# Non-homogeneous fourth order elliptic inequalities with a convolution term
motivated by the suspension bridge problem

Zhe Yu111ORCID: 0000-0002-2261-4981

###### Abstract

We are concerned with the study of the twin non-local inequalities featuring
non-homogeneous differential operators

$\displaystyle-\Delta^{2}u+\lambda\Delta
u\geq(K_{\alpha,\beta}*u^{p})u^{q}\quad\text{ in }\mathbbm{R}^{N}(N\geq 1),$

and

$\displaystyle\;\;\,\Delta^{2}u-\lambda\Delta
u\geq(K_{\alpha,\beta}*u^{p})u^{q}\quad\text{ in }\mathbbm{R}^{N}(N\geq 1),$

with parameters $\lambda,p,q>0$, $0\leq\alpha\leq N$ and $\beta>\alpha-N$. In
the above inequalities the potential $K_{\alpha,\beta}$ is given by
$K_{\alpha,\beta}(x)=|x|^{-\alpha}\log^{\beta}(1+|x|)$ while
$K_{\alpha,\beta}*u^{p}$ denotes the standard convolution operator in
$\mathbbm{R}^{N}$. We discuss the existence and non-existence of non-negative
solutions in terms of $N,p,q,\lambda,\alpha$ and $\beta$.

Keywords: Higher order differential operators; suspension bridge problem; non-
local inequalities; existence and nonexistence of solutions

2020 AMS MSC: 35J30; 35A23; 35A01; 42B37; 35B33

##  1 Introduction

This paper is concerned with the study of non-negative solutions to the
following inequalities involving the bi-harmonic operator and with the
convolution term:

$-\Delta^{2}u+\lambda\Delta u\geq(K_{\alpha,\beta}*u^{p})u^{q}\quad\text{in
}\;\;\mathbbm{R}^{N}(N\geq 1),$ ($P^{-}$)

and

$\;\;\,\Delta^{2}u-\lambda\Delta
u\geq(K_{\alpha,\beta}*u^{p})u^{q}\quad\text{in }\;\;\mathbbm{R}^{N}(N\geq
1).$ ($P^{+}$)

The above inequalities assume that $N\geq 1$, $\lambda,p,q>0$ and

$K_{\alpha,\beta}(x)=\lvert x\rvert^{-\alpha}\log^{\beta}(1+\lvert
x\rvert),\quad\text{where }0\leq\alpha\leq N,\,\beta>\alpha-N.$ (1.1)

We note that $K_{\alpha,\beta}(x)\in L^{1}_{loc}(\mathbbm{R}^{N})$ and

$\displaystyle K_{\alpha,\beta}(x)\simeq|x|^{-\alpha+\beta}\mbox{ as }|x|\to
0,$ $\displaystyle K_{\alpha,\beta}(x)\simeq|x|^{-\alpha}\log(|x|)\mbox{ as
}|x|\to\infty.$

For $x\in\mathbbm{R}^{N}$ large, we observe that $K_{\alpha,\beta}$ is a
decreasing function of the radial variable $|x|$. The quantity
$K_{\alpha,\beta}*u^{p}$ represents the convolution operation in
$\mathbbm{R}^{N}$ given by

$(K_{\alpha,\beta}\ast
u^{p})(x)=\int_{\mathbbm{R}^{N}}\frac{u^{p}(y)\log^{\beta}(1+\lvert
x-y\rvert)}{\lvert x-y\rvert^{\alpha}}dy\,,\quad\text{where
}x\in\mathbbm{R}^{N}.$

This paper will examine classical non-negative solutions of ($P^{-}$)
(respectively of ($P^{+}$)), that is, functions
$u\in\mathcal{C}^{4}(\mathbbm{R}^{N})$ such that

$u(x)\geq 0\;,\;\;\big{(}K_{\alpha,\beta}*u^{p}\big{)}(x)<\infty\quad\mbox{for
all }\;x\in\mathbbm{R}^{N},$ (1.2)

and $u$ satisfies ($P^{-}$) (respectively of ($P^{+}$)) pointwise in
$\mathbbm{R}^{N}$.

For $\alpha\in(0,N)$ and $\beta=0$ the potential $K_{\alpha,0}$ becomes the
standard Riesz potential which plays a crucial role in potential theory and
partial differential equations. Beyond this setting, the case $\alpha=N$ is
related to the exponential transform of a domain as introduced in [16] (see
also [29]). Precisely, given a domain $D\subset\mathbbm{R}^{N}$, we set

$E_{D}(x)=\exp\left\\{-\frac{2}{\lvert\partial
B_{1}\rvert}\int_{C}K_{N,0}(x-y)dy\right\\}\text{ for all
}x\in\mathbbm{R}^{N}\setminus\overline{D}.$

By polarization, this definition is extended to points $x\in D$ by

$H_{D}(x)=\lim\limits_{R\to\infty}\frac{1}{R^{2}E_{{B_{R}}\setminus D(x)}}.$

We ascertain from [16] that $E_{\Omega}$ is superharmonic in
$\mathbbm{R}^{N}\setminus D$, while $H_{D}$ is subharmonic in $D$. Also,
$H_{D}$ has explicit expression for common domains $D$ whose boundary is
described by quadratic polynomials. In particular, when $D$ is the unit ball
$B_{1}$ in $\mathbbm{R}^{N}$, one has $H_{B_{1}}=(1-\lvert x\rvert^{2})^{-1}$.
The exponential transform $E_{D}$ was used in [17] to study the regularity of
free boundaries in two dimensions. Furthermore, the potential of type
$K_{N,\beta}$ are related to logarithmic Besov spaces [4, 21].

The study of varying non-homogeneous higher order differential operators has
attracted a significant attention in recent years. Results involving the
operator $\Delta^{2}+\Delta$ can be found in [8, 20, 23]. In the present work,
we study a class of differential inequalities involving the operators
$-\Delta^{2}+\lambda\Delta$ and $\Delta^{2}-\lambda\Delta$, $\lambda>0$. The
operator $\Delta^{2}u-\lambda\Delta u$ appears in Mechanics, in the study of
suspension bridges in the early 20th century. In dimension $N=1$, we have

$\Delta^{2}u-\lambda\Delta u=u_{xxxx}-\lambda u_{xx},$

and the suspension bridges equation is

$u_{xxxx}-(H+h(u))u_{xx}+\frac{q}{H}h(u)=p(x),\quad\text{where }0\leq x\leq
L.$ (1.3)

In above equation, $L$ is the length of the beam, $q$ and $p(x)$ are the dead
and live loads per unit length applied to the beam, and the non-local term
$h(u)$ is the additional tension in the cable produced by the live load $p$.
The equation (1.3) is called the Melan equation in suspension bridges theory
introduced in 1913 [22] (see also [2, 9, 11] for further results on this topic
as well as [1, 10] for suspension bridge equation with non-local terms). In
1980s, the authors demonstrated that the nonlinear feature of such non-
homogeneous operator provided a good fit in the study of travelling waves in
suspension bridges in [24] and [18]. Later, in [19] and [25], the problem with
a more general nonlinear non-negative term $f(x,u)$, such that

$\left\\{\begin{split}&\Delta^{2}u-\lambda\Delta u=f(x,u)&\quad\text{in
}&B_{R},\\\ &\Delta u=0,u=0&\quad\text{on }&\partial B_{R},\end{split}\right.$

is proven to have solutions in $\mathbbm{R}^{N}$.

Local and non-local inequalities involving the operator
$\Delta^{2}-\lambda\Delta$ can be found in [5, 6]. For instance, the local
case

$\Delta^{2}u-\lambda\Delta u\geq|u|^{q}\quad\text{in
}\mathcal{D}^{\prime}(\mathbbm{R}^{N}),$

was discussed in [5]. It is shown in [5] that the above inequality has no
solutions whenever $1<q\leq\frac{N}{N-2}$. Also, the non-local inequality

$\Delta^{2}u-\lambda\Delta u\geq(|x|^{-\alpha}*|u|^{p})|u|^{q}\quad\text{in
}\mathcal{D}^{\prime}(\mathbbm{R}^{N}),$ (1.4)

was investigated in [5] where it was obtained (see [5, Theorem 8.7]) that if
$p\geq 1$, $q>1$, $\alpha\in(0,N)$ then (1.4) has a non-negative solution if
and only if

$\min\\{p,q\\}>\frac{N-\alpha}{N-2}\quad\mbox{ and }\quad
p+q>\frac{2N-\alpha}{N-2}.$ (1.5)

The approach in [5] relies essentially on integral representation formulae for
non-homogeneous differential equations. In contrast, our approach is based on:

  * •

Obtaining local properties of solutions. We show that any non-negative
solution $u$ of ($P^{-}$) satisfies $\Delta u\geq 0$ in $\mathbbm{R}^{N}$ and
any non-negative solution $u$ of ($P^{+}$) satisfies $-\Delta u\geq 0$ in
$\mathbbm{R}^{N}$.

  * •

We use Harnack inequalities [31] in order to derive estimates for the non-
existence of a solution.

  * •

We provide further integral estimates involving logarithmic terms (see Lemma
2.1 below) that extends previous results in the literature (see [3, 13, 15,
27]).

We shall see that the presence of the logarithmic term in ($P^{+}$) leads to
different conditions for the existence of a non-negative solution.

We first discuss the non-existence of non-negative solutions to ($P^{-}$). To
the best of our knowledge, this is the first result concerning the
differential operator $-\Delta^{2}+\lambda\Delta$.

######  Theorem 1.1.

(Non-existence for ($P^{-}$))
Assume that $0<\alpha<N$, $p,q>0$ and $\lambda>0$. Then ($P^{-}$) has no non-
negative solutions in $\mathbbm{R}^{N}$ if one of following conditions hold:

  * (i)

$p\geq 1$;

  * (ii)

$p<1$ and $u$ is bounded;

  * (iii)

$u$ is radial.

From Theorem 1.1, we notice that in most of the cases related to $p\geq 1$ and
$p<1$, the inequality ($P^{-}$) has no non-negative solutions.

The corresponding non-existence part for ($P^{+}$) is presented in our next
result below.

######  Theorem 1.2.

(Non-existence for ($P^{+}$))
Assume $p,q>0$ and $\lambda>0$. Then ($P^{+}$) has no non-negative solutions
in $\mathbbm{R}^{N}$ if one of following conditions hold:

  1. (i)

$1\leq N\leq 2;$

  2. (ii)

$N\geq 3$, $0\leq\alpha\leq 2\quad$and $1\leq p<\frac{N-\alpha}{N-2};$

  3. (iii)

$N\geq 3$, $0\leq\alpha\leq 2$, $p=\frac{N-\alpha}{N-2}\quad$and
$\beta\geq-1;$

  4. (iv)

$N\geq 3$, $p\geq 1$, $p+q<\frac{2N-\alpha}{N-2}\quad$and $\beta>\alpha-N;$

  5. (v)

$N\geq 3$, $p\geq 1$, $p+q=\frac{2N-\alpha}{N-2}\quad$and
$\beta>\frac{1}{p+q}-1;$

  6. (vi)

$N\geq 3$, $0\leq\alpha<2\quad$and $1\leq q<\frac{N-\alpha}{N-2};$

  7. (vii)

$N\geq 3$, $0\leq\alpha<2$, $q=\frac{N-\alpha}{N-2}\quad$and
$\beta>\frac{1}{q}-1;$

  8. (viii)

$N\geq 3$, $0\leq\alpha<2$, $p=\frac{N}{N-2}$,
$q=\frac{N-\alpha}{N-2}\quad$and $\beta>-2+\frac{1}{q};$

  9. (ix)

$N\geq 3$, $0\leq\alpha<2$, $p=\frac{N-\alpha}{N-2}$,
$q=\frac{N}{N-2}\quad$and $\beta>-2+\frac{1}{q}.$

We next turn to existence results of non-negative solutions to ($P^{+}$).
Under the assumption of $N\geq 3$ and $\alpha,\beta$ satisfy (1.1), we provide
essentially optimal conditions in terms of $p,q,\alpha,\beta$ and $\lambda$
for the existence of a non-negative solution. Precisely, we have:

######  Theorem 1.3.

(Existence for ($P^{+}$))
Assume $0\leq\alpha<N$, $\beta>\alpha-N$, $p>0$ and $q>0$. Then, there exists
$\lambda>0$ so that ($P^{+}$) has non-negative solutions in $\mathbbm{R}^{N}$
if one of following conditions hold:

  1. (i)

$p>\frac{N-\alpha}{N-2}$, $q>\frac{N-\alpha}{N-2}$,
$p+q>\frac{2N-\alpha}{N-2}\quad$and $\beta>\alpha-N;$

  2. (ii)

$p=\frac{N-\alpha}{N-2}$, $q>\frac{N}{N-2}\quad$and $\beta<-1;$

  3. (iii)

$p>\frac{N}{N-2}$, $q=\frac{N-\alpha}{N-2}\quad$and $\beta<-1$;

  4. (iv)

$p>\frac{N-\alpha}{N-2}$, $q>\frac{N-\alpha}{N-2}$,
$p+q=\frac{2N-\alpha}{N-2}\quad$and $\beta<-1$;

  5. (v)

$p=\frac{N-\alpha}{N-2}$, $q=\frac{N}{N-2}\quad$and $\beta<-2$;

  6. (vi)

$p=\frac{N}{N-2}$, $q=\frac{N-\alpha}{N-2}\quad$and $\beta<-2.$

The outcome of Theorem 1.2 and Theorem 1.3 can be extended to the extremal
case $\alpha=N$. In this setting, condition (1.1) yields $\beta>0$. We find
that Theorem 1.3(i) provides the optimal condition for the existence of non-
negative solutions to ($P^{+}$). More exactly, we have

######  Theorem 1.4.

Assume $N\geq 3$, $p\geq 1$, $q>0$, $\alpha=N$ and $\beta>0$. There exists
$\lambda>0$ so that ($P^{+}$) has a non-negative solution in $\mathbbm{R}^{N}$
if and only if

$p+q>\frac{N}{N-2}.$

Moreover, for $p,\,q\geq 1$, the results in Theorem 1.2 and Theorem 1.3 are
nearly optimal. Precisely, we conclude that conditions (i)-(vi) in Theorem 1.3
are optimal for the existence of a non-negative solution if $0\leq\alpha<N$
and $\beta<-2$ or $\beta>-1+\frac{1}{q}$.

######  Corollary 1.5.

Assume $N\geq 3$, $p,q\geq 1$.

  1. (i)

If $0\leq\alpha<N$ and $\beta<-2$ then, there exists $\lambda>0$ so that
($P^{+}$) has a non-negative solution in $\mathbbm{R}^{N}$ if and only if

$p,\,q\geq\frac{N-\alpha}{N-2}\quad\text{and}\quad
p+q\geq\frac{2N-\alpha}{N-2}.$

  2. (ii)

If $0\leq\alpha<N$ and $\beta>-1+\frac{1}{q}$ then, there exists $\lambda>0$
so that ($P^{+}$) has a non-negative solution in $\mathbbm{R}^{N}$ if and only
if

$p,\,q>\frac{N-\alpha}{N-2}\quad\text{and}\quad p+q>\frac{2N-\alpha}{N-2}.$

Finally, we leave open the following cases (see Table 1 below) which
essentially involve a small interval that contains the exponent
$\beta>\alpha-N$.

Open cases for ($P^{+}$) $p$ | $q$ | $p+q$ | $\beta$
---|---|---|---
$p>\max\left\\{1,\frac{N-\alpha}{N-2}\right\\}$ |  $q>\frac{N-\alpha}{N-2}$ |  $p+q=\frac{2N-\alpha}{N-2}$ |  $-1\leq\beta\leq-1+\frac{1}{p+q}$
$1\leq p=\frac{N-\alpha}{N-2}$ |  $q=\frac{N}{N-2}$ |  $p+q=\frac{2N-\alpha}{N-2}$ |  $-2\leq\beta\leq-2+\frac{1}{q}$
$p=\frac{N}{N-2}$ |  $q=\frac{N-\alpha}{N-2}$ |  $p+q=\frac{2N-\alpha}{N-2}$ |  $-2\leq\beta\leq-2+\frac{1}{q}$
$p>\frac{N}{N-2}$ |  $1<q=\frac{N-\alpha}{N-2}$ |  $p+q>\frac{2N-\alpha}{N-2}$ |  $-1\leq\beta\leq-1+\frac{1}{q}$
$p>\frac{N}{N-2}$ |  $0<q\leq 1$ |  $p+q=\frac{2N-\alpha}{N-2}$ |  $\alpha-N<\beta\leq-1+\frac{1}{p+q}$
$p>\frac{N}{N-2}$ |  $0<q\leq 1$ |  $p+q>\frac{2N-\alpha}{N-2}$ |  $\beta>\alpha-N$

Table 1: The range of exponents not covered in Theorem 1.2 and Theorem 1.3.

The remainder of this paper is organized as follows. Section 2 contains some
preliminary results, including various integral estimates involving the
potential $K_{\alpha,\beta}$. Section 3 and Section 4 provide the proofs of
Theorem 1.1 and Theorem 1.2 respectively. Section 5 and Section 6 are devoted
to the proofs of Theorem 1.3 and 1.4. The summary of exponents
$p,q,\alpha,\beta$ for which Theorem 1.2, Theorem 1.3 and Theorem 1.4 yield
the existence, non-existence and open cases for a non-negative solutions of
($P^{+}$), assuming a proper $\lambda>0$ large, is presented in the Table 2
below.

Existence & Non-existence for ($P^{+}$) ${p\geq 1}$ | $q>0$ | $p+q$ | $\alpha$ | $\beta>\alpha-N$ | Exist/Non-exist | Result
---|---|---|---|---|---|---
$p<\frac{N-\alpha}{N-2}$ | $q>0$ | $p+q>0$ | $\alpha\in[0,2]$ | $\beta>\alpha-N$ | Non-exist | Theorem 1.2(ii)
$p=\frac{N-\alpha}{N-2}$ | $0<q<\frac{N}{N-2}$ | $p+q<\frac{2N-\alpha}{N-2}$ | $\alpha\in[0,2]$ | $\beta>\alpha-N$ | Non-exist | Theorem 1.2(iv)
$q=\frac{N}{N-2}$ | $p+q=\frac{2N-\alpha}{N-2}$ | $\alpha\in[0,2]$ | $\beta<-2$ | Exist | Theorem 1.3(v)
$\beta>-2+\frac{1}{q}$ | Non-exist | Theorem 1.2(iii)&(ix)
$-2\leq\beta\leq-2+\frac{1}{q}$ | ? | ?
$q>\frac{N}{N-2}$ | $p+q>\frac{2N-\alpha}{N-2}$ | $\alpha\in[0,2]$ | $\beta\geq-1$ | Non-exist | Theorem 1.2(iii)
$\beta<-1$ | Exist | Theorem 1.3(ii)
$\frac{N-\alpha}{N-2}<p<\frac{N}{N-2}$ | $0<q\leq\frac{N-\alpha}{N-2}$ | $p+q<\frac{2N-\alpha}{N-2}$ | $\alpha\in[0,2]$ | $\beta>\alpha-N$ | Non-exist | Theorem 1.2(iv)
$q>\frac{N-\alpha}{N-2}$ | $p+q=\frac{2N-\alpha}{N-2}$ | $\alpha\in[0,2]$ | $\beta>-1+\frac{1}{p+q}$ | Non-exist | Theorem 1.2(v)
$-1<\beta\leq-1+\frac{1}{p+q}$ | ? | ?
$\beta<-1$ | Exist | Theorem 1.3(iv)
$p+q>\frac{2N-\alpha}{N-2}$ | $\alpha\in[0,2]$ | $\beta>\alpha-N$ | Exist | Theorem 1.3(i)
$p=\frac{N}{N-2}$ | $0<q<\frac{N-\alpha}{N-2}$ | $p+q<\frac{2N-\alpha}{N-2}$ | $\alpha\in[0,2]$ | $\beta>\alpha-N$ | Non-exist | Theorem 1.2(iv)
$q=\frac{N-\alpha}{N-2}$ | $p+q=\frac{2N-\alpha}{N-2}$ | $\alpha\in[0,2]$ | $\beta>-2+\frac{1}{q}$ | Non-exist | Theorem 1.2(viii)
$-2\leq\beta\leq-2+\frac{1}{q}$ | ? | ?
$\alpha\in[0,N)$ | $\beta<-2$ | Exist | Theorem 1.3(vi)
$q>\frac{N-\alpha}{N-2}$ | $p+q>\frac{2N-\alpha}{N-2}$ | $\alpha\in[0,N]$ | $\beta>\alpha-N$ | Exist | Theorem 1.3(i)
$p>\frac{N}{N-2}$ | $0<q<\frac{N-\alpha}{N-2}$ | $p+q<\frac{2N-\alpha}{N-2}$ | $\alpha\in[0,N)$ | $\beta>\alpha-N$ | Non-exist | Theorem 1.2(iv)
$0<q<1$ | $p+q=\frac{2N-\alpha}{N-2}$ | $\alpha\in[0,N)$ | $\beta>-1+\frac{1}{p+q}$ | Non-exist | Theorem 1.2(v)
$p+q=\frac{2N-\alpha}{N-2}$ | $\alpha\in[0,N)$ | $\alpha-N<\beta\leq-1+\frac{1}{p+q}$ | ? | ?
$p+q>\frac{2N-\alpha}{N-2}$ | $\alpha\in[0,N)$ | $\beta>\alpha-N$ | ? | ?
$1<q<\frac{N-\alpha}{N-2}$ | $p+q>0$ | $\alpha\in[0,2)$ | $\beta>\alpha-N$ | Non-exist | Theorem 1.2(vi)
$q=\frac{N-\alpha}{N-2}$ | $p+q>\frac{2N-\alpha}{N-2}$ | $\alpha\in[0,2)$ | $\beta>-1+\frac{1}{q}$ | Non-exist | Theorem 1.2(vii)
$-1\leq\beta\leq-1+\frac{1}{q}$ | ? | ?
$\beta<-1$ | Exist | Theorem 1.3(iii)
$q>\frac{N-\alpha}{N-2}$ | $p+q>\frac{2N-\alpha}{N-2}$ | $\alpha\in[0,N]$ | $\beta>\alpha-N$ | Exist | Theorem 1.3(i)&1.4

Table 2: The range of exponents for the existence, non-existence and open
cases of a non-negative solution in Theorem 1.2, 1.3 and 1.4.

##  2 Preliminary Results

In this section we collect some auxiliary results for proving Theorem 1.2,
Theorem 1.3 and Corollary 1.5. Our first result in this sense extends the
integral estimates obtained in [12, 13, 14, 15, 32].

######  Lemma 2.1.

Let $f\in L^{1}_{loc}(\mathbbm{R}^{N})$, $f\geq 0$ and let $K_{\alpha,\beta}$
be given by (1.1).

  1. (i)

There exists a constant $C>0$ such that

$(K_{\alpha,\beta}*f)(x)\geq C\lvert x\rvert^{-\alpha}\log^{\beta}(1+\lvert
x\rvert)\quad\text{in }\mathbbm{R}^{N}\setminus B_{1}.$ (2.1)

Furthermore, if $f(x)\geq c\lvert x\rvert^{-N}$ in $\mathbbm{R}^{N}\setminus
B_{1}$ for some $c>0$ and $\beta\leq 0$, then, there exists a constant $C>0$
such that

$(K_{\alpha,\beta}*f)(x)\geq C|x|^{-\alpha}\log^{1+\beta}(1+\lvert
x\rvert)\quad\text{in }\mathbbm{R}^{N}\setminus B_{1}.$ (2.2)

  2. (ii)

Suppose

$f(x)\geq c\lvert x\rvert^{-\sigma}\log^{\kappa}(1+\lvert
x\rvert)\quad\text{in }\mathbbm{R}^{N}\setminus B_{1}\quad\text{for some
}c>0,\kappa\in\mathbbm{R}\text{ and }\sigma\geq 0.$

Then, for all $x\in\mathbbm{R}^{N}\setminus B_{1}$ one has:

$\begin{split}(K_{\alpha,\beta}*f)(x)=\infty\quad&\text{if
}\;\;N-\alpha-\sigma>0,\\\ &\text{or }\;N-\alpha-\sigma=0,\quad
1+\beta+\kappa\geq 0.\end{split}$

Furthermore, there exists a constant $C>0$, such that for all
$x\in\mathbbm{R}^{N}\setminus B_{1}$ one has:

$(K_{\alpha,\beta}*f)(x)\geq C\left\\{\begin{split}&\lvert
x\rvert^{N-\alpha-\sigma}\log^{\beta+\kappa}(1+\lvert x\rvert)&\text{if
}N-\alpha-\sigma<0,&\\\ &\log^{1+\beta+\kappa}(1+\lvert x\rvert)&\text{if
}N-\alpha-\sigma=0,&\quad 1+\beta+\kappa<0.\end{split}\right.$

  3. (iii)

Suppose

$f(x)\leq c(A+\lvert x\rvert)^{-\sigma}\log^{\kappa}(A+\lvert
x\rvert)\quad\text{in }\mathbbm{R}^{N},$

for some $c>0$, $\sigma\geq N-\alpha$, $A>1$ and $\kappa\in\mathbbm{R}$. Then,
there exists $C=C(N,\alpha,\beta,\sigma,\kappa,A)>0$ such that for all
$x\in\mathbbm{R}^{N}$ one has following estimates:
If $\alpha<N$, then

$(K_{\alpha,\beta}*f)(x)\leq
C\begin{cases}\log^{1+\beta+\kappa}(A+|x|)&\text{if
}\sigma=N-\alpha,\,1+\beta+\kappa<0,\\\\[5.69046pt] (A+\lvert
x\rvert)^{N-\alpha-\sigma}\log^{\beta+\kappa}(A+\lvert x\rvert)&\text{if
}N-\alpha<\sigma<N,\\\\[5.69046pt] (A+\lvert
x\rvert)^{-\alpha}\log^{\beta}(A+\lvert x\rvert)&\text{if }\sigma>N\text{ or
}\sigma=N,\,\kappa<-1,\\\\[5.69046pt] (A+\lvert
x\rvert)^{-\alpha}\log^{1+\beta+\kappa}(A+\lvert x\rvert)&\text{if
}\sigma=N,\,\kappa>-1,\\\\[5.69046pt]
(A+|x|)^{-\alpha}\log^{\beta}(A+|x|)\log(\log(e+|x|))&\text{if
}\sigma=N,\,\kappa=-1.\\\\[5.69046pt] \end{cases}$

If $\alpha=N$, then

$(K_{\alpha,\beta}*f)(x)\leq
C\begin{cases}(A+|x|)^{-\sigma}\log^{1+\beta+\kappa}(A+\lvert
x\rvert)&\text{if }0<\sigma<N,\\\\[5.69046pt]
(A+|x|)^{-N}\log^{1+\beta+\kappa}(A+\lvert x\rvert)&\text{if
}\sigma=N,\,\kappa>-1\\\\[5.69046pt] (A+|x|)^{-N}\log^{\beta}(A+|x|)&\text{if
}\sigma>N\text{ or }\sigma=N,\,\kappa<-1,\\\\[5.69046pt]
(A+|x|)^{-N}\log^{\beta}(A+|x|)\log(\log(e+|x|))&\text{if
}\sigma=N,\,\kappa=-1.\\\\[5.69046pt] \end{cases}$

###### Proof.

(i) Let $x\in\mathbbm{R}^{N}\setminus B_{1}$. Then, for all $y\in
B_{\frac{1}{2}}$ one has

$\frac{|x|}{2}\leq|x|-|y|\leq|x-y|\leq|x|+|y|\leq\frac{3}{2}|x|.$

Thus

$K_{\alpha,\beta}(x-y)\geq c|x|^{-\alpha}\log^{\beta}(1+|x|)\quad\text{for
some }c>0.$

It follows that

$\displaystyle(K_{\alpha,\beta}*f)(x)$
$\displaystyle\geq\int_{|y|\leq\frac{1}{2}}K_{\alpha,\beta}(x-y)f(y)dy$
$\displaystyle\geq
c|x|^{-\alpha}\log^{\beta}(1+|x|)\int_{|y|\leq\frac{1}{2}}f(y)dy$
$\displaystyle\geq C|x|^{-\alpha}\log^{\beta}(1+|x|)\quad\text{for all
}x\in\mathbbm{R}^{N}\setminus B_{1}.$

Furthermore, let $f(x)\geq c|x|^{-N}$ in $\mathbbm{R}^{N}\setminus B_{1}$ and
$\beta\leq 0$. Then, for all $1\leq|y|\leq|x|$, we have $|x-y|\leq 2|x|$ and

$K_{\alpha,\beta}(x-y)\geq c|x|^{-\alpha}\log^{\beta}(1+|x|)\quad\text{for
some }c>0.$

Thus, by Coarea formula, we estimate

$\displaystyle(K_{\alpha,\beta}*f)(x)$ $\displaystyle\geq\int_{|x-y|\leq
2|x|}K_{\alpha,\beta}(x-y)f(y)dy$ $\displaystyle\geq
C|x|^{-\alpha}\log^{\beta}(1+|x|)\int_{1<|y|\leq|x|}|y|^{-N}dy$
$\displaystyle=C|x|^{-\alpha}\log^{\beta}(1+|x|)\int_{1}^{|x|}t^{-1}dt$
$\displaystyle\geq C|x|^{-\alpha}\log^{1+\beta}(1+|x|)\quad\text{for all
}x\in\mathbbm{R}^{N}\setminus B_{1}.$

(ii) Let $|y|\geq 2|x|\geq 2$. Then,

$\frac{|y|}{2}\leq|y|-|x|\leq|x-y|\leq|x|+|y|\leq\frac{3}{2}|y|.$ (2.3)

Thus,

$K_{\alpha,\beta}(x-y)\geq c|y|^{-\alpha}\log^{\beta}(1+|y|)\quad\text{for
some }c>0.$ (2.4)

Using (2.4) and Coarea formula, for all $|x|>1$ we estimate

$\displaystyle(K_{\alpha,\beta}*f)(x)$ $\displaystyle\geq C\int_{|y|\geq
2|x|}|y|^{-\alpha-\sigma}\log^{\beta+\kappa}(1+|y|)dy$
$\displaystyle=C\int_{2|x|}^{\infty}t^{N-\alpha-\sigma-1}\log^{\beta+\kappa}(1+t)dt$
$\displaystyle\geq\left\\{\begin{aligned} &\infty\quad&\text{if
}\sigma<N-\alpha&,\\\ &\infty\quad&\text{if
}\sigma=N-\alpha&,\;1+\beta+\kappa\geq 0,\\\
&C|x|^{N-\alpha-\sigma}\log^{\beta+\kappa}(1+|x|)\quad&\text{if
}\sigma>N-\alpha&,\\\ &C\log^{1+\beta+\kappa}(1+|x|)\quad&\text{if
}\sigma=N-\alpha&,\;1+\beta+\kappa<0.\end{aligned}\right.$

(iii) Let $x\in\mathbbm{R}^{N}$. We split the convolution integral as follows:

$\begin{split}(K_{\alpha,\beta}*f)(x)&=I_{1}+I_{2}+I_{3}\\\
&=\left\\{\int_{|y|\geq
2|x|}+\int_{\frac{|x|}{2}\leq|y|<2|x|}+\int_{|y|<\frac{|x|}{2}}\right\\}K_{\alpha,\beta}(x-y)f(y)dy.\end{split}$
(2.5)

To estimate each component of (2.5), we divide our analysis into two cases.
Case 1: $|x|>1$.
For $|y|\geq 2|x|$, we use (2.3) and we have

$\frac{|y|}{2}\leq|x-y|\leq\frac{3|y|}{2}.$ (2.6)

Thus,

$K_{\alpha,\beta}(x-y)\leq c|y|^{-\alpha}\log^{\beta}(1+|y|)\quad\text{for
some }c>0.$ (2.7)

Hence,

$I_{1}\leq C\int_{|y|\geq
2|x|}|y|^{-\alpha-\sigma}\log^{\beta+\kappa}(A+|y|)dy=C\int_{2|x|}^{\infty}t^{N-\alpha-\sigma-1}\log^{\beta+\kappa}(A+t)dt.$
(2.8)

If $\sigma=N-\alpha$ and $1+\beta+\kappa<0$, we use the fact that
$t>\frac{A+t}{A+1}$ whenever $t>1$. Then, from (2.8) one has

$\displaystyle I_{1}$ $\displaystyle\leq
C\int_{2|x|}^{\infty}t^{-1}\log^{\beta+\kappa}(A+t)dt$ $\displaystyle\leq
C(A+1)\int_{2|x|}^{\infty}\frac{\log^{\beta+\kappa}(A+t)}{A+t}dt$
$\displaystyle\leq C\log^{1+\beta+\kappa}(A+|x|).$ (2.9)

If $\sigma>N-\alpha$, then, from (2.8) and the inequality

$|x|\geq\frac{A+|x|}{A+1}\quad\mbox{ for all }|x|>1,$ (2.10)

we obtain

$\displaystyle I_{1}$ $\displaystyle\leq
C|x|^{N-\alpha-\sigma}\log^{\beta+\kappa}(A+|x|)$ (2.11) $\displaystyle\leq
C\Big{(}\frac{A+|x|}{A+1}\Big{)}^{N-\alpha-\sigma}\log^{\beta+\kappa}(A+|x|)$
$\displaystyle\leq C(A+|x|)^{N-\alpha-\sigma}\log^{\beta+\kappa}(A+|x|).$

Combine (2.9) and (2.11) to conclude

$I_{1}\leq C\left\\{\begin{split}&\log^{1+\beta+\kappa}(A+|x|)\quad&\text{if
}N-\alpha-\sigma=0&,1+\beta+\kappa<0,\\\
&(A+|x|)^{N-\alpha-\sigma}\log^{\beta+\kappa}(A+|x|)\quad&\text{if
}N-\alpha-\sigma<0&.\end{split}\right.$ (2.12)

To estimate $I_{2}$, we have $\frac{|x|}{2}\leq|y|<2|x|$, which implies
$|x-y|<3|x|$. It follows that

$\displaystyle I_{2}$ $\displaystyle\leq
C(A+|x|)^{-\sigma}\log^{\kappa}(A+|x|)\int_{|x-y|<3|x|}|x-y|^{-\alpha}\log^{\beta}(1+|x-y|)dy$
$\displaystyle\leq
C(A+|x|)^{-\sigma}\log^{\kappa}(A+|x|)\int_{0}^{3|x|}t^{N-\alpha-1}\log^{\beta}(1+t)dt.$
(2.13)

If $\alpha<N$, then from (2) one has

$\displaystyle I_{2}$ $\displaystyle\leq
C(A+|x|)^{-\sigma}\log^{\kappa}(A+|x|)|x|^{N-\alpha}\log^{\beta}(1+|x|)$
$\displaystyle\leq C(A+|x|)^{N-\alpha-\sigma}\log^{\beta+\kappa}(A+|x|).$
(2.14)

If $\alpha=N$, then (1.1) yields $\beta>0$ and from (2) together with
$\log(1+t)\leq t$ we deduce

$\displaystyle I_{2}$ $\displaystyle\leq
C(A+|x|)^{-\sigma}\log^{\kappa}(A+|x|)\int_{0}^{3|x|}t^{-1}\log^{\beta}(1+t)dt$
$\displaystyle\leq
C(A+|x|)^{-\sigma}\log^{\kappa}(A+|x|)\left\\{\int_{0}^{1}t^{\beta-1}dt+2\int_{1}^{3|x|}\frac{\log^{\beta}(1+t)}{1+t}dt\right\\}$
$\displaystyle\leq C(A+|x|)^{-\sigma}\log^{1+\beta+\kappa}(A+|x|).$ (2.15)

Thus, (2) and (2) we have

$I_{2}\leq
C\begin{cases}(A+|x|)^{N-\alpha-\sigma}\log^{\beta+\kappa}(A+|x|)&\mbox{ if
}\alpha<N,\\\\[5.69046pt] (A+|x|)^{-\sigma}\log^{1+\beta+\kappa}(A+|x|)&\mbox{
if }\alpha=N.\end{cases}$ (2.16)

Finally, to estimate $I_{3}$, for $|y|<\frac{|x|}{2}$, one has
$\frac{|x|}{2}<|x-y|<\frac{3|x|}{2}$. Thus,

$K_{\alpha,\beta}(x-y)\leq c|x|^{-\alpha}\log^{\beta}(1+|x|)\quad\text{for
some }c>0.$

It follows that

$I_{3}\leq
C|x|^{-\alpha}\log^{\beta}(1+|x|)\int_{|y|<\frac{|x|}{2}}(A+|y|)^{-\sigma}\log^{\kappa}(A+|y|)dy.$

Using (2.10) again, we further estimate:

$\displaystyle I_{3}$ $\displaystyle\leq
C(A+|x|)^{-\alpha}\log^{\beta}(A+|x|)\int_{0}^{\frac{|x|}{2}}(A+t)^{N-\sigma-1}\log^{\kappa}(A+t)dt$
$\displaystyle\leq
C(A+|x|)^{-\alpha}\log^{\beta}(A+|x|)\left\\{1+\int_{\frac{1}{2}}^{\frac{|x|}{2}}(A+t)^{N-\sigma-1}\log^{\kappa}(A+t)dt\right\\}$
$\displaystyle\leq C\left\\{\begin{aligned}
&(A+|x|)^{N-\alpha-\sigma}\log^{\beta+\kappa}(A+|x|)&\text{if }\sigma<N&,\\\
&(A+|x|)^{-\alpha}\log^{\beta}(A+|x|)&\text{if }\sigma>N&\text{ or
}\sigma=N,\,\kappa<-1,\\\
&(A+|x|)^{-\alpha}\log^{1+\beta+\kappa}(A+|x|)&\text{if
}\sigma=N&,\,\kappa>-1,\\\
&(A+|x|)^{-\alpha}\log^{\beta}(A+|x|)\log(\log(e+|x|))&\text{if
}\sigma=N&,\,\kappa=-1.\\\ \end{aligned}\right.$ (2.17)

Case 2: $|x|\leq 1$.
For $I_{1}$, we see that (2.6) and (2.7) still hold and then

$\displaystyle I_{1}$ $\displaystyle\leq\int_{|y|\geq
2|x|}|y|^{-\alpha}\log^{\beta}(1+|y|)(A+|y|)^{-\sigma}\log^{\kappa}(A+|y|)dy$
$\displaystyle\leq
C\int_{0}^{\infty}t^{N-\alpha-1}(A+t)^{-\sigma}\log^{\beta}(1+t)\log^{\kappa}(A+t)dt$

Using the fact that $ct\leq\log(1+t)\leq t$ for all $0<t\leq 1$, if either

  * •

$\sigma>N-\alpha$;

  * •

or $\sigma=N-\alpha$ and $\beta+\kappa<0$,

we conclude

$\displaystyle I_{1}$ $\displaystyle\leq
C\int_{0}^{1}t^{N-\alpha+\beta-1}(A+t)^{-\sigma}\log^{\kappa}(A+t)dt+C\int_{1}^{\infty}t^{N-\alpha-\sigma-1}\log^{\beta+\kappa}(1+t)dt$
$\displaystyle\leq
C\int_{0}^{1}t^{N-\alpha+\beta-1}dt+C\int_{1}^{\infty}t^{N-\alpha-\sigma-1}\log^{\beta+\kappa}(1+t)dt$
$\displaystyle\leq C.$

For $I_{2}$, we observe that the estimate (2) remains valid for the case
$\alpha<N$ whenever $|x|\leq 1$. For the case $\alpha=N$, we note that (1.1)
yields $\beta>0$ and we deduce

$\begin{split}I_{2}&\leq
C(A+|x|)^{-\sigma}\log^{\kappa}(A+|x|)\int_{0}^{3}t^{-1}\log^{\beta}(1+t)dt\\\
&\leq C(A+|x|)^{-\sigma}\log^{\kappa}(A+|x|)\int_{0}^{3}t^{\beta-1}dt\leq
C.\end{split}$

For $I_{3}$, from $|y|<\frac{|x|}{2}\leq\frac{1}{2}$ one has
$|x-y|<\frac{3}{2}$. It follows that

$\begin{split}I_{3}&\leq\int_{|x-y|<\frac{3}{2}}|x-y|^{-\alpha}\log^{\beta}(1+|x-y|)(A+|y|)^{-\sigma}\log^{\kappa}(A+|y|)dy\\\
&\leq C\int_{|x-y|<\frac{3}{2}}|x-y|^{-\alpha}\log^{\beta}(1+|x-y|)dy\\\ &\leq
C\int_{0}^{\frac{3}{2}}t^{N-\alpha-1}\log^{\beta}(1+t)dt\\\ &\leq
C\int_{0}^{\frac{3}{2}}t^{N-\alpha+\beta-1}dt\leq C.\end{split}$

Thus, we observe that $I_{1}$, $I_{2}$ and $I_{3}$ are bounded by a positive
constant. Hence, (2.12), (2.16) and (2) hold for $|x|\leq 1$ as well with a
larger constant $C>0$. Therefore, combining (2.5), (2.12), (2.16) and (2), we
reach the conclusion for all $x\in\mathbbm{R}^{N}$. ∎

We finish the presentation of this Section with the following useful result.

######  Lemma 2.2.

  1. (i)

There are no positive, non-constant superharmonic functions in
$\mathbbm{R}^{N}$, $N=1,2$.

  2. (ii)

Any positive superharmonic function $u\in\mathcal{C}^{2}(\mathbbm{R}^{N})$,
$N\geq 3$, satisfies

$u(x)\geq C|x|^{2-N}\quad\text{in }\mathbbm{R}^{N}\setminus B_{1},$ (2.18)

for some positive constant $C>0$.

If $N=1$, the proof of Lemma 2.2(i) is obvious. If $N=2$, Lemma 2.2(i) is
given in [28, Theorem 29, page 30]. Finally, if $N\geq 3$ the proof of Lemma
2.2(ii) follows from a standard maximum principle argument. It is also a
consequence of [30, Lemma 2.3].

##  3 Proof of Theorem 1.1

Assume $u\in\mathcal{C}^{4}(\mathbbm{R}^{N})$, $u\geq 0$ and $u\not\equiv 0$
is a solution of ($P^{-}$). We claim that $\Delta u\geq 0$ in
$\mathbbm{R}^{N}$. Letting $v=\Delta u$, then ($P^{-}$) implies

$-\Delta v+\lambda v\geq 0\quad\text{in }\;\;\mathbbm{R}^{N}.$ (3.1)

Assume by contradiction that $v(x_{0})<0$ for some $x_{0}\in\mathbbm{R}^{N}$.
Replacing $v(x)$ by $v(x+x_{0})$, we may assume that $v(0)<0$. Denote by
$\overline{v}(r)$ the spherical average of $v(x)$ over $\partial B_{r}$, so
that

$\overline{v}(r)=\fint_{\partial B_{r}}v(x)d\sigma(x)\quad\mbox{ for all
}r\geq 0.$

It follows from (3.1) that

$-\Delta\overline{v}(r)+\lambda\overline{v}(r)\geq 0\quad\text{for all }r\geq
0.$ (3.2)

Since $\overline{v}(0)=v(0)<0$, we may define

$\rho:=\sup\\{r>0:\overline{v}(t)<0\quad\text{for all }\;0<t<r\\}>0.$

We claim that $\rho=\infty$. Assuming the contrary, we have $\rho<\infty$ and
by continuity argument it follows $\overline{v}(\rho)=0$. Then, (3.2) implies

$\Delta\overline{v}(r)\leq\lambda\overline{v}(r)<0\quad\mbox{for all }0\leq
r<\rho.$

This means that $(r^{N-1}\overline{v}^{\prime})^{\prime}<0$ for all $0\leq
r<\rho$. Thus, the mapping $[0,\rho]\ni r\longmapsto
r^{N-1}\overline{v}^{\prime}(r)$ is decreasing which yields
$\overline{v}^{\prime}<0$ in $[0,\rho]$. In particular, this implies
$\overline{v}(\rho)<\overline{v}(0)<0$ which contradicts
$\overline{v}(\rho)=0$. This proves $\rho=\infty$. Hence,

$\overline{v}(r)=\Delta\overline{u}(r)\leq\overline{v}(0)=-c\quad\mbox{for all
}r\geq 0,\quad\text{where }c>0\text{ is a constant.}$

This yields

$\big{(}r^{N-1}\overline{u}^{\prime}(r)\big{)}^{\prime}\leq-
cr^{N-1}\quad\mbox{for all }r\geq 0.$

Integrating twice in the above estimate we obtain

$\overline{u}(r)<-\frac{cr^{2}}{2N}+\overline{u}(0)\quad\mbox{for all }r\geq
0.$

Therefore, $\overline{u}(r)<0$ for $r$ large, which contradicts our assumption
that $u(x)\geq 0$ for any $x\in\mathbbm{R}^{N}$. We conclude that $\Delta
u\geq 0$ in $\mathbbm{R}^{N}$.
(i) Take $r_{0}>0$ so that $\sup_{B_{r_{0}}}u>0$. If $p>1$, then for
$R>2r_{0}$, by the weak Harnack Inequality (see [31, Theorem 1.3]), we have

$\left(\fint_{B_{R}}u^{p}\right)^{\frac{1}{p}}\geq
c\sup\limits_{B_{r_{0}}}u\geq C,\quad\text{where }c,C>0\text{ are constants}.$

It follows that

$\int_{B_{R}}u^{p}\geq CR^{N}.$

If $p=1$, by using $\Delta u\geq 0$ in $\mathbbm{R}^{N}$ and taking the
average, we have $(r^{N-1}\overline{u}^{\prime})^{\prime}\geq 0$ for all
$r\geq 0$. This means that $\overline{u}^{\prime}\geq 0$ and $\overline{u}$ is
increasing. Thus, $\overline{u}(r)\geq\overline{u}(r_{0})=c>0$ for all $r\geq
r_{0}$ and $c>0$ is a constant independent of $r$. Then, for any $R>2r_{0}$,
we have

$\begin{split}\int_{B_{R}}u&\geq\int_{B_{R}\setminus
B_{r_{0}}}u=\sigma_{N}\int_{r_{0}}^{R}r^{N-1}\overline{u}(r)dr\\\ &\geq
C\int_{r_{0}}^{R}r^{N-1}dr=c(R^{N}-r_{0}^{N})\geq CR^{N},\end{split}$

where $C$ is a positive constant independent of $R$. Hence, for all $p\geq 1$,
we conclude

$\int_{B_{R}}u^{p}\geq CR^{N}.$ (3.3)

Then, using (3.3), we estimate

$\begin{split}(K_{\alpha,\beta}*u^{p})(0)&=\int_{\mathbbm{R}^{N}}K_{\alpha,\beta}(y)u^{p}(y)dy\\\
&\geq\int_{B_{R}}\frac{u^{p}(y)\log^{\beta}(1+\lvert y\rvert)}{\lvert
y\rvert^{\alpha}}dy\\\
&\geq\frac{\log^{\beta}(1+R)}{R^{\alpha}}\int_{B_{R}}u^{p}(y)dy\\\ &\geq
CR^{N-\alpha}\log^{\beta}(1+R)\to\infty\quad\text{as }R\to\infty.\\\
\end{split}$

Hence, $(K_{\alpha,\beta}*u^{p})(0)=+\infty$ which contradicts condition
(1.2). This shows that ($P^{-}$) has no solutions whenever $p\geq 1$.
(ii) Assume that $p<1$ and $u(x)$ is a non-negative bounded solution of
($P^{-}$) such that $u\leq\sup_{\mathbbm{R}^{N}}u=M<\infty$. By applying the
weak Harnack inequality as before for the exponent $p+1$, we obtain

$\int_{B_{R}}u^{p}\geq\frac{1}{M}\int_{B_{R}}u^{p+1}\geq CR^{N}.$

for some positive constant $C$ independent of $R$. It follows that

$\begin{split}(K_{\alpha,\beta}*u^{p})(0)&=\int_{\mathbbm{R}^{N}}K_{\alpha,\beta}(y)u^{p}(y)dy\\\
&\geq\int_{B_{R}}\frac{u^{p}(y)\log^{\beta}(1+\lvert y\rvert)}{\lvert
y\rvert^{\alpha}}dy\\\ &\geq
CR^{N-\alpha}\log^{\beta}(1+R)\to\infty\quad\text{as }R\to\infty.\end{split}$

Thus, ($P^{-}$) has no bounded solutions whenever $p<1$.
(iii) Assume $u(x)\geq 0$ is a radial solution of ($P^{-}$) and $u\not\equiv
0$. From the previous section, we know that $\Delta u\geq 0$ in
$\mathbbm{R}^{N}$. It follows that

$r^{N-1}\Delta u=(r^{N-1}u^{\prime})^{\prime}\geq 0\quad\text{for all }r\geq
0.$

Then $u^{\prime}\geq 0$ and $u$ is an increasing function. Thus $u(x)\geq
u(x_{0})=c>0$ whenever $|x|\geq|x_{0}|=r$, for some $r>0$. Moreover,

$\begin{split}(K_{\alpha,\beta}*u^{p})(0)&=\int_{\mathbbm{R}^{N}}K_{\alpha,\beta}(y)u^{p}(y)dy\\\
&\geq c^{p}\int_{\mathbbm{R}^{N}\setminus
B_{r}}|y|^{-\alpha}\log^{\beta}(1+|y|)dy\\\
&=c^{p}\int_{r}^{\infty}t^{N-\alpha-1}\log^{\beta}(1+t)dt=\infty\quad\text{since
}\alpha,\beta\text{ satisfy}\;\eqref{ptt}.\end{split}$

Hence, $(K_{\alpha,\beta}*u^{p})(0)=+\infty$ which contradicts condition
(1.2). This shows that ($P^{-}$) has no radial solutions and concludes our
proof. ∎

##  4 Proof of Theorem 1.2

Assume $u\in\mathcal{C}^{4}(\mathbbm{R}^{N})$, $u\geq 0$ and $u\not\equiv 0$
is a solution of ($P^{+}$). We claim that $-\Delta u\geq 0$ in
$\mathbbm{R}^{N}$. Letting $w=-\Delta u$, from ($P^{+}$) we deduce

$-\Delta w+\lambda w\geq 0\quad\text{ in }\;\;\mathbbm{R}^{N}.$ (4.1)

Assume by contradiction that $w(x_{0})<0$ for some $x_{0}\in\mathbbm{R}$.
Replacing $w(x)$ by $w(x+x_{0})$, we may assume that $w(0)<0$. Again, denote
by $\overline{w}(r)$ the spherical average of $w(x)$ over $\partial B_{r}(0)$.
It follows from (4.1) that

$-\Delta\overline{w}(r)+\lambda\overline{w}(r)\geq 0\quad\mbox{for all }r\geq
0.$ (4.2)

We next apply a similar argument as in Theorem 1.1 to obtain
$\overline{w}\leq\overline{w}(0)<0$. Therefore,

$-\Delta\overline{u}(r)=\overline{w}(r)\leq\overline{w}(0)=-c\quad\mbox{for
all }r\geq 0,$

where $c>0$ is a constant. Integrating twice in the above inequality we obtain

$\overline{u}(r)\geq\frac{cr^{2}}{2N}+\overline{u}(0)\quad\mbox{for all }r\geq
0.$

By Coarea formula, we get

$\begin{split}(K_{\alpha,\beta}*u^{p})(0)&=\int_{0}^{\infty}\int_{\partial
B_{r}}\frac{u^{p}(y)\log^{\beta}(1+\lvert y\rvert)}{\lvert
y\rvert^{\alpha}}d\sigma(y)dr\\\
&=C\int_{0}^{\infty}r^{N-\alpha-1}\log^{\beta}(1+r)\fint_{\partial
B_{r}}u^{p}(y)d\sigma(y)dr\\\ \end{split}$

We further apply Jensen’s inequality to $u^{p}(y)$ where $p\geq 1$ to obtain:

$\begin{split}(K_{\alpha,\beta}*u^{p})(0)&\geq
C\int_{0}^{\infty}r^{N-\alpha-1}\log^{\beta}(1+r)\left(\fint_{\partial
B_{r}}u(y)d\sigma(y)\right)^{p}dr\\\
&>C\int_{0}^{\infty}r^{N-\alpha-1+2p}\log^{\beta}(1+r)=\infty,\end{split}$

which contradicts the condition (1.2). Thus, we conclude that $-\Delta u\geq
0$ in $\mathbbm{R}^{N}$.
(i) Since $-\Delta u\geq 0$, $u\geq 0$ and $u\neq 0$ in $\mathbbm{R}^{N}$, by
the maximum principle we have $u>0$. Since $1\leq N\leq 2$, by Lemma 2.2(i) we
deduce $u$ is constant. This is a clear contradiction, since constant
functions $u$ do not satisfy (1.2). Thus, ($P^{+}$) has no non-negative
solutions if $N=1,2$.
(ii) Assume $N\geq 3$. From $-\Delta u\geq 0$ in $\mathbbm{R}^{N}$ and
$u\not\equiv 0$, we deduce $u>0$ in $\mathbbm{R}^{N}$. Thus, by (2.18) in
Lemma 2.2(ii), we deduce $u(x)\geq c\lvert x\rvert^{2-N}$ in
$\mathbbm{R}^{N}\setminus B_{1}$, for some $c>0$. It follows that

$\displaystyle(K_{\alpha,\beta}*u^{p})(0)$
$\displaystyle\geq\int_{\mathbbm{R}^{N}\setminus
B_{1}}K_{\alpha,\beta}(y)u^{p}(y)dy$ $\displaystyle\geq
C\int_{\mathbbm{R}^{N}\setminus B_{1}}\lvert
y\rvert^{(2-N)p-\alpha}\log^{\beta}(1+\lvert y\rvert)dy$ $\displaystyle\geq
C\int_{1}^{\infty}r^{N-1-\alpha-p(N-2)}\log^{\beta}(1+r)dr.$ (4.3)

Hence, $(K_{\alpha,\beta}*u^{p})(0)=+\infty$ if $N-\alpha-(N-2)p>0$ which
contradicts condition (1.2). This shows that ($P^{+}$) has no solutions
whenever $1\leq p<\frac{N-\alpha}{N-2}$.

(iii) Assume $N\geq 3$ and $0\leq\alpha\leq 2$. Let $p=\frac{N-\alpha}{N-2}$.
From (4), we have

$\begin{split}(K_{\alpha,\beta}*u^{p})(0)&\geq\int_{\mathbbm{R}^{N}\setminus
B_{1}}K_{\alpha,\beta}(y)u^{p}(y)dy\\\ &\geq
C\int_{1}^{\infty}r^{N-1-\alpha-p(N-2)}\log^{\beta}(1+r)dr\\\
&=C\int_{1}^{\infty}r^{-1}\log^{\beta}(1+r)dr=\infty\quad\text{if
}\beta\geq-1.\end{split}$

Thus, ($P^{+}$) has no solutions whenever $p=\frac{N-\alpha}{N-2}$ and
$\beta\geq-1$.

(iv) Let $\psi\in C_{c}^{\infty}(\mathbbm{R}^{N})$, $0\leq\psi\leq 1$,
supp$(\psi)=\overline{B}_{2}\subset\mathbbm{R}^{N}$ and $\psi\equiv 1$ on
$\overline{B}_{1}$. For $R\geq 1$, let
$\varphi(x)=\psi^{4}\left(\frac{x}{R}\right)$. We take $\varphi^{2}$ as the
test function in ($P^{+}$). Observe that

$\Delta\varphi^{2}(x)=\Delta\left(\psi^{8}\left(\frac{x}{R}\right)\right)=\frac{8}{R^{2}}\psi^{6}\left(\frac{x}{R}\right)\left\\{7|\nabla\psi|^{2}\left(\frac{x}{R}\right)+\psi\left(\frac{x}{R}\right)\Delta\psi\left(\frac{x}{R}\right)\right\\}.$

Then,

$|\Delta\varphi^{2}|\leq\frac{C}{R^{2}}\psi^{6}\leq\frac{C}{R^{2}}\varphi\quad\text{in
}\mathbbm{R}^{N}.$

Similarly, we find

$|\Delta^{2}\varphi^{2}|\leq\frac{C}{R^{4}}\varphi\quad\text{in
}\mathbbm{R}^{N}.$

Thus,

$|\Delta^{2}(\varphi^{2})-\lambda\Delta\varphi^{2}|\leq\frac{C}{R^{2}}\varphi\quad\text{in
}\mathbbm{R}^{N},$ (4.4)

where $C>0$ is a positive constant independent of $R$. Using $\varphi^{2}$ as
a test function in ($P^{+}$), we have

$\displaystyle\int_{\mathbbm{R}^{N}}(K_{\alpha,\beta}*u^{p})u^{q}\varphi^{2}$
$\displaystyle\leq\int_{B_{2R}}\varphi^{2}(\Delta^{2}u-\lambda\Delta u)$
$\displaystyle=\int_{B_{2R}}u(\Delta^{2}\varphi^{2}-\lambda\Delta\varphi^{2})$
$\displaystyle\leq\int_{B_{2R}}u|\Delta^{2}\varphi^{2}-\lambda\Delta\varphi^{2}|$
$\displaystyle\leq\frac{C}{R^{2}}\int_{B_{2R}}u\varphi.$ (4.5)

Let $x\in\text{supp}(\varphi^{2})=B_{2R}$. Then, for $R>1$ large we have

$\displaystyle(K_{\alpha,\beta}*u^{p})(x)$
$\displaystyle\geq\int_{B_{2R}}u^{p}(y)|x-y|^{-\alpha}\log^{\beta}(1+|x-y|)\varphi
dy$
$\displaystyle\geq(4R)^{-\alpha}\log^{\beta}(1+4R)\int_{B_{2R}}u^{p}\varphi.$
(4.6)

From (4) and (4), we deduce

$\int_{B_{2R}}u\varphi\geq
CR^{2-\alpha}\log^{\beta}(1+4R)\left(\int_{B_{2R}}u^{p}\varphi\right)\left(\int_{B_{2R}}u^{q}\varphi^{2}\right).$
(4.7)

We claim next that

$\int_{B_{2R}}u^{p}\varphi\geq
CR^{-N(p-1)}\left(\int_{B_{2R}}u\varphi\right)^{p},$ (4.8)

for some $C>0$. Indeed, if $p=1$, then (4.8) clearly holds. Assume next $p>1$
and let $p^{\prime}>1$ be its Hölder’s conjugate. We apply the Hölder’s
inequality to obtain

$\int_{B_{2R}}u\varphi\leq\left(\int_{B_{2R}}u^{p}\varphi\right)^{\frac{1}{p}}\left(\int_{B_{2R}}\varphi\right)^{\frac{1}{p^{\prime}}}\leq
CR^{\frac{N}{p^{\prime}}}\left(\int_{B_{2R}}u^{p}\varphi\right)^{\frac{1}{p}}.$

This yields (4.8) which now combined with (4.7) gives

$C^{\prime}\geq
R^{2-\alpha-N(p-1)}\log^{\beta}(1+4R)\left(\int_{B_{2R}}u\varphi\right)^{p-1}\left(\int_{B_{2R}}u^{q}\varphi^{2}\right),$
(4.9)

where $C^{\prime}>0$ is a constant. Since $-\Delta u\geq 0$ in
$\mathbbm{R}^{N}$, we have $u(x)\geq c|x|^{2-N}$ in $\mathbbm{R}^{N}\setminus
B_{1}$. Using this estimate in (4.9), for $R>1$ large, we get

$\begin{split}C^{\prime}&\geq
CR^{2-\alpha-N(p-1)+2(p-1)+N-q(N-2)}\log^{\beta}(1+4R)\\\
&=CR^{2N-\alpha-(N-2)(p+q)}\log^{\beta}(1+4R),\end{split}$ (4.10)

where $C$ is a positive constant independent of $R$. Thus, the above estimate
yields a contradiction if the exponent of $R$ is positive. Therefore,
($P^{+}$) has no solutions if $p+q<\frac{2N-\alpha}{N-2}$ and
$\beta>\alpha-N$.

(v) Assume $p+q=\frac{2N-\alpha}{N-2}$. If $\beta>0$, the estimate (4.10)
still yields a contradiction since (4.10) is equivalent to $C^{\prime}\geq
C\log^{\beta}(1+4R)$ for $\beta>0$ and $R>1$ large.
It remains to raise a contradiction in the case where
$\frac{1}{p+q}-1<\beta\leq 0$. We also note that we may assume
$p>\frac{N-\alpha}{N-2}$, since, otherwise, if $p\leq\frac{N-\alpha}{N-2}$ and
$\beta\geq-1$ we know from part (ii) and (iii) above that ($P^{+}$) has no
non-negative solutions. Thus, we may assume $p>\frac{N-\alpha}{N-2}$. Recall
that by (2.18) in Lemma 2.2(ii) we have $u(x)\geq c|x|^{2-N}$ in
$\mathbbm{R}^{N}\setminus B_{1}$, where $c>0$. Thus, for $|x|>1$ we have

$\begin{split}(K_{\alpha,\beta}*u^{p})(x)&\geq\int_{|y|\geq|x|}u^{p}(y)|x-y|^{-\alpha}\log^{\beta}(1+|x-y|)dy\\\
&\geq C\int_{|y|\geq|x|}u^{p}(y)(2|y|)^{-\alpha}\log^{\beta}(1+2|y|)dy\\\
&\geq C\int_{|y|\geq|x|}|y|^{-\alpha-p(N-2)}\log^{\beta}(1+2|y|)dy\\\ &\geq
C|x|^{N-\alpha-p(N-2)}\log^{\beta}(1+2|x|).\end{split}$

Combine the above estimate into ($P^{+}$) to obtain

$\begin{split}\Delta^{2}u-\lambda\Delta u&\geq(K_{\alpha,\beta}*u^{p})u^{q}\\\
&\geq C|x|^{N-\alpha-(p+q)(N-2)}\log^{\beta}(1+|x|)\\\ &\geq
C|x|^{-N}\log^{\beta}|x|\quad\text{if }|x|>2\text{ large}.\end{split}$

Thus, $\Delta^{2}u-\lambda\Delta u\geq C|x|^{-N}\log^{\beta}|x|$ in
$\mathbbm{R}^{N}\setminus B_{2}$. Let $v=-\Delta u\geq 0$. Then

$-\Delta v+\lambda v\geq C|x|^{-N}\log^{\beta}|x|\quad\text{in
}\mathbbm{R}^{N}\setminus B_{2}.$

Let $z(x)=C|x|^{-N}\log^{\beta}|x|$ which satisfies

$-\Delta z+\lambda z\leq C|x|^{-N}\log^{\beta}|x|\quad\text{if
}|x|>r_{0}>2\text{ large}.$

Let $c>0$ be such that $v\geq cz$ on $\partial B_{r_{0}}$ and $\varepsilon>0$.
Then, there exists $R_{\varepsilon}>r_{0}>2$ so that

$v+\varepsilon\geq cz\quad\text{in }\mathbbm{R}^{N}\setminus
B_{R_{\varepsilon}}.$

By the maximum principle, we have $v+\varepsilon\geq cz$ in
$B_{R_{\varepsilon}}\setminus B_{r_{0}}$. It follows that

$v+\varepsilon\geq cz\quad\text{in }\mathbbm{R}^{N}\setminus B_{r_{0}}.$
(4.11)

Since $\varepsilon>0$ was arbitrary, (4.11) yields $v\geq cz$ in
$\mathbbm{R}^{N}\setminus B_{r_{0}}$, so that for some constant $C>0$ we have

$-\Delta u\geq C|x|^{-N}\log^{\beta}|x|\quad\text{in }\mathbbm{R}^{N}\setminus
B_{r_{0}}.$

Let now $w(x)=|x|^{2-N}\log^{\theta}|x|$, where

$\frac{-\beta}{p+q-1}<\theta<1+\beta.$

Then, by direct computation one has

$\begin{split}-\Delta
w&=\theta(N-2)|x|^{-N}\log^{\theta-1}|x|+\theta(\theta-1)|x|^{-N}\log^{\theta-2}|x|\\\
&\leq C|x|^{-N}\log^{\beta}|x|\leq-\Delta u\quad\text{in
}\mathbbm{R}^{N}\setminus B_{r_{0}}.\end{split}$

As above, let $c>0$ so that $u\geq cw$ on $\partial B_{r_{0}}$. Take
$\varepsilon>0$. Then, there exists $R_{\varepsilon}>r_{0}>0$ so that
$u+\varepsilon\geq cw$ in $\mathbbm{R}^{N}\setminus B_{R_{\varepsilon}}$. By
the maximum principle, we have $u+\varepsilon\geq cw$ in
$B_{R_{\varepsilon}}\setminus B_{r_{0}}$, thus $u+\varepsilon\geq cw$ in
$\mathbbm{R}^{N}\setminus B_{r_{0}}$. Since $\varepsilon>0$ was arbitrary, it
follows that

$u\geq cw\geq c|x|^{2-N}\log^{\theta}|x|\quad\text{in
}\mathbbm{R}^{N}\setminus B_{r_{0}}.$ (4.12)

Finally, we use (4.12) into the estimate (4.9) to obtain

$\begin{split}C^{{}^{\prime}}&\geq
R^{2-\alpha-N(p-1)}\log^{\beta}R\left(\int_{B_{2R}\setminus
B_{r_{0}}}u\right)^{p-1}\left(\int_{B_{2R}\setminus B_{r_{0}}}u^{q}\right)\\\
&\geq CR^{2-\alpha-N(p-1)+2(p-1)+N-q(N-2)}\log^{\beta+(p-1+q)\theta}R\\\
&=C\log^{\beta+(p+q-1)\theta}R\to\infty\quad\text{as }R\to\infty.\end{split}$
(4.13)

Hence, ($P^{+}$) has no non-negative solutions if $p+q=\frac{2N-\alpha}{N-2}$
and $\beta>\frac{1}{p+q}-1$.

(vi-vii) Assume $N\geq 3$, $q>1$ and $0\leq\alpha<2$. We fix $\delta\in(1,q)$
and let $k\geq 1$ be such that

$k>\frac{4}{\delta-1}.$ (4.14)

Let $\psi\in C_{c}^{\infty}(\mathbbm{R}^{N})$, $0\leq\psi\leq 1$,
supp$(\psi)=\overline{B}_{2}\subset\mathbbm{R}^{N}$ and $\psi\equiv 1$ on
$\overline{B}_{1}$. For $R\geq 1$, let
$\varphi(x)=\psi^{k}\left(\frac{x}{R}\right)$. We use $\varphi^{\delta}$ as a
test function in ($P^{+}$) and obtain

$\int_{B_{2R}}(\Delta^{2}u-\lambda\Delta
u)\varphi^{\delta}\geq\int_{B_{2R}}(K_{\alpha,\beta}*u^{p})u^{q}\varphi^{\delta}.$
(4.15)

We next estimate from above the left-hand side integral in (4.15). Applying a
similar argument to that in part (iv) where we deduced the estimate (4.4),
from (4.14) we have

$|\Delta^{2}(\varphi^{\delta})-\lambda\Delta\varphi^{\delta}|\leq\frac{C}{R^{2}}\varphi\quad\text{in
}\mathbbm{R}^{N},$

and thus

$\displaystyle\int_{B_{2R}}(\Delta^{2}u-\lambda\Delta
u)\varphi^{\delta}=\int_{B_{2R}}u(\Delta^{2}(\varphi^{\delta})-\lambda\Delta\varphi^{\delta})\leq\frac{C}{R^{2}}\int_{B_{2R}}u\varphi.$
(4.16)

where $C>0$ is a positive constant independent of $R$. To estimate the right-
hand side integral in (4.15), we first apply Lemma 2.1(i) to obtain

$(K_{\alpha,\beta}*u^{p})(x)\geq
C|x|^{-\alpha}\log^{\beta}(1+|x|)\quad\text{in }\mathbbm{R}^{N}\setminus
B_{1}.$

Hence, for $R>1$ large we have

$\int_{B_{2R}}(K_{\alpha,\beta}*u^{p})u^{q}\varphi^{\delta}\geq
CR^{-\alpha}\log^{\beta}(1+R)\int_{B_{2R}}u^{q}\varphi^{\delta}.$ (4.17)

Combining now (4.15), (4.16) and (4.17) we derive

$\int_{B_{2R}}u\varphi\geq
CR^{2-\alpha}\log^{\beta}(1+R)\int_{B_{2R}}u^{q}\varphi^{\delta}.$ (4.18)

By Hölder’s inequality we obtain

$\int_{B_{2R}}u\varphi\leq\left(\int_{B_{2R}}u^{q}\varphi^{\delta}\right)^{\frac{1}{q}}\left(\int_{B_{2R}}\varphi^{\frac{q-\delta}{q-1}}\right)^{\frac{1}{q^{\prime}}}\leq
CR^{\frac{N}{q^{\prime}}}\left(\int_{B_{2R}}u^{q}\varphi^{\delta}\right)^{\frac{1}{q}}.$

This implies

$\int_{B_{2R}}u^{q}\varphi^{\delta}\geq
CR^{-N(q-1)}\left(\int_{B_{2R}}u\varphi\right)^{q}.$

Plugging this last estimate into (4.18) we deduce

$C^{\prime}\geq
R^{2-\alpha-N(q-1)}\log^{\beta}(1+R)\left(\int_{B_{2R}}u\varphi\right)^{q-1}.$
(4.19)

Using $u(x)\geq c|x|^{2-N}$ in $\mathbbm{R}^{N}\setminus B_{1}$ (which holds
thanks to Lemma 2.2(ii)), it follows that

$\displaystyle C^{\prime}$ $\displaystyle\geq
CR^{2-\alpha-N(q-1)}\log^{\beta}(1+R)\left(\int_{B_{2R}\setminus
B_{1}}|x|^{2-N}dx\right)^{q-1}$ $\displaystyle\geq
CR^{2-\alpha-N(q-1)+2(q-1)}\log^{\beta}(1+R)$
$\displaystyle=CR^{N-\alpha-(N-2)q}\log^{\beta}(1+R)\quad\text{for all }R>1,$

where $C,C^{\prime}>0$ are positive constants independent of $R$. Thus, we
conclude that ($P^{+}$) does not have non-negative solutions if $1\leq
q<\frac{N-\alpha}{N-2}$ or $q=\frac{N-\alpha}{N-2}$ and $\beta>0$.
If $q=\frac{N-\alpha}{N-2}$ and $-1+\frac{1}{q}<\beta\leq 0$, then
$-\frac{\beta}{q-1}<1+\beta$ and we chose $\theta>0$ such that

$0\leq-\frac{\beta}{q-1}<\theta<1+\beta<1.$

As in part (v), the estimate (4.12) holds which we now use it into (4.19) to
obtain

$\displaystyle C^{\prime}$ $\displaystyle\geq
CR^{2-\alpha-N(q-1)}\log^{\beta}(1+R)\left(\int_{B_{2R}\setminus
B_{1}}|x|^{2-N}\log^{\theta}|x|dx\right)^{q-1}$ $\displaystyle\geq
CR^{2-\alpha-N(q-1)+2(q-1)}\log^{\beta+(q-1)\theta}R$
$\displaystyle=C\log^{\beta+(q-1)\theta}R\quad\text{for all }R>1.$

This leads to a contradiction as $R\to\infty$, since $\beta+(q-1)\theta>0$.

(viii) Assume $p=\frac{N}{N-2}$, $q=\frac{N-\alpha}{N-2}$ and
$-2+\frac{1}{q}<\beta\leq 0$. Then, using Lemma 2.2(ii) we have

$u^{p}(x)\geq c|x|^{-p(N-2)}=c|x|^{-N}\quad\mbox{ in }\mathbbm{R}^{N}\setminus
B_{1}.$

By Lemma 2.1(i) it follows that $K_{\alpha,\beta}*u^{p}$ satisfies the
improved estimate

$(K_{\alpha,\beta}*u^{p})(x)\geq
C|x|^{-\alpha}\log^{1+\beta}(1+|x|)\quad\text{in }\mathbbm{R}^{N}\setminus
B_{1}.$ (4.20)

Therefore, retaking the estimates (4.15)-(4.18) in the proof of part (vi-vii)
and (4.20), for all $R>1$ we have

$\displaystyle\int_{B_{2R}}u\varphi$ $\displaystyle\geq
CR^{2}\int_{B_{2R}}\varphi^{\delta}(\Delta^{2}u-\lambda\Delta u)$
$\displaystyle\geq
CR^{2}\int_{B_{2R}}(K_{\alpha,\beta}*u^{p})u^{q}\varphi^{\delta}$
$\displaystyle\geq
CR^{2-\alpha}\log^{1+\beta}(1+R)\int_{B_{2R}}u^{q}\varphi^{\delta}$
$\displaystyle\geq
CR^{2-\alpha-N(q-1)}\log^{1+\beta}(1+R)\left(\int_{B_{2R}}u\varphi\right)^{q}.$
(4.21)

We next divide our analysis into two separate cases.
Case 1: If $-1<\beta<0$. Since $u(x)\geq c|x|^{2-N}$ in $B_{2R}\setminus
B_{1}$, it follows that

$\displaystyle C^{\prime}$ $\displaystyle\geq
CR^{2-\alpha-N(q-1)}\log^{1+\beta}(1+R)\left(\int_{B_{2R}}u\varphi\right)^{q-1}$
$\displaystyle\geq CR^{2-\alpha-N(q-1)+2(q-1)}\log^{1+\beta}R$
$\displaystyle=C\log^{1+\beta}R\quad\text{for all }R>1.$

where $C,C^{\prime}>0$ are positive constants independent of $R$. Thus, since
$1+\beta>0$, we conclude that ($P^{+}$) does not have non-negative solutions
in Case 1.
Case 2: If $-2+\frac{1}{q}<\beta\leq-1$. We choose $\theta_{0}>0$ such that

$0<\frac{-1-\beta}{q-1}<\theta_{0}<2+\beta\leq 1.$ (4.22)

Using the estimate (4.12) into (4) we obtain:

$\displaystyle C^{\prime}$ $\displaystyle\geq
CR^{2-\alpha-N(q-1)}\log^{1+\beta}(1+R)\left(\int_{B_{2R}}|x|^{2-N}\log^{\theta_{0}}|x|\right)^{q-1}$
$\displaystyle\geq CR^{2-\alpha-N(q-1)+2(q-1)}\log^{1+\beta+(q-1)\theta_{0}}R$
$\displaystyle=C\log^{1+\beta+(q-1)\theta_{0}}R\quad\text{for all $R$ large}.$

where $C,C^{\prime}>0$ are positive constants independent of $R$. Thus, since
$1+\beta+(q-1)\theta_{0}>0$, we conclude that ($P^{+}$) does not have non-
negative solutions in Case 2, thereby completes our proof by combining Case 1
and Case 2.

(ix) Assume $p=\frac{N-\alpha}{N-2}$, $q=\frac{N}{N-2}$ and
$-2+\frac{1}{q}<\beta\leq 0$. Let us note that from part (iii) above,
($P^{+}$) has no solutions if $\beta\geq-1$. Thus, we shall assume in the
following that $-2+\frac{1}{q}<\beta\leq-1$. Using Lemma 2.2(ii) we have

$u^{p}(x)\geq c|x|^{-p(N-2)}=c|x|^{-(N-\alpha)}\quad\mbox{ in
}\mathbbm{R}^{N}\setminus B_{1}.$

Thus, for all $|y|\geq 2|x|>2$ and $\frac{|y|}{2}\leq|x-y|\leq\frac{3|y|}{2}$,
one has

$\displaystyle(K_{\alpha,\beta}*u^{p})(x)$ $\displaystyle\geq C\int_{|y|\geq
2|x|}|y|^{-\alpha-p(N-2)}\log^{\beta}(1+|y|)dy$ $\displaystyle=C\int_{|y|\geq
2|x|}|y|^{-N}\log^{\beta}(1+|y|)dy$ $\displaystyle\geq
C\log^{1+\beta}(1+|x|)\quad\text{for all }x\in\mathbbm{R}^{N}\setminus B_{1}.$
(4.23)

Retaking the estimates (4.15)-(4.18) in the proof of part (vi-vii) and (4),
for all $R>1$ we have

$\displaystyle\int_{B_{2R}}u\varphi$ $\displaystyle\geq
CR^{2}\int_{B_{2R}}(\Delta^{2}u-\lambda\Delta u)\varphi^{\delta}$
$\displaystyle\geq
CR^{2}\int_{B_{2R}}(K_{\alpha,\beta}*u^{p})u^{q}\varphi^{\delta}$
$\displaystyle\geq
CR^{2}\log^{1+\beta}(1+R)\int_{B_{2R}}u^{q}\varphi^{\delta}$
$\displaystyle\geq
CR^{2-N(q-1)}\log^{1+\beta}(1+R)\left(\int_{B_{2R}}u\varphi\right)^{q}.$
(4.24)

Similarly, we choose $\theta_{0}>0$ that satisfies (4.22). Using the estimate
(4.12) into (4) we find:

$\displaystyle C^{\prime}$ $\displaystyle\geq
CR^{2-N(q-1)}\log^{1+\beta}(1+R)\left(\int_{B_{2R}}|x|^{2-N}\log^{\theta_{0}}|x|\right)^{q-1}$
$\displaystyle\geq CR^{2-N(q-1)+2(q-1)}\log^{1+\beta+(q-1)\theta_{0}}R$
$\displaystyle=C\log^{1+\beta+(q-1)\theta_{0}}R\quad\text{for all $R$ large}.$

where $C,C^{\prime}>0$ are positive constants independent of $R$. Thus, since
$1+\beta+(q-1)\theta_{0}>0$, we conclude that ($P^{+}$) does not have non-
negative solutions in this case and thus complete the proof of our Theorem.∎

##  5 Proof of Theorem 1.3

Assume $N\geq 3$. To construct a solution
$u\in\mathcal{C}^{4}(\mathbbm{R}^{N})$, we consider

$u(x)=\Phi(x)*w^{-\gamma}\log^{\tau}w\quad\text{in }\mathbbm{R}^{N},$ (5.1)

where $w(x)=(A+|x|^{2})^{\frac{1}{2}}$ for some $A>e$ and
$\Phi(x)=\frac{1}{(N-2)\sigma_{N}}|x|^{-(N-2)}$ is the fundamental solution of
the Laplace operator.

######  Lemma 5.1.

Assume $2<\gamma\leq N$, $-1<\tau<1$ and $A>e$. Let $u$ be given by (5.1).
Then, there exists $\lambda^{*}=\lambda(N,A,\gamma,\tau)>0$ so that for all
$\lambda>\lambda^{*}$ large one has

$\Delta^{2}u-\frac{\lambda}{2}\Delta u\geq 0\quad\text{in }\mathbbm{R}^{N}.$

In particular, one has

$\Delta^{2}u-\lambda\Delta u\geq-\frac{\lambda}{2}\Delta
u=\frac{\lambda}{2}w^{-\gamma}\log^{\tau}w\quad\text{in }\mathbbm{R}^{N}.$
(5.2)

###### Proof.

From (5.1) and Theorem 1.1 in [5], we have

$-\Delta u(x)=w(x)^{-\gamma}\log^{\tau}w(x)\quad\text{in }\mathbbm{R}^{N}.$
(5.3)

Let $v=-\Delta u$, by computing directly, we have

$\displaystyle v^{\prime}=\,$ $\displaystyle-\gamma
rw^{-\gamma-2}\log^{\tau}w+\tau rw^{-\gamma-2}\log^{\tau-1}w,$ $\displaystyle
v^{\prime\prime}=\,$ $\displaystyle-\gamma w^{-\gamma-2}\log^{\tau}w+\tau
w^{-\gamma-2}\log^{\tau-1}w+\gamma(\gamma+2)r^{2}w^{-\gamma-4}\log^{\tau}w$
$\displaystyle-\tau(2\gamma+2)r^{2}w^{-\gamma-4}\log^{\tau-1}w+\tau(\tau-1)r^{2}w^{-\gamma-4}\log^{\tau-2}w.$

Since $r^{2}=w^{2}-A$, we find

$\displaystyle-$ $\displaystyle\Delta
v=-v^{\prime\prime}-\frac{N-1}{r}v^{\prime}$ $\displaystyle=$
$\displaystyle\gamma(N-\gamma-2)w^{-\gamma-2}\log^{\tau}w+\tau(2\gamma+2-N)w^{-\gamma-2}\log^{\tau-1}w+A\gamma(\gamma+2)w^{-\gamma-4}\log^{\tau}w$
$\displaystyle-A\tau(2\gamma+2)w^{-\gamma-4}\log^{\tau-1}w+\tau(1-\tau)(w^{2}-A)w^{-\gamma-4}\log^{\tau-2}w.$
(5.4)

Then, combining (5.2), (5.3) and (5), we get

$\displaystyle\Delta^{2}u-\frac{\lambda}{2}\Delta u=-\Delta
v+\frac{\lambda}{2}v$ $\displaystyle=$ $\displaystyle
w^{-\gamma-2}\log^{\tau-1}w\left\\{\left[\gamma(N-\gamma-2)+\frac{\lambda
w^{2}}{2}\right]\log w+\tau(2\gamma+2-N)\right\\}$
$\displaystyle+Aw^{-\gamma-4}\log^{\tau-1}w\left\\{\gamma(\gamma+2)\log
w-\tau(2\gamma+2)\right\\}+\tau(1-\tau)(w^{2}-A)w^{-\gamma-4}\log^{\tau-2}w.$

Hence,

$\displaystyle\frac{\Delta^{2}u-\frac{\lambda}{2}\Delta
u}{w^{-\gamma-2}\log^{\tau-1}w}=$
$\displaystyle\left[\gamma(N-\gamma-2)+\frac{\lambda w^{2}}{2}\right]\log
w+\tau(2\gamma+2-N)$ $\displaystyle+A\frac{\gamma(\gamma+2)\log
w-\tau(2\gamma+2)}{w^{2}}+\frac{\tau(1-\tau)}{\log
w}\Big{(}1-\frac{A}{w^{2}}\Big{)}.$

The leading term in the right-hand side of the above equality is
$\frac{\lambda w^{2}}{2}\log w$ and it is the only term containing $\lambda$.
Thus, we may take the proper constant $\lambda^{*}=\lambda(N,A,\gamma,\tau)>0$
such that

$\Delta^{2}u-\frac{\lambda}{2}\Delta u\geq 0\quad\text{in }\mathbbm{R}^{N}.$

From the above, the inequality (5.2) follows immediately. ∎

######  Lemma 5.2.

Assume $0\leq\alpha<N$, $\beta>\alpha-N$, $p,q>0$, $2<\gamma\leq N$ and
$-1<\tau<1$. If $u(x)$ is the function defined in (5.1), then for all
$x\in\mathbbm{R}^{N}$ one has

$u(x)\leq c\left\\{\begin{split}&w^{-(\gamma-2)}\log^{\tau}w&\text{if
}&2<\gamma<N,\\\ &w^{-(N-2)}\log^{1+\tau}w&\text{if
}&\gamma=N,\end{split}\right.$

for some constant $c>0$. Furthermore, there exists
$C=C(N,\alpha,\beta,A,p,q,\gamma,\tau)>0$ such that

$\displaystyle(K_{\alpha,\beta}*u^{p})u^{q}(x)$ $\displaystyle\leq
C\begin{cases}\displaystyle w^{-(\gamma-2)q}\log^{1+\beta+\tau(p+q)}w&\text{if
}\displaystyle\gamma<N,\,p=\frac{N-\alpha}{\gamma-2},\,\beta+\tau
p<-1,\\\\[8.5359pt] \displaystyle
w^{N-\alpha-(\gamma-2)(p+q)}\log^{\beta+\tau(p+q)}w&\text{if
}\displaystyle\gamma<N,\,\frac{N-\alpha}{\gamma-2}<p<\frac{N}{\gamma-2},\\\\[8.5359pt]
w^{-\alpha-(\gamma-2)q}\log^{1+\beta+\tau(p+q)}w&\text{if
}\displaystyle\gamma<N,\,p=\frac{N}{\gamma-2},\,\tau p>-1\\\\[8.5359pt]
w^{-\alpha-(\gamma-2)q}\log^{\beta+\tau q}w&\text{if
}\displaystyle\gamma<N,\,p>\frac{N}{\gamma-2},\\\\[8.5359pt]
w^{-(N-2)q}\log^{1+\beta+(1+\tau)(p+q)}w&\text{if
}\displaystyle\gamma=N,\,p=\frac{N-\alpha}{N-2},\,\beta+(1+\tau)p<-1,\\\\[8.5359pt]
w^{N-\alpha-(N-2)(p+q)}\log^{\beta+(1+\tau)(p+q)}w&\text{if
}\displaystyle\gamma=N,\,\frac{N-\alpha}{N-2}<p<\frac{N}{N-2},\\\\[8.5359pt]
w^{-\alpha-(N-2)q}\log^{1+\beta+(1+\tau)(p+q)}w&\text{if
}\displaystyle\gamma=N,\,p=\frac{N}{N-2},\\\\[8.5359pt]
w^{-\alpha-(N-2)q}\log^{\beta+(1+\tau)q}w&\text{if
}\displaystyle\gamma=N,\,p>\frac{N}{N-2}.\end{cases}$ (5.5)

###### Proof.

Recall from (1.1) that

$K_{\alpha,\beta}(x)=|x|^{-\alpha}\log^{\beta}(1+|x|)\quad\text{in
}\mathbbm{R}^{N},$

where $0\leq\alpha\leq N,\beta>\alpha-N$. From (5.1), one has

$u=K_{N-2,0}*w^{-\gamma}\log^{\tau}w,$

where $2<\gamma\leq N$, $-1<\tau<1$ and $w(x)=(A+|x|^{2})^{\frac{1}{2}}$ with
$A>e$ large. We first apply Lemma 2.1(iii) for $\alpha=N-2$, $\beta=0$ and
$f=w^{-\gamma}\log^{\tau}w$ so that $\sigma=\gamma$, $\kappa=\tau$. Thus, for
all $x\in\mathbbm{R}^{N}$, $u(x)$ satisfies the following estimate:

$u(x)\leq c\left\\{\begin{split}&w^{-(\gamma-2)}\log^{\tau}w&\text{if
}&2<\gamma<N\\\ &w^{-(N-2)}\log^{1+\tau}w&\text{if
}&\gamma=N\end{split}\right.\quad\text{ in }\mathbbm{R}^{N}.$ (5.6)

In the case $2<\gamma<N$ one has

$u(x)\leq cw^{-(\gamma-2)}\log^{\tau}w\quad\text{in }\mathbbm{R}^{N}.$

We next apply Lemma 2.1(iii) again for $f=u^{p}\leq
cw^{-(\gamma-2)p}\log^{\tau p}w$ so that $\sigma=(\gamma-2)p$, $\kappa=\tau
p$. Then, for all $x\in\mathbbm{R}^{N}$ one has

$\displaystyle(K_{\alpha,\beta}*u^{p})(x)\leq c\left\\{\begin{aligned}
&\log^{1+\beta+\tau p}w&\text{if }&(\gamma-2)p=N-\alpha,\,\beta+\tau p<-1,\\\
&w^{N-\alpha-(\gamma-2)p}\log^{\beta+\tau p}w&\text{if
}&N-\alpha<(\gamma-2)p<N,\\\ &w^{-\alpha}\log^{1+\beta+\tau p}w&\text{if
}&(\gamma-2)p=N,\,\tau p>-1,\\\ &w^{-\alpha}\log^{\beta}w&\text{if
}&(\gamma-2)p>N.\\\ \end{aligned}\right.$ (5.7)

Similarly, in the case $\gamma=N$ one has

$u(x)\leq cw^{-(N-2)}\log^{1+\tau}w\quad\text{in }\mathbbm{R}^{N}.$

We apply Lemma 2.1(iii) for $f=u^{p}\leq cw^{-(N-2)p}\log^{(1+\tau)p}w$ so
that $\sigma=(N-2)p$, $\kappa=(1+\tau)p$. Thus, for all $x\in\mathbbm{R}^{N}$,
we have

$\displaystyle(K_{\alpha,\beta}*u^{p})(x)$ $\displaystyle\leq
c\left\\{\begin{aligned} &\log^{1+\beta+(1+\tau)p}w&\text{if
}&(N-2)p=N-\alpha,\,\beta+(1+\tau)p<-1,\\\
&w^{N-\alpha-(N-2)p}\log^{\beta+(1+\tau)p}w&\text{if }&N-\alpha<(N-2)p<N,\\\
&w^{-\alpha}\log^{1+\beta+(1+\tau)p}w&\text{if }&(N-2)p=N,\\\
&w^{-\alpha}\log^{\beta}w&\text{if }&(N-2)p>N.\\\ \end{aligned}\right.$ (5.8)

Hence, we combine (5.7)-(5.8) to obtain

$\displaystyle(K_{\alpha,\beta}*u^{p})(x)\leq C\left\\{\begin{aligned}
&\log^{1+\beta+\tau p}w&\text{if
}&\gamma<N,\,p=\frac{N-\alpha}{\gamma-2},\,\beta+\tau p<-1,\\\
&w^{N-\alpha-(\gamma-2)p}\log^{\beta+\tau p}w&\text{if
}&\gamma<N,\,\frac{N-\alpha}{\gamma-2}<p<\frac{N}{\gamma-2},\\\
&w^{-\alpha}\log^{1+\beta+\tau p}w&\text{if
}&\gamma<N,\,p=\frac{N}{\gamma-2},\,\tau p>-1\\\
&w^{-\alpha}\log^{\beta}w&\text{if }&\gamma<N,\,p>\frac{N}{\gamma-2},\\\
&\log^{1+\beta+(1+\tau)p}w&\text{if
}&\gamma=N,\,p=\frac{N-\alpha}{N-2},\,\beta+(1+\tau)p<-1,\\\
&w^{N-\alpha-(N-2)p}\log^{\beta+(1+\tau)p}w&\text{if
}&\gamma=N,\,\frac{N-\alpha}{N-2}<p<\frac{N}{N-2},\\\
&w^{-\alpha}\log^{1+\beta+(1+\tau)p}w&\text{if
}&\gamma=N,\,p=\frac{N}{N-2},\\\ &w^{-\alpha}\log^{\beta}w&\text{if
}&\gamma=N,\,p>\frac{N}{N-2}.\end{aligned}\right.$

From the above estimates and (5.6) we deduce (5.5). ∎

######  Lemma 5.3.

Assume $\alpha=N$, $\beta>0$, $p,\,q>0$, $2<\gamma\leq N$ and $\tau=0$. If
$u(x)$ is the function defined in (5.1), then for all $x\in\mathbbm{R}^{N}$
one has

$u(x)\leq c\left\\{\begin{split}&w^{-(\gamma-2)}&\text{if }&2<\gamma<N,\\\
&w^{-(N-2)}\log w&\text{if }&\gamma=N,\end{split}\right.$

for some constant $c>0$. Furthermore, Thus, there exists
$C=C(N,\beta,A,p,q,\gamma)>0$ such that

$(K_{\alpha,\beta}*u^{p})u^{q}(x)\leq
C\begin{cases}w^{-(\gamma-2)(p+q)}\log^{1+\beta}w&\text{if
}\gamma<N,\,0<p<\frac{N}{\gamma-2},\\\\[5.69046pt]
w^{-N-(N-2)q}\log^{\beta+q}w&\text{if }\gamma=N,\,p>\frac{N}{N-2}.\end{cases}$
(5.9)

###### Proof.

We first apply a similar argument to that in (5.6) with $\tau=0$ to obtain the
following estimates.

$u(x)\leq c\left\\{\begin{split}&w^{-(\gamma-2)}&\text{if }&2<\gamma<N\\\
&w^{-(N-2)}\log w&\text{if }&\gamma=N\end{split}\right.\quad\text{ in
}\mathbbm{R}^{N}.$ (5.10)

Since $\alpha=N$ and $\beta>0$, one has

$K_{N,\beta}(x)=|x|^{-N}\log^{\beta}(1+|x|)\quad\text{in }\mathbbm{R}^{N}.$

We apply Lemma 2.1(iii) for $K_{N,\beta}$ and $f=u^{p}$, where $u(x)$
satisfies (5.10). Thus, for all $x\in\mathbbm{R}^{N}$ one has

$\displaystyle(K_{N,\beta}*u^{p})(x)\leq c\left\\{\begin{aligned}
&w^{-(\gamma-2)p}\log^{1+\beta}w&\text{if }&0<\gamma<N,0<(\gamma-2)p<N,\\\
&w^{-N}\log^{\beta}w&\text{if }&\gamma=N,(N-2)p>N.\\\ \end{aligned}\right.$
(5.11)

Combining now (5.10) and (5.11) we deduce (5.9). ∎

We are now ready to proceed with the construction of a positive solution to
($P^{+}$) for varying cases regarding the value of $p,q,p+q$ and $\beta$.

$\displaystyle\textbf{Case 1:}\quad p>\frac{N-\alpha}{N-2},\quad
q>\frac{N-\alpha}{N-2},\quad
p+q>\frac{2N-\alpha}{N-2}\quad\text{and}\quad\beta>\alpha-N.$

We divide our analysis into two separate cases.
Case 1a: $\frac{N-\alpha}{N-2}<p\leq\frac{N}{N-2}$. This means
$N-\alpha<(N-2)p\leq N$ and we also have $(N-2)(p+q)>2N-\alpha$. Thus, we
choose $\gamma<N$ close to $N$ so that

$N>(\gamma-2)p>N-\alpha\quad\text{and}\quad(\gamma-2)(p+q)>2N-\alpha>N+\gamma-\alpha.$
(5.12)

Take $u$ as defined in (5.1), with $\gamma$ satisfying (5.12) and $\tau=0$.
Then, by the second estimate in (5.5) one has

$\displaystyle(K_{\alpha,\beta}*u^{p})u^{q}(x)\leq
cw^{N-\alpha-(\gamma-2)(p+q)}\log^{\beta}w\leq cw^{-\gamma}\quad\text{in
}\mathbbm{R}^{N}.$ (5.13)

Combining (5.13) with (5.2) in Lemma 5.1 we find

$\Delta^{2}u-\lambda\Delta u\geq\frac{\lambda}{2}w^{-\gamma}\geq
C(K_{\alpha,\beta}*u^{p})u^{q}\quad\text{in }\mathbbm{R}^{N}.$

Case 1b: $p>\frac{N}{N-2}$. Let $u$ be given by (5.1) where $\gamma=N$ and
$\tau=0$. Since $p>\frac{N}{N-2}$ and $q>\frac{N-\alpha}{N-2}$, we apply the
last estimate in (5.5) to obtain

$\displaystyle(K_{\alpha,\beta}*u^{p})u^{q}(x)\leq
cw^{-\alpha-(N-2)q}\log^{\beta+q}w\leq cw^{-N}.$ (5.14)

Combining (5.14) with (5.2) in Lemma 5.1 we find

$\Delta^{2}u-\lambda\Delta u\geq\frac{\lambda}{2}w^{-N}\geq
C(K_{\alpha,\beta}*u^{p})u^{q}\quad\text{in }\mathbbm{R}^{N}.$

Thus, for suitable constants $C=C(N,\alpha,\beta,\lambda,p,q,A)>0$ in the case
1a and 1b separately, $Cu$ is a solution of ($P^{+}$).

$\displaystyle\textbf{Case 2:}\quad p=\frac{N-\alpha}{N-2},\quad
q>\frac{N}{N-2}\quad\text{and}\quad\beta<-1.$

Let $u$ be given by (5.1) where $\gamma=N$ and $\tau>-1$ close to $-1$ so that

$\beta+(1+\tau)p<-1.$

Since $p=\frac{N-\alpha}{N-2}$, by applying the fifth estimate in (5.5) one
has

$\displaystyle(K_{\alpha,\beta}*u^{p})u^{q}(x)\leq
cw^{-(N-2)q}\log^{1+\beta+(1+\tau)(p+q)}w\leq cw^{-N}\log^{\tau}w\quad\text{in
}\mathbbm{R}^{N}.$ (5.15)

Combining (5.15) with (5.2) in Lemma 5.1 we find

$\Delta^{2}u-\lambda\Delta u\geq\frac{\lambda}{2}w^{-N}\log^{\tau}w\geq
C(K_{\alpha,\beta}*u^{p})u^{q}\quad\text{in }\mathbbm{R}^{N}.$

Thus, for a suitable constant $C=C(N,\alpha,\beta,\lambda,p,q,A)>0$, $Cu$ is a
solution of ($P^{+}$).

$\displaystyle\textbf{Case 3:}\quad p>\frac{N}{N-2},\quad
q=\frac{N-\alpha}{N-2}\quad\text{and}\quad\beta<-1.$

Let $u$ be given by (5.1) where $\gamma=N$ and $\tau>-1$ close to $-1$ so that

$\tau>\beta+(1+\tau)q.$

Since $p>\frac{N}{N-2}$ and $q=\frac{N-\alpha}{N-2}$, by the last estimate in
(5.5) one has

$\displaystyle(K_{\alpha,\beta}*u^{p})u^{q}(x)\leq
cw^{-\alpha-(N-2)q}\log^{\beta+(1+\tau)q}w\leq
cw^{-N}\log^{\tau}w\quad\text{in }\mathbbm{R}^{N}.$ (5.16)

Combining (5.16) with (5.2) in Lemma 5.1 we find

$\Delta^{2}u-\lambda\Delta u\geq\frac{\lambda}{2}w^{-N}\log^{\tau}w\geq
C(K_{\alpha,\beta}*u^{p})u^{q}\quad\text{in }\mathbbm{R}^{N}.$

Thus, for a suitable constant $C=C(N,\alpha,\beta,\lambda,p,q,A)>0$, $Cu$ is a
solution of ($P^{+}$).

$\displaystyle\textbf{Case 4:}\quad p>\frac{N-\alpha}{N-2},\quad
q>\frac{N-\alpha}{N-2},\quad
p+q=\frac{2N-\alpha}{N-2}\quad\text{and}\quad\beta<-1.$

Let $u$ be given by (5.1) where $\gamma=N$ and $\tau>-1$ close to $-1$ so that

$\tau>\beta+(1+\tau)(p+q).$

Since $p,q>\frac{N-\alpha}{N-2}$ and $p+q=\frac{2N-\alpha}{N-2}$, we have
$N-\alpha<(N-2)p<N$. Then, by the sixth estimate in (5.5) one has

$\displaystyle(K_{\alpha,\beta}*u^{p})u^{q}(x)\leq
cw^{N-\alpha-(N-2)(p+q)}\log^{\beta+(1+\tau)(p+q)}w\leq cw^{-N}\log^{\tau}w.$
(5.17)

Combining (5.17) with (5.2) in Lemma 5.1 we find

$\Delta^{2}u-\lambda\Delta u\geq\frac{\lambda}{2}w^{-N}\log^{\tau}w\geq
C(K_{\alpha,\beta}*u^{p})u^{q}\quad\text{in }\mathbbm{R}^{N}.$

Thus, for a suitable constant $C=C(N,\alpha,\beta,\lambda,p,q,A)>0$, $Cu$ is a
solution of ($P^{+}$).

$\displaystyle\textbf{Case 5:}\quad p=\frac{N-\alpha}{N-2},\quad
q=\frac{N}{N-2}\quad\text{and}\quad\beta<-2.$

Let $u$ be given by (5.1) where $\gamma=N$ and $\tau>-1$ close to $-1$ so that

$\tau>1+\beta+(1+\tau)(p+q).$

Since $p=\frac{N-\alpha}{N-2}$, by the fifth estimate in (5.5) one has

$\displaystyle(K_{\alpha,\beta}*u^{p})u^{q}(x)\leq
cw^{-(N-2)q}\log^{1+\beta+(1+\tau)(p+q)}w\leq cw^{-N}\log^{\tau}w.$ (5.18)

Combining (5.18) with (5.2) in Lemma 5.1 we find

$\Delta^{2}u-\lambda\Delta u\geq\frac{\lambda}{2}w^{-N}\log^{\tau}w\geq
C(K_{\alpha,\beta}*u^{p})u^{q}\quad\text{in }\mathbbm{R}^{N}.$

Thus, for a suitable constant $C=C(N,\alpha,\beta,\lambda,p,q,A)>0$, $Cu$ is a
solution of ($P^{+}$).

$\displaystyle\textbf{Case 6:}\quad p=\frac{N}{N-2},\quad
q=\frac{N-\alpha}{N-2}\quad\text{and}\quad\beta<-2.$

Let $u$ be given by (5.1) where $\gamma=N$ and $\tau>-1$ close to $-1$ so that

$\tau>1+\beta+(1+\tau)(p+q).$

Since $p=\frac{N}{N-2}$, by the seventh estimate in (5.5) one has

$\displaystyle(K_{\alpha,\beta}*u^{p})u^{q}(x)\leq
cw^{-\alpha-(N-2)q}\log^{1+\beta+(1+\tau)(p+q)}w\leq cw^{-N}\log^{\tau}w.$
(5.19)

Combining (5.19) with (5.2) in Lemma 5.1 we find

$\Delta^{2}u-\lambda\Delta u\geq\frac{\lambda}{2}w^{-N}\log^{\tau}w\geq
C(K_{\alpha,\beta}*u^{p})u^{q}\quad\text{in }\mathbbm{R}^{N}.$

Thus, for a suitable constant $C=C(N,\alpha,\beta,\lambda,p,q,A)>0$, $Cu$ is a
solution of ($P^{+}$). ∎

##  6 Proof of Theorem 1.4

Assume $N\geq 3$, $\alpha=N$, $\beta>0$ and $q>0$. Since $p\geq 1$, we split
our proof into two cases: $1\leq p\leq\frac{N}{N-2}$ and $p>\frac{N}{N-2}$.
Case 1: $1\leq p\leq\frac{N}{N-2}$. This means $N-2\leq(N-2)p\leq N$ and we
also have $(N-2)(p+q)>N$. Thus, we choose $\gamma<N$ close to $N$ so that

$(\gamma-2)p<N\quad\text{and}\quad(\gamma-2)(p+q)>N>\gamma.$ (6.1)

Let $u$ be the function defined in (5.1), with $\gamma$ satisfying (6.1) and
$\tau=0$. Then, by the first estimate in (5.9) one has

$(K_{\alpha,\beta}*u^{p})u^{q}(x)\leq w^{-(\gamma-2)(p+q)}\log^{1+\beta}w\leq
cw^{-\gamma}\quad\text{in }\mathbbm{R}^{N}.$ (6.2)

Combining (6.2) with (5.2) in Lemma 5.1 we find

$\Delta^{2}u-\lambda\Delta u\geq\frac{\lambda}{2}w^{-\gamma}\geq
C(K_{\alpha,\beta}*u^{p})u^{q}\quad\text{in }\mathbbm{R}^{N}.$

Case 2: $p>\frac{N}{N-2}$. Let $u$ be given by (5.1) where $\gamma=N$ and
$\tau=0$. Since $p>\frac{N}{N-2}$ and $q>0$, we apply the second estimate in
(5.9) to obtain

$(K_{\alpha,\beta}*u^{p})u^{q}(x)\leq w^{-N-(N-2)q}\log^{\beta+q}w\leq
cw^{-N}\quad\text{in }\mathbbm{R}^{N}.$ (6.3)

Combining (6.3) with (5.2) in Lemma 5.1 we find

$\Delta^{2}u-\lambda\Delta u\geq\frac{\lambda}{2}w^{-N}\geq
C(K_{\alpha,\beta}*u^{p})u^{q}\quad\text{in }\mathbbm{R}^{N}.$

Thus, for suitable constants $C=C(N,\beta,\lambda,p,q,A)>0$ in the Case 1 and
Case 2 separately, $Cu$ is a solution of ($P^{+}$). ∎

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