infty}(\mathbb{R}^{N})}:=\sup_{|h|>0}\bigg{\|}\frac{\delta^{2}_{h}u}{|h|^{\beta}}\bigg{\|}_{L^{m}(\mathbb{R}^{N})}\mbox{
for }\beta\in(0,2).$

Now, we prove our improved interior Hölder regularity result for local weak
solutions.
Sketch of the proof of Theorem 2.1: We first consider the case
$\sigma\in(0,s_{1})$. For $qs_{2}<ps_{1}+2(1-s_{1})$, we claim that, for every
$4h_{0}<R\leq R_{0}-5h_{0}$, there holds:

$\displaystyle\sup_{0<|h|<h_{0}}\bigg{\|}\frac{\delta^{2}_{h}u}{|h|^{s_{1}}}\bigg{\|}^{m+1}_{L^{m+1}(B_{R-4h_{0}})}\leq
CK_{2}(u,m)\Big{(}\sup_{0<|h|<h_{0}}\bigg{\|}\frac{\delta^{2}_{h}u}{|h|^{s_{1}}}\bigg{\|}^{m}_{L^{m}(B_{R+4h_{0}})}+1\Big{)},$
(3.1)

where $m\geq p$, $h_{0}=R_{0}/10$, $C=C(N,h_{0},p,m,s_{1})>0$ (which depends
inversely on $h_{0}$) and

$\displaystyle
K_{2}(u,m,R_{0}):=1+T_{p-1}(u;R_{0})^{p-1}+T_{q-1}(u;R_{0})^{q-1}+\|u\|^{\frac{m(p-1)}{p-2}}_{L^{\infty}(B_{R_{0}})}+\|u\|^{q-1}_{L^{\infty}(B_{R_{0}})}+\|f\|_{L^{\infty}(B_{R_{0}})}.$

Indeed, for $2\leq q\leq p<\infty$, (3.1) is proved in [7, Proposition 3.9].
For the other case, we set $S_{1}=\\{(x,y)\in B_{R}\times B_{R}\;:\;|x-y|\leq
1\\}$ and $S_{2}=(B_{R}\times B_{R})\setminus S_{1}$. Then, the proof, in this
case, runs similarly by noting the following (using the same notations of the
Proposition),

$\displaystyle\left(\iint_{S_{1}}+\iint_{S_{2}}\right)\frac{|u(x)-u(y)|^{q-2}}{|x-y|^{N+qs_{2}}}\big{|}\eta^{\frac{p}{2}}(x)-\eta^{\frac{p}{2}}(y)\big{|}^{2}\frac{|\delta_{h}u(x)|^{\beta+1}}{|h|^{1+\nu\beta}}dxdy$
$\displaystyle\leq
C\|u\|^{q-p+1}_{L^{\infty}(B_{R_{0}})}\iint_{S_{1}}\frac{|u(x)-u(y)|^{p-2}}{|x-y|^{N+ps_{1}+\alpha-2}}\frac{|\delta_{h}u(x)|^{\beta}}{|h|^{1+\nu\beta}}dxdy+C\|u\|^{q-1}_{L^{\infty}(B_{R_{0}})}\int_{B_{R}}\frac{|\delta_{h}u(x)|^{\beta}}{|h|^{1+\nu\beta}}dx$
$\displaystyle\leq
CK_{2}(u)[u]^{m}_{W^{\frac{s_{1}(p-2-\epsilon)}{p-2},m}(B_{R+h_{0}})}+CK_{2}(u)\Big{(}\int_{B_{R}}\frac{|\delta_{h}u(x)|^{\frac{\beta
m}{m-p+2}}}{|h|^{\frac{(1+\nu\beta)m}{m-p+2}}}dx+1\Big{)}$

where we have used Hölder’s and Young’s inequality together with the fact that
$qs_{2}\leq ps_{1}+\alpha$ with $\alpha<2(1-s_{1})$ and
$\epsilon\in(0,\frac{2-\alpha}{s_{1}}-2)$. Thus, $\tilde{I}_{11}(q)$ (hence
$I_{11}(q)$) is estimated as similar to $\tilde{I}_{11}(p)$. Set

$\displaystyle s_{1}-\sigma>\frac{N}{p+i_{\infty}},\quad
h_{0}=\frac{\bar{R}_{0}}{64i_{\infty}}\quad\mbox{for some
}i_{\infty}\in\mathbb{N}$

and define the following sequences

$\displaystyle m_{i}=p+i,\quad
R_{i}=\frac{7\bar{R}_{0}}{8}-4(2i+1)h_{0}\quad\mbox{for all
}i=0,\dots,i_{\infty}.$

We take $\psi\in C_{c}^{\infty}(B_{(5\bar{R}_{0})/8})$ such that

$0\leq\psi\leq 1,\quad\psi=1\mbox{ in
}B_{\bar{R}_{0}/2},\quad|\nabla\psi|\mbox{ and }|\nabla^{2}\psi|\leq C.$

Using the discrete Leibniz rule on $\delta^{2}_{h}$, we obtain

$\displaystyle[u\psi]_{\mathcal{B}^{s_{1},m_{i_{\infty}}}_{\infty}(\mathbb{R}^{N})}\leq
C\Big{[}\sup_{0<|h|<h_{0}}\bigg{\|}\frac{\delta^{2}_{h}u}{|h|^{s_{1}}}\bigg{\|}_{L^{m_{i_{\infty}}}(B_{(3\bar{R}_{0})/4})}+\|u\|_{L^{m_{i_{\infty}}}(B_{(3\bar{R}_{0})/4})}\Big{]}.$

The first term of the right hand side on the above expression is estimated on
account of (3.1). Therefore, employing the embedding result of the Besov
spaces into the Hölder spaces, we get that $u\in
C^{0,\sigma}_{\mathrm{loc}}(\Omega)$, for all $\sigma\in(0,s_{1})$. For the
case $2\leq p<q$ and $ps_{1}<qs_{2}$, we proceed exactly as above by
interchanging the role of $(p,s_{1})$ with $(q,s_{2})$ and the corresponding
spaces. In this case, $\tilde{I}_{11}(p)$ is estimated as above by choosing
$\epsilon>0$ such that $\epsilon<\frac{2}{s_{2}}-q+p-2$.
The higher regularity result follows by using the above almost $s_{1}$ (or
$s_{2}$)-Hölder continuity result and proceeding on the similar lines of the
proof of [1, Theorem 5.2] (with minor modification as in the proof above).
$\square$

###  3.2 Boundary regularity and maximum principle

In this subsection, we prove the boundary behavior of the weak solutions. For
$\alpha,\rho>0$ and $\kappa\geq 0$, we set

$\displaystyle d_{e}(x)=\begin{cases}d(x)&\mbox{if }x\in\Omega,\\\
-d(x)&\mbox{if }x\in(\Omega^{c})_{\rho},\\\
-\rho&\mbox{otherwise},\end{cases}\quad\overline{w}_{\rho}(x)=\begin{cases}(d_{e}(x)+\kappa^{1/\alpha})_{+}^{\alpha}\quad&\mbox{if
}x\in\Omega\cup(\Omega^{c})_{\rho},\\\ 0\qquad&\mbox{otherwise},\end{cases}$

where
$(\Omega^{c})_{\rho}:=\\{x\in\Omega^{c}:\mathrm{dist}(x,\partial\Omega)<\rho\\}$.
Sketch of the proof of Theorem 2.2: We proceed as below.

  * (a)

By flattening the boundary $\partial\Omega$ and using suitable
$C^{1,1}(\mathbb{R}^{N},\mathbb{R}^{N})$ diffeoemorphisms, we prove that:
there exist $\kappa_{1},\varrho_{1}>0$ such that for all
$\kappa\in[0,\kappa_{1})$ and $\varrho\in(0,\varrho_{1})$,

$\displaystyle(-\Delta)_{p}^{s_{1}}\overline{w}_{\rho}\begin{cases}\geq
C_{1}(d+\kappa^{1/\alpha})^{-(ps_{1}-\alpha(p-1))}\quad\mbox{for all
}\alpha\in(0,s_{1}),\\\ =h\quad\mbox{for all }\alpha\in[s_{1},1)\mbox{ with
}\alpha\neq p^{\prime}s_{1}\end{cases}\quad\mbox{weakly in }\Omega_{\varrho},$

where $C_{1}>0$ is a constant and $h\in L^{\infty}(\Omega_{\varrho_{1}})$
(both are independent of $\kappa\in(0,1)$). Further, for all $\kappa>0$ and
$\alpha\in(0,s_{1})$,
$\overline{w}_{\rho}\in\widetilde{W}^{s_{1},p}(\Omega_{\varrho_{1}})$, and for
$k=0$, $\overline{w}_{\rho}\in\widetilde{W}^{s_{1},p}(\Omega_{\varrho_{1}})$,
whenever $\alpha>s_{1}-1/p$.

  * (b)

For $\Gamma>1$, $\max\\{s_{1}-1/p,s_{2}-1/q\\}<\alpha<s_{1}$ and $\varrho>0$
(sufficiently small), we have, weakly in $\Omega_{\varrho}$,

$\displaystyle(-\Delta)_{p}^{s_{1}}(\Gamma
d^{\alpha})+(-\Delta)_{q}^{s_{2}}(\Gamma d^{\alpha})$ $\displaystyle\geq
C_{5}\Gamma^{p-1}d^{-(ps_{1}-\alpha(p-1))}-\Gamma^{q-1}\|h\|_{L^{\infty}(\Omega_{\varrho})}$
$\displaystyle\geq C_{6}\Gamma^{p-1}d^{-(ps_{1}-\alpha(p-1))}.$

Then, employing the weak comparison principle in $\Omega_{\varrho}$, for
suitable $\Gamma$, we get that $u\leq\Gamma d^{\alpha}$ in $\Omega$.
Subsequently, we perform a similar process for $-u$ also.

  * (c)

The proof of the Hölder continuity can be completed by taking into account
Theorem 2.1 and the boundary behavior presented in Step (b). $\square$

######  Remark 3.

We remark that in Theorem 2.2, the choice of $\sigma$ can be optimal (that is,
$\sigma=s_{1}$) for the case $s_{1}=s_{2}$ or $s_{1}>q^{\prime}s_{2}$. Indeed,
for $s_{1}=s_{2}$, we can show that the barrier function as constructed in [9,
Lemma 4.3] satisfies $(-\Delta)_{q}^{s_{1}}w\geq 0$ weakly in
$B_{r}(e_{N})\setminus\overline{B_{1}}$. Thus, for appropriate choice of
$\Gamma>1$, the Step (b) above can be improved. Similar arguments apply to the
case $s_{1}>q^{\prime}s_{2}$ with a careful reading of the proof of [7, Lemma
3.12].

Proof of Corollary 2.3: When $f(x):=f(x,u)$, on account of Theorem 3.2 and
Remark 2, we observe that

$|f(x,u)|\leq C_{0}(1+|u|^{p^{*}_{s_{1}}-1})\leq
C_{0}(1+\|u\|_{L^{\infty}(\Omega)}^{p^{*}_{s_{1}}-1})=:K>0.$

Thus, the required result, in this case, follows from Theorem 2.2. $\square$

Next we state our strong maximum principle. The proof is contained in [6] and
done by proving that continuous weak super-solutions are viscosity super-
solutions.

######  Theorem 3.4.

Suppose that $1<q\leq p<\infty$. Let $g\in C(\mathbb{R})\cap BV_{\rm
loc}(\mathbb{R})$ and let $u\in W^{s_{1},p}_{0}(\Omega)\cap
C(\overline{\Omega})$ be such that

$\displaystyle(-\Delta)_{p}^{s_{1}}u+(-\Delta)_{q}^{s_{2}}u+g(u)\geq
g(0)\quad\mbox{weakly in }\Omega.$

Further, assume that $u\not\equiv 0$ with $u\geq 0$ in $\Omega$. Then, there
exists $c_{1}>0$ such that $u\geq c_{1}{\rm
dist}(\cdot,\partial\Omega)^{s_{1}}$ in $\Omega$.

Sketch of the proof of Theorem 2.4: By continuity and the fact that
$u\not\equiv v$, we can find $x_{0}\in\Omega$, $\rho,\epsilon>0$ such that
$B_{\rho}(x_{0})\subset\Omega$ and

$\displaystyle\sup_{B_{\rho}(x_{0})}v<\inf_{B_{\rho}(x_{0})}u-\epsilon/2.$
(3.2)

For $\Gamma>1$ and for all $x\in\mathbb{R}^{N}$, we define

$\displaystyle w_{\Gamma}(x)=\begin{cases}\Gamma v(x)&\mbox{if }x\in
B_{\rho/2}^{c}(x_{0})\\\ u(x)&\mbox{if }x\in B_{\rho/2}(x_{0}).\end{cases}$

Taking into account the nonlocal super-position principle [8, Lemma 2.5], we
have, weakly in $\Omega\setminus B_{\rho}(x_{0})$,

$\displaystyle(-\Delta)_{p}^{s_{1}}w_{\Gamma}+(-\Delta)_{q}^{s_{2}}w_{\Gamma}$
$\displaystyle\leq(-\Delta)_{p}^{s_{1}}u+(-\Delta)_{q}^{s_{2}}u+(\Gamma^{p-1}-1)K+(\Gamma^{p-1}-\Gamma^{q-1})K_{1}-C_{1}\epsilon^{p-1}-C_{2}\epsilon^{q-1}.$

We can choose $\Gamma>1$ (close to $1$) to employ the weak comparison
principle ([8, Proposition 2.6]), consequently, we get $w_{\Gamma}\leq u$ in
$\Omega$. Hence, using (3.2) and Theorem 3.4, we obtain $u\geq\Gamma v>v$ in
$\Omega$, and $\frac{u-v}{d^{s_{1}}}\geq\frac{(\Gamma-1)v}{d^{s_{1}}}\geq C>0$
in $\Omega$. $\square$

##  4 Applications

We consider the problem ($\mathcal{P}$) with the choice $f(x):=f(x,u)$, where
$f:\Omega\times\mathbb{R}\to\mathbb{R}$ is a Carathéodory function satisfying
the following:

  * (A1)

$|f(x,t)|\leq C_{0}(1+|t|^{r-1})$, for a.a. $x\in\Omega$ and all
$t\in\mathbb{R}$, where $C_{0}>0$ is a constant and $r\in(1,p^{*}_{s_{1}}]$.

  * (A2)

For a.a. $x\in\Omega$, $f(x,t)t\leq 0$, for all $t\in[-\varsigma,\varsigma]$
($\varsigma>0$) and $f(x,t)t\geq-c_{1}t^{p}$ ($c_{1}>0$) for all
$t\in\mathbb{R}$.

  * (A3)

For $F(x,t):=\int_{0}^{t}f(x,\tau)d\tau$,
$\lim_{|t|\to\infty}\frac{F(x,t)}{|t|^{p}}=\infty$ uniformly for a.a.
$x\in\Omega$.

  * (A4)

Let $r\in(p,p^{*}_{s_{1}})$, there exists
$\nu\in\big{(}(r-p)\max\\{N/(ps_{1}),1\\},p^{*}_{s_{1}}\big{)}$ such that

$\displaystyle\lim_{|t|\to\infty}\frac{f(x,t)t-pF(x,t)}{|t|^{\nu}}>0\quad\mbox{uniformly
a.e. }x\in\Omega.$

One example for $f$ satisfying (A1)-(A4) is given by
$f(x,u)=-c_{1}|t|^{p-2}t+|t|^{r-2}t$. The Euler functional
$\mathcal{J}:W^{s_{1},p}_{0}(\Omega)\rightarrow\mathbb{R}$ associated to
problem ($\mathcal{P}$) is given by

$\mathcal{J}(u)=\frac{1}{p}\|u\|_{W^{s_{1},p}_{0}(\Omega)}^{p}+\frac{1}{q}\|u\|_{W^{s_{2},q}_{0}(\Omega)}^{q}-\int_{\Omega}F(x,u)dx.$

First we prove the following Sobolev versus Hölder minimizer result.

######  Theorem 4.1.

Suppose that $2\leq q\leq p<\infty$ and (A1) holds. Let $u_{0}\in
W^{s,p}_{0}(\Omega)$, then for all $\alpha\in(0,s_{1})$, the following are
equivalent

  1. (i)

there exists $\sigma>0$ such that $\mathcal{J}(u_{0}+v)\geq\mathcal{J}(u_{0})$
for all $v\in W^{s_{1},p}_{0}(\Omega)$,
$\|v\|_{W^{s_{1},p}_{0}(\Omega)}\leq\sigma$,

  2. (ii)

there exists $\omega>0$ such that $\mathcal{J}(u_{0}+v)\geq\mathcal{J}(u_{0})$
for all $v\in W^{s_{1},p}_{0}(\Omega)\cap C^{0,\alpha}(\overline{\Omega})$
with $\|v\|_{C^{\alpha}(\overline{\Omega})}\leq\omega$.

Proof.  From $(ii)$ and the density argument, we get that
$\langle\mathcal{J}^{\prime}(u_{0}),\phi\rangle=0$ for all $\phi\in
W^{s_{1},p}_{0}(\Omega)$. Consequently, Theorem 2.2 implies that $u_{0}\in
C^{0,\alpha}(\overline{\Omega})$. To prove $(i)$, on the contrary assume that
there exists $\tilde{u}_{n}\in W^{s_{1},p}_{0}(\Omega)$ such that
$\tilde{u}_{n}\to u_{0}$ in $W^{s_{1},p}_{0}(\Omega)$ and
$\mathcal{J}(\tilde{u}_{n})<\mathcal{J}(u_{0})$ for all $n\in\mathbb{N}$. Set

$\mathcal{K}(v)=\frac{1}{p^{*}_{s_{1}}}\int_{\Omega}|v|^{p^{*}_{s_{1}}},\
\varepsilon_{n}:=\mathcal{K}(\tilde{u_{n}}-u_{0})\ \mbox{and }S_{n}:=\\{u\in
W^{s_{1},p}_{0}(\Omega)\ :\ \mathcal{K}(u-u_{0})\leq\varepsilon_{n}\\}.$

By the continuous embedding $W^{s_{1},p}_{0}(\Omega)\hookrightarrow
L^{p^{*}_{s_{1}}}(\Omega)$, we see that $\varepsilon_{n}\to 0$ and hence
$S_{n}$ is a closed convex subset of $W^{s_{1},p}_{0}(\Omega)$. Next, for all
$t\in\mathbb{R}$ and $k>0$, set $[t]_{k}=\mathrm{sign}(t)\min\\{|t|,k\\}$ and
$f_{k}(x,t):=f(x,[t]_{k})$ with $F_{k}(x,t):=\int_{0}^{t}f_{k}(x,\tau)d\tau$.
Then, on account of the Lebesgue dominated convergence theorem, for fixed
$n\in\mathbb{N}$ and
$\sigma_{n}\in(0,\mathcal{J}(u_{0})-\mathcal{J}(\tilde{u}_{n}))$, there exists
$k_{n}>\|u_{0}\|_{L^{\infty}(\Omega)}+1$ such that

$\displaystyle\Big{|}\int_{\Omega}F_{n}(x,\tilde{u}_{n})dx-\int_{\Omega}F(x,\tilde{u}_{n})dx\Big{|}<\sigma_{n},$

where $F_{n}=F_{k_{n}}$. Furthermore, we define

$\displaystyle\mathcal{J}_{n}(u)=\frac{1}{p}\|u\|_{W^{s_{1},p}_{0}(\Omega)}^{p}+\frac{1}{q}\|u\|_{W^{s_{2},q}_{0}(\Omega)}^{q}-\int_{\Omega}F_{n}(x,u)\leavevmode\nobreak\
dx.$

From the structure of the function $F_{n}$, it is clear that there exists a
minimizer $u_{n}\in S_{n}$ for $\mathcal{J}_{n}$. Moreover, by the choice of
$\sigma_{n}$ and $k_{n}$, we see that

$\displaystyle\mathcal{J}_{n}(u_{n})\leq\mathcal{J}_{n}(\tilde{u}_{n})\leq\mathcal{J}(\tilde{u}_{n})+\sigma_{n}<\mathcal{J}(u_{0})=\mathcal{J}_{n}(u_{0}).$
(4.1)

It is clear that $\mathcal{J}_{n}$ is Gǎteaux differentiable at $u_{n}$.
Therefore, there exists $\mu_{n}\leq 0$ such that

$(\mathcal{P}_{n})\left\\{\begin{array}[]{rllll}(-\Delta)^{s_{1}}_{p}u_{n}+(-\Delta)^{s_{2}}_{q}u_{n}&=f_{n}(x,u_{n})+\mu_{n}|u_{n}-u_{0}|^{p^{*}_{s_{1}}-2}(u_{n}-u_{0})\;\text{
in }\Omega,\\\ u_{n}&=0\quad\text{ in
}\mathbb{R}^{N}\setminus\Omega.\end{array}\right.$

If $\inf_{n}\mu_{n}:=l>-\infty$, then from the fact that $u_{0}\in
L^{\infty}(\Omega)$, we have

$\displaystyle|f_{n}(x,u_{n})+\mu_{n}|u_{n}-u_{0}|^{p^{*}_{s_{1}}-2}(u_{n}-u_{0})|\leq
C(1+|u_{n}|^{p^{*}_{s_{1}}-1}).$

If $\inf_{n}\mu_{n}:=-\infty$, there exists $M>0$ (independent of $n$) such
that

$f_{n}(x,t)+\mu_{n}|t-u_{0}(x)|^{p^{*}_{s_{1}}-2}(t-u_{0}(x))<0\quad\mbox{for
a.a. }x\in\Omega,\mbox{ and all }t\in(M,\infty).$

This implies that $u_{n}\leq M$ for all $n\in\mathbb{N}$. Further, since
$\mathcal{J}^{\prime}(u_{0})=0$, we take
$w=|u_{n}-u_{0}|^{\kappa-1}(u_{n}-u_{0})$ as a test function and using [11,
Lemma 2.3] (consequently, the difference of terms involving $A_{q}$, below, is
non-negative), we have

$\displaystyle(C\kappa^{p-1})^{-1}\|(u_{n}-u_{0})^{\frac{p-1+\kappa}{p}}\|^{p}_{W^{s_{1},p}_{0}(\Omega)}$
$\displaystyle\leq
A_{p}(u_{n},[u_{n}-u_{0}]^{\kappa})-A_{p}(u_{0},[u_{n}-u_{0}]^{\kappa})$
$\displaystyle\quad+A_{q}(u_{n},[u_{n}-u_{0}]^{\kappa})-A_{q}(u_{n},[u_{n}-u_{0}]^{\kappa})$
$\displaystyle=\int_{\Omega}(f_{n}(x,u_{n})-f(x,u_{0}))[u_{n}-u_{0}]^{\kappa}+\mu_{n}\int_{\Omega}|u_{n}-u_{0}|^{p^{*}_{s_{1}}-1+\kappa}.$

Noting the uniform bound $\|u_{n}\|_{L^{\infty}(\Omega)}\leq M$ and using
Hölder’s inequality, and subsequently passing to the limit $\kappa\to\infty$,
we obtain

$\displaystyle-\mu_{n}\|u_{n}-u_{0}\|^{p^{*}_{s_{1}}-1}_{L^{\infty}(\Omega)}\leq
C$

where $C>0$ is a constant independent of $n$. Thus, in all the cases, we
obtain

$|f_{n}(x,u_{n})+\mu_{n}|u_{n}-u_{0}|^{p^{*}_{s_{1}}-2}(u_{n}-u_{0})|\leq
C(1+|u_{n}|^{p^{*}_{s_{1}}-1})\quad\mbox{for all }n\in\mathbb{N}.$

Moreover, from the construction of $u_{n}$, it is clear that
$\\{\|u_{n}\|_{W^{s_{1},p}_{0}(\Omega)}\\}_{n}$ remains bounded. Then,
applying Corollary 3.3, we have $\|u_{n}\|_{L^{\infty}(\Omega)}\leq C$, where
$C>0$ is a constant independent of $n$. Consequently, from Corollary 2.3, we
deduce that $\|u_{n}\|_{C^{\alpha}(\overline{\Omega})}\leq C$, for some
positive constant $C$ independent of $n$, and all $\alpha\in(0,s_{1})$.
Therefore, by Arzela-Ascoli’s theorem, $u_{n}\to u_{0}$ in
$C^{0,\alpha}(\overline{\Omega})$, for all $\alpha<s_{1}$. Thus, for
sufficiently large $n$, we have
$\|u_{n}-u_{0}\|_{C^{\alpha}(\overline{\Omega})}\leq\omega$, and since $u_{n}$
is uniformly bounded in $L^{\infty}(\Omega)$,
$\mathcal{J}_{n}(u_{n})=\mathcal{J}(u_{n})$, for sufficiently large $n$. This
along with (4.1) contradicts the fact that $u_{0}$ is a minimizer for
$\mathcal{J}$ in $W^{s_{1},p}_{0}(\Omega)\cap
C^{0,\alpha}(\overline{\Omega})$.
The proof of the other implication is standard. $\square$

######  Theorem 4.2.

Suppose that $2\leq q\leq p<\infty$. Then, there exist at least three non-
trivial solutions $u\in W^{s_{1},p}_{0}(\Omega)\cap
C^{0,\alpha}(\overline{\Omega})$, for all $\alpha\in(0,s_{1})$, to problem
($\mathcal{P}$) with $f(x):=f(x,u)$ satisfying (A1)-(A4). Moreover, if $u$ is
non-negative and $s_{1}\neq q^{\prime}s_{2}$, then $u\geq cd^{s_{1}}$ in
$\Omega$.

Sketch of the Proof: We consider the truncation of the nonlinear term as
$f_{\pm}(x,t)=f(x,\pm t^{\pm})$ with
$F_{\pm}(x,t):=\int_{0}^{t}f_{\pm}(x,\tau)d\tau$ and the corresponding Euler
functionals as $\mathcal{J}_{\pm}$. By using Theorem 4.1, we can prove that
$0$ is a local minimizer for $\mathcal{J}_{+}$ in $W^{s_{1},p}_{0}(\Omega)$
topology and it satisfies the mountain pass geometry. Thus, we obtain a
positive solution $u_{+}$ to problem ($\mathcal{P}$). Similar procedure yields
a negative solution $u_{-}$. Subsequently, by using topological tools (such as
critical groups and Morse theory, see [5] for the linear operator case), we
establish a third solution of undetermined sign nature. $\square$

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