# Nehari manifold approach for fractional Kirchhoff problems with extremal
value of the parameter

P. K. Mishra
Department of Mathematics
Indian Institute of Technology Bhilai
492015, Raipur, Chhattisgarh, India<EMAIL_ADDRESS>and  V. M. Tripathi
Department of Mathematics
Indian Institute of Technology Bhilai
492015, Raipur, Chhattisgarh, India<EMAIL_ADDRESS>
###### Abstract.

In this work we study the following nonlocal problem

$\left\\{\begin{aligned}
M(\|u\|^{2}_{X})(-\Delta)^{s}u&=\lambda{f(x)}|u|^{\gamma-2}u+{g(x)}|u|^{p-2}u&&\mbox{in}\
\ \Omega,\\\ u&=0&&\mbox{on}\ \
\mathbb{R}^{N}\setminus\Omega,\end{aligned}\right.$

where $\Omega\subset\mathbb{R}^{N}$ is open and bounded with smooth boundary,
$N>2s,s\in(0,1),M(t)=a+bt^{\theta-1},\;t\geq 0$ with $\theta>1,a\geq 0$ and
$b>0$. The exponents satisfy $1<\gamma<2<{2\theta<p<2^{*}_{s}=2N/(N-2s)}$
(when $a\neq 0$) and $2<\gamma<2\theta<p<2^{*}_{s}$ (when $a=0$). The
parameter $\lambda$ involved in the problem is real and positive. The problem
under consideration has nonlocal behaviour due to the presence of nonlocal
fractional Laplacian operator as well as the nonlocal Kirchhoff term
$M(\|u\|^{2}_{X})$, where
$\|u\|^{2}_{X}=\iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^{2}}{\left|x-y\right|^{N+2s}}dxdy$.
The weight functions $f,g:\Omega\to\mathbb{R}$ are continuous, $f$ is positive
while $g$ is allowed to change sign. In this paper an extremal value of the
parameter, a threshold to apply Nehari manifold method, is characterized
variationally for both degenerate and non-degenerate Kirchhoff cases to show
an existence of at least two positive solutions even when $\lambda$ crosses
the extremal parameter value by executing fine analysis based on fibering maps
and Nehari manifold.

###### Key words and phrases:

Nehari manifold, variational methods, extremal parameter, concave-convex,
multiplicity

###### Key words and phrases:

Nehari manifold (primary) and variational methods and extremal parameter and
concave-convex and multiplicity

###### 2010 Mathematics Subject Classification:

35R11, 35A15, 49J35

##  1\. Introduction

In the paper, we study the following nonlocal problem

($P_{\lambda}$) $\left\\{\begin{aligned}
M(\|u\|^{2}_{X})(-\Delta)^{s}u&=\lambda{f(x)}|u|^{\gamma-2}u+{g(x)}|u|^{p-2}u&&\mbox{in}\
\ \Omega,\\\ u&=0&&\mbox{on}\ \
\mathbb{R}^{N}\setminus\Omega,\end{aligned}\right.$

where
$\|u\|^{2}_{X}=\iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^{2}}{\left|x-y\right|^{N+2s}}dxdy$,
$\Omega\subset\mathbb{R}^{N}$ is open and bounded with smooth boundary,
$N>2s,s\in(0,1),M(t)=a+bt^{\theta-1},\;t\geq 0$ with $p/2>\theta>1,a\geq 0$
and $b>0$. The exponents satisfy $1<\gamma<2<{2\theta<p<2^{*}_{s}}$ (when
$a\neq 0$) and $2<\gamma<2\theta<p<2^{*}_{s}$ (when $a=0$), where
$2^{*}_{s}=2N/(N-2s)$ is fractional Sobolev critical exponent. Here
$(-\Delta)^{s}$ is well know fractional Laplacian operator which is defined,
up to a normalization constant, as

$(-\Delta)^{s}\varphi(x)=\int_{\mathbb{R}^{N}}\frac{2\varphi(x)-\varphi(x+y)-\varphi(x-y)}{|y|^{N+2s}}dy,\quad
x\in\mathbb{R}^{N},$

for any $\varphi\in C_{0}^{\infty}(\Omega)$. The weight functions
$f,g:\Omega\to\mathbb{R}$ are continuous and satisfy following assumptions:

  * (F)

$f\in C(\Omega)\cap L^{\infty}(\Omega)$ and there exists some $f_{0}>0$ such
that $f(x)>f_{0}$ for all $x$ near $\partial\Omega$,

  * (G)

$g\in C(\Omega)\cap L^{\infty}(\Omega)$ with $0\not\equiv
g^{+}=\max\\{0,g(x)\\}.$

The Kirchhoff term is called degenerate if $a=0$ and non-degenerate otherwise.
The problem is non-local due to the presence of the operator formed by the
nonlocal Kirchhoff term and fractional Laplacian operator. The nonlocal nature
of the operator does not allow us to compare the equation in ($P_{\lambda}$)
pointwise. The problem ($P_{\lambda}$) has a variational structure and the
suitable functional space (inspired from [22]) to look for solutions can be

$X=\big{\\{}u\in H^{s}(\mathbb{R}^{N}):\,\,u=0\mbox{ a.e. in
}\mathbb{R}^{N}\setminus\Omega\big{\\}},$

where $H^{s}(\mathbb{R}^{N})$ is the fractional Sobolev space (see [17] for
more details). We recall that $X$ is a Hilbert space endowed with the
following norm

$\left\|u\right\|_{X}:=\left(\iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^{2}}{\left|x-y\right|^{N+2s}}dxdy\right)^{1/2}.$

######  Definition 1.1.

A function $u\in X$ is said to be a (weak) solution of ($P_{\lambda}$) if for
every $\phi\in X$

$\displaystyle(a+b\|u\|^{2(\theta-1)}_{X})$
$\displaystyle\iint_{\mathbb{R}^{2N}}\frac{(u(x)-u(y))(\phi(x)-\phi(y))}{|x-y|^{N+2s}}dxdy$
$\displaystyle-{\lambda}\int_{\Omega}fu^{\gamma-1}\phi
dx-\int_{\Omega}gu^{p-1}\phi dx=0.$

To study the problem via variatonal methods, we define the associated energy
functional $\mathcal{E}_{\lambda}:X\to\mathbb{R}$ as

$\mathcal{E}_{\lambda}(u)=\frac{a}{2}\|u\|^{2}_{X}+\frac{b}{2\theta}\|u\|^{2\theta}_{X}-\frac{\lambda}{\gamma}\int_{\Omega}f|u|^{\gamma}dx-\frac{1}{p}\int_{\Omega}g|u|^{p}dx.$

One can see that $\mathcal{E}_{\lambda}(u)$ is of class $C^{1}(X)$ and
solutions of ($P_{\lambda}$), in the light of above Definition 1.1, can be
viewed as the critical points of $\mathcal{E}_{\lambda}(u)$. The class of
problem under consideration has widely studied in recent past because of their
vast application in many areas of science. The above class of problems can be
seen as stationary state of the following problem

$\left\\{\begin{aligned} u_{tt}-M\left(\int_{\Omega}|\nabla
u|^{2}dx\right)\Delta u&=F(x,u)&&\mbox{in}\ \ \Omega,\\\ u&=0&&\mbox{on}\ \
\partial\Omega,\end{aligned}\right.$

which was firstly introduced by Kirchhoff to deal with free transversal
oscillations of elastic strings (see [14]). The term $M$ measures the tension
in the string caused by any change in the length of the string during
vibration and is directly proportional to the Sobolev norm of the displacement
of the string. The degenerate case, arises if the initial tension of the
string is zero, physically a very realistic model. We refer the interested
readers to the servey [19] for recent advances in nonlocal Kirchhoff problems.

The class of nonlinearity under consideration in this paper is referred as
concave-convex nonlinearity and attracted many researchers to explore after
the pioneering work of Ambrosetti et al. in [1] where, authors have
established global multiplicity result. The work on this class of nonlinearity
can be found in [2, 4, 6, 25, 26] and references therein for both local and
nonlocal problems. In particular, [4] and [2] contain the study of quasilinear
and fractional counter part of [1], respectively. Precisely the problem
involving non-local fractional operator of the type

$(-\Delta)^{s}u=\lambda fu^{q-1}+gu^{p-1},\;u>0,\ \ \textrm{in}\ \Omega,\ \
u=0\ \textrm{on}\ \mathbb{R}^{n}\setminus\Omega,$

has been studied in [10]( see [25, 26] for $s=1$) for existence of at least
two solutions when $1<q<2$ and ${1<p<2^{*}_{s}-1}$ via Nehari manifold
technique introduced by Pokhozhaev (see [11]) and Nehari (see [15, 16]). The
multiplicity results obtained obtained in [10] (also in [25, 26]) heavily
relies on the fact that the nonlinearity exhibit the convex-concave behaviour
which is essential for fibering maps to have two critical points. In this
paper, we have shown that the degenerate fractional Kirchhoff operator allows
us to break this structure of concave-convex nonlinearity in order to show
multiple solutions via Nehari manifold technique.

Another class of nonlocal problems involves the operator formed with the
fusion of Kirchhoff term and fractional Laplacian. In [9] (see the Appendix)
authors have given a motivation for this class of problem by modelling a
fractional Kirchhoff problem coming out of string vibrations. These Kirchhoff
counterparts have also occupied the enough space in the literature due to its
vast applications in applied science, see for reference [3, 5, 8, 9, 18, 20,
21, 27] and references therein . We also refer readers to [7, 18] for non-
degenerate structure of Kirchhoff term where results have been established
with a control on the parameter $\lambda<\Lambda$ for some small $\Lambda>0$
via Nehari manifold minimization argument. The control on the parameter
$\lambda$ is natural as for $\lambda\geq\Lambda$ it is not obvious to show
that the constrained minimizers obtained in different decompositions of Nehari
set are the critical points of the associated energy functional of the problem
because $\mathcal{N}_{\lambda}$ is no longer a manifold.

In this paper, inspired from the work of [13, 23], we have included this
delicate case in our study when $\lambda\geq\lambda^{*}_{a,b}$ where
$\lambda^{*}_{a,b}$ is termed as extremal parameter introduced in [12] while
studying generalized Rayleigh quotient. The value $\lambda^{*}_{a,b}$ is
extremal in the sense that the Nehari set will no longer remain a manifold if
the parameter $\lambda$ crosses this threshold $\lambda^{*}_{a,b}$. By
overcoming the above technical difficulty, for $\epsilon>0$ sufficiently
small, we have shown existence of at least two positive solutions by
constrained minimization on Nehari manifold based on fibering analysis for
$\lambda\in(0,\lambda^{*}_{a,b}+\epsilon)$.

The work of this paper is divided in the following order. In section [2], we
have provided framework of Nehari setup and section [3] includes some
technical results together with our main result of the paper. In section [4],
[5] and [6], we have shown the existence of minimizers for the energy
functional when $\lambda<\lambda^{*}_{a,b},\lambda=\lambda^{*}_{a,b}$ and
$\lambda>\lambda^{*}_{a,b}$ respectively. We have attempted to address both
the degenerate and non-degenerate Kirchhoff cases collectively and
distinguished whenever it’s required.

##  2\. Nehari manifold structure

The main objective of this paper is to look for critical points of
$\mathcal{E}_{\lambda}$ over $X$. The critical points of
$\mathcal{E}_{\lambda}$ can be obtained via minimization of
$\mathcal{E}_{\lambda}$. Since $\mathcal{E}_{\lambda}$, fails to be bounded
from below in $X$ and therefore, we constrained the minimization problem in a
proper subset of $X$, namely Nehari manifold (a natural constraint as it
contains all the critical points of $\mathcal{E}_{\lambda}$) defined as

$\mathcal{N}_{\lambda}=\left\\{u\in
X\setminus\\{0\\}:D_{u}\mathcal{E}_{\lambda}(u)(u)=0\right\\}.$

It can be observed that $\mathcal{N}_{\lambda}$ is a disjoint union of the
following sets,

$\displaystyle\mathcal{N}_{\lambda}^{+}=\\{u\in\mathcal{N}_{\lambda}:$
$\displaystyle
D_{uu}\mathcal{E}_{\lambda}(u)(u,u)>0\\},\mathcal{N}_{\lambda}^{-}=\\{u\in\mathcal{N}_{\lambda}:D_{uu}\mathcal{E}_{\lambda}(u)(u,u)<0\\},$
$\displaystyle\mathcal{N}_{\lambda}^{0}=\\{u\in\mathcal{N}_{\lambda}:$
$\displaystyle D_{uu}\mathcal{E}_{\lambda}(u)(u,u)=0\\}.$

Consider the fibering functions $\psi_{\lambda,u}:[0,\infty)\to\mathbb{R}$ for
every $u\in X$, defined as $\psi_{\lambda,u}(t)=\mathcal{E}_{\lambda}(tu)$,
that is,

$\psi_{\lambda,u}(t)=\frac{at^{2}}{2}\|u\|^{2}_{X}+\frac{bt^{2\theta}}{2\theta}\|u\|^{2\theta}_{X}-\frac{\lambda
t^{\gamma}}{\gamma}\int_{\Omega}f|u|^{\gamma}dx-\frac{t^{p}}{p}\int_{\Omega}g|u|^{p}dx.$

Note that $u\in\mathcal{N}_{\lambda}$ if and only if
$\psi^{\prime}_{\lambda,u}(1)=0$ and, in general, $tu\in\mathcal{N}_{\lambda}$
if and only if $\psi^{\prime}_{\lambda,u}(t)=0$. Therefore, we can read Nehari
submanifolds in terms of $t=1$ being local maxima or local minima or saddle
point of $\psi_{\lambda,u}(t)$ as follows

$\displaystyle\mathcal{N}_{\lambda}^{+}$
$\displaystyle=\\{u\in\mathcal{N}_{\lambda}:\psi_{\lambda,u}^{\prime\prime}(1)>0\\},\;\mathcal{N}_{\lambda}^{-}=\\{u\in\mathcal{N}_{\lambda}:\psi_{\lambda,u}^{\prime\prime}(1)<0\\},$
$\displaystyle\mathcal{N}_{\lambda}^{0}$
$\displaystyle=\\{u\in\mathcal{N}_{\lambda}:\psi_{\lambda,u}^{\prime\prime}(1)=0\\}.$

The presence of sign changing weights as well as the nature of Kirchhoff term
can govern the behaviour of fibering functions and therefore we split the
discussion based on the sign of the integral involving sign changing weight
$g(x)$ and the nature of Kirchhoff term. For, we define the following

$\displaystyle\mathcal{C}^{+}$ $\displaystyle=\\{u\in
X:\int_{\Omega}g|u|^{p}dx>0\\}\;\;\textrm{and}\;\;\mathcal{C}^{-}=\\{u\in
X:\int_{\Omega}g|u|^{p}dx\leq 0\\}.$

In the next section we give a characterization of extremal parameter
$\lambda^{*}_{a,b}$ for the both degenerate and non-degenerate Kirchhoff term.

##  3\. Characterization of extremal parameter

For $u\in\mathcal{C}^{+}$, by solving following system of two nonlinear
equations in variables $t$ and $\lambda$

(3.1) $\left\\{\begin{aligned}
a\|tu\|^{2}_{X}+b\|tu\|^{2\theta}_{X}-\lambda\int_{\Omega}f|tu|^{\gamma}dx-t^{p}\int_{\Omega}g(x)|u|^{p}dx&=0,\\\
2a\|tu\|^{2}_{X}+2b\theta\|tu\|^{2\theta}_{X}-\lambda\gamma\int_{\Omega}f|tu|^{\gamma}dx-p\
t^{p}\int_{\Omega}g(x)|u|^{p}dx&=0,\end{aligned}\right.$

we can find $(t_{a,b}(u),\lambda_{a,b}(u))$ uniquely. In fact, by eliminating
$\lambda$ from above equations we can see $t_{a,b}(u)$ as unique zero of the
following scalar function

$m_{u}(t)=a(2-\gamma)t^{2}\|u\|^{2}_{X}+b(2\theta-\gamma)t^{2\theta}\|u\|^{2\theta}_{X}-{(p-\gamma)t^{p}}\int_{\Omega}g(x)|u|^{p}dx,$

and hence we have unique $\lambda_{a,b}(u)$, given as

$\lambda_{a,b}(u)=\frac{a(p-2)\|t_{a,b}(u)u\|^{2}_{X}+2b(p-2\theta)\|t_{a,b}(u)u\|^{2\theta}_{X}}{(p-\gamma)\int_{\Omega}f|t_{a,b}(u)u|^{\gamma}dx}.$

In the degenerate Kirchhoff case, we can get an explicit expression for the
solution of the above system (3.1) as

$t_{0,1}(u):=t(u)=\left(\frac{(2\theta-\gamma)\|u\|^{2\theta}_{X}}{{(p-\gamma)}\int_{\Omega}g(x)|u|^{p}dx}\right)^{\frac{1}{p-2\theta}}$

and

$\displaystyle\lambda_{0,1}(u):=\lambda(u)$
$\displaystyle=\left(\frac{p-2\theta}{p-\gamma}\frac{\|u\|^{2\theta}_{X}}{\int_{\Omega}f|u|^{\gamma}dx}\right)\left(\frac{2\theta-\gamma}{p-\gamma}\frac{\|u\|^{2\theta}_{X}}{\int_{\Omega}g(x)|u|^{p}dx}\right)^{\frac{2\theta-\gamma}{p-2\theta}}.$

For $a\geq 0,b>0$ define the extremal value

(3.2) $\lambda^{*}_{a,b}=\inf_{X\cap C^{+}}\lambda_{a,b}(u)$

In particular, when $a=0,b=1$, we denote $\lambda^{*}=\lambda^{*}_{0,1}$.

The following proposition is direct consequence of compact fractional Sobolev
embeddings.

######  Proposition 3.1.

The function $\lambda_{a,b}(u)$ is continuous, 0-homogeneous and unbounded
from above. Moreover, $\lambda^{*}_{a,b}>0$ and there exists
$u\in\mathcal{C}^{+}$ such that $\lambda^{*}_{a,b}=\lambda_{a,b}(u)$.

######  Remark 3.1.

The function $\lambda_{a,b}(u)$ is weakly lower semicontinuous also. For the
degenerate case, it follows directly form weak lower semi-continuity of the
norm. For non-degerate case, let $u_{n}\rightharpoonup u$ in
$\mathcal{C}^{+}$. If $\|u_{n}\|_{X}\rightarrow\|u\|_{X},$ then from
uniqueness of the solution of the system in (3.1), we get $t(u_{n})\rightarrow
t(u)$ and $\lambda(u_{n})\rightarrow\lambda(u)$. If not, then we have
$\|u\|_{X}<\liminf_{n\rightarrow\infty}\|u_{n}\|$. Therefore,
$0=\psi^{\prime}_{\lambda_{a,b},u}(t(u))<\liminf_{n\rightarrow\infty}\psi^{\prime}_{\lambda_{a,b},u_{n}}(t(u)),$
i.e., $\psi^{\prime}_{\lambda,u_{n}}(t(u))>0$ for large value of $n$. Thus
graph of $\psi_{\lambda_{a,b},u_{n}}(t)>0$ will be of the type fig 2(b) for
large values of $n$. Equivalently, $\lambda_{a,b}(u)<\lambda_{a,b}(u_{n})$ for
large $n$ or $\lambda_{a,b}(u)\leq\liminf_{n\to\infty}\lambda_{a,b}(u_{n})$.

In the following Proposition, we show that Nehari decompositions are non-
empty.

######  Proposition 3.2.

For a given $u\in X$ and $a>0,b>0$ with $1<\gamma<2$ or $a=0,b>0$ with
$2<\gamma<2\theta$ there are the following two cases:

  1. (i)

If $u\in\mathcal{\mathcal{missing}}{C}^{-}$, then for any $\lambda>0$, there
exists a unique $t_{\lambda}^{+}(u)>0$ such that
$t_{\lambda}^{+}u\in\mathcal{N}_{\lambda}^{+}.$

  2. (ii)

If $u\in\mathcal{C}^{+}$, then we have following three situations.

    1. (a)

$0<\lambda<\lambda_{a,b}(u)$, there exist unique
$0<t_{\lambda}^{+}(u)<t_{\lambda}^{-}(u)$ such that
$t_{\lambda}^{+}u\in\mathcal{N}_{\lambda}^{+}$ and
$t_{\lambda}^{-}u\in\mathcal{N}_{\lambda}^{-}$.

    2. (b)

$\lambda=\lambda_{a,b}(u)$, then there exists unique $t^{0}_{\lambda}(u)>0$
such that $t^{0}_{\lambda}u\in\mathcal{N}_{\lambda}^{0}$.

    3. (c)

$\lambda>{\lambda_{a,b}}(u)$, then $tu\not\in\mathcal{N}_{\lambda}$ for any
$t>0$. In particular, $u\not\in\mathcal{N}_{\lambda}.$

###### Proof.

Proof of this proposition is divided for degenerate and non-degenerate cases
respectively as follows.
Case 1:(Degenrate Kirchhoff case) To see the graph of fibering map
$\psi_{\lambda,u}$ define a map $\Phi_{u}:\mathbb{R}^{+}\rightarrow\mathbb{R}$
such that

$\Phi_{u}(t)=t^{2\theta-\gamma}\|u\|^{2\theta}_{X}-t^{p-\gamma}\int_{\Omega}g|u|^{p}dx,$

which has a unique critical point when $u\in\mathcal{C}^{+}$ and can be
obtained by solving the following equation

$\Phi^{\prime}_{u}(t)=(2\theta-\gamma)t^{2\theta-\gamma-1}\|u\|^{2\theta}_{X}-(p-\gamma)t^{p-\gamma-1}\int_{\Omega}g|u|^{p}dx.$

Moreover, as $\lim_{t\rightarrow 0^{+}}\Phi_{u}(t)=0$ and
$\lim_{t\rightarrow\infty}\Phi_{u}(t)=-\infty$ the graph of $\Phi_{u}$ be as
shown in the fig 1(b). When $u\in\mathcal{C}^{-}$, $\Phi^{\prime}_{u}(t)>0$
for all $t\geq 0$ and hence no critical point for $\Phi_{u}(t)$ leading to the
graph as shown in fig 1(a).

Case 2:(Non-degenrate Kirchhoff case) For a given $u\in X\setminus 0$, define
a map $\Phi_{u}:\mathbb{R}^{+}\rightarrow\mathbb{R}$ such that

$\Phi_{u}(t)=at^{2-\gamma}\|u\|^{2}_{X}+bt^{2\theta-\gamma}\|u\|^{2\theta}_{X}-t^{p-\gamma}\int_{\Omega}g|u|^{p}dx.$

Note that $tu\in\mathcal{N}_{\lambda}$ (or $t$ is a critical point of
$\psi_{\lambda,u}$) if and only if the following equation has a scalar
solution in $t$

(3.3) $\Phi_{u}(t)=\lambda\int_{\Omega}f|u|^{\gamma}dx$

In order to look for solution of above scalar equation (3.3), we analyze the
behaviour of $\Phi_{u}(t)$. For,

$\Phi^{\prime}_{u}(t)=a(2-\gamma)t^{1-\gamma}\|u\|^{2}_{X}+b(2\theta-\gamma)t^{2\theta-\gamma-1}\|u\|^{2\theta}_{X}-(p-\gamma)t^{p-\gamma-1}\int_{\Omega}g|u|^{p}dx.$

When $u\in\mathcal{C}^{-}$, $\Phi^{\prime}_{u}(t)>0$ for all $t\geq 0$ and
hence (3.3) has unique solution for all values of $\lambda>0$. Consequently,
unique projection of $u$, namely $t_{\lambda}^{+}u$ in
$\mathcal{N}_{\lambda}^{+}$ (as
$\psi_{\lambda,u}^{\prime\prime}(t)=t^{1+\gamma}\Phi^{\prime}_{u}(t)$) In
order to see the behaviour of $\Phi_{u}(t)$ when $u\in\mathcal{C}^{+}$, we can
rewrite $\Phi^{\prime}_{u}(t)=t^{1-\gamma}\mathcal{H}_{u}(t)$, where

$\mathcal{H}_{u}(t)=a(2-\gamma)\|u\|^{2}_{X}+b(2\theta-\gamma)t^{2\theta-2}\|u\|^{2\theta}_{X}-(p-\gamma)t^{p-2}\int_{\Omega}g|u|^{p}dx.$

Then

$\mathcal{H}^{\prime}_{u}(t)=b(2\theta-2){(2\theta-\gamma)}t^{2\theta-1}\|u\|^{2\theta-2}_{X}-(p-2)(p-\gamma)t^{p-1}\int_{\Omega}g|u|^{p}dx.$

Thus one can observe that there exist a unique critical point $t^{*}>0$ such
that $\mathcal{H}^{\prime}_{u}(t^{*})=0$, where

$t^{*}=\displaystyle{\left(\frac{b(2\theta-2){(2\theta-\gamma)}\|u\|^{2\theta}_{X}}{(p-2)(p-\gamma)\int_{\Omega}g|u|^{p}dx}\right)^{\frac{1}{p-2\theta}}}.$

When $u\in\mathcal{C}^{+}$, $\mathcal{H}^{\prime}_{u}(t)>0$ as $t\rightarrow
0^{+}$ and $\mathcal{H}^{\prime}_{u}(t)\rightarrow-\infty$ as
$t\rightarrow\infty$. As $\mathcal{H}_{u}(t)>0$, as $t\rightarrow 0^{+}$ and
$\mathcal{H}_{u}(t)\rightarrow-\infty$ as $t\rightarrow\infty$, there exists
unique $t_{*}>t^{*}>0$ such that $\mathcal{H}_{u}(t_{*})=0$. Therefore
$\Phi_{u}(t)$ has global maximum at unique point $t=t_{*}$. Consequently,
equation (3.3) has exactly two solutions for suitable control over $\lambda$.
In otherwords, there are unique projections of $u$, namely $t_{\lambda}^{+}u$
and $t_{\lambda}^{-}u$ such that
$t_{\lambda}^{+}u\in\mathcal{N}_{\lambda}^{+}$ and
$t_{\lambda}^{-}u\in\mathcal{N}_{\lambda}^{-}$ for $\lambda<\lambda_{a,b}(u)$.

The result of this proposition is depicted in the figure fig 2 above. Observe
that fig 2$(a)$ corresponds to the part $(i)$ and fig 2$(b)-2(d)$ are related
with parts $(ii)(a)-(c)$ respectively. ∎

######  Remark 3.2.

It is clear in light of the above results, that
$\mathcal{N}_{\lambda}^{-}\neq\emptyset$ and
$\mathcal{N}_{\lambda}^{+}\neq\emptyset$ for all $\lambda>0$. Moreover for
$0<\lambda<\lambda^{*}_{a,b}$, $\mathcal{N}_{\lambda}^{0}=\emptyset$. By
setting, $N(u)=D_{u}\mathcal{E}_{\lambda}(u)u$, we have
$D_{u}N(u)(u)=D_{uu}\mathcal{E}_{\lambda}(u)(u,u)+D_{u}\mathcal{E}_{\lambda}(u)u=D_{uu}\mathcal{E}_{\lambda}(u)(u,u)\neq
0$ as $u\in\mathcal{N}_{\lambda}$ and $\mathcal{N}_{\lambda}^{0}=\emptyset$
for $\lambda<\lambda^{*}_{a,b}$. Therefore from implicit function theorem one
can view $\mathcal{N}_{\lambda}$ as $C^{1}$ manifold of codimension one when
$\lambda<\lambda^{*}_{a,b}$. But once
$\mathcal{N}^{0}_{\lambda}\neq\emptyset$, we can have a
$u\in\mathcal{N}_{\lambda}$ such that $D_{uu}\mathcal{E}_{\lambda}(u)(u,u)=0$,
in that case the set $\mathcal{N}_{\lambda}$ will be no longer a manifold. In
coming section we will see that Nehari decomposition
$\mathcal{N}^{0}_{\lambda}\neq\emptyset$ when $\lambda$ crosses the extremal
value.

The main results of this paper are stated in the form of the following
theorems.

######  Theorem 3.1.

Let $g<0$ near $\partial\Omega$, $M$ be defined as $M(t)=a+bt^{\theta-1}$ with
$\theta>1$ satisfying $2\theta<p<2^{*}_{s}$ and $a>0$ then for
$\gamma\in(1,2)$ there exist at least two positive solutions for the problem
($P_{\lambda}$) for $\lambda\in(0,\lambda^{*}_{a,b}+\epsilon)$, where
$\epsilon>0$ is sufficiently small.

In the sequal for the degenerate Kirchhoff case we established the
multiplicity results even when the non-linearity looses the concave-convex
behaviour with no control on the sign of $g$ near boundary. Precisely, we have
the following result.

######  Theorem 3.2.

Let $M(t)=t^{\theta-1},\theta>1$ satisfying $2\theta<p<2^{*}_{s}$, then for
$\gamma\in(2,2\theta)$ there exist two positive solutions for ($P_{\lambda}$)
when $\lambda\in(0,\lambda^{*}+\bar{\epsilon})$, where $\bar{\epsilon}>0$ is
sufficiently small.

######  Remark 3.3.

The results obtained in this paper contributes to the literature in the
following way. We have complimented the work of [10, 23] for nonlocal
fractional Kirchhoff problem and the work of [10] has further extended for the
parameter lying beyond the extremal parameter value. Even when the
nonlinearity lacks the sublinear-superlinear combination, we have shown
multiplicity results for degenerate Kirchhoff case.

Observe that, from Proposition 3.2, $u$ may has its projection in
$\mathcal{N}_{\lambda}^{-}$ only if $u\in\mathcal{C}^{+}$ but
$\mathcal{N}_{\lambda}^{+}$ can have projections of $u$ irrespective of $u$
lying in $\mathcal{C}^{+}$ or not. We require this distinction on later stages
while studying the minimization problems. Therefore, for each $\lambda>0$, we
define the following sets noting this distinction

(3.4) $\displaystyle\mathcal{\hat{N}}_{\lambda}$ $\displaystyle=\left\\{u\in
X\setminus\\{0\\}:u\in\mathcal{C}^{+}\ \textrm{and}\
\lambda<\lambda_{a,b}(u)\right\\},$
$\displaystyle\hat{\mathcal{N}}^{+}_{\lambda}$ $\displaystyle=\left\\{u\in
X\setminus\\{0\\}:u\in\mathcal{C}^{-}\right\\}.$

######  Remark 3.4.

One can observe that when
$u\in\mathcal{\hat{N}}_{\lambda}\cup\hat{\mathcal{N}}^{+}_{\lambda}$ then
$tu\in\mathcal{\hat{N}}_{\lambda}\cup\hat{\mathcal{N}}^{+}_{\lambda}$ as
$\lambda_{a,b}(u)$ is $0-$ homogeneous and thus we can say that the set
$\mathcal{\hat{N}}_{\lambda}\cup\hat{\mathcal{N}}^{+}_{\lambda}$ represents a
cone generated by $\mathcal{N}_{\lambda}^{+}\cup\mathcal{N}_{{\lambda}}^{-}$,
that is,

$\mathcal{\hat{N}}_{\lambda}\cup\hat{\mathcal{N}}^{+}_{\lambda}=\\{tu:t>0,u\in\mathcal{N}_{\lambda}^{+}\cup\mathcal{N}_{\lambda}^{-}\\}.$

To prove our main result, for all $\lambda>0$, let us consider the following
minimization problems

$\hat{\mathcal{J}}^{-}_{\lambda}=\textrm{inf}\\{\mathcal{J}^{-}_{\lambda}(u):u\in\mathcal{N}^{-}_{\lambda}\\}\
\ \textrm{and}\ \
\hat{\mathcal{J}}^{+}_{\lambda}=\textrm{inf}\\{\mathcal{J}^{+}_{\lambda}(u):u\in\mathcal{N}^{+}_{\lambda}\\},$

where functions
$\mathcal{J}^{-}_{\lambda}:\hat{\mathcal{N}}_{\lambda}\rightarrow\mathbb{R}$
and
$\mathcal{J}^{+}_{\lambda}:\mathcal{\hat{N}}_{\lambda}\cup\hat{\mathcal{N}}^{+}_{\lambda}\rightarrow\mathbb{R}$
are defined as follows

$\mathcal{J}^{-}_{\lambda}(u)=\mathcal{E}_{\lambda}(t^{-}_{\lambda}(u)u)\ \
\textrm{and}\ \ \
\mathcal{J}^{+}_{\lambda}(u)=\mathcal{E}_{\lambda}(t^{+}_{\lambda}(u)u).$

We have the following observations about ${\mathcal{J}}^{\pm}_{\lambda}$ as
$1<\gamma<p$ and $2<2\theta<p$ .

######  Proposition 3.3.

The functional ${\mathcal{J}}^{\pm}_{\lambda}$ are coercive on Nehari set.

The following lemma is about the behaviour of energy functional with respect
to the parameter $\lambda$.

######  Lemma 3.1.

Let $u\in X\setminus\\{0\\}$. Let $I$ be an open interval in $\mathbb{R}^{+}$
such that $t^{\pm}_{\lambda}$ are well defined for all $\lambda\in I$. Then,

  1. (i)

the functions $I\ni\lambda\rightarrow t^{\pm}_{\lambda}(u)$ are $C^{1}$.
Moreover, $I\ni\lambda\rightarrow t^{-}_{\lambda}(u)$ is decreasing while
$I\ni\lambda\rightarrow t^{+}_{\lambda}(u)$ is increasing.

  2. (ii)

the functions $I\ni\lambda\rightarrow{\mathcal{J}}^{\pm}_{\lambda}(u)$ are
$C^{1}$ and decreasing.

###### Proof.

The proof follows from implicit function theorem. ∎

######  Remark 3.5.

When $I=(0,\lambda^{*}_{a,b})$, both the claims in Lemma 3.1 remain true,
independent of $u\in X$.

In order to prove our main result we have designed the following three
sections dealing with the case
$\lambda<\lambda^{*}_{a,b},\lambda=\lambda^{*}_{a,b}$ and
$\lambda>\lambda^{*}_{a,b}$ respectively.

##  4\. Existence of solutions when $0<\lambda<\lambda^{*}_{a,b}$

In order to prove Theorem 3.1 first we have proved existence of at least two
solutions of ($P_{\lambda}$) for $\lambda\in(0,\lambda^{*}_{a,b})$. For that,
we begin by showing that the functional $\mathcal{E}_{\lambda}$ achieves its
minimizers in the Nehari decomposition $\mathcal{N}^{-}_{\lambda}$ and
$\mathcal{N}^{+}_{\lambda}$ for all $\lambda\in(0,\lambda^{*}_{a,b})$ for both
degenerate and non-degenerate cases. We close this section by verifying that
these minimizers are the solutions to the problem in $\eqref{p}$.

######  Lemma 4.1.

For each $0<\lambda<\lambda^{*}_{a,b}$, there exists
$v_{\lambda}\in\mathcal{N}^{+}_{\lambda}$ such that
$\mathcal{J}^{+}_{\lambda}(v_{\lambda})=\hat{\mathcal{J}}^{+}_{\lambda}$.

###### Proof.

Let $\\{v_{n}\\}\subset\mathcal{N}^{+}_{\lambda}$ be an arbitrary minimizing
sequence for ${\mathcal{J}}^{+}_{\lambda}$, that is
${\mathcal{J}}^{+}_{\lambda}(v_{n})\rightarrow\hat{\mathcal{J}}^{+}_{\lambda}$.
First we claim that $\\{v_{n}\\}$ is bounded. In fact, when
$v_{n}\in\mathcal{N}^{+}_{\lambda}$ using Hölder inequality, we have

(4.1)
$b(p-2\theta)\|v_{n}\|^{2\theta}_{X}<a(p-2)\|v_{n}\|^{2}_{X}+b(p-2\theta)\|v_{n}\|^{2\theta}_{X}<\lambda(p-\gamma)S^{-\frac{\gamma}{2}}\|f\|_{\frac{2^{*}_{s}}{2^{*}_{s}-\gamma}}\|v_{n}\|^{\gamma}_{X},$

implies that sequence $\\{v_{n}\\}$ is bounded. Moreover, one can observe that
when $v\in\mathcal{N}^{-}_{\lambda}$, we have

$\displaystyle\mathcal{E}_{\lambda}(v)$
$\displaystyle=a\left(\frac{1}{2}-\frac{1}{p}\right)\|v_{n}\|^{2}_{X}+b\left(\frac{1}{2\theta}-\frac{1}{p}\right)\|v_{n}\|^{2\theta}_{X}-\left(\frac{1}{\gamma}-\frac{1}{p}\right)\lambda\int_{\Omega}f|v_{n}|^{\gamma}dx$
$\displaystyle<a(p-2)\left(\frac{\gamma-2}{2p\gamma}\right)\|v\|^{2}_{X}+b(p-2\theta)\left(\frac{\gamma-2\theta}{2p\theta\gamma}\right)\|v\|^{2\theta}_{X}.$

For for non-degenerate case (when $\gamma\in(1,2)$), we get
$\mathcal{E}_{\lambda}(v)<0.$ and hence, $\hat{\mathcal{J}}^{+}_{\lambda}<0$.
For degenerate case (when $\gamma\in(2,2\theta)$), we get

$\mathcal{E}_{\lambda}(v)<(p-2\theta)\left(\frac{\gamma-2\theta}{2p\theta\gamma}\right)\|v\|^{2\theta}_{X}<0,$

and hence $\hat{\mathcal{J}}^{+}_{\lambda}<0$. Moreover,
$\hat{\mathcal{J}}^{+}_{\lambda}>-\infty$ from definition of
$\mathcal{E}_{\lambda}$. Thus up to a subsequence, $v_{n}\rightharpoonup
v_{\lambda}\geq 0$ in $X$. Let us prove $v_{\lambda}\not\equiv 0$. If
$v_{\lambda}\equiv 0$, then
$0=\mathcal{E}_{\lambda}(v_{\lambda})\leq\displaystyle\liminf_{n\to\infty}\mathcal{E}_{\lambda}(v_{n})=\hat{\mathcal{J}}^{+}_{\lambda}<0$,
a contradiction and hence, $v_{\lambda}\not\equiv 0$. Observe that,
irrespective of $v_{\lambda}\in\mathcal{C}^{\pm}$, there exists
$t^{+}_{\lambda}(v)>0$ such that
$t^{+}_{\lambda}(v_{\lambda})v_{\lambda}\in\mathcal{N}^{+}_{\lambda}$ and
$\psi_{\lambda,v_{\lambda}}$ is decreasing in
$(0,t^{+}_{\lambda}(v_{\lambda}))$ with
$\psi^{\prime}_{\lambda,v_{\lambda}}(t^{+}_{\lambda}(v_{\lambda}))=0$.

Our claim is $v_{n}\rightarrow v_{\lambda}$ in $X$. Suppose on contrary
$v_{n}\rightharpoonup v$ then $v_{n}\rightarrow v$ in $L^{q}(\Omega)$ where
$q\in[1,2^{*}_{s})$ and $|v_{n}(x)|\leq h(x)$ a.e. in $\Omega$ for some
$h(x)\in L^{q}(\Omega)$. Under the assumptions $(F)$ and $(G)$, using Hölder
inequality and Lebesgue dominated convergence theorem, we obtain

$\int_{\Omega}f(|v_{n}|^{\gamma}-|v_{\lambda}|^{\gamma})\rightarrow
0,\;\;\int_{\Omega}g(x)|v_{n}(x)|^{p}\rightarrow\int_{\Omega}g(x)|v_{\lambda}(x)|^{p}.$

From weak lower semicontinuity of norm,
$\|v_{\lambda}\|_{X}<\liminf_{n\to\infty}\|v_{n}\|_{X}$. Using all this
information, we have

$\displaystyle\liminf_{n\rightarrow\infty}\psi^{\prime}_{\lambda,v_{n}}(t^{+}_{\lambda}(v_{\lambda}))=\liminf_{n\rightarrow\infty}\left[a(t^{+}_{\lambda})(v_{\lambda})\|v_{n}\|^{2}_{X}+b(t^{+}_{\lambda})^{2\theta-1}(v_{\lambda})\|v_{n}\|^{2\theta}_{X}\right]$
$\displaystyle-\liminf_{n\to\infty}\left[(t^{+}_{\lambda}(v_{\lambda}))^{\gamma-1}\lambda\int_{\Omega}f|v_{n}|^{\gamma}dx+(t^{+}_{\lambda}(v_{\lambda}))^{p-1}\int_{\Omega}g(x)|u_{n}(x)|^{p}\right]$
$\displaystyle>a(t^{+}_{\lambda})(v_{\lambda})\|v_{\lambda}\|^{2}_{X}+b(t^{+}_{\lambda})^{2\theta-1}(v_{\lambda})\|v_{\lambda}\|^{2\theta}_{X}-(t^{+}_{\lambda}(v_{\lambda}))^{\gamma-1}\lambda\int_{\Omega}f|v_{\lambda}|^{\gamma}dx$
$\displaystyle-(t^{+}_{\lambda}(v_{\lambda}))^{p-1}\int_{\Omega}g(x)|u_{\lambda}(x)|^{p}$
$\displaystyle=\psi^{\prime}_{\lambda,v_{\lambda}}(t^{+}_{\lambda}(v_{\lambda}))=0.$

Therefore, $\psi^{\prime}_{\lambda,v_{n}}(t^{+}_{\lambda}(v_{\lambda}))>0$,
for sufficiently large $n$. Since $v_{n}\in\mathcal{N}^{+}_{\lambda}$, by
possible fibering map it is easy to see that
$\psi^{\prime}_{\lambda,v_{n}}(t)<0$ for $0<t<1$ and
$\psi^{\prime}_{\lambda,v_{n}}(1)=0$ ($t^{+}_{\lambda}(v_{n})=1$), therefore
$t^{+}_{\lambda}(v_{\lambda})>1=t^{+}_{\lambda}(v_{n})$ for large $n$. Thus,
we have

$\hat{\mathcal{J}}^{+}_{\lambda}\leq{\mathcal{E}_{\lambda}}((t^{+}_{\lambda}(v_{\lambda})v_{\lambda}))=\mathcal{J}^{+}_{\lambda}(v_{\lambda})<\liminf_{n\rightarrow\infty}\mathcal{J}^{+}_{\lambda}(v_{n})=\hat{\mathcal{J}}^{+}_{\lambda},$

which is an absurd. Hence $v_{n}\rightarrow v_{\lambda}$ in $X$. As a
consequence of strong convergence, we have

$\lim_{n\rightarrow\infty}\psi^{{}^{\prime}}_{\lambda,v_{n}}(1)=\psi^{{}^{\prime}}_{\lambda,v_{\lambda}}(1)=0\
\ \textrm{and}\ \
\lim_{n\rightarrow\infty}\psi^{{}^{\prime\prime}}_{\lambda,v_{n}}(1)=\psi^{{}^{\prime\prime}}_{\lambda,v_{\lambda}}(1)\geq
0,$

and since $\mathcal{N}^{0}_{\lambda}=\emptyset$ for
$0<\lambda<\lambda^{*}_{a,b}$, we have
$v_{\lambda}\in\mathcal{N}^{+}_{\lambda}$ and
$\mathcal{J}^{+}_{\lambda}(v_{\lambda})=\hat{\mathcal{J}}^{+}_{\lambda}.$ ∎

######  Lemma 4.2.

For each $0<\lambda<\lambda^{*}_{a,b}$, there exists
$u_{\lambda}\in\mathcal{N}^{-}_{\lambda}$ such that
$\mathcal{J}^{-}_{\lambda}(u_{\lambda})=\hat{\mathcal{J}}^{-}_{\lambda}$.

###### Proof.

The proof is similar to Lemma 4.1. ∎

In order to show that the minimizers obtained in Lemma 4.1 and Lemma 4.2 are
the weak solutions of ($P_{\lambda}$), we make use of Theorem 2.3 of [6] as
for $\lambda<\lambda_{a,b}$, the set $\mathcal{N}^{0}_{\lambda}=\emptyset$.
Therefore the above non-trival minimizers of $\mathcal{E}_{\lambda}$ are
critical points of $\mathcal{E}_{\lambda}$ in $X$, equivalently weak solutions
of ($P_{\lambda}$).

##  5\. Existence of solutions when $\lambda=\lambda^{*}_{a,b}$

In this section, we consider the case when the parameter $\lambda$ takes the
extremal value, that is, $\lambda=\lambda^{*}_{a,b}$. Contrary to the case of
$\lambda<\lambda^{*}_{a,b}$, the set $\mathcal{N}^{0}_{\lambda^{*}_{a,b}}$ is
non-empty. Hence the study of $\mathcal{N}^{0}_{\lambda^{*}_{a,b}}$ becomes
important. In fact, we also explore the intersection of solution set of
$(P_{\lambda^{*}_{a,b}})$ with $\mathcal{N}^{0}_{\lambda^{*}_{a,b}}$ in this
section.

######  Proposition 5.1.

The set $\mathcal{N}_{\lambda^{*}_{a,b}}^{0}\neq\emptyset$ and
$\mathcal{N}_{\lambda^{*}_{a,b}}^{0}=\\{u\in\mathcal{N}_{\lambda^{*}_{a,b}}\cap\mathcal{C}^{+}:\lambda_{a,b}(u)=\lambda^{*}_{a,b}\\}$
. Moreover, $\mathcal{N}_{\lambda}^{0}\neq\emptyset$ for all
$\lambda\geq\lambda^{*}_{a,b}$.

###### Proof.

From Proposition 3.1, the minimization problem in (3.2) has a solution in
$X_{0}\cap\mathcal{C}^{+}$, that is, $\lambda_{a,b}(u)=\lambda^{*}_{a,b}$ for
some $u\in X\cap\mathcal{C}^{+}$ . Using Proposition 3.2 there exists
$t_{\lambda^{*}_{a,b}}^{0}=t_{\lambda^{*}_{a,b}}^{0}(u)>0$ such that
$t_{\lambda^{*}_{a,b}}^{0}u\in\mathcal{N}^{0}_{\lambda^{*}_{a,b}=\lambda_{a,b}(u)}$
and hence it’s nonempty. Since $\lambda_{a,b}(u)$ is unbounded above,
therefore for all $\lambda\geq\lambda^{*}_{a,b}$ there exists
$u\in\mathcal{C}^{+}$ such that $\lambda=\lambda_{a,b}(u)$ and by definition
of $\lambda_{a,b}(u)$, $t(u)u\in\mathcal{N}^{0}_{\lambda_{a,b}(u)=\lambda}$,
hence $\mathcal{N}^{0}_{\lambda_{a,b}}\neq\emptyset$ for all
$\lambda\geq\lambda^{*}_{a,b}$. ∎

Next we are going to prove an important Lemma which will help us to conclude
that the minimizers of the energy functional are not in
$\mathcal{N}^{0}_{\lambda^{*}_{a,b}}$, in spite of
$\mathcal{N}^{0}_{\lambda^{*}_{a,b}}$ being non-empty.

######  Proposition 5.2.

For any $a>0$ and $b>0$, the problem $(P_{\lambda^{*}_{a,b}})$ has no solution
in $\mathcal{N}^{0}_{\lambda^{*}_{a,b}}$.

###### Proof.

In order to give a proof of this Proposition, we first prove that for each
$u\in\mathcal{N}^{0}_{\lambda^{*}_{a,b}}$, we have

$\displaystyle-2(a+\theta
b\|u\|^{2\theta-2}_{X})(-\Delta)^{s}u-\lambda^{*}_{a,b}\gamma
f|u|^{\gamma-2}u-pg|u|^{p-2}u=0.$

In fact, for $t(u)u\in\mathcal{N}_{\lambda}$, we have

$at^{2}\|u\|^{2}_{X}+bt^{2\theta}\|u\|^{2\theta}_{X}-{\lambda}t^{\gamma}\int_{\Omega}f|u|^{\gamma}dx-t^{p}\int_{\Omega}g|u|^{p}dx=0.$

Differentiating above equation with respect to $u$ then for all $w\in X$, we

$\displaystyle t^{\prime}(u)$ $\displaystyle(2at(u)\|u\|^{2}_{X}+2\theta
b(t(u))^{2\theta-1}\|u\|^{2\theta}_{X}-\lambda(u)\gamma(t(u))^{\gamma-1}\int_{\Omega}f|u|^{\gamma}dx-p(t(u))^{p-1}\int_{\Omega}g|u|^{p}dx)$
$\displaystyle-2a(t(u))^{2}(-\Delta)^{s}uw-2\theta
b(t(u))^{2\theta}\|u\|^{2\theta-2}_{X}(-\Delta)^{s}uw$
$\displaystyle-\lambda(u)\gamma(t(u))^{\gamma}\int_{\Omega}f|u|^{\gamma-2}uwdx-(t(u))^{p}p\int_{\Omega}g|u|^{p-2}uwdx=0.$

As $u\in\mathcal{N}^{0}_{\lambda^{*}_{a,b}=\lambda(u)}$ implies $t(u)=1$, we
get

$-2(a+\theta b\|u\|^{2\theta-2}_{X})(-\Delta)^{s}u-\lambda^{*}_{a,b}\gamma
f|u|^{\gamma-2}u-pg|u|^{p-2}u=0.$

Now if $u\in X$ solves the problem $(P_{\lambda^{*}_{a,b}})$, we have

$-(a+b\|u\|^{2\theta-2}_{X})(-\Delta)^{s}u-\lambda^{*}_{a,b}f|u|^{\gamma-2}u-g|u|^{p-2}u=0.$

Now solving last two equations by eliminating $\|u\|^{2\theta-2}_{X}$, we get

$-2a(\theta-1)(-\Delta)^{s}u=\lambda^{*}_{a,b}(2\theta-\gamma)f|u|^{\gamma-2}u+(2\theta-p)g|u|^{p-2}u,$

which implies,

$-(-\Delta)^{s}u=\lambda^{*}_{a,b}\frac{2\theta-\gamma}{2a(\theta-1)}f|u|^{\gamma-2}u+\frac{2\theta-p}{2a(\theta-1)}g|u|^{p-2}u.$

Therefore near $\partial\Omega$, we have $-(-\Delta)^{s}u\geq 0$ leading to

$\displaystyle\lambda^{*}_{a,b}\gamma f|u|^{\gamma-2}u+pg|u|^{p-2}u$
$\displaystyle=-\left(2a+2b\theta\|u\|^{2\theta-2}_{X}\right)(-\Delta)^{s}u$
$\displaystyle\geq-(2a+2b\|u\|^{2\theta-2}_{X})(-\Delta)^{s}u$
$\displaystyle=2\lambda^{*}_{a,b}f|u|^{\gamma-2}u+2g|u|^{p-2}u$

or

$\left(\frac{p-2}{2-\gamma}\right)g|u|^{p-\gamma}\geq\lambda^{*}_{a,b}f>0.$

Now using $u(x)\rightarrow 0$ as $x\rightarrow\partial\Omega$, left hand side
of above inequality goes to zero while right hand side is bounded away from
zero near boundary in the light of the assumption $(F)$ to give an absurd. ∎

######  Remark 5.1.

In degenerate Kirchhoff case, after elimination of $(-\Delta)^{s}$ term, we
get

(5.1) ${g|u|^{p-\gamma}=\lambda^{*}\frac{2\theta-\gamma}{p-2\theta}f.}$

Now, using $(F)$ and choosing $x$ very close to boundary of $\Omega$, one can
observe that left hand side of (5.1) goes to 0, while the right hand side of
(5.1) is finite and bounded away from zero, which gives us an absurd.

As a consequence of above Proposition, we have the following corollary.

######  Corollary 5.1.

The set $\mathcal{N}_{\lambda^{*}_{a,b}}^{0}$ is compact.

In order to find solutions of $(P_{\lambda^{*}})$ we take a sequence
$\lambda_{n}$ such that $\lambda_{n}\uparrow\lambda^{*}_{a,b}$. As
$(P_{\lambda_{n}})$ has solutions
$u_{\lambda_{n}}\in\mathcal{N}^{-}_{\lambda_{n}}$ and
$v_{\lambda_{n}}\in\mathcal{N}^{+}_{\lambda_{n}}$ we expect strong limits of
these solution sequences to give rise to solutions of
$(P_{\lambda^{*}_{a,b}})$. Based on the definition in equation (3.4), we have
the following couple of propositions which are required to study the limiting
behaviour in case of $\lambda\uparrow\lambda^{*}_{a,b}$.

######  Proposition 5.3.

There holds,

$\overline{\mathcal{N}^{+}_{\lambda^{*}_{a,b}}\cup\mathcal{N}^{-}_{\lambda^{*}_{a,b}}}=\mathcal{N}^{+}_{\lambda^{*}_{a,b}}\cup\mathcal{N}^{-}_{\lambda^{*}_{a,b}}\cup\mathcal{N}^{0}_{\lambda^{*}_{a,b}}\cup\\{0\\}.$

###### Proof.

The proof of this proposition follows by taking a sequence
$v_{n}\in\mathcal{N}^{-}_{\lambda^{*}_{a,b}}$ converging strongly to $v$ in
$X$ and using compact sobolev embeddings. ∎

######  Remark 5.2.

Remark 3.4 together with Proposition 5.3 leads to the following observation

$\overline{\mathcal{\hat{N}}_{\lambda^{*}_{a,b}}\cup\hat{\mathcal{N}}^{+}_{\lambda^{*}_{a,b}}}=\mathcal{\hat{N}}_{\lambda^{*}_{a,b}}\cup\hat{\mathcal{N}}^{+}_{\lambda^{*}_{a,b}}\cup\\{tu:t>0,t\in\mathcal{N}^{0}_{\lambda^{*}_{a,b}}\\}\cup\\{0\\}.$

Now we have necessary mathematical background to show the existence of at
least two positive solutions of the problem $(P_{\lambda^{*}_{a,b}})$ taking
advantage of multiplicity of solutions ($P_{\lambda}$) when
$\lambda<\lambda^{*}_{a,b}$ via limiting behaviour.

Define the map
${t_{\lambda^{*}_{a,b}}}:\overline{\hat{\mathcal{N}}_{\lambda^{*}_{a,b}}}\setminus\\{0\\}\rightarrow\mathbb{R}$
as

$t_{\lambda^{*}_{a,b}}(u)=\begin{cases}t^{-}_{\lambda^{*}_{a,b}}(u)&u\in\hat{\mathcal{N}}_{\lambda^{*}_{a,b}}\\\
t^{0}_{\lambda^{*}_{a,b}}(u)&\textrm{otherwise}\end{cases}$

and
$s_{\lambda^{*}_{a,b}}:\overline{\hat{\mathcal{N}}_{\lambda^{*}_{a,b}}\cup\hat{\mathcal{N}}^{+}_{\lambda^{*}_{a,b}}}\rightarrow\mathbb{R}$
by

$s_{\lambda^{*}_{a,b}}(v)=\begin{cases}t^{+}_{\lambda^{*}_{a,b}}(v)&v\in\hat{\mathcal{N}}_{\lambda^{*}_{a,b}}\cup\hat{\mathcal{N}}^{+}_{\lambda^{*}_{a,b}}\\\
t^{0}_{\lambda^{*}_{a,b}}(v)&\textrm{otherwise}.\end{cases}.$

The following result can be adopted from [23].

######  Proposition 5.4.

The following conclusions hold:

  1. (i)

The functions $s_{\lambda^{*}_{a,b}}$ and $t_{\lambda^{*}_{a,b}}$ are
continuous. Moreover, once $u\notin\hat{\mathcal{N}}^{+}_{\lambda^{*}_{a,b}}$,
we have

$\lim_{\lambda\uparrow\lambda^{*}_{a,b}}t^{-}_{\lambda}(u)=t_{\lambda^{*}_{a,b}}(u),\
\
\lim_{\lambda\uparrow\lambda^{*}_{a,b}}t^{+}_{\lambda}(u)=s_{\lambda^{*}_{a,b}}(u)$

and

$\lim_{\lambda\uparrow\lambda^{*}_{a,b}}\mathcal{E}_{\lambda}(t^{-}_{\lambda}(u)u)=\mathcal{E}_{{\lambda}^{*}_{a,b}}(t_{\lambda^{*}_{a,b}}(u)u),\
\
\lim_{\lambda\uparrow\lambda^{*}_{a,b}}\mathcal{E}_{\lambda}(t^{+}_{\lambda}(u)u)=\mathcal{E}_{{\lambda}^{*}_{a,b}}(s_{\lambda^{*}_{a,b}}(u)u).$

  2. (ii)

The set $\mathcal{N}^{0}_{\lambda^{*}_{a,b}}$ has empty interior in
$\mathcal{N}_{\lambda^{*}_{a,b}}$.

Now we are ready to discuss following minimization problems

$\hat{\mathcal{E}}^{-}_{\lambda^{*}_{a,b}}=\inf\\{\mathcal{E}_{\lambda^{*}_{a,b}}(t_{\lambda^{*}_{a,b}}(u)u):u\in\mathcal{N}^{-}_{\lambda^{*}_{a,b}}\cup\mathcal{N}^{0}_{\lambda^{*}_{a,b}}\\},$

and

$\hat{\mathcal{E}}^{+}_{\lambda^{*}_{a,b}}=\inf\\{\mathcal{E}_{\lambda^{*}_{a,b}}(s_{\lambda^{*}_{a,b}}(v)v):v\in\mathcal{N}^{+}_{\lambda^{*}_{a,b}}\cup\mathcal{N}^{0}_{\lambda^{*}_{a,b}}\\}.$

From Proposition 5.4 it follows that
$\hat{\mathcal{J}}^{\pm}_{\lambda^{*}_{a,b}}=\hat{\mathcal{E}}^{\pm}_{\lambda^{*}_{a,b}}$.
Next we observe behaviour of
$(0,\lambda^{*}_{a,b}]\ni\lambda\mapsto\hat{\mathcal{J}}^{\pm}_{\lambda}$ in
following proposition when $\lambda\uparrow\lambda^{*}_{a,b}$.

######  Proposition 5.5.

The function $\lambda\mapsto\hat{\mathcal{J}}^{\pm}_{\lambda}$ is decreasing
for all $\lambda\in(0,\lambda^{*}_{a,b}].$ Moreover,

$\lim_{\lambda\uparrow\lambda^{*}_{a,b}}\hat{\mathcal{J}}^{\pm}_{\lambda}=\hat{\mathcal{J}}^{\pm}_{\lambda^{*}_{a,b}}.$

###### Proof.

We know from Lemma 3.1 that $\mathcal{J}^{\pm}_{\lambda}$ is decreasing on
$(0,\lambda^{*}_{a,b})$, from there we can conclude that
$\hat{\mathcal{J}}^{\pm}_{\lambda}$ is decreasing on $(0,\lambda^{*}_{a,b})$.
To show that $\mathcal{J}^{\pm}_{\lambda}$ is decreasing on
$(0,\lambda^{*}_{a,b}]$ take $\lambda\in(0,\lambda^{*}_{a,b})$ arbitrary. Then
for all
$u\in\mathcal{N}^{-}_{\lambda^{*}_{a,b}}\cup\mathcal{N}^{0}_{\lambda^{*}_{a,b}}$
using decreasing behaviour of $\mathcal{J}^{-}_{\lambda}$ and Proposition 5.4,
we have

$\displaystyle\hat{\mathcal{J}}^{-}_{\lambda^{*}_{a,b}}=\hat{\mathcal{E}}^{-}_{\lambda^{*}_{a,b}}\leq\mathcal{E}_{\lambda^{*}_{a,b}}(t_{\lambda^{*}_{a,b}}(u)u)$
$\displaystyle={\lim_{{\Lambda}\downarrow\lambda^{*}_{a,b}}}\mathcal{E}_{\Lambda}(t^{-}_{{\Lambda}}(u)u)=\lim_{{\Lambda}\downarrow\lambda^{*}_{a,b}}\mathcal{J}^{-}_{{\Lambda}}(u)$
$\displaystyle<\lim_{{\Lambda}\downarrow\lambda^{*}_{a,b}}\mathcal{J}^{-}_{\lambda^{*}_{a,b}}(u)=\mathcal{J}^{-}_{\lambda^{*}_{a,b}}(u)<\mathcal{J}^{-}_{\lambda}(u),$

and hence
$\hat{\mathcal{J}}^{-}_{\lambda^{*}_{a,b}}\leq\hat{\mathcal{J}}^{-}_{\lambda}$.
Similarly it follows that
$\hat{\mathcal{J}}^{+}_{\lambda^{*}_{a,b}}\leq\hat{\mathcal{J}}^{+}_{\lambda}$.

To prove remaining part take $\lambda_{n}\in(0,\lambda^{*}_{a,b}]$ such that
$\lambda_{n}\uparrow\lambda^{*}_{a,b}$. Then
$\hat{\mathcal{J}}^{-}_{\lambda_{n}}\geq\hat{\mathcal{J}}^{-}_{\lambda^{*}_{a,b}}$.
Assume on contrary $\hat{\mathcal{J}}^{-}_{\lambda_{n}}\rightarrow
J>\hat{\mathcal{J}}^{-}_{\lambda^{*}_{a,b}}$ and suppose that there exists
$\delta>0$ such that, $J-\hat{\mathcal{J}}^{-}_{\lambda^{*}_{a,b}}\geq\delta$.
Choose $\beta>0$ such that $2\beta<\delta$ and
$u(\beta)\in\mathcal{N}^{-}_{\lambda^{*}_{a,b}}$ such that
$\mathcal{J}^{-}_{\lambda^{*}_{a,b}}(u(\beta))-\hat{\mathcal{J}}^{-}_{\lambda^{*}_{a,b}}\leq\beta$.
Now, using continuity of $\mathcal{J}^{-}_{\lambda}$ when
$\lambda<\lambda^{*}_{a,b}$, we have

$0\leq\mathcal{J}^{-}_{\lambda_{n}}(u(\beta))-\mathcal{J}^{-}_{\lambda^{*}_{a,b}}(u(\beta))\leq\beta$

for large values of $n$. Thus,

$\displaystyle\hat{\mathcal{J}}^{-}_{\lambda_{n}}$
$\displaystyle\leq\mathcal{J}^{-}_{\lambda_{n}}(u(\beta))\leq\mathcal{J}^{-}_{\lambda^{*}_{a,b}}(u(\beta))+\beta\leq\hat{\mathcal{J}}^{-}_{\lambda^{*}_{a,b}}+2\beta\leq
J-\delta+2\beta<J.$

Therefore, for $n$ sufficiently large we have $J<J$, an absurd. Hence
$J=\hat{\mathcal{J}}^{-}_{\lambda^{*}_{a,b}}$. Similarly, we can show that
$\lim_{\lambda\uparrow\lambda^{*}_{a,b}}\hat{\mathcal{J}}^{+}_{\lambda}=\hat{\mathcal{J}}^{+}_{\lambda^{*}_{a,b}}.$
∎

######  Lemma 5.1.

There exists $u_{\lambda^{*}_{a,b}}\in\mathcal{N}^{-}_{\lambda^{*}_{a,b}}$ and
$v_{\lambda^{*}_{a,b}}\in\mathcal{N}^{+}_{\lambda^{*}_{a,b}}$ such that
$\hat{\mathcal{J}}^{-}_{\lambda^{*}_{a,b}}=\mathcal{J}^{-}_{\lambda^{*}_{a,b}}(u_{\lambda^{*}_{a,b}})$
and
$\hat{\mathcal{J}}^{+}_{\lambda^{*}_{a,b}}=\mathcal{J}^{+}_{\lambda^{*}_{a,b}}(v_{\lambda^{*}_{a,b}})$.

###### Proof.

We already know from Lemma 4.1, 4.2 that we have
$u_{\lambda_{n}}\in\mathcal{N}^{-}_{\lambda_{n}}$ and
$v_{\lambda_{n}}\in\mathcal{N}^{+}_{\lambda_{n}}$ such that
$\mathcal{J}^{-}_{\lambda}(u_{\lambda_{n}})=\hat{\mathcal{J}}^{-}_{\lambda_{n}}$
and
$\mathcal{J}^{+}_{\lambda_{n}}(v_{\lambda_{n}})=\hat{\mathcal{J}}^{+}_{\lambda_{n}}$
respectively when $\lambda_{n}<\lambda^{*}_{a,b}$ for each $n\in\mathbb{N}$.
To show the existence of minimizers for $\mathcal{E}_{\lambda^{*}_{a,b}}$ we
are looking to pass the limit $\lambda_{n}\uparrow\lambda^{*}_{a,b}$ in
$\hat{\mathcal{J}}_{\lambda_{n}}^{\pm}$. We know for each
$\lambda_{n}<\lambda^{*}_{a,b}$, we can get a sequence
$\\{u_{\lambda_{n}}\\}\subset\mathcal{N}_{\lambda_{n}}^{-}$ such that
$\hat{\mathcal{J}}^{-}_{\lambda_{n}}=\mathcal{J}^{-}_{\lambda_{n}}(u_{\lambda_{n}})$.
Moreover, $\\{u_{\lambda_{n}}\\}$ is bounded, otherwise, if
$\|u_{\lambda_{n}}\|_{X}\rightarrow\infty$, then
$\infty>\lim\hat{\mathcal{J}}^{-}_{\lambda_{n}}\geq\infty$, which is an
absurd. Therefore, up to a subsequence, we get

$\left\\{\begin{array}[]{ll}u_{\lambda_{n}}\rightharpoonup
u_{\lambda^{*}_{a,b}}\hskip 36.98866pt\ \textrm{in}\ X,\\\
u_{\lambda_{n}}\rightarrow u_{\lambda^{*}_{a,b}}\ \textrm{in}\ \ \ \ \ \ \ \
\textrm{in}\ L^{q}(\Omega)\ \forall\ {q\in[1,2^{*}_{s})},\\\
u_{\lambda_{n}}\rightarrow u_{\lambda^{*}_{a,b}}\hskip
42.67912pt\textrm{a.e.}\ \textrm{in}\ \Omega\\\ \end{array}\right.$

with $u_{\lambda^{*}_{a,b}}\geq 0$. Our claim is
$u_{\lambda^{*}_{a,b}}\not\equiv 0$ and
$u_{\lambda^{*}_{a,b}}\in\mathcal{C}^{+}$. Otherwise, we would have
$\|u_{\lambda_{n}}\|_{X}\rightarrow 0$, an absurd. Now, once $u_{\lambda_{n}}$
is solution of $(P_{\lambda_{n}})$, for all $\phi\in X$, we have

$(a+b\|u_{\lambda_{n}}\|^{2(\theta-1)}_{X})\iint_{\mathbb{R}^{2N}}\frac{(u_{\lambda_{n}}(x)-u_{\lambda_{n}}(y))(\phi(x)-\phi(y))}{|x-y|^{N+2s}}dxdy$
$-\lambda_{n}\int_{\Omega}fu_{\lambda_{n}}^{\gamma-1}\phi
dx-\int_{\Omega}gu^{p-1}_{\lambda_{n}}\phi dx=0.$

Taking $u_{\lambda_{n}}-u_{\lambda^{*}_{a,b}}$ as a text function in above
equation and suppose $\|u_{\lambda_{n}}\|_{X}\rightarrow\alpha>0$, we get

$\displaystyle(a+$ $\displaystyle
b\|u_{\lambda_{n}}\|^{2(\theta-1)}_{X})\iint_{\mathbb{R}^{2N}}\frac{|u_{\lambda_{n}}(x)-u_{\lambda_{n}}(y)|^{2}}{|x-y|^{N+2s}}dxdy$
$\displaystyle-(a+b\|u_{\lambda_{n}}\|^{2(\theta-1)}_{X})\iint_{\mathbb{R}^{2N}}\frac{(u_{\lambda_{n}}(x)-u_{\lambda_{n}}(y))(u_{\lambda^{*}_{a,b}}(x)-u_{\lambda^{*}_{a,b}}(y))}{|x-y|^{N+2s}}dxdy$
$\displaystyle-\lambda_{n}\int_{\Omega}fu_{\lambda_{n}}^{\gamma-1}(u_{\lambda_{n}}(x)-u_{\lambda^{*}_{a,b}}(x))dx-\int_{\Omega}gu^{p-1}_{\lambda_{n}}(u_{\lambda_{n}}-u_{\lambda^{*}_{a,b}})(x)dx=0.$

Using weak lower semi-continuity of norms, we get

$\displaystyle(a+b\alpha^{2(\theta-1)})(\lim_{n\rightarrow\infty}\|u_{\lambda_{n}}\|_{X}^{2}-\|u_{\lambda^{*}_{a,b}}\|_{X}^{2})=0.$

Therefore,
$\lim_{n\rightarrow\infty}\|u_{\lambda_{n}}\|^{2}_{X}=\|u_{\lambda^{*}_{a,b}}\|^{2}_{X}$
( because $a+b\alpha^{2\theta-2}>0$), which implies that
$u_{\lambda_{n}}\rightarrow u_{\lambda^{*}_{a,b}}$ in $X$. From strong
convergence, we have

$\psi^{\prime}_{\lambda^{*},u_{\lambda^{*}_{a,b}}}(1)=\lim_{n\to\infty}\psi^{\prime}_{\lambda_{n},u_{\lambda_{n}}}(1)=0\
\ \textrm{and}\ \
\psi^{\prime\prime}_{\lambda^{*}_{a,b},u_{\lambda^{*}_{a,b}}}(1)=\lim_{n\to\infty}\psi^{\prime\prime}_{\lambda_{n},u_{\lambda_{n}}}(1)\leq
0.$

Also when $u_{\lambda_{n}}\in\mathcal{N}^{-}_{\lambda_{n}}$, we have

$0<b(2\theta-\gamma)\|u_{\lambda^{*}_{a,b}}\|^{2\theta}_{X}=b(2\theta-\gamma)\lim_{n\rightarrow\infty}\|u_{\lambda_{n}}\|^{2\theta}_{X}\leq(p-\gamma)\int_{\Omega}gu^{p}_{\lambda^{*}_{a,b}}dx,$

thus,
$u_{\lambda^{*}_{a,b}}\in\mathcal{N}^{-}_{\lambda^{*}_{a,b}}\cup\mathcal{N}^{0}_{\lambda^{*}_{a,b}}.$
From strong convergence of $\\{u_{\lambda_{n}}\\}$ and Proposition 5.5, we
have

$\mathcal{E}_{\lambda^{*}_{a,b}}(u_{\lambda^{*}_{a,b}})=\lim_{n\rightarrow\infty}\mathcal{E}_{\lambda_{n}}(u_{\lambda_{n}})=\lim_{\lambda_{n}\uparrow\lambda^{*}_{a,b}}\hat{\mathcal{J}}_{\lambda_{n}}^{-}=\hat{\mathcal{J}}^{-}_{\lambda^{*}_{a,b}}.$

We claim that
$u_{\lambda^{*}_{a,b}}\not\in\mathcal{N}^{0}_{\lambda^{*}_{a,b}}$. If not, we
get a contradiction from Proposition 5.2. Therefore $u_{\lambda^{*}_{a,b}}$ is
a constrained minimizer of $\mathcal{E}_{\lambda^{*}_{a,b}}$ forced to be in
$\mathcal{N}^{-}_{\lambda^{*}_{a,b}}$.

In order to achieve second minimizer $v_{\lambda^{*}_{a,b}}$ of the energy
functional $\mathcal{E}_{\lambda^{*}_{a,b}}$ , taking sequence
$\\{v_{\lambda_{n}}\\}\subset\mathcal{N}^{+}_{\lambda_{n}}$ where
$\lambda_{n}\uparrow\lambda^{*}_{a,b}$ and following same arguments as above
we get the second minimizer $v_{\lambda^{*}_{a,b}}$ of
$\mathcal{E}_{\lambda^{*}_{a,b}}$ such that

$\mathcal{E}_{\lambda^{*}_{a,b}}(v_{\lambda^{*}_{a,b}})=\lim_{n\rightarrow\infty}\mathcal{E}_{\lambda}(v_{\lambda_{n}})=\lim_{n\rightarrow\infty}\hat{{\mathcal{J}}}^{+}_{\lambda_{n}}=\hat{\mathcal{J}}^{+}_{\lambda^{*}_{a,b}},$

forced to be in $\mathcal{N}^{+}_{\lambda^{*}_{a,b}}$ following the same
argument as above. This completes the proof.

∎

In this case, i.e., $\lambda=\lambda^{*}_{a,b}$ the non-trivial minimizers
$u_{\lambda^{*}_{a,b}},v_{\lambda^{*}_{a,b}}$ obtained in Lemma 5.1 are
avoided to be in $\mathcal{N}^{0}_{\lambda^{*}_{a,b}}$ as in Proposition 5.2
and hence again from Theorem 2.3 of [6] are non trivial weak solutions of
$(P_{\lambda^{*}_{a,b}})$.

##  6\. Existence of solutions when $\lambda>\lambda^{*}_{a,b}$

Since, $\mathcal{N}_{\lambda^{*}_{a,b}}^{0}\not\equiv\emptyset$, the
minimizers in $\mathcal{N}_{\lambda}^{\pm}$ may not be the critical points of
the associated energy functional of the problem ($P_{\lambda}$). Therefore, we
look for the minimizers of associated energy functional over subsets of
$\mathcal{N}^{+}_{\lambda^{*}_{a,b}}$ and
$\mathcal{N}^{-}_{\lambda^{*}_{a,b}}$ (defined below) which are strictly
separated from $\mathcal{N}^{0}_{\lambda^{*}_{a,b}}$. Subsequently,
projections of these minimizers lying in $\mathcal{N}_{\lambda}^{\pm}$ for
$\lambda\in(\lambda^{*}_{a,b},\lambda^{*}_{a,b}+\epsilon)$ for sufficiently
small $\epsilon>0$ turn out to be the desired critical points.

######  Proposition 6.1.

Let $0<c^{+}<c^{-}$, $a\geq 0,b>0$ and
$\lambda_{n}\downarrow\lambda^{*}_{a,b}$

  * (i)

if $u_{n}\in\mathcal{N}^{-}_{\lambda^{*}_{a,b}}$ which satisfy
$c^{+}\leq\|u_{n}\|_{X}\leq c^{-}$ for all $n\in\mathbb{N}$ and

$\displaystyle a(t_{\lambda_{n}}^{-}(u_{n}))^{2}\|u_{n}\|^{2}_{X}$
$\displaystyle+b(2\theta-1)(t_{\lambda_{n}}^{-}(u_{n}))^{2\theta}\|u_{n}\|_{X}^{2\theta}-\lambda(\gamma-1)(t_{\lambda_{n}}^{-}(u_{n}))^{\gamma}$
$\displaystyle\lambda_{n}\int_{\Omega}f|u_{n}|^{\gamma}dx-(p-1)(t_{\lambda_{n}}^{-}(u_{n}))^{p}\int_{\Omega}g|u_{n}|^{p}dx\rightarrow
0$

as $n\rightarrow\infty$, then
$\textrm{dist}(u_{n},\mathcal{N}^{0}_{\lambda^{*}_{a,b}})\rightarrow 0$.

  * (ii)

if $v_{n}\in\mathcal{N}^{+}_{\lambda^{*}_{a,b}}$ which satisfy
$c^{+}\leq\|v_{n}\|_{X}\leq c^{-}$ for all $n\in\mathbb{N}$ and

$\displaystyle a(t_{\lambda_{n}}^{+}(v_{n}))^{2}\|v_{n}\|^{2}_{X}$
$\displaystyle+b(2\theta-1)(t_{\lambda_{n}}^{+}(v_{n}))^{2\theta}\|v_{n}\|_{X}^{2\theta}-\lambda(\gamma-1)(t_{\lambda_{n}}^{+}(v_{n}))^{\gamma}$
$\displaystyle\lambda_{n}\int_{\Omega}f|v_{n}|^{\gamma}dx-(p-1)(t_{\lambda_{n}}^{-}(v_{n}))^{p}\int_{\Omega}g|v_{n}|^{p}dx\rightarrow
0$

as $n\rightarrow\infty$, then
$\textrm{dist}(v_{n},\mathcal{N}^{0}_{\lambda^{*}_{a,b}})\rightarrow 0$.

###### Proof.

$(i)$ For $u_{n}\in\mathcal{N}^{-}_{\lambda^{*}_{a,b}}$, we have
$\displaystyle\int_{\Omega}g|u_{n}|^{p}dx\geq c^{+}$. Our claim is
$\int_{\Omega}f|u_{n}|^{\gamma}dx\geq c^{+}$. Therefore from Proposition 3.2,
there exist two critical points
$t_{\lambda_{n}}^{-}=t_{\lambda_{n}}^{-}(u_{n})$ and
$t_{\lambda_{n}}^{+}(u_{n})=t_{\lambda_{n}}^{+}$ for
$\psi_{\lambda_{n},u_{n}}$ for large $n$ satisfying
$t_{\lambda_{n}}^{+}(u_{n})<t_{\lambda_{n}}^{-}(u_{n})$ such that
$t_{\lambda_{n}}^{+}u_{n}\in\mathcal{N}_{\lambda_{n}}^{+}$ and
$t_{\lambda_{n}}^{-}u_{n}\in\mathcal{N}_{\lambda_{n}}^{-}$. That is,

(6.1) $\left\\{\begin{aligned}
&a(t_{\lambda_{n}}^{-})^{2}\|u_{n}\|^{2}_{X}+b(t_{\lambda_{n}}^{-})^{2\theta}\|u_{n}\|_{X}^{2\theta}-\lambda_{n}(t_{\lambda_{n}}^{-})^{\gamma}\int_{\Omega}f|u_{n}|^{\gamma}dx\\\
&\quad-(t_{\lambda_{n}}^{-})^{p}\int_{\Omega}g|u_{n}|^{p}dx=0,\\\
&a(t_{\lambda_{n}}^{-})^{2}\|u_{n}\|^{2}_{X}+b(2\theta-1)(t_{\lambda_{n}}^{-})^{2\theta}\|u_{n}\|_{X}^{2\theta}-\lambda_{n}(\gamma-1)(t_{\lambda_{n}}^{-})^{\gamma}\int_{\Omega}f|u_{n}|^{\gamma}dx\\\
&\quad-(t_{\lambda_{n}}^{-})^{p}(p-1)\int_{\Omega}g(x)|u_{n}|^{p}dx=o(1),\\\
&a(t_{\lambda_{n}}^{+})^{2}\|u_{n}\|^{2}_{X}+b(t_{\lambda_{n}}^{+})^{2}\|u_{n}\|_{X}^{2}-\lambda_{n}(t_{\lambda_{n}}^{+})^{\gamma}\int_{\Omega}f|u_{n}|^{\gamma}dx\\\
&\quad-(t_{\lambda_{n}}^{+})^{p}\int_{\Omega}|u_{n}|^{p}dx=0.\end{aligned}\right.$

Getting the values of $\int_{\Omega}f|u_{n}|^{\gamma}dx$ and
$\displaystyle\int_{\Omega}g(x)|u_{n}|^{p}dx$ by solving first and third
equations of (6.1) and substituting the values in second equation of (6.1) and
using simple calculus, we get

(6.2) $\displaystyle a\|u_{n}\|_{X}^{2}(t^{-}_{\lambda_{n}})^{2}$
$\displaystyle\left(1-(\gamma-1)\left(\frac{1-\left(\frac{t^{+}_{\lambda_{n}}}{t^{-}_{\lambda_{n}}}\right)^{2-p}}{1-\left(\frac{t^{+}_{\lambda_{n}}}{t^{-}_{\lambda_{n}}}\right)^{\gamma-p}}\right)-{(p-1)}\left(\frac{1-\left(\frac{t^{+}_{\lambda_{n}}}{t^{-}_{\lambda_{n}}}\right)^{2-\gamma}}{1+\left(\frac{t^{+}_{\lambda_{n}}}{t^{-}_{\lambda_{n}}}\right)^{p-\gamma}}\right)\right)$
$\displaystyle+b\|u_{n}\|_{X}^{2\theta}(t^{-}_{\lambda_{n}})^{2\theta}\left((2\theta-1)-(\gamma-1)\left(\frac{1-\left(\frac{t^{+}_{\lambda_{n}}}{t^{-}_{\lambda_{n}}}\right)^{2\theta-p}}{1-\left(\frac{t^{+}_{\lambda_{n}}}{t^{-}_{\lambda_{n}}}\right)^{\gamma-p}}\right)\right.$
$\displaystyle\left.\hskip
99.58464pt-{(p-1)}\left(\frac{1-\left(\frac{t^{+}_{\lambda_{n}}}{t^{-}_{\lambda_{n}}}\right)^{2\theta-\gamma}}{1-\left(\frac{t^{+}_{\lambda_{n}}}{t^{-}_{\lambda_{n}}}\right)^{p-\gamma}}\right)\right)=o(1).$

Since both term in left hand side of above equation are positive and entire
term converges to zero implies both the term will converge to zero separately.
Suppose $t^{-}_{\lambda_{n}}\rightarrow\alpha$,
$t^{+}_{\lambda_{n}}\rightarrow\beta$. Taking limit $n\rightarrow\infty$ in
(6.2) we get, $t_{\lambda_{n}}^{-}(u_{n})$$\setminus$
$t_{\lambda_{n}}^{+}(u_{n})\rightarrow 1$,i.e. $\alpha=\beta$, as $1$ is the
only zero of

$m(t)=(2p-2\theta-2)t^{\gamma-p}+(\gamma-p)t^{2-p}+(\gamma-p)t^{2\theta-p}+2-2\gamma+2\theta.$

Once $t^{+}_{\lambda_{n}}u_{n}\in\mathcal{N}^{+}_{\lambda_{n}}$, from (4.1) we
have $\int_{\Omega}f|u_{n}|^{\gamma}\geq c$. Thus

$\displaystyle
a\|{\alpha}u_{n}\|^{2}_{X}+b\|{\alpha}u_{n}\|_{X}^{2\theta}-\lambda^{*}_{a,b}\int_{\Omega}f|{\alpha}u_{n}|^{\gamma}dx-\displaystyle\int_{\Omega}g(x)|{\alpha}u_{n}|^{p}dx=o(1),$
$\displaystyle
a\|{\alpha}u_{n}\|^{2}_{X}+b(2\theta-1)\|{\alpha}u_{n}\|_{X}^{2\theta}-\lambda^{*}_{a,b}(\gamma-1)\int_{\Omega}f|{\alpha}u_{n}|^{\gamma}dx$
$\displaystyle\quad-(p-1)\displaystyle\int_{\Omega}g(x)|{\alpha}u_{n}|^{p}dx$
$\displaystyle=o(1).$

Solving these two we get

$\frac{a(p-2)\|{\alpha}u_{n}\|^{2}_{X}+b(p-2\theta)\|{\alpha}u_{n}\|^{2\theta}_{X}}{(p-\gamma)f|{\alpha}u_{n}|^{\gamma}dx}=\lambda^{*}_{a,b}+o(1),$
$\frac{a(\gamma-2)\|{\alpha}u_{n}\|^{2}_{X}+b(\gamma-2\theta)\|{\alpha}u_{n}\|^{2\theta}_{X}}{(\gamma-p)\displaystyle\int_{\Omega}g(x)|{\alpha}u_{n}|^{p}dx}=1+o(1).$

Therefore, from expression of $\lambda(u)$, we have

$\lambda({\alpha}u_{n})=\lambda(u_{n})=(1+o(1))^{\frac{2\theta-\gamma}{p-2\theta}}(\lambda^{*}+o(1))$

leading to $\lambda(u_{n})\rightarrow\lambda^{*}$. Hence $u_{n}$ is a bounded
minimizing sequence for $\lambda^{*}$. Now, by following same argument as in
proof of Proposition 3.1, up to a subsequence, we obtain $u_{n}\rightarrow
u\in\mathcal{N}^{0}_{\lambda^{*}}$ and thus
$\textrm{dist}(u_{n},\mathcal{N}^{0}_{\lambda^{*}})\rightarrow 0$ as
$n\rightarrow\infty$. Similarly from expression of $\lambda_{a,b}(u_{n})$, we
have $\lambda_{a,b}(u_{n})\rightarrow\lambda^{*}_{a,b}$ and thus
$\textrm{dist}(u_{n},\mathcal{N}^{0}_{\lambda^{*}_{a,b}})\rightarrow 0$.

$(ii)$ Follows similarly from item $(i)$. ∎

Consider the sets

$\mathcal{N}^{-}_{\lambda^{*}_{a,b},d,c^{-}}=\\{u\in\mathcal{N}^{-}_{\lambda^{*}_{a,b}}:\textrm{dist}(u,\mathcal{N}^{0}_{\lambda^{*}_{a,b}})\geq
d,\|u\|_{X}\leq c^{-}\\},$

and

$\mathcal{N}^{+}_{\lambda^{*}_{a,b},d,c^{+}}=\\{v\in\mathcal{N}^{+}_{\lambda^{*}_{a,b}}:\textrm{dist}(v,\mathcal{N}^{0}_{\lambda^{*}_{a,b}})\geq
d,\|v\|_{X}\geq c^{+}\\},$

where $c^{\pm}\ \textrm{and}\ d$ are positive constants.

In light of above proposition one can observe that elements of
$\mathcal{N}^{\pm}_{\lambda,d,c^{\pm}}$ can be projected over
$\mathcal{N}^{\pm}_{\lambda}$ when $\lambda\downarrow\lambda^{*}_{a,b}$ i.e.,
if $u\in\mathcal{N}^{\pm}_{\lambda,d,c^{\pm}}$ there exists $\epsilon>0$ such
that $u\in\hat{\mathcal{N}}_{\lambda}$ or in
$\hat{\mathcal{N}}_{\lambda}\cup\hat{\mathcal{N}}^{+}_{\lambda}$ for all
$\lambda\in(\lambda^{*}_{a,b},\lambda^{*}_{a,b}+\epsilon)$. Also it is
important to observe that
$\textrm{dist}(\mathcal{M}^{\pm}_{\lambda^{*}_{a,b}},\mathcal{N}^{0}_{\lambda^{*}_{a,b}})>0$,
where

$\mathcal{M}^{\pm}_{\lambda}=\\{u\in\mathcal{N}^{\pm}_{\lambda}:\mathcal{J}^{\pm}_{\lambda}(u)=\hat{\mathcal{J}}^{\pm}_{\lambda}\\}.$

In fact,
$\textrm{dist}(\mathcal{M}^{\pm}_{\lambda^{*}_{a,b}},\mathcal{N}^{0}_{\lambda^{*}_{a,b}})\to
0$ we can get a contradiction to the fact that no solution of
$(P_{\lambda^{*}_{a,b}})$ can be in $\mathcal{N}^{0}_{\lambda^{*}_{a,b}}$.

Clearly the sets $\mathcal{M}^{\pm}_{\lambda}\neq\emptyset$ for all
$\lambda\in(0,\lambda^{*}_{a,b}]$. Now define

$d^{+}_{\lambda^{*}_{a,b}}=\textrm{dist}(\mathcal{M}^{+}_{\lambda^{*}_{a,b}},\mathcal{N}^{0}_{\lambda^{*}_{a,b}})\
\ \textrm{and}\ \
d^{-}_{\lambda^{*}_{a,b}}=\textrm{dist}(\mathcal{M}^{-}_{\lambda^{*}_{a,b}},\mathcal{N}^{0}_{\lambda^{*}_{a,b}}).$

Now choose $c^{-}_{\lambda^{*}_{a,b}}<c^{-}$ such that $\|u\|_{X}\leq
c^{-}_{\lambda^{*}_{a,b}}$ for all $u\in\mathcal{M}^{-}_{\lambda^{*}_{a,b}}$
and $d^{-}\in(0,d^{-}_{\lambda^{*}_{a,b}})$. With such controls for all
$\lambda\in(\lambda^{*}_{a,b},\lambda^{*}_{a,b}+\epsilon)$ we will study
following minimization problem

(6.3)
$\hat{{\mathcal{J}}}^{-}_{\lambda,d^{-},c^{-}}=\textrm{inf}\\{\mathcal{J}^{-}_{\lambda}(u):u\in\mathcal{N}^{-}_{\lambda^{*}_{a,b},d^{-},c^{-}}\\}.$

In a similar way, we define with the choice of
$d^{+}<d^{+}_{\lambda^{*}_{a,b}}$ and $c^{+}<c^{+}_{\lambda^{*}_{a,b}}$ where
$\|u\|_{X}\geq c^{+}_{\lambda^{*}_{a,b}}$ for all
$u\in\mathcal{M}^{+}_{\lambda^{*}_{a,b}}$,

$\hat{{\mathcal{J}}}^{+}_{\lambda,d^{+},c^{+}}=\textrm{inf}\\{\mathcal{J}^{+}_{\lambda}(v):v\in\mathcal{N}^{+}_{\lambda^{*}_{a,b},d^{+},c^{+}}\\}$

for all $\lambda\in(\lambda^{*}_{a,b},\lambda^{*}_{a,b}+\epsilon)$. With such
choice of $c^{\pm},d^{\pm}$ we can observe that
$\mathcal{M}^{-}_{\lambda^{*}_{a,b}}\subset\mathcal{N}^{-}_{\lambda^{*}_{a,b},d^{-},c^{-}}$
as, if $u\in\mathcal{M}^{-}_{\lambda^{*}_{a,b}}$ we have
$\textrm{dist}(u,\mathcal{N}^{0}_{\lambda^{*}_{a,b}})=d^{-}_{\lambda^{*}}>d^{-}$
implies $u\in\mathcal{N}^{-}_{\lambda^{*}_{a,b},d^{-},c^{-}}$. Similarly
$\mathcal{M}^{+}_{\lambda^{*}_{a,b}}\subset\mathcal{N}^{+}_{\lambda^{*}_{a,b},d^{+},c^{+}}$.

######  Lemma 6.1.

For the above choices of $c^{-},d^{-}$ there exits $\epsilon^{-}>0$ such that
the functional $\mathcal{J}^{-}_{\lambda}$ constrained to
$\mathcal{N}^{-}_{\lambda^{*}_{a,b},d^{-},c^{-}}$ has a minimizer
${u({\lambda})}\in$ $\mathcal{N}^{-}_{\lambda^{*}_{a,b},d^{-},c^{-}}$ for all
$\lambda\in(\lambda^{*}_{a,b},\lambda^{*}_{a,b}+\epsilon^{-})$.

###### Proof.

For each $\lambda\in(\lambda^{*}_{a,b},\lambda^{*}_{a,b}+\epsilon)$, take a
minimizing sequence
$u_{n}(\lambda)\in\mathcal{N}^{-}_{\lambda^{*}_{a,b},d^{-},c^{-}}$ for
$\hat{\mathcal{J}}^{-}_{\lambda,d^{-},c^{-}}$. From Proposition 6.1, we have
$u_{n}(\lambda)\in\hat{\mathcal{N}}_{\lambda}$ and we can get $\delta<0$ such
that
$(t^{-}_{\lambda}(u_{n}(\lambda)))^{2}\psi_{\lambda,u_{n}}(t^{-}_{\lambda}(u_{n}(\lambda)))<\delta$
for all $\lambda\in(\lambda^{*}_{a,b},\lambda^{*}_{a,b}+\epsilon)$. Since
$u_{n}(\lambda)$ is bounded, hence up to a subsequence
$u_{n}(\lambda)\rightharpoonup u(\lambda)$ for some ${u(\lambda)}\not\equiv 0$
in $X$. We target to show that, the weak limit ${u(\lambda)}$ belongs to
$\hat{\mathcal{N}}_{\lambda}$ for all
$\lambda\in(\lambda^{*}_{a,b},\lambda^{*}_{a,b}+\epsilon^{-})$. As
$u_{n}(\lambda)\in\hat{\mathcal{N}}_{\lambda}$ for all
$\lambda\in(\lambda^{*}_{a,b},\lambda^{*}_{a,b}+\epsilon^{-})$, there exists
unique scalar $t^{-}_{\lambda}(u_{n}(\lambda))>0$ for all
$\lambda\in(\lambda^{*}_{a,b},\lambda^{*}_{a,b}+\epsilon^{-})$. Moreover,
since
$t^{-}_{\lambda}(u_{n}(\lambda))<t^{-}_{\lambda^{*}_{a,b}}(u_{n}(\lambda))=1$
we get, $t^{-}_{\lambda}(u_{n}(\lambda))\rightarrow t({\lambda})\in(0,1)$. We
claim that there exists $\epsilon^{-}>0$ such that
$u(\lambda)\in\hat{\mathcal{N}}_{\lambda}$ for all
$\lambda\in(\lambda^{*}_{a,b},\lambda^{*}_{a,b}+\epsilon^{-})$. Suppose on
contrary $u(\lambda)\notin\hat{\mathcal{N}}_{\lambda}$, thus there exists
sequence $\lambda_{k}\downarrow\lambda^{*}_{a,b}$ such that
$u(\lambda_{k})\notin\hat{\mathcal{N}}_{\lambda_{k}}$ for large value of $k$.
To get contradiction for this fact, assuming a minimizing sequence
$u_{n,k}\equiv t^{-}_{\lambda_{k}}(u_{n}(\lambda_{k}))u_{n}(\lambda_{k})$ for
$\hat{\mathcal{J}}^{-}_{\lambda_{k},d^{-},c^{-}}$ for large $k\in\mathbb{N}$
(using $\eqref{eq 6.1}$) and we show that it is also a minimizing sequence for
$\hat{\mathcal{J}}^{-}_{\lambda^{*}_{a,b}}$. In the squeal we first prove
following claim.

Claim- When $\lambda\downarrow\lambda^{*}_{a,b}$ function
$\hat{\mathcal{J}}^{-}_{\lambda,d^{-},c^{-}}$ is decreasing and

(6.4)
$\displaystyle\lim_{\lambda\downarrow\lambda^{*}_{a,b}}\hat{\mathcal{J}}^{-}_{\lambda,d^{-},c^{-}}=\hat{\mathcal{J}}^{-}_{\lambda^{*}_{a,b}}.$

In order to prove this claim, using decreasing behaviour of
$\mathcal{J}^{-}_{\lambda}(u)$ from Proposition (3.1) $(ii)$ for all
$u\in\hat{\mathcal{N}}^{-}_{\lambda^{*}_{a,b},d^{-},c^{-}}$, we have
$\hat{\mathcal{J}}^{-}_{\lambda,d^{-},c^{-}}\leq\hat{\mathcal{J}}^{-}_{\lambda^{\prime},d^{-},c^{-}}$,
where $\lambda^{*}_{a,b}<\lambda^{\prime}<\lambda<\lambda^{*}_{a,b}+\epsilon$.
Also when $u_{\lambda^{*}_{a,b}}\in\mathcal{M}^{-}_{\lambda^{*}_{a,b}}$ we
have
$\hat{\mathcal{J}}^{-}_{\lambda,d^{-},c^{-}}\leq\mathcal{J}^{-}_{\lambda}(u_{\lambda^{*}_{a,b}})<\mathcal{J}^{-}_{\lambda^{*}_{a,b}}(u_{\lambda^{*}_{a,b}})=\hat{\mathcal{J}}^{-}_{\lambda^{*}_{a,b}}$
for all $\lambda\in(\lambda^{*}_{a,b},\lambda^{*}_{a,b}+\epsilon)$. To prove
(6.4) suppose on contrary that there exists a sequence
$\lambda_{n}\downarrow\lambda^{*}_{a,b}$ or
$\lambda_{n}\in(\lambda^{*}_{a,b},\lambda^{*}_{a,b}+\epsilon)$ for large $n$,
such that

$\lim_{n\rightarrow\infty}\hat{\mathcal{J}}^{-}_{\lambda_{n},d^{-},c^{-}}=\mathcal{J}<\hat{\mathcal{J}}^{-}_{\lambda^{*}_{a,b}}.$

Also from equation (6.3) we can have a sequence
$\\{u_{k}\\}\subset\mathcal{N}^{-}_{\lambda^{*}_{a,b},d^{-},c^{-}}$ such that
for a given $\epsilon>0$ there exists $n_{0},k_{0}\in\mathbb{N}$ such that

(6.5)
$|\mathcal{J}^{-}_{\lambda_{n}}(u_{k})-\hat{{\mathcal{J}}}^{-}_{\lambda_{n},d^{-},c^{-}}|<\frac{\epsilon}{2},\
\ \textrm{for all}\ n\geq n_{0},\ k\geq k_{0}.$

Using continuity of $t^{-}_{\lambda}$ for
$\lambda\in(\lambda^{*}_{a,b},\lambda^{*}_{a,b}+\epsilon)$ following Lemma 3.1
$(i)$, we have

(6.6)
$|\mathcal{J}^{-}_{\lambda_{n}}(u_{k})-\mathcal{J}^{-}_{\lambda^{*}_{a,b}}(u_{k})|<\frac{\epsilon}{2},\
\ \textrm{for all}\ n\geq n_{1}.$

Taking $n\geq n_{2}=\textrm{max}\\{n_{0},n_{1}\\}$ and using (6.5) and (6.6),
we get

$\displaystyle\hat{\mathcal{J}}^{-}_{\lambda^{*}_{a,b}}-\hat{\mathcal{J}}^{-}_{\lambda_{n},d^{-},c^{-}}$
$\displaystyle<\mathcal{J}^{-}_{\lambda^{*}_{a,b}}(u_{k})-\hat{\mathcal{J}}^{-}_{\lambda_{n},d^{-},c^{-}}$
$\displaystyle\leq|\mathcal{J}^{-}_{\lambda^{*}_{a,b}}(u_{k})-\mathcal{J}^{-}_{\lambda_{n}}(u_{k})|+|\mathcal{J}^{-}_{\lambda_{n}}(u_{k})-\hat{\mathcal{J}}^{-}_{\lambda_{n},d^{-},c^{-}}|$
$\displaystyle\leq\frac{\epsilon}{2}+\frac{\epsilon}{2}=\epsilon,$

implies,

$\hat{\mathcal{J}}^{-}_{\lambda^{*}_{a,b}}<\hat{\mathcal{J}}^{-}_{\lambda_{n},d^{-},c^{-}}+\epsilon\
\textrm{for}\ \textrm{all}\ n\geq n_{2}.$

Therefore, $\hat{\mathcal{J}}^{-}_{\lambda^{*}_{a,b}}\leq\mathcal{J}$ as
$n\rightarrow\infty$ a contradiction. Thus
$\lim_{\lambda\downarrow\lambda^{*}_{a,b}}\hat{\mathcal{J}}^{-}_{\lambda,d^{-},c^{-}}=\hat{\mathcal{J}}^{-}_{\lambda^{*}_{a,b}}$,
which complete the proof of the claim.

As a consequence of this claim, we have

(6.7)
$|\hat{\mathcal{J}}^{-}_{\lambda^{*}_{a,b}}-\mathcal{J}^{-}_{\lambda_{k}}(u_{n,k})|\leq|\hat{\mathcal{J}}^{-}_{\lambda^{*}_{a,b}}-\hat{\mathcal{J}}^{-}_{\lambda_{k},d^{-},c^{-}}|+|\mathcal{J}^{-}_{\lambda_{k}}(u_{n,k})-\hat{\mathcal{J}}^{-}_{\lambda_{k},d^{-},c^{-}}|\rightarrow
0$

as $n\rightarrow\infty$ followed by $k\to\infty$. Therefore, up to a
subsequence, $u_{{n,k}}\rightharpoonup u$ in $X\setminus\\{0\\}$. Now we claim
that $u_{n,k}\rightarrow u$ in $X\setminus\\{0\\}$. Suppose on contrary, by
weak lower semi continuity of norm, we get

$\liminf_{n,k\rightarrow\infty}\psi^{\prime}_{\lambda_{k},u_{n,k}}(t_{\lambda^{*}_{a,b}}(u))>\psi^{\prime}_{\lambda^{*}_{a,b},u}(t_{\lambda^{*}_{a,b}}(u))=0.$

Hence $t_{\lambda^{*}_{a,b}}(u)<t^{-}_{\lambda_{k}}(u_{n,k})$ for sufficiently
large $n,k$. Therefore, from (6.7), we get

$\displaystyle\mathcal{E}_{\lambda^{*}_{a,b}}(t_{\lambda^{*}_{a,b}}(u)u)$
$\displaystyle<\liminf_{n,k\to\infty}\mathcal{E}_{\lambda_{k}}(t_{\lambda^{*}_{a,b}}(u)u_{n,k}$
$\displaystyle<\liminf_{n,k\to\infty}\mathcal{E}_{\lambda_{k}}(t^{-}_{\lambda_{k}}(u_{n,k}))u_{n,k}=\hat{\mathcal{J}}^{-}_{\lambda^{*}_{a,b}}$

which is an absurd, therefore $u_{n,k}\rightarrow u$ in $X\setminus\\{0\\}$.
Now, $u_{n,k}-u(\lambda)\rightharpoonup u(\lambda_{k})-u(\lambda)$, implies

$\|u(\lambda_{k})-u(\lambda)\|_{X}\leq\liminf_{n\rightarrow\infty}\|u_{n,k}-u(\lambda)\|_{X},$

for large $k$. Thus $u(\lambda_{k})\to
u(\lambda)\in\mathcal{N}^{-}_{\lambda^{*}_{a,b},d^{-},c^{-}}$ and consequently
$u(\lambda_{k})\in\hat{\mathcal{N}}_{\lambda_{k}}$ for large $k$, which is a
contradiction. Thus $u_{n}(\lambda)\rightarrow u(\lambda)$ in $X$ and
$u(\lambda)\in\hat{\mathcal{N}}_{\lambda}$ for all
$\lambda\in(\lambda^{*}_{a,b},\lambda^{*}_{a,b}+\epsilon^{-})$. Subsequently,
$t^{-}_{\lambda}(u_{n}(\lambda))u_{n}(\lambda)\rightarrow
t(\lambda)u(\lambda)$ and
$u(\lambda)\in\mathcal{N}^{-}_{\lambda^{*}_{a,b},d^{-},c^{-}}$ with

$\mathcal{J}^{-}_{\lambda,d^{-},c^{-}}=\mathcal{J}^{-}_{\lambda}(u(\lambda))$

that is, $u(\lambda)$ is minimizer for $\mathcal{J}^{-}_{\lambda,d^{-},c^{-}}$
for all $\lambda\in(\lambda^{*}_{a,b},\lambda^{*}_{a,b}+\epsilon^{-})$.

∎

######  Lemma 6.2.

For the above choices of $c^{+},d^{+}$ there exits $\epsilon^{+}>0$ such that
the functional $\mathcal{J}^{+}_{\lambda}$ constrained to
$\mathcal{N}^{+}_{\lambda^{*}_{a,b},d^{+},c^{+}}$ has a minimizer
${v({\lambda})}\in$ $\mathcal{N}^{+}_{\lambda^{*}_{a,b},d^{+},c^{+}}$ for all
$\lambda\in(\lambda^{*}_{a,b},\lambda^{*}_{a,b}+\epsilon^{+})$.

###### Proof.

The proof of the Lemma is similar to Lemma 6.1 above. ∎

Now we choose $\epsilon=\textrm{min}\\{\epsilon^{-},\epsilon^{+}\\}$ and
rename the minimizers obtained above as $u_{\lambda}=u(\lambda)$ and
$v_{\lambda}=v(\lambda)$ for all
$\lambda\in(\lambda^{*}_{a,b},\lambda^{*}_{a,b}+\epsilon)$. For
$\lambda\in(\lambda^{*}_{a,b},\lambda^{*}_{a,b}+\epsilon)$ the minimization
problem posed in $\mathcal{N}^{\pm}_{\lambda^{*}_{a,b},d^{\pm},c^{\pm}}$
ensures that minimizers $u_{\lambda}$ and $v_{\lambda}$ are separated from
$\mathcal{N}^{0}_{\lambda^{*}_{a,b}}$. Therefore, for $\epsilon>0$
sufficiently small chosen above, $u_{\lambda}\in\mathcal{N}^{-}_{\lambda}$ and
$v_{\lambda}\in\mathcal{N}^{+}_{\lambda}$ for
$\lambda\in(\lambda^{*}_{a,b},\lambda^{*}_{a,b}+\epsilon)$. Finally, invoking
Theorem 2.3 of [6], we get nontrivial weak solutions of ($P_{\lambda}$) for
$\lambda\in(\lambda^{*}_{a,b},\lambda^{*}_{a,b}+\epsilon)$.

Proof of Theorem 3.1 and Theorem 3.2 : The proof of Theorem 3.1 and Theorem
3.2 follows by combining the arguments of previous sections. Eventually, we
get the nontrivial weak solutions of ($P_{\lambda}$) for
$\lambda\in(0,\lambda^{*}_{a,b}+\epsilon)$. Next, we show that the weak
solutions are positive. In fact, due to the presence of nonlocal fractional
operator, we have $\|u\|\neq\|\ |u|\ \|$ in $X$. As a consequence,
$\mathcal{E}_{\lambda}(|u|)\neq\mathcal{E}_{\lambda}(u)$. To overcome this
situation, we define the perturbed problem with nonlinearity as $\lambda
f(x)(u^{+})^{\gamma-1}+g(x)(u^{+})^{p-1}$ and corresponding energy functional
as follows

$\mathcal{E}^{+}_{\lambda}(u):=\frac{a}{2}\|u\|^{2}_{X}+\frac{b}{2\theta}\|u\|^{2\theta}_{X}-\frac{\lambda}{\gamma}\int_{\Omega}f(u^{+})^{\gamma}dx-\frac{1}{p}\int_{\Omega}g(u^{+})^{p}dx.$

It is easy to see that the critical points of $\mathcal{E}_{\lambda}$ are also
the critical points of $\mathcal{E}^{+}_{\lambda}$. Consequently the weak
solutions of the perturbed problem. Now testing the solutions of perturbed
problem with test function $\phi=u^{-}$ and using

$(u(x)-u(y))(u^{-}(x)-u^{-}(y))\leq-|u^{-}(x)-u^{-}(y)|^{2}$

we get, $\|u^{-}\|^{2}_{X}=0$, thus $u$ is a non-negative solution of
($P_{\lambda}$). The positivity of the solutions follows via maximum principle
(see [24], Proposition 2.2.8) which completes the proof.

## Acknowledgments

The research of the first author is supported by Science and Engineering
Research Board, Govt. of India, grant SRG/2021/001076.

## Conflict of interest

The authors declare that they have no conflict of interest.

## References

  * [1] Ambrosetti, A., Brezis, H., Cerami, G.: Combined effects of concave and convex nonlinearities in some elliptic problems. J. Funct. Anal. 122, 519-543 (1994)
  * [2] Ambrosetti, A., Garcia-Azorero J., Peral, T.: Existence and multiplicity results for some nonlinear elliptic equations. A survey. Rend. Mat. Appl. 20, 167–198 (2000)
  * [3] Autuori, G., Fiscella, A., Pucci, P.: Stationary Kirchhoff problems involving a fractional elliptic operator and a critical nonlinearity. Nonlinear Anal. 125, 699–714 (2015)
  * [4] Barrios, B., Colorado, E., Servadei, R., Soria, F.: A critical fractional equation with concave-convex power nonlinearities. Ann. Inst. H. Poincaré. 32, 875–900 (2015)
  * [5] Binlin, Z., Fiscella A., Liang S.: Infinitely many solutions for critical degenerate Kirchhoff type equations involving the fractional $p-$Laplacian. Appl. Math. Optim. 80, 63–80 (2019)
  * [6] Brown, K. J., Zhang, Y.: The Nehari manifold for a semilinear elliptic problem with a sign changing weight function. J. Differential Equations. 193, 481-499 (2003)
  * [7] Chen, C., Kuo, Y., Wu, T.: The Nehari manfiold for Kirchhoff problem involving sign-changing weight functions. J. Differential Equations. 250, 1876-1908 (2011)
  * [8] Fiscella, A., Mishra, P.K.: The Nehari manifold for fractional Kirchhoff problems involving singular and critical terms. Nonlinear Anal. 186, 6–32 (2019)
  * [9] Fiscella, A., Valdinoci, E.: A critical Kirchhoff type problem involving a nonlocal operator. Nonlinear Anal. 94, 156–170 (2014)
  * [10] Goyal, S., Sreenadh, K.: Nehari manifold for non-local elliptic operator with concave–convex nonlinearities and sign-changing weight functions. Proc. Indian Acad. Sci. Math. Sci. 125, 545–558 (2015)
  * [11] Pokhozhaev, S. I.: The fibration method for solving nonlinear boundary value problems. Tr. Mat. Inst. Steklova. 192, 146-163 (1990)
  * [12] Il’yasov, Y.: On extreme values of Nehari manifold method via nonlinear Rayleigh’s quotient. Topol. Methods Non- linear Anal. 49, 683–714 (2017)
  * [13] Il’yasov, Y., Silva, K.: On the branch of positive solution for $p-$Laplacian problem at the extreme value of the Nehari manifold method. Proc. Amer. Math. Soc. 146, 2924-2935 (2018)
  * [14] Kirchhoff, G.: Mechanik, Teubner, Leipzig, 1883
  * [15] Nehari, Z.: Characteristic values associated with a class of non-linear second-order differential equations. Acta Math. 105, 141–175 (1961) https://doi .org /10 .1007 /BF02559588
  * [16] Nehari, Z.: On a class of nonlinear second-order differential equations. Trans. Amer. Math. Soc. 95, 101–123 (1960)
  * [17] Nezza, E. D., Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev space. Bull. Sci. Math. 136, 521-573 (2012)
  * [18] do Ó, J. M., He, X., Mishra, P.K.: Fractional Kirchhoff problem with critical indefinite nonlinearity. Math. Nachr. 292, 615-632 (2019) .
  * [19] Pucci, P., Rădulescu, V.D.: Progress in Nonlinear Kirchhoff Problems [Editorial]. Nonlinear Anal. 186, 1–5 (2019)
  * [20] Pucci, P., Saldi, S.: Critical stationary Kirchhoff equations in $\mathbb{R}^{N}$ involving nonlocal operators. Rev. Mat. Iberoam. 32, 1–22 (2016)
  * [21] Pucci, P., Xiang, M., Zhang, B.: Existence and multiplicity of entire solutions for fractional p-Kirchhoff equations. Adv. Nonlinear Anal. 5, 27–55 (2016)
  * [22] Servadei, R., Valdinoci, E.: The Brezis-Nirenberg result for the fractional Laplacian. Trans.Amer. Math. Soc. 367, 67-102 (2015)
  * [23] Silva, K., Macedo, A.: Local minimizer over the Nehari manifold for a class of concave-convex probelms with sign changing nonlinearity. J. Differential Equations. 265, 1894-1921 (2018)
  * [24] Silvestre, L.: Regularity of the obstacle problem for a fractional power of the Laplace operator. Comm. Pure Appl. Math. 60, 67-112 (2007)
  * [25] Wu, T. F.: Multiplicity results for a semilinear elliptic equation involving sign-changing weight function. Rocky Mountain J. Math. 39, 995–1012 (2009)
  * [26] Wu, T. F.: On semilinear elliptic equations involving concave-convex nonlinearities and sign-changing weight function. J. Math. Anal. Appl. 318, 253–270 (2006)
  * [27] Xiang, M.Q., Molica Bisci, G., Tian, G.H., Zhang, B.L.: Infinitely many solutions for the stationary Kirchhoff problems involving the fractional $p-$Laplacian. Nonlinearity. 29, 357–374 (2016)