#  Existence and nonexistence results of polyharmonic boundary value problems
with supercritical growth

Abdellaziz Harrabi<EMAIL_ADDRESS>Foued Mtiri
<EMAIL_ADDRESS>Wafa Mtaouaa<EMAIL_ADDRESS>Mathematics Department,
Northern Border University, Arar, Saudi Arabia.  Institut supérieur de
Mathématiques Appliquées et de l’Informatique, Kairouan, Tunisia.  Abdus Salam
International Centre for Theoretical Physics, Trieste, Italy.  Mathematics
Department, Faculty of Sciences and Arts, King Khalid University, Muhayil
Asir, Saudi Arabia.  ANLIG, UR13ES32, Faculty of Sciences of Tunis, Elmanar
University, Tunisia.  Mathematics Department, Faculty of Sciences and Arts,
King Khalid University, Muhayil Asir, Saudi Arabia.  AGTS, LR11ES53,
Université de Sfax, Tunisie.

###### Abstract

We establish some existence results of polyharmonic boundary value problems
with supercritical growth. Our approach is based on truncation argument as
well as $L^{\infty}$-bounds. Also, by virtue of Pucci-serrin’s variational
identity [19], we provide some nonexistence results.

###### keywords:

variational identity, polyharmonic equation, supercritical growth, boot strap
argument..

##  1 Introduction

Consider the following polyharmonic equations

$(P_{m}):\quad(-\Delta)^{m}u=g(u)\quad\mbox{in }\;\Omega,$

under the Dirichlet boundary conditions

$\displaystyle u=\frac{\partial
u}{\partial\nu}=\ldots=\frac{\partial^{m-1}u}{\partial\nu^{m-1}}=0\quad\mbox{on
}\;\partial\Omega,$ (1.1)

or the Navier boundary conditions

$\displaystyle u=\Delta u=\ldots=\Delta^{m-1}u=0\quad\mbox{on
}\;\partial\Omega,$ (1.2)

where $N\geq 2m+1$, $\Omega\subset\mathbb{R}^{N}$ is a bounded smooth domain
and $g\in C(\mathbb{R}).$ satisfying:

$\displaystyle|g(s)|\leq
C(1+|s|^{\frac{N+2m}{N-2m}}),\quad\forall\,\,s\in\mathbb{R}.$ (1.3)

We define the main $m$-order differential operator $D^{m}$ by

$D^{m}u=\left\\{\begin{array}[]{llll}\nabla\Delta^{j-1}u&\mbox{ if
$r=2j-1$},\\\ \Delta^{j}u&\mbox{ if $r=2j.$ }\end{array}\right.$

The appropriate functional spaces of the variational setting to the above
problems are respectively

$H_{0}^{m}(\Omega):=\left\\{v\in
H^{m}(\Omega);\;\nabla^{j}u=0\;\mbox{on}\;\partial\Omega,\;\mbox{for}\;j=0,1,..,m-1\right\\},$

and

$H_{\vartheta}^{m}(\Omega):=\left\\{v\in
H^{m}(\Omega);\;\Delta^{j}v=0\;\mbox{on}\;\partial\Omega,\;\mbox{for}\;j<\frac{m}{2}\right\\}.$

We denote indifferently $H_{\vartheta}^{m}(\Omega)$ and $H_{0}^{m}(\Omega)$ by
$H_{m}$ which is a Hilbert space equipped with the following scalar product
(see [10])

$\displaystyle\int_{\Omega}D^{m}u\cdot
D^{m}vdx=\displaystyle\int_{\Omega}\nabla\Delta^{j-1}u\cdot\nabla\Delta^{j-1}vdx\quad\mbox{if
}\;m=2j-1,\\\
\quad\quad\displaystyle\int_{\Omega}\Delta^{j}u\Delta^{j}vdx\quad\mbox{if
}\;m=2j.$

Set $G(s)=\displaystyle\int^{s}_{0}g(t)dt$. If $g$ satisfies (1.3), then the
associate energy functional of the boundary value problem $(P_{m})$ is defined
by

$I(u)=\frac{1}{2}\int_{\Omega}|D^{m}u|^{2}dx-\int_{\Omega}G(u)dx,\;\;\,\forall\,u\in
H_{m},$

and belongs to $C^{1}(H_{m})$. So, $u\in H_{m}$ is a weak solution of problem
$(P_{m})$ if and only if $u$ is a critical point of $I$, that is

$\int_{\Omega}D^{m}u\cdot
D^{m}vdx=\displaystyle\int_{\Omega}{g}(u)vdx,\;\;\forall\;v\in H_{m}.$

If in addition we assume that $g\in C^{0,a}_{loc}(\mathbb{R})$, then any weak
solution of the boundary value problem $(P_{m})$ belongs in
$C^{2m}(\overline{\Omega})$ (see the appendix of [15] for more
details)(1)(1)(1) This regularity allows us to recuperate the last part of the
Navier boundary conditions (1.2). The Palais-Smale compactness condition plays
a central role in the critical points theory and it is satisfied in the most
part of literature under the following standard assumptions [2]:

  1. (AR)

Ambrosetti-Rabinowitz condition: There exist $\beta>2$ and $s_{0}>0$ such that

$g(s)s\geq\beta G(s)>0\;\;\mbox{ for all }\,|s|>s_{0}.$

  2. (SCP)

Subcritical polynomial growth condition: There exist $1<p<\frac{N+2m}{N-2m}$
and a positive constant such that

$|g(s)|\leq C(1+|s|^{p})\;\;\mbox{ for all }\;s\in\mathbb{R}.$

These assumptions have been relaxed into the following large subcritical
growth conditions [15]:

  1. $(G_{1})$:

There exist $C>0$ and $s_{0}>0$ such that

$C|g(s)|^{\frac{2N}{N+2m}}\leq sg(s)-2G(s),\,\;\;\forall\;\;|s|>s_{0},$

and

  1. $(G_{2})$:

$\lim\limits_{s\to\infty}\dfrac{g(s)}{|s|^{\frac{N+2m}{N-2m}}}=0.$

In this paper, we consider

$\left\\{\begin{array}[]{llll}(-\Delta)^{m}u=f(u)+\lambda|u|^{p-1}u,\;\mbox{
in }\;\Omega;\\\ u\mbox{ satisfies }\eqref{D};\mbox{ or
}\eqref{N},\end{array}\right.$ $None$

where $\lambda$ is a positive real parameter and

$\displaystyle 1<p<\frac{N+2m}{N-2m}.$ (1.4)

We examine the effect of the parameter $\lambda$ to study the existence and
the nonexistence of regular solutions of $(E_{m,\lambda})$. In [4], Brezis and
Nirenberg considered the problem $E_{1,\lambda}$ with
$f(s)=|s|^{\frac{4}{N-2}}s$. They used a variant of the mountain pass theorem
without the Palais-Smale condition to prove the existence of positive solution
for all $\lambda>0$ if $N\geq 4,$ and $N=3$ with $3<p<5,$ or $\lambda$ large
enough if $N=3$ and $1<p\leq 3$.

When the variational approach cannot be employed, the question of existence of
solutions may be dealt via topological methods, or bifurcation theory, or
truncation technique [1, 6, 11, 14, 15, 16, 21, 23]. The proof of existence is
essentially reduced to deriving $L^{\infty}$-bounds for a priori solutions or
solutions of the corresponding perturbed equations. We shall employ truncation
procedure together with minimax argument to obtain existence results of
problems $(E_{m,\lambda}),$ for $\lambda$ large enough where we only  assume
that $f\in C(\mathbb{R})$ satisfies the following assumption near $0$:

  1. $(H_{0})$

: $\lim\limits_{s\to 0}\frac{f(s)}{s}=L\in[-\infty,\;\lambda_{1})$ and there
exist $\nu\in[0,1)$ and $C_{1}>0$ such that

$\displaystyle|f(s)|\leq C_{1}|s|^{1-\nu},\quad\;\forall\;\;|s|\leq 1.$ (1.5)

Note that if $L\in(-\infty,\;\lambda_{1})$, then (1.5) holds with $\nu=0.$

Our main existence result reads as follows:

###### Theorem 1.1

Assume that $p$ verifies (1.4) and $f\in C(\mathbb{R})$ satisfies $(H_{0})$.
Then, there exist $\underline{\nu}=\underline{\nu}(\mathbb{N},p,m)>0$ and
$\overline{\lambda}=\overline{\lambda}(\nu_{0},\Omega,N,p,f,m)>0$ such that
for all $\nu<\underline{\nu}$ and $\lambda\geq\overline{\lambda}$:

  1. 1.

The problem $(E_{m,\lambda})$ admits a nontrivial solution belonging in
$C^{2m-1}(\overline{\Omega})$ and $\|u\|_{L^{\infty}(\Omega)}<1.$ Moreover, if
$f\in C^{0,a}\left((-1,1)\right)$ for some $a\in(0,1)$, then $u\in$
$C^{2m}(\overline{\Omega}).$

  2. 2.

If we assume in addition that $f(s)s>0,\,\ \,\forall\,\ |s|\leq 1\,\
\,\mbox{and}\,\ \,s\neq 0$. Then the problem $(E_{m,\lambda})$ with the Navier
boundary conditions (1.2), admits two nontrivial regular solutions
$u_{-}<0<u_{+}$.

Consider the following Dirichlet boundary value problem

$(-\Delta)^{m}u=|u|^{q-1}u+\lambda|u|^{p-1}u\;\,\,\mbox{in}\,\,\,\Omega,\,\,\,\,u\mbox{
satisfies }\eqref{D},$ (1.6)

When $\Omega$ is a star shaped domain and $m=1$, the first relevant
nonexistence result follows from Pohozaev’s identity for all
$q\geq\frac{N+2}{N-2}$ and $\lambda\leq 0$ (see [5, 18]). For more general
higher order Dirichlet boundary value problem, Pucci and Serrin [19]
established a variational identity allowing further nonexistence results . In
particular they proved

###### Theorem A

  * 1.

Let $u\in C^{2m}(\Omega)\cap C^{2m-1}(\overline{\Omega})$ be a solution of the
Dirichlet boundary value problem $(P_{m})$. Assume that there exists
$q>\frac{N+2m}{N-2m}$ such that $g(s)s-(q+1)G(s)\geq 0$ for all
$s\in\mathbb{R}$. Then $u\equiv 0.$

  * 2.

Moreover, if $u$ is a solution of problem (1.6) with $\lambda<0,$
$q\geq\frac{N+2m}{N-2m};$ or $\lambda=0$ and $q>\frac{N+2m}{N-2m}.$ Then
$u\equiv 0.$

The critical growth $q=\frac{N+2m}{N-2m}$ is more difficult even for the case
$\lambda=0,$ which has been achieved if $m=2$ for only positive solutions of
(1.6) [17] or nodal solutions in a ball [13]. When $\lambda>0$, $m=1$ and
$\Omega$ is a ball, Brezis and Nirenberg [4] proved that nontrivial positive
bounded solutions of (1.6) only exist in the interval
$(\frac{\lambda_{1}}{4},\lambda_{1})$ for $N=3$ and $(0,\lambda_{1})$ for all
$N\geq 4$ (see also [3] concerning radial solutions with prescribed number of
nodes). For the higher order case $m\geq 2$, similar result was stated in only
a so called critical dimension $2m+1\leq N\leq 4m-1$ [8, 12, 20]. The reader
may consult [9, 10] for further comments including also some nonexistence
results involving the Navier boundary value biharmonic problem.

In the following, we assume that

$\displaystyle 1\leq p<\frac{N+2m}{N-2m},$ (1.7)

and

  * $(H_{1})$

: There exist $q>\frac{N+2m}{N-2m}$, $s_{0}>0$ and $C_{0}>0$ such that

$f(s)s-(q+1)F(s)\geq 0,\quad\mbox{for all}\quad
s\in\mathbb{R}\quad\mbox{and}\quad f(s)s\geq C_{0}|s|^{q+1},\quad\mbox{for
all}\quad|s|\leq s_{0}.$

We exhibit a $\underline{\lambda}>0$ such that a necessary condition for a
nontrivial solution of the Dirichlet boundary value problem $(E_{m,\lambda})$
to exist is $\lambda>\underline{\lambda}$. This extends the nonexistence
results stated in Theorem A.

###### Theorem 1.2

Let $\Omega$ be a smooth star shaped domain. Assume that $p$ verifies (1.7)
and $f\in C(\mathbb{R})$ satisfies $(H_{1})$. Then, there exists
$\underline{\lambda}=\underline{\lambda}(p,q,m,s_{f},N,\Omega)>0$ such that
for any $\lambda<\underline{\lambda},$ the problem $(E_{m,\lambda})$ with the
Dirichlet boundary conditions (1.1), has no nontrivial regular solution.

###### Remark 1.1

  * 1.

Assume that $p$ verifies (1.4) and $f\in C(\mathbb{R})$ satisfies $(H_{0})$
with $L\in[0,\lambda_{1}),$ and $(H_{1})$. Thanks to Theorems 1.1 and 1.2 we
can find $\overline{\lambda}>0$ and $\underline{\lambda}>0$ such that the
Dirichlet boundary value problem $(E_{m,\lambda})$ admits a nontrivial
solution if $\lambda>\overline{\lambda}$ and has no a nontrivial solution if
$\lambda<\underline{\lambda}.$

  * 2.

The following nonlinearities satisfy $(H_{0})$-$(H_{1})$

$f(s):=|s|^{q-1}s\exp(as),\,\ a\geq 0\quad\mbox{ and }\quad
f(s):=Ls\exp((q+1)s)\quad\mbox{ with }\quad q>\frac{N+2m}{N-2m}\mbox{ and
}L\in[0,\lambda_{1}).$

Next, consider the following problem

$\left\\{\begin{array}[]{llll}(-\Delta)^{m}u=\alpha^{-1}f(\alpha
u)+|u|^{p-1}u\;\mbox{in }\;\Omega;\\\ u\mbox{ satisfies }\eqref{D};\mbox{ or
}\eqref{N},\end{array}\right.$ $None$

where $\alpha\in(0,1)$. From Theorems 1.1 and 1.2, we derive the following
Corollary.

###### Corollary 1.1

Assume that $f\in C(\mathbb{R})$ and $p$ satisfy respectively $(H_{0})$ and
(1.4). Then there exist $\underline{\nu}=\underline{\nu}(,N,m,p)>0$ and
$\underline{\alpha}=\underline{\alpha}(f,N,m,\Omega)>0$ such that for any
$0\leq\nu\leq\underline{\nu}$ and $0\leq\alpha\leq\underline{\alpha},$ we have

  1. 1.

The problem $(Z_{m,\alpha})$ admits a nontrivial solution belonging in
$C^{2m-1}(\overline{\Omega})$ and $\|u\|_{L^{\infty}(\Omega)}<\alpha^{-1}.$
Moreover, if $f\in C^{0,a}\left((-1,1)\right)$ for some $a\in(0,1)$, then
$u\in$ $C^{2m}(\overline{\Omega}).$

  2. 2.

If we assume in addition that $f(s)s>0,\,\forall|s|\leq 1\,\ \,\mbox{and}\,\
\,s\neq 0$. Then the Navier boundary value problem $(Z_{m,\alpha})$ admits two
nontrivial regular solutions $u_{-}<0<u_{+}.$

  3. 3.

Let $\Omega$ be a star shaped domain. Assume that $f\in C(\mathbb{R})$ and $p$
satisfy respectively $(H_{1})$ and (1.7). Then there exists
$\overline{\alpha}=\overline{\alpha}(p,q,m,f,N,\Omega)>0$ such that for any
$\alpha>\overline{\alpha},$ the Dirichlet boundary value problem
$(Z_{m,\alpha})$ has no nontrivial regular solution.

The proof of Corollary 1.1 follows from a simple scaling argument. In fact,
set $v=\alpha^{-1}u$ with $\alpha=\lambda^{\frac{-1}{p-1}}$, it is easy to see
that $v$ is a solution of $(Z_{m,\alpha})$ if and only if $u$ is a solution of
$(E_{m,\lambda})$.

###### Remark 1.2

  * 1.

Point 1 of this Corollary improves the existence result stated in [1].

  * 2.

Set $f(s)=\frac{-|s|^{2-\nu}\exp(s)}{s}$. Then $f$ satisfies $(H_{0})$ with
$L=-\infty$ and Corollary 1.1 implies that the following problem has a
nontrivial solution

$(-\Delta)^{m}u=-\frac{|s|^{2-\nu}\exp(\alpha
s)}{\alpha^{\nu}s}+|u|^{p-1}u\;\;\;\mbox{in }\;\;\;\Omega,$

for $\nu$ and $\alpha$ small enough. Point out that the above equation cannot
be seen as a small perturbation of the following subcritical problem

$(-\Delta)^{m}u=|u|^{p-1}u\;\;\;\mbox{in }\;\;\;\Omega.$

This paper is organized as follows: In section 2, we provide the proofs of
Theorems 1.1. Section 3 is devoted to the proofs of Theorems 1.2.

##  2  Proof of Theorem 1.1

In the following $C$ denotes always generic positive constants depending on
$(N,\Omega,p,m,C_{1})$ only, which could be changed from one line to another.

Proof of point $1$. According to assumption $(H_{0})$, we can find
$0<\epsilon_{0}<\inf(1,\lambda_{1})$ and $0<s^{\prime}_{0}<1$ such that

$\displaystyle
F(s)\leq\frac{\lambda_{1}-\epsilon_{0}}{2}s^{2}\quad\;\forall\;\;|s|\leq
s^{\prime}_{0},$ (2.8)

and

$\displaystyle|f(s)s|\leq C|s|^{2-\nu},\quad\mbox{ and }\quad|F(s)|\leq
C|s|^{2-\nu}\quad\;\forall\;\;|s|\leq s^{\prime}_{0}.$ (2.9)

Observe that we may rewrite the Poincaré inequality as follows

$\frac{1}{2}\int_{\Omega}|D^{m}u|^{2}dx-\frac{\lambda_{1}-\epsilon_{0}}{2}\int_{\Omega}u^{2}dx\geq\frac{\epsilon_{0}}{2\lambda_{1}}\int_{\Omega}|D^{m}u|^{2}dx,\quad\forall\;\;u\in
H_{m}.$ (2.10)

Let $\theta\in C^{1}(\mathbb{R})$ the cut-off function such that
$\theta(s)=1\;\;\mbox{if
}\;\;|s|\leq\frac{s^{\prime}_{0}}{2},\quad\theta(s)=0\;\;\mbox{if
}\;\;|s|\geq\ s^{\prime}_{0},\,\mbox{ and }0\leq\theta\leq 1\;\forall
s\in\mathbb{R}.$ For $\alpha\in(0,1)$, set

$f_{\alpha}(s)=\frac{\theta(\alpha s)}{\alpha}f(\alpha s),\quad
F_{\alpha}(s)=\int_{0}^{s}f_{\alpha}(t)ds,\quad
g_{\alpha}(t)=f_{\alpha}(s)+|s|^{p-1}s,$

and
$G_{\alpha}(s)=\displaystyle\int_{0}^{s}g_{\alpha}(t)dt=F_{\alpha}(s)+\frac{|s|^{p+1}}{p+1}.$
In view of the definition of $\theta$ and (2.8) (respectively (2.9)), we
deduce then

$0\leq F_{\alpha}(s)=\frac{F(\alpha
s)}{\alpha^{2}}\leq\frac{\lambda_{1}-\epsilon_{0}}{2}s^{2},\;\;\;\;\forall\;\;(\alpha,s)\in(0,1)\times\mathbb{R},$
(2.11)

respectively

$|f_{\alpha}(s)|\leq C\alpha^{-\nu},\;\;|f_{\alpha}(s)s|\leq
C\alpha^{-\nu}\quad\mbox{ and }\quad|F_{\alpha}(s)|\leq
C\alpha^{-\nu},\;\;\,\forall\;\;(\alpha,s)\in(0,1)\times\mathbb{R}.$ (2.12)

Consider the truncated problems under the Navier or the Dirichlet boundary
conditions

$\left\\{\begin{array}[]{llll}(-\Delta)^{m}u=g_{\alpha}(u)\;\;\;\mbox{in
}\;\Omega;\\\ u\mbox{ satisfies }\eqref{D};\mbox{ or
}\eqref{N}.\end{array}\right.$ $None$

The associate energy functional of problems $(E_{m,\alpha})$ is

$I_{\alpha}(u)=\frac{1}{2}\int_{\Omega}|D^{m}u|^{2}dx-\int_{\Omega}G_{\alpha}(u)dx,\quad
u\in H_{m}.$

As $g_{\alpha}(s)=|s|^{p-1}s$ for all $|s|\geq\frac{s^{\prime}_{0}}{\alpha}$,
then $g_{\alpha}$ satisfies $(AR).$ Also (2.9) implies

$\displaystyle(SCP)_{\alpha}:\quad|g_{\alpha}(s)|\leq
C\alpha^{-\nu}(1+|s|^{p})\quad\mbox{for all}\quad s\in\mathbb{R}.$

Consequently, $I_{\alpha}\in C^{1}(H_{m})$ and satisfies the Palais-Smale
condition. From (2.10) and (2.11), one concludes

$I_{\alpha}(u)\geq\frac{\epsilon_{0}}{2\lambda_{1}}\int_{\Omega}|D^{m}u|^{2}dx-\frac{1}{p+1}\int_{\Omega}|u|^{p+1}dx.$

We apply Sobolev’s inequality, there holds

$I_{\alpha}(u)\geq\frac{\epsilon_{0}}{2\lambda_{1}}\int_{\Omega}|D^{m}u|^{2}dx-C\left(\int_{\Omega}|D^{m}u|^{2}dx\right)^{\frac{p+1}{2}}.$

Hence, there exist two constants $\rho,\,\beta>0$ such that
$(I_{\alpha})_{\mid\partial B_{\rho}}\geq\beta.$ Fix $w_{0}\in H_{m}$,
$w_{0}\neq 0$ and set $v_{\nu}=b_{0}\alpha^{-\frac{\nu}{p+1}}w_{0}$,
$b_{0}>0$. As $p+1>2$ and $\alpha\in(0,1)$, from (2.9), we can find $b_{0}>0$
large enough such that

$I_{\alpha}(v_{\nu})\leq\alpha^{-\nu}\left(\frac{b_{0}^{2}}{2}\int_{\Omega}|D^{m}w_{0}|^{2}dx-\frac{b_{0}^{p+1}}{p+1}\int_{\Omega}w_{0}^{p+1}dx+C\right)\leq
0.$

According to the mountain pass theorem, $I_{\alpha}$ admits a nontrivial
critical point $u_{\alpha}\in H_{m}$ satisfying

$I_{\alpha}(u_{\alpha})=\inf\limits_{\gamma\in\Gamma}\sup\limits_{t\in[0,1]}I_{\alpha}(\gamma(t))\quad\mbox{where
}\quad\Gamma=\left\\{\gamma\in
C([0,1],H_{m});\;\gamma(0)=0,\gamma(1)=v_{\nu}\right\\}.$

Using again (2.9), we derive

$\displaystyle
I_{\alpha}(u_{\alpha})\leq\sup_{t\in[0,1]}I_{\alpha}(tv_{\nu})\leq\frac{t^{2}}{2}\int_{\Omega}|D^{m}v_{\nu}|^{2}dx-\int_{\Omega}F_{\alpha}(tv_{\nu})dx\leq
C\alpha^{-\nu}.$ (2.13)

As

$I^{\prime}_{\alpha}(u_{\alpha})u_{\alpha}=\displaystyle\int_{\Omega}|D^{m}u_{\alpha}|^{2}dx-\displaystyle\int_{\Omega}{g}(u_{\alpha})u_{\alpha}dx=0,$

we obtain

$\displaystyle\begin{split}&\;I_{\alpha}(u_{\alpha})=I_{\alpha}(u_{\alpha})-\frac{1}{p+1}I_{\alpha}^{\prime}(u_{\alpha})u_{\alpha}\\\
&=\left(\frac{1}{2}-\frac{1}{p+1}\right)\int_{\Omega}|D^{m}u_{\alpha}|^{2}dx-\int_{\Omega}F_{\alpha}(u_{\alpha})dx+\frac{1}{p+1}\int_{\Omega}u_{\alpha}f_{a}(u_{\alpha})dx\leq
C\alpha^{-\nu}.\end{split}$

According to (2.13) and (2.9), there holds

$\displaystyle\int_{\Omega}|D^{m}u_{\alpha}|^{2}dx\leq C\alpha^{-\nu}.$ (2.14)

Now, we need the following Lemma based on a boot-strap argument:

###### Lemma 2.1

$u_{\alpha}\in C^{2m-1}(\overline{\Omega})$ and there exist $C>0$ and
$\gamma>0$ depending respectively only on $(\Omega,N,m,p,C)$ and $(N,p,m)$
such that

$\displaystyle\|u_{\alpha}\|_{C^{2m-1,\alpha^{\prime}}(\Omega)}\leq
C\alpha^{-\nu\gamma}.$ (2.15)

Precisely if $\frac{2N}{p(N-2m)}<\frac{N}{2m},$ then
$\gamma=\frac{2mp^{3}}{N(p-1)}\left(\frac{2m}{N(p-1)}-\frac{(N-2m}{2N}\right)^{-1}.$

Proof. Sobolev’s inequality gives

$\|u_{\alpha}\|_{L^{\frac{2N}{N-2m}}(\Omega)}\leq
A\|D^{m}u_{\alpha}\|^{2}_{L^{2}(\Omega)},$

which combined with $(SCP)_{\alpha}$ imply

$\displaystyle\|g_{\alpha}(u_{\alpha})\|_{L^{q_{1}}(\Omega)}\leq
C\alpha^{-\nu}\left(1+\|D^{m}u_{\alpha}\|^{p}_{L^{2}(\Omega)}\right),\quad\mbox{where}\quad
q_{1}=\frac{2N}{p(N-2m)}.$ (2.16)

We invoke now Rellich-Kondrachov’s theorem and $L^{q}$-$W^{2m,q}$ regularity
[10] (2)(2)(2) Note that the $L^{q}$-$W^{2m,q}$ regularity is also valid under
the Navier boundary conditions (see [14]).. Precisely, if
$g_{\alpha}(u_{\alpha})\in L^{q}(\Omega)$ for some $q>1$, we have

$\displaystyle\|u_{\alpha}\|_{L^{q^{*}}(\Omega)}\leq
C\|u_{\alpha}\|_{W^{2m,q}(\Omega)}\leq
C\|g_{\alpha}(u_{\alpha})\|_{L^{q}(B_{1})},$ (2.17)

where

$q^{*}=\frac{qN}{N-2mq}\quad\mbox{if}\;\;2mq<N,\quad\mbox{ and }\quad
q^{*}=p\frac{N+1}{2m}\quad\mbox{ if }\;\;q=\frac{N}{2m},$

and

$\displaystyle\|u_{\alpha}\|_{C^{2m-1,a^{\prime}}(\Omega)}\leq
C\|u_{\alpha}\|_{W^{2m,q}(\Omega)}\leq
C\|g_{\alpha}(u_{\alpha})\|_{L^{q}(B_{1})},\quad\mbox{ if }\;\;2mq>N.$ (2.18)

So, if $q>p$, $(SCP)_{\alpha}$ and (2.17) imply

$\displaystyle\|g_{\alpha}\|_{L^{\frac{q^{*}}{p}}(B_{\frac{1}{2}})}\leq
C\alpha^{-\nu}\left(1+\|g\|^{p}_{L^{q^{*}}(B_{1})}\right),\quad\quad\mbox{ if
}\;\;2mq\leq N.$ (2.19)

Consequently, if $2mq_{1}>N$ (respectively $2mq_{1}=N$, the desired estimate
(2.15) follows from (2.16) and (2.18) (respectively (2.19) with $q=q_{1}$,
(2.18)-(2.19) with $q=p\frac{N+1}{2m}$ and (2.16)).

Let us now give more attention to the difficult case $2mq_{1}<N$. Firstly, we
have

$q_{1}^{*}=\frac{q_{1}N}{N-2mq_{1}}=\frac{2N}{p(N-2m)-4m}>\frac{2N}{N-2m}>p\quad\mbox{as}\quad
p(N-2m)<N+2m.$

Set $q_{2}=\frac{q_{1}^{*}}{p}$, $q_{k+1}=\frac{q_{k}^{*}}{p}$. We claim that
there exists $k_{0}\in\mathbb{N}^{*}$ such that

$\displaystyle 2mq_{k_{0}+1}>N\quad\mbox{and}\quad 2mq_{k_{0}}<N.$ (2.20)

Indeed, working by contradiction and suppose that $2mq_{k}<N$ for all
$k\in\mathbb{N}^{*}$. Recall that
$q^{*}=\frac{qN}{N-2mq}\;\;\mbox{if}\;\;2mq<N$, then
$\frac{1}{q_{k+1}}=\frac{p}{q_{k}}-\frac{2mp}{N}$. From a direct calculation,
we obtain

$\displaystyle\frac{1}{q_{k+1}}=\frac{p^{k}}{q_{1}}-\frac{2mp}{N}\sum_{j=0}^{k-1}p^{j}=p^{k}\left(\frac{1}{q_{1}}-\frac{2mp}{N(p-1)}\right)+\frac{2mp}{N(p-1)}.$
(2.21)

As $p$ verifies (1.4) and $q_{1}=\frac{2N}{p(N-2m)}$, then
$\frac{1}{q_{1}}-\frac{2mp}{N(p-1)}<0$. Consequently $\frac{1}{q_{k}}$
converges to $-\infty$, so we reach a contradiction since we assume that
$\frac{1}{q_{k}}>\frac{1}{2mN}$. Set

$\beta=\frac{2mp}{N(p-1)}\left(\frac{2mp}{(p-1)}-\frac{1}{q_{1}}\right)^{-1}.$

From (2.21), we may see that

$\displaystyle p^{k_{0}}<\beta\quad\mbox{and}\quad p^{k_{0}+1}>\beta.$ (2.22)

Therefore, iterating (2.19) and using (2.22), there holds

$\displaystyle\|g_{\alpha}(u_{\alpha})\|_{L^{q_{k_{0}+1}}(\Omega)}\leq
C\alpha^{-\nu^{p^{k_{0}}}}\left(1+\|g_{\alpha}(u_{\alpha})\|^{p^{k_{0}}}_{L^{q_{1}}(\Omega)}\right)\leq
C\alpha^{-2\nu^{p^{k_{0}}}}\left(1+\|D^{m}u_{\alpha}\|^{p^{k_{0}+1}}_{L^{2}(\Omega)}\right).$

From (2.20) we have $mq_{k_{0}+1}>N$ , then the last inequality combined with
(2.18), (2.16) and (2.14) imply

$\displaystyle\|u_{\alpha}\|_{C^{2m-1,a^{\prime}}(\Omega)}\leq\|g_{\alpha}(u_{\alpha})\|_{L^{q_{k_{0}+1}}(B_{1})}\leq
C\alpha^{-2\nu^{p^{k_{0}}}}\left(1+\|D^{m}u_{\alpha}\|^{p^{k_{0}+1}}_{L^{2}(\Omega)}\right)\leq
A\alpha^{-2\nu^{2p^{k_{0}+1}}}.$

Therefore, $u_{\alpha}\in C^{2m-1,a^{\prime}}((\overline{\Omega})$ and the
desired inequality (2.15) follows from (2.21).

Set $\underline{\nu}=\gamma^{-1}$ and take if $\nu<\underline{\nu}$, then from
(2.15) we can choose $\alpha_{0}\in(0,1)$ small enough such that

$C\alpha\alpha^{-\gamma\nu}<\frac{s^{\prime}_{0}}{2}\quad\mbox{ so that
}\quad|\alpha u_{\alpha}|<\frac{s^{\prime}_{0}}{2}\quad\mbox{ and }\quad
g_{\alpha}(u_{\alpha})=\frac{1}{\alpha}f(\alpha
u_{\alpha})+|u_{\alpha}|^{p-1}u_{\alpha},\quad\forall\;0<\alpha<\alpha_{0}.$

Set $v_{\alpha}=\alpha u_{\alpha},\;$ $\overline{\lambda}=\alpha_{0}^{1-p}$
and $\lambda=\alpha^{1-p}$, we obtain

$(-\Delta)^{m}v_{\alpha}=\alpha
g_{\alpha}(u_{\alpha})=f(v_{\alpha})+\lambda|v_{\alpha}|^{p-1}v_{\alpha}.$

So, $v_{\alpha}\in C^{2m-1,\alpha^{\prime}}((\overline{\Omega})$ is a solution
of problem $(E_{m,\lambda})$ with $\lambda=\alpha^{1-p}$ and
$\|v_{\alpha}\|_{L^{\infty}(\Omega)}\leq 1.$ This means that Theorem 1.1 holds
for $\lambda_{p}=\alpha_{0}^{1-p}.$ Moreover, if $f\in
C^{0,a}\left((-1,1)\right)$ then $g_{\alpha}(u)\in
C^{0,\inf(\alpha,\alpha^{\prime})}(\overline{\Omega})$ and from Schauder
regularity [10], we derive that $v_{\alpha}\in
C^{2m,\inf(\alpha,\alpha^{\prime})}(\overline{\Omega}).$

Proof of point 2. Recall first the following Lemma [10]:

###### Lemma 2.2

Let $u\in C^{2m}(\overline{\Omega})$ be a solution of

$\displaystyle(-\Delta)^{m}u=h(x),\mbox{ in }\Omega,\,\ \ \mbox{$u$ satisfies
\eqref{N}},$

where $h\in C(\overline{\Omega})$ is a nonnegative function such that
$h(x_{0})>0$ for some $x_{0}\in\Omega$. Then, $u(x)>0$ for all $x\in\Omega.$

Set $s_{+}=\max(0,s)$, $s_{-}=\max(0,-s)$ and

$g^{+}_{\alpha}(s)=f_{\alpha}(s_{+})+s_{+}^{p}\quad\mbox{ respectively }\quad
g^{-}_{\alpha}(s)=f_{\alpha}(-s_{-})-s_{-}^{p}.$

Following the proof of point $1$, we can show that there exists
$\overline{\lambda}>0$ such that

$\displaystyle(-\Delta)^{m}u=f(u_{+})+\lambda u_{+}^{p},\ \ \mbox{ in
}\Omega,\,\ \ \mbox{$u$ satisfies \eqref{D}},$

respectively

$\displaystyle(-\Delta)^{m}u=f(-u_{-})-\lambda u_{-}^{p},\ \ \mbox{ in
}\Omega,\,\ \ \mbox{$u$ satisfies \eqref{D}},$

admits a nontrivial solution $u_{+}\in C^{2m-1}(\overline{\Omega})$ with
(respectively $u_{-}C^{2m-1}(\overline{\Omega})$) for all
$\lambda\geq\overline{\lambda}$. Since we assume that $f(s)s>0$ for all $s\neq
0$ then $g^{+}_{\alpha}\geq 0$ (respectively $-g^{-}_{\alpha}\geq 0$).
According to Lemma 2.2, we deduce that

$u_{-}<0<u_{+}\,\ \ \mbox{ on}\,\ \ \Omega.$

The proof of Theorem 1.1 is thereby completed.

##  3 Proof of Theorem 1.2

As above, $C$ denotes always generic positive constants depending on
$(N,\Omega,p,q,m,f)$. According to assumption $(H_{1})$, we get

$\displaystyle\left(1-\frac{2N}{(q+1)(N-2m)}\right)f(s)s\leq
f(s)s-\frac{2N}{N-2m}F(s),\;\;\;\forall\;s\in\mathbb{R},$ (3.1)

Also, we can find a positive constant $C=C(s_{0},q)$ such that

$\displaystyle C|s|^{q+1}\leq f(s)s,\;\;\;\forall\;s\in\mathbb{R}.$ (3.2)

In fact, as $\leq f(s)s,\;\forall\;|s|\leq s_{0},$ we have $F(\pm s_{0})>0$.
From $(H_{1})$ we have $\left(\frac{F(s)}{s^{q+1}}\right)^{\prime}\geq 0$ for
all $s>0$. Therefore,

$f(s)s\geq F(s)\geq\frac{F(s_{0})}{s_{0}^{q+1}}s^{q+1},\;\,\forall\;s\geq
s_{0}.$

Observe that the nonlinearity $-f(-s)$ satisfies also $(H_{1})$. Then
inequality (3.2) follows. Let $\Omega$ be a star shaped smooth bounded domain.
Then, there exists $y\in\mathbb{R}^{n}$ such that $(x-y)\cdot\nu(x)\geq 0$,
for all $x\in\partial\Omega$. Consider $u\in C^{2m}(\Omega)\cap
C^{2m-1}(\Omega)$ a solution of $(P_{m})$ and $a\in\mathbb{R}$. Applying
Pucci-Serrin’s variational identity (see [19] page 702), we obtain

$\displaystyle\int_{\Omega}\Bigg{[}\left(\frac{N}{2}-a-m\right)|D^{m}u|^{2}+aug(u)-NG(u)\Bigg{]}dx=\frac{-1}{2}\int_{\partial\Omega}|D^{m}u|^{2}(\nu\cdot(x-y))ds.$

Fix $a=\frac{N-2m}{2}$, we derive

$\displaystyle\begin{split}\int_{\Omega}\left(g(u)u-\frac{2N}{N-2m}G(u)\right)dx\leq
0.\end{split}$ (3.3)

We multiply $(P_{m})$ by $u$, then Poincaré’s and Sobolev’s inequalities imply

$\displaystyle
C\left(\int_{\Omega}|u|^{p+1}dx\right)^{\frac{2}{p+1}}\leq\int_{\Omega}|D^{m}u|^{2}dx=\int_{\Omega}g(u)udx.$
(3.4)

If $u\in C^{2m}(\Omega)\cap C^{2m-1}(\Omega)$ is a nontrivial solution of
problem $(E_{m,\lambda})$ under the Dirichlet boundary conditions (1.1), then
$u$ is also a solution of $(P_{m})$ with $g(s)=f(s)+\lambda|s|^{p-1}s$. Hence,
(3.3) and (3.1) imply

$\displaystyle\;\left(1-\frac{2N}{(q+1)(N-2m)}\right)\int_{\Omega}f(u)udx$
$\displaystyle\;\leq\int_{\Omega}\left(f(u)u-\frac{2N}{N-2m}F(u)\right)dx\leq\lambda\left(\frac{2N}{(p+1)(N-2m)}-1\right)\int_{\Omega}|u|^{p+1}dx.$

By (3.2), we get then

$\displaystyle\;C\left(1-\frac{2N}{(q+1)(N-2m)}\right)\int_{\Omega}|u|^{q+1}dx$
$\displaystyle\;\leq\left(1-\frac{2N}{(q+1)(N-2m)}\right)\int_{\Omega}f(u)udx\leq\lambda\left(\frac{2N}{(p+1)(N-2m)}-1\right)\int_{\Omega}|u|^{p+1}dx.$

As $\frac{2N}{(p+1)(N-2m)}-1>0,$ we deduce that
$\displaystyle\int_{\Omega}|u|^{q+1}dx=0$ if $\lambda\leq 0$. Then $u\equiv
0$.

If $\lambda>0$, from Hölder’s inequality and (3.1), we obtain

$\displaystyle\left(\int_{\Omega}|u|^{p+1}dx\right)^{\frac{q+1}{p+1}}\leq
C\int_{\Omega}|u|^{q+1}dx\leq C\int_{\Omega}f(u)udx\leq
C\lambda\int_{\Omega}|u|^{p+1}dx.$ (3.5)

Therefore,

$\displaystyle\int_{\Omega}|u|^{p+1}dx\leq C_{1}\lambda^{\frac{p+1}{q-p}},$
(3.6)

where $C_{1}$ is a positive constant depending on $(N,\Omega,p,q,m,f)$. In
view of (3.4) and (3.5), we get

$\displaystyle\left(\int_{\Omega}|u|^{p+1}dx\right)^{\frac{2}{p+1}}\leq
C\int_{\Omega}|D^{m}u|^{2}dx\leq C\lambda\int_{\Omega}|u|^{p+1}dx.$ (3.7)

If $p=1$, we get $\lambda>\frac{1}{C}$ . If $p$ satisfies (1.4), from (3.7),
one has

$\displaystyle C_{2}\lambda^{-\frac{p+1}{p-1}}\leq\int_{\Omega}|u|^{p+1}dx,$

where $C_{2}$ is a positive constant depending on $(N,\Omega,p,q,m,f)$.
Combining the last inequality with (3.6), we deduce

$\lambda^{\frac{p+1}{q-p}+\frac{p+1}{p-1}}\geq\frac{C_{2}}{C_{1}}.$

In conclusion, the problem $(E_{m,\lambda})$ under the Dirichlet boundary
conditions (1.1), has no nontrivial regular solution if $p$ satisfies (1.4)
and $\lambda<\underline{\lambda}\mbox{ where
}\underline{\lambda}=\left(\frac{C_{2}}{C_{1}}\right)^{\delta}\quad\mbox{with}\quad\delta=\left(\frac{p+1}{q-p}+\frac{p+1}{p-1}\right)^{-1}$
(respectively if $p=1$ and $\lambda<\underline{\lambda}$ with
$\underline{\lambda}=\frac{1}{C}$).

## Acknowledgment

The authors extend their appreciation to the Deanship of Scientific Research
at King Khalid University, Abha, KSA for funding this work through Research
Group under grant number (R.G.P-2 / 121/ 42).

## References

  * [1] C.O. Alves and M.A.S. Souto, _On existence of solution for a class of semilinear elliptic equations with nonlinearities that lies between different powers,_ Abstract and App Analysis, 2008, Article ID 578417.
  * [2] A. Ambrosetti and P. H. Rabinowitz, _Dual variational methods in critical point theory and applications,_ J. Funct. Anal. 14 (1973), 349
  * [3] F. V. Atkinson, H. Brezis, and L. A. Peletier,_Nodal solutions of elliptic equations with critical Sobolev exponents,_ J. Differential Equations. 85 (1990), 151-170.
  * [4] H. Brezis and L. Nirenberg, _Positive solutions of nonlinear elliptic equations involving critical exponents,_ Comm. Pure Appl. Math. 36 (1983), 437-477.
  * [5] C. Budd, J. Norbury, _Semilinear elliptic equations and supercritical growth,_ J. Differential Equations 68 (1987) no. 2, 169–197.
  * [6] D.G. De Figueiredo, P. L. Lions and R. Nussbaum, _A priori estimates and existence of positive solutions of semilinear elliptic equations,_ J. Math. Pures Appl. 61 (1982), 41-63.
  * [7] D.G. De Figueiredo and J. Yang, _On a semilinear elliptic problem without (PS) condition,_ J. Differntial Equations 187 (2003), 412-428.
  * [8] F. Gazzola, _Critical growth problems for polyharmonic operators,_ Proceedings of the Royal Society of Edinburgh: Section A Mathematics. 128, no. (2), (1996), 251-263. doi:10.1017/S0308210500012774
  * [9] F. Gazzola, H. C. Grunau and M. Squassina, _Existence and nonexistence results for critical growth biharmonic elliptic equation,_ Cal Var 18 (2003), 117–143.
  * [10] F. Gazzola, H. C. Grunau and G. Sweers, _Polyharmonic Boundary Value Problems,_ Lecture Notes in Mathematics, vol.1991, Springer, 2010.
  * [11] B. Gidas and J. Spruck , _A priori bounds for positive solutions of nonlinear elliptic equations,_ Comm. Partial Differential Equations. 6 (1981), 883-901.
  * [12] H. C. Grunau, On a conjecture of P. Pucci and J. Serrin, Analysis. 16, no. 4, (1996), 399-404.
  * [13] H. C. Grunau, ”Polyharmonische Dirichletprobleme: Positivität, kritische Exponenten und kritische Dimensionen”, Habilitationsschrift, Universität Bayreuth, (1996).
  * [14] H. Hajlaoui, A. Harrabi, _A priori estimates and existence of positive solutions forhigher-order elliptic equations,_ J. Math.Anal.Appl.426 (2015) 484–504.
  * [15] A. Harrabi, _On the Palais–Smale condition,_ J. Funct. Anal. 267 (2014) 2995–3015.
  * [16] A. Harrabi, S. Rebhi and A. Selmi, _Existence of radial solutions with prescribed number of zeros of elliptic solutions and their Morse index,_ J. Differntial Equations 251(2011), 2409-2430.
  * [17] P. Oswald, _On a priori estimates for positive solutions of a semilinear biharmonic equation in a ball,_ Comment. Math. Univ. Carolinae 26 (1985), 565–577.
  * [18] S.I. Pohozaev, _Eigenfunctions of the equation $\Delta u+\lambda f(u)=0$_. Sov. Math. Doklady 5 (1965), 1408-1411.
  * [19] P. Pucci and J. Serrin, _A general variational identity_ , Indiana Univ. Math. J. 35 (1986), 681-703.
  * [20] P. Pucci and J. Serrin, _Critical exponents and critical dimensions for polyharmonic operators,_ J. Math.Pures Appl. 69 (1990), 55–83.
  * [21] M. Ramos and P. Rodrigues _On a fourth order superlinear elliptic problem,_ Electron. J. Diff. Eqns, Conf. 06 (2001), 243-255. 114-133.
  * [22] B. Sirakov, _Existence results and a priori bounds for higher order elliptic equations and systems,_ J.math.Pures.Appl. 9 89 (2008), no. 2, 114-133.
  * [23] R. Soranzo, _A priori estimates and existence of positive solutions of superlinear polyharmonic equation,_ Dynamic System and Appl. 3 (1994), 465-488.