Fractional N-Laplacian Problems Defined on the One-Dimensional Subspace

Abstract

The research of the fractional Orlicz-Sobolev space and the fractional N-Laplacian operators will give the development of nonlinear elasticity theory, electro rheological fluids, non-Newtonian fluid theory in a porous medium as well as Probability and Analysis as they proved to be accurate models to describe different phenomena in Physics, Finance, Image processing and Ecology. We study the number of weak solutions for one-dimensional fractional N-Laplacian systems in the product of the fractional Orlicz-Sobolev spaces, where the corresponding functionals of one-dimensional fractional N-Laplacian systems are even and symmetric. We obtain two results for these problems. One result is that these problems have at least one nontrivial solution under some conditions. The other result is that these problems also have infinitely many weak solutions on the same conditions. We use the variational approach, critical point theory and homology theory on the product of the fractional Orlicz-Sobolev spaces.

1. Introduction

Let $(a,b)\subset R^1$ and $0<s<1$ be a fractional parameter. Let N denote the dimension of the domain $\mathsf{\Omega }$, N denote the left-hand side attached character of the N-function and n denote the dimension of the space $\left\{(u_1,\cdots ,u_n)\right\}\subset R^n$. Let g be an odd, increasing homeomorphism from $[0,\infty )$ onto $[0,\infty )$. Let G be the function defined by

$$G(x)={\int }_0^{x}g(t)dt\mathrm{for}\mathrm{all}x\ge 0.$$

G is a Young function and also a N-function (The Young function G is defined as follows: G is the Young function if and only if G satisfies that $G(0)=0$, ${\lim }_{x\to +\infty }G(x)=+\infty$ and G is convex. The N-function G is defined as follows: G is the N-function if and only if ${\lim }_{x\to 0}\frac{G(x)}{x}=0$, ${\lim }_{x\to \infty }\frac{G(x)}{x}=+\infty$, $G(x)=0$ if and only if $x=0$). Let $\alpha (x)$, $p(x)$, $m(x),\gamma (x)\in C([a,b],R)$ be measurable functions with $\alpha (x)>0$, $1<p(x)<\infty$, $m(x)>0$ and $\gamma (x)>1$. Let D be an open subset in $R^n$, $n\ge 2$, with compact complement $C=R^n\backslash D$ containing $\theta =(0,\cdots ,0)$. Let $|·|$ be denoted as a $R^n$ norm.

We consider the number of weak solutions $u=(u_1,\cdots ,u_n)\in C^1([a,b],D)$ for one-dimensional N-Laplacian systems with measurable positive coefficient functions

$$\left\{\begin{array}{ccc}& {(-\Delta )}_g^s{u}_1(x)+{\mathrm{grad}}_{u_1}{(\alpha (x)|u(x)|}^{p(x)})+{\mathrm{grad}}_{u_1}(m(x)\frac{1}{{\left|u(x)\right|}^{\gamma (x)}})\hfill & =0\mathrm{in}(a,b),\hfill \\ & ⋮\hfill & ⋮\hfill & \\ & {(-\Delta )}_g^s{u}_n(x)+{\mathrm{grad}}_{u_n}{(\alpha (x)|u(x)|}^{p(x)})+{\mathrm{grad}}_{u_n}(m(x)\frac{1}{{\left|u(x)\right|}^{\gamma (x)}})\hfill & =0\mathrm{in}(a,b)\hfill \\ & u_i(a)=u_i(b),u_i^{\prime }(a)=u_i^{\prime }(b)\mathrm{for}\mathrm{all}1\le i\le n,\hfill \end{array}\right.$$

(1)

where ${\mathrm{grad}}_{u_i}=\frac{\partial }{\partial u_i}$ and ${(-\Delta )}_g^s{u}_i(x)$ is the fractional N-Laplacian operator defined as follows: for each $x\in R$ and any $u_i$ in the fractional Orlicz-Sobolev space $W^s{L}_G((a,b),R)$,

$${(-\Delta )}_g^s{u}_i(x)=\mathrm{P.V.}{\int }_a^{b}g(\frac{|u_i(x)-u_i(y)|}{{|x-y|}^s})\frac{u_i(x)-u_i(y)}{|u_i(x)-u_i(y)|}\frac{dy}{{|x-y|}^{1+s}}.$$

(2)

Here $\mathrm{P}.\mathrm{V}.$ denotes the Cauchy principle value. For $0<s<1$, ${(-\Delta )}_g^s$ is called the fractional N-Laplacian operator and for $u=(u_1,\cdots ,u_n)\in W^s{L}_G((a,b),R^n)$,

$${(-\Delta )}_g^{s}u(x)=({(-\Delta )}_g^s{u}_1(x),\cdots ,{(-\Delta )}_g^s{u}_n(x)).$$

We assume that

(i)

$sup\left\{\right|\alpha (x)|u(x){|}^{p(x)}+m(x)\frac{1}{{\left|u(x)\right|}^{\gamma (x)}}|$

$$\begin{gathered} +\parallel {\mathrm{grad}}_u{(\alpha (x)|u(x)|}^{p(x)})+{\mathrm{grad}}_u(m(x)\frac{1}{{\left|u(x)\right|}^{\gamma (x)}}){\parallel }_{R^n}|(x,u)\in (a,b)\times (R^n\backslash B_{R_0})\}< \\ +\infty \end{gathered}$$

for some $R_0>0$.

(ii)

${\alpha (x)\left|u(x)\right|}^{p(x)})+m(x)\frac{1}{{\left|u(x)\right|}^{\gamma (x)}}\ge \frac{A}{d^2(u,C)}\mathrm{for}(x,u)\in (a,b)\times U$

for some neighborhood U of C, the distance function $d(u,C)$ from u to C and a constant $A>0$.

The corresponding functionals of (1)

$${\int }_a^b{\int }_a^b\left[{\mathsf{\Sigma }}_{i=1}^{n}G(\frac{|u_i(x)-u_i(y)|}{{|x-y|}^s})\right]\frac{dxdy}{|x-y|}+{\int }_a^b\alpha (x){\left|u(x)\right|}^{p(x)}dx+{\int }_a^{b}m(x)\frac{1}{{\left|u\right|}^{\gamma (x)}}dx$$

are even and symmetric.

Our problems contain the Orlicz space, the fractional Orlicz-Sobolev space and the fractional N-Laplacian operators. The Orlicz space, the fractional Orlicz-Sobolev space and the fractional N-Laplacian operators arise in the non-linear elasticity theory, electro rheological fluids, non-Newtonian fluid theory, a porous medium and the context of stochastic Lvy processes with jumps. In recent years, Probability and Analysis in the context of Physics, Finance, Image processing and Ecology have been provided the mathematical models containing the Orlicz space, the fractional Orlicz-Sobolev space and the fractional N-Laplacian operators to describe different phenomena.

For The Orlicz space, the fractional Orlicz-Sobolev space and the fractional N-Laplacian operators, we refer to [1,2,3,4,5,6,7,8,9,10,11,12,13,14].

For the singular elliptic systems and the singular problems involving fractional N-Laplacian, we refer to [15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35].

Let $0<s<1$, $p:\mathsf{\Omega }\times \mathsf{\Omega }\to (1,\infty )$ be a continuous function with $1<p(x,y)<\infty$. The fractional Sobolev spaces with variable exponent defined on one-dimensional subspace $(a,b)$ are defined as

$$W^{s,p(x,y)}((a,b),R)=\{u\in L^{p(x,y)}((a,b),R|{\int }_a^b{\int }_a^b\frac{{|u(x)-u(y)|}^{p(x,y)}}{{\lambda }^{p(x,y)}{|x-y|}^{1+sp(x,y)}}dxdy<\infty ,$$

$$\mathrm{for}\mathrm{some}\lambda >0\}$$

with the norm

$${\parallel u\parallel }_{s,p(x,y)}=inf\{\lambda >0|{\int }_a^b{\int }_a^b\frac{{|u(x)-u(y)|}^{p(x,y)}}{{\lambda }^{p(x,y)}{|x-y|}^{1++sp(x,y)}}dxdy\le 1\}.$$

when $g(t)={\left|t\right|}^{p(x,y)-2}t$ in (2), (2) reduced to the fractional $p(·)-$ Laplacian operator with variable exponent

$${(-\Delta )}_{p(·)}^s{u}_i(x)=\mathrm{P.V.}{\int }_a^b\frac{|u_i(x)-u_i{(y)|}^{p(x,y)-2}(u_i(x)-u_i(y))}{{|x-y|}^{1+sp(x,y)}}dy,x\in (a,b).$$

(3)

Nonsingular $p(·)-$ Laplacian boundary value problems like the following

$$-{\mathrm{div}|\nabla u|}^{p(x)-2}\nabla u=\lambda f(x,u),$$

$$u=0\partial \mathsf{\Omega },$$

we refer to [2,13,36,37,38,39,40,41,42,43,44,45,46]. In case of $p-$ Laplacian with $p(x)=p$ a constant, we refer to [3,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63]. When $0<s<1$, $g(t)=t$ in (2), (2) reduced to the usual fractional Laplacian operator ${(-\Delta )}^s$. For the fractional Laplacian operator, see [45,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87].

We deal with the fractional Orlicz spaces with variable exponent, the fractional Orlicz-Sobolev spaces with variable exponent and their corresponding nonhomogeneous fractional N-Laplacian operators as the adequate function spaces. For the theory of Orlicz and Orlicz-Sobolev spaces and the fractional N-Laplacian operator, we also refer the readers to [1,10,60,88,89].

When $s=1$, $g(t)={\left|t\right|}^{p-2}t$ and $\mathsf{\Omega }\subset R^n$, (1) reduce to the $p-$Laplacian problems. We [90] proved the existence of at least one solution for the $p-$Laplacian boundary value problem with two singular and subcritical nonlinearities

$$\left\{\begin{array}{cc}& -{\mathrm{div}(|\nabla u|}^{p-2}{\nabla u)=a|u|}^{p-2}u-{\mathrm{grad}}_u\frac{1}{{(|u|}^2+{\left|v\right|}^2{)}^q}-{\mathrm{grad}}_u\frac{1}{(|u-e_1{|}^2+|v-e_2{{|}^2)}^r}+\frac{2\alpha }{\alpha +\beta }{\left|u\right|}^{\alpha -1}{\left|v\right|}^{\beta }\hfill \\ & \mathrm{in}\mathsf{\Omega },\hfill \\ & -{\mathrm{div}(|\nabla v|}^{p-2}{\nabla v)=b|v|}^{p-2}v-{\mathrm{grad}}_v\frac{1}{{(|u|}^2+{\left|v\right|}^2{)}^q}-{\mathrm{grad}}_v\frac{1}{(|u-e_1{|}^2+|v-e_2{{|}^2)}^r}+\frac{2\beta }{\alpha +\beta }{\left|u\right|}^{\alpha }{\left|v\right|}^{\beta -1}\hfill \\ & \mathrm{in}\mathsf{\Omega },\hfill \end{array}\right.$$

where a, b, p, q, r, $\alpha$ and $\beta$ are real constants, and $1<p<\infty$, $q,r>1$ and $p<\alpha +\beta <p^{*}$, where $p^{*}$ is critical exponent such that

$$p^{*}=\left\{\begin{array}{cc}& \frac{np}{n-p}\mathrm{if}n>p,\hfill \\ & \infty \mathrm{if}n\le p.\hfill \end{array}\right.$$

When $s=1$, $g(t)={\left|t\right|}^{p-2}t$, $n=1$ and $\mathsf{\Omega }=(a,b)\subset R^1$, (1) reduce to the one-dimensional $p-$Laplacian problems. For the one-dimensional singular problems involving $p-$Laplacian

$$-(|u^{\prime }{{|}^{p-2}{u}^{\prime })}^{\prime }=m(x)u^{-\gamma }in\mathsf{\Omega },$$

$$u>0in\mathsf{\Omega },$$

$$u=0on\partial \mathsf{\Omega },$$

we refer to [24,30,91,92,93,94,95,96,97,98,99,100,101]. For the one-dimensional fractional $p-$Laplacian problems, we refer to [102]. For the Kirchhoff equations involving fractional $p-$Laplacian, we refer to [60,76,77,103,104,105].

Let $G^{-1}$ be the function defined by

$$G^{-1}(x)={\int }_0^x{g}^{-1}(t)dt\mathrm{for}\mathrm{all}x\ge 0.$$

The function $G^{-1}$ is called the complementary function of G and satisfies

$$G^{-1}(x)=sup\{yx-G(y)|y\ge 0\}\mathrm{for}\mathrm{all}x\ge 0.$$

$G^{-1}$ satisfies that

$$\underset{x\to 0}{lim}\frac{G^{-1}(x)}{x}=0,\underset{x\to \infty }{lim}\frac{G^{-1}(x)}{x}=+\infty ,$$

i.e., $G^{-1}$ is an N-function. Moreover, by the Young’s inequality,

$$xy\le G(x)+G^{-1}(y),\mathrm{for}\mathrm{all}x,y\ge 0.$$

(4)

From now on we shall denote by $C^1(S^1,D)$ the subset of $C^1([a,b],D)$ satisfying the conditions

$$u(a)=u(b),u^{\prime }(a)=u^{\prime }(b).$$

Orlicz space $L_G(S^1,R)$ defined by N-function G is defined as

$$L_G(S^1,R)=\left\{u\right|u:S^1\to R\mathrm{is}a\mathrm{measurable}\mathrm{function}\mathrm{with}$$

$${\parallel u\parallel }_{L_G}=sup\{{\int }_a^{b}uvdx|{\int }_a^b{G}^{-1}(|v|)dx\le 1\}<\infty \}.$$

$L_G(S^1,R)$ is a Banach space with a norm ${\parallel u\parallel }_{L_G}$ equivalent to the Luxemburg norm

$${\parallel u\parallel }_G=inf\{\nu >0|{\int }_a^{b}G(|\frac{u(x)}{\nu }|)\le 1\}.$$

$H\ddot{o}lder$ inequality holds in the Orlicz space $L_G(S^1,R)$ (see [19]):

$${\int }_a^b\left|uv\right|dx\le {2\parallel u\parallel }_{L_G}{\parallel v\parallel }_{L_{G^{-1}}}\mathrm{for}\mathrm{all}u,v\in L_G(S^1,R).$$

Orlicz-Sobolev space $W^s{L}_G(S^1,R)$ is defined as

$$W^s{L}_G(S^1,R)=\{u\in L_G(S^1,R):{\int }_a^b{\int }_a^{b}G(\frac{|u(x)-u(y)|}{{|x-y|}^s})\frac{dxdy}{|x-y|}<\infty \}$$

with the norm

$${\parallel u\parallel }_{s,G}={\parallel u\parallel }_G+{\left[u\right]}_{s,G},$$

where ${\left[u\right]}_{s,G}$ is the Gagliardo semi-norm defined by

$${\left[u\right]}_{s,G}=inf\{\mu >0|{\int }_a^b{\int }_a^{b}G(\frac{|u(x)-u(y)|}{{\mu |x-y|}^s})\frac{dxdy}{|x-y|}\le 1\}.$$

In [88], for any $0<s<1$ and G a Young function such that G and $G^{-1}$ satisfy that

$$G(2t)\le C_{1}G(t)\mathrm{and}{G}^{-1}(2t)\le C_2{G}^{-1}(t),\forall t\ge 0,C_1,C_2>0,$$

(5)

$W^s{L}_G(S^1,R)$ is reflexive and separable. Moreover $C_0^{\infty }(S^1,R)$ is dense in $W^s{L}_G(S^1,R)$ in the norm ${\parallel ·\parallel }_{s,G}$. Let the Orlicz-Sobolev space $W_0^s{L}_G(S^1,R)$ be the closure of $C_0^{\infty }(S^1,R)$ in $W^s{L}_G(S^1,R)$. The space $W_0^s{L}_G(S^1,R)$ is also reflexive. By Lemma 5.7 in [4], the norm ${\parallel u\parallel }_{s,G}$ is equivalent to ${\left[u\right]}_{s,G}$.

Let

$$L_G(S^1,R^n)=L_G(S^1,R)\times \cdots \times L_G(S^1,R)$$

be the $n-$cartesian product space of $L_G(S^1,R)$ equipped with the norm

$${\parallel u\parallel }_{L_G(S^1,R^n)}=\sum _{i=1}^n{\parallel u_i\parallel }_{L_G(S^1,R)},u=(u_1,\cdots ,u_n)$$

and

$$H=W^s{L}_G(S^1,R^n)=W^s{L}_G(S^1,R)\times \cdots \times W^s{L}_G(S^1,R)$$

equipped with the norm

$${\parallel u\parallel }_H=\sum _{i=1}^n{\parallel u_i\parallel }_{s,G}.$$

For any G a Young function such that G and $G^{-1}$ satisfy $(5)$, $W^s{L}_G(S^1,R^n)$ is reflexive and separable. Moreover $C_0^{\infty }(S^1,R^n)$ is dense in $W^s{L}_G(S^1,R^n)$ in the norm ${\parallel ·\parallel }_{s,G}$. Let $W_0^s{L}_G(S^1,R^n)$ be the closure of $C_0^{\infty }(S^1,R^n)$ in $W^s{L}_G(S^1,R^n)$.

Zhikov [106] observed that smooth functions are not dense in $W^{1,p(x)}(S^1,R)$ without additional assumptions on the exponent $p(x)$. However, when the exponent $p(x)$ is log-$H\ddot{o}lder$ continuous, i.e., there is a constant B such that

$$|p(x)-p(y)|\le \frac{B}{-\log |x-y|}$$

for every x, $y\in (a,b)$ with $|x-y|\le \frac{1}{2}$, then smooth functions are dense in $W^{1,p(x)}(S^1,R)$. and there is no confusion in defining the Sobolev space with zero boundary values, $W_0^{1,p(x)}$, as the completion of $C_0^{\infty }(S^1,R)$ with respect to the norm ${\parallel u\parallel }_{W^{1,p(x)}(S^1,R)}$ [39,93].

Let us set

$$g_0=\underset{t>0}{inf}\frac{tg(t)}{G(t)}{g}^0=\underset{t>0}{sup}\frac{tg(t)}{G(t)}$$

and assume that

$$1<g_0\le \frac{tg(t)}{G(t)}\le g^0<\infty \forall t\ge 0.$$

(6)

By Proposition 2.3 of [6], it implies that each G satisfies the ${\Delta }_2-$condition, i.e., there exists a constant $C>0$ such that

$$G(2t)\le CG(t),t\ge 0.$$

Assume that G is a function such that

$$G:t\in [0,\infty )\mapsto G(\sqrt{t})\mathrm{is}\mathrm{convex}.$$

(7)

Weak solutions $u=(u_1,\cdots ,u_n)\in W^s{L}_G(S^1,D)$ satisfy

$${\int }_a^b{[(-\Delta )}_g^{s}u(x)·\varphi (x)dx+{\mathrm{grad}}_u{(\alpha (x)|u(x)|}^{p(x)})·\varphi (x)+{\mathrm{grad}}_u(m(x)\frac{1}{{\left|u\right|}^{\gamma (x)}})·\varphi (x)]dx=0$$

$\forall \varphi \in W^s{L}_G(S^1,D)$.

Theorem 1.

Let $0<s<1$, $s{g}_0<1$, $p\in C([a,b],R)$ with $p(x)>1$ be log-$H\ddot{o}lder$ continuous. Let $\alpha (x)$, $p(x)$, $m(x),\gamma (x)\in C([a,b],R)$ be measurable functions with $\alpha (x)>0$, $1<p(x)<\infty$, $m(x)>0$ and $\gamma (x)>1$. We assume that (5)–(7) and

$$1<p(x)\le p_{+}<g_0^{*}=\frac{g_0}{1-s{g}_0}$$

hold, where $p_{+}={sup}_{x\in [a,b]}p(x)$ and $p_{-}={inf}_{x\in [a,b]}p(x).$ Then (1) has at least one nontrivial weak solution.

Theorem 2.

Under the assumptions of Theorem 1, (1) has infinitely many weak solutions.

We use variational approach, minimax method in critical point theory on the loop space $W^s{L}_G(S^1,D)$ and homology theory. In Section 2, we introduce some preliminaries. In Section 3, we obtain some variational results on the potential and prove that the associated functional J of (1) satisfies the $(P.S.)$ condition on the loop subspace $W^s{L}_G(S^1,D)$. In Section 4, we prove Theorems 1 and 2 by using minimax method, critical point theory and homology theory.

2. Fractional Orlicz-Sobolev Space

For the variational setting, we need some properties on $L^{p(x)}(S^1,R)$, $L^{p(x)}(S^1,R^n)$, $L_G(S^1,R)$, $L_G(S^1,R^n)$, $W^s{L}_G(S^1,R)$ and $W^s{L}_G(S^1,R^n)$ which are introduced in Section 1 and can be found in [107,108].

Lemma 1.

([107]) The space $L^{p(x)}(S^1,R)$ is a separable, uniformly convex Banach space, and its conjugate space is $L^{p^{\prime }(x)}(S^1,R)$, where $\frac{1}{p(x)}+\frac{1}{p^{\prime }(x)}=1$. Let $u\in L^{p(x)}(S^1,R)$ and $v\in L^{p^{\prime }(x)}(S^1,R)$. Then we have

$$|{\int }_a^{b}uvdx|\le \left(\frac{1}{p_{-}}+\frac{1}{{(p^{\prime })}_{-}}\right){\parallel u\parallel }_{L^{p(x)}(S^1,R)}{\parallel v\parallel }_{L^{p^{\prime }(x)}(S^1,R)}\le {2\parallel u\parallel }_{L^{p(x)}(S^1,R)}{\parallel v\parallel }_{L^{p^{\prime }(x)}(S^1,R)}.$$

Lemma 2.

Let $p_{+}<\infty$ and set

$$\tau (u)={\int }_a^b{\left|u\right|}^{p(x)}dxforallu\in L^{p(x)}(S^1,R).$$

Then

(i) 

$\tau (u)>1$⇔${\parallel u\parallel }_{L^{p(x)}(S^1,R)}>1$,

$\tau (u)=1$⇔${\parallel u\parallel }_{L^{p(x)}(S^1,R)}=1,$

$\tau (u)<1$⇔${\parallel u\parallel }_{L^{p(x)}(S^1,R)}<1.$

(ii) 

If${\parallel u\parallel }_{L^{p(x)}(S^1,R)}>1$, then${\parallel u\parallel }_{L^{p(x)}(S^1,R)}^{p_{-}}\le \tau (u)\le {\parallel u\parallel }_{L^{p(x)}(S^1,R)}^{p_{+}}$.

(iii) 

If${\parallel u\parallel }_{L^{p(x)}(S^1,R)}<1$, then${\parallel u\parallel }_{L^{p(x)}(S^1,R)}^{p_{+}}\le \tau (u)\le {\parallel u\parallel }_{L^{p(x)}(S^1,R)}^{p_{-}}$.

Lemma 3.

([88]) Let $0<s<1$ and G be a $N-$function. Then $W^s{L}_G(S^1,R^n)$ is a reflexive and separable Banach space. Furthermore $C_0^{\infty }(S^1,R^n)$ is dense in $W^s{L}_G(S^1,R^n)$ in the norm ${\parallel ·\parallel }_{s,G}$.

Let $u_i\in W^s{L}_G(S^1,R)$ and set

$${\left[u_i\right]}_{s,G}=inf\{\sigma >0|{\int }_a^b{\int }_a^{b}G(\frac{|u_i(x)-u_i(y)|}{{\sigma |x-y|}^s})\frac{dxdy}{|x-y|}\le 1\},1\le i\le n.$$

Lemma 4.

([109]) (Generalized $Poincar$ inequality on the Orlicz-Sobolev space)

Let $0<s<1$ and G be a Young function. Then

$$\parallel u_i{\parallel }_G\le {\nu }_i{\left[u_i\right]}_{s,G},\forall u_i\in W^s{L}_G(S^1,R),1\le i\le n$$

(8)

for some constants ${\nu }_i>0$, $1\le i\le n$. That is, the embedding

$$W^s{L}_G(S^1,R)\hookrightarrow L_G(S^1,R)$$

is continuous. Furthermore ${\left[u_i\right]}_{s,G}$ is a norm of $W^s{L}_G(S^1,R)$ equivalent to $\parallel u_i{\parallel }_{s,G}$.

Let $u=(u_1,\cdots ,u_n)\in W^s{L}_G(S^1,R^n)$, $u_i\in W^s{L}_G(S^1,R)$ and set

$${\left[u\right]}_{s,G}=\sum _{i=1}^n{\left[u_i\right]}_{s,G}.$$

(9)

Combining Lemma 4 and (9), we obtain the following lemma:

Lemma 5.

(Generalized $Poincar$ inequality on the product of the Orlicz-Sobolev space)

Let $0<s<1$ and G be a Young function. Then

$${\parallel u\parallel }_G\le \nu {\left[u\right]}_{s,G},\forall u\in W^s{L}_G(S^1,R^n)$$

for some constant $\nu >0$. That is, the embedding

$$W^s{L}_G(S^1,R^n)\hookrightarrow L_G(S^1,R^n)$$

is continuous. Moreover ${\left[u\right]}_{s,G}$ is a norm of $W^s{L}_G(S^1,R^n)$ equivalent to ${\parallel ·\parallel }_{s,G}$.

Lemma 6.

([110]) Let $u_i\in W^s{L}_G(S^1,R)$. Then

$$\begin{array}{ccc}\hfill \parallel u_i{\parallel }_{s,G}^{g_0}& \le & {\int }_a^b{\int }_a^{b}G(\frac{|u_i(x)-u_i(y)|}{{|x-y|}^s})\frac{dxdy}{|x-y|}\le \parallel u_i{\parallel }_{s,G}^{g^0},if{\parallel u_i\parallel }_{s,G}>1,1\le i\le n,\hfill \\ \hfill \parallel u_i{\parallel }_{s,G}^{g^0}& \le & {\int }_a^b{\int }_a^{b}G(\frac{|u_i(x)-u_i(y)|}{{|x-y|}^s})\frac{dxdy}{|x-y|}\le \parallel u_i{\parallel }_{s,G}^{g_0},if{\parallel u_i\parallel }_{s,G}<1.\hfill \end{array}$$

(10)

Proof. 

The proof follows from (6) and Theorem 3.11 of [110]. □

Let us define the functional $Q_{s,G}:W^s{L}_G(S^1,R^n)\to R$ by

$$Q_{s,G}(u)={\int }_a^b{\int }_a^b\left[{\mathsf{\Sigma }}_{i=1}^{n}G(\frac{|u_i(x)-u_i(y)|}{{|x-y|}^s})\right]\frac{dxdy}{|x-y|},$$

$$u=(u_1,\cdots ,u_n)\in W^s{L}_G(S^1,R^n),u_i\in W^s{L}_G(S^1,R).$$

Combining Lemma 6 and (9), we obtain that:

Lemma 7.

Let $u\in W^s{L}_G(S^1,R^n)$. Then

$${\parallel u\parallel }_{s,G}^{g_0}\le Q_{s,G}(u)\le {\parallel u\parallel }_{s,G}^{g^0},if{\parallel u\parallel }_{s,G}>1,$$

$${\parallel u\parallel }_{s,G}^{g^0}\le Q_{s,G}(u)\le {\parallel u\parallel }_{s,G}^{g_0},if{\parallel u\parallel }_{s,G}<1.$$

Lemma 8.

([109]) Let $0<s<1$, $s{g}_0<1$ and G be a N-function. Then the embeddings

$$W^{s,g_0}(S^1,R)\hookrightarrow L^{r(x)}(S^1,R)$$

and

$$W^s{L}_G(S^1,R)\hookrightarrow L^{r(x)}(S^1,R)$$

are continuous and compact for all $1\le r(x)<g_0^{*}$.

Moreover

$$\parallel u_i{\parallel }_{L^{r(x)}(S^1,R)}\le C_i{\left[u_i\right]}_{s,G}forall1\le r(x)<g_0^{*}$$

for some constants $C_i$, $1\le i\le n$.

Proof. 

By [109], the embedding $W^{s,g_0}(S^1,R)\hookrightarrow L^{r(x)}(S^1,R)$ is continuous and compact for all $1\le r(x)<g_0^{*}$. By (6), the embedding $W^s{L}_G(S^1,R)\hookrightarrow W^{s,g_0}(S^1,R)$ is continuous. Combining these facts, we obtain that the embedding $W^s{L}_G(S^1,R)\hookrightarrow L^{r(x)}(S^1,R)$ is continuous and compact for all $1\le r(x)<g_0^{*}$. □

Lemma 9.

Let $0<s<1$, $s{g}_0<1$ and G be a N-function. Then the embeddings

$$W^{s,g_0}(S^1,R^n)\hookrightarrow L^{r(x)}(S^1,R^n)$$

and

$$W^s{L}_G(S^1,R^n)\hookrightarrow L^{r(x)}(S^1,R^n)$$

are continuous and compact for all $1\le r(x)<g_0^{*}$.

Moreover

$${\parallel u\parallel }_{L^{r(x)}(S^1,R^n)}\le C{\left[u\right]}_{s,G}forall1\le r(x)<g_0^{*}.$$

(11)

for some constant $C>0$.

Lemma 10.

Weakly convergent sequence $\{u_k=(u_{k_1},\cdots ,u_{k_n})\}$ converging to $u=(u_1,\cdots ,u_n)$ in $W^s{L}_G(S^1,R^n)$ satisfying

$$\underset{k\to +\infty }{lim}sup<Q_{s,G}^{\prime }(u_k),u_k-u>\le 0.$$

converges strongly to u in $W^s{L}_G(S^1,R^n)$.

Proof. 

Since the sequence $\left\{u_k\right\}$ converges weakly to u in $W^s{L}_G(S^1,R^n)$ and

${lim}_{k\to +\infty }sup<Q^{\prime }(u_k),u_k-u>\le 0$, by (5), we have

$$\begin{array}{ccc}& & {\int }_a^b{\int }_a^b\left[{\mathsf{\Sigma }}_{i=1}^{n}g(\frac{|u_{k_i}(x)-u_{k_i}(y)|}{{|x-y|}^s})\frac{u_{k_i}(x)-u_{k_i}(y)}{|u_{k_i}(x)-u_{k_i}(y)|}\frac{u_{k_i}(x)-u_{k_i}(y)}{{|x-y|}^s}\right]\frac{dxdy}{|x-y|}\hfill \\ \hfill & & \le {\int }_a^b{\int }_a^b\left[{\mathsf{\Sigma }}_{i=1}^{n}g(\frac{|u_i(x)-u_i(y)|}{{|x-y|}^s})\frac{u_i(x)-u_i(y)}{|u_i(x)-u_i(y)|}\frac{u_i(x)-u_i(y)}{{|x-y|}^s}\right]\frac{dxdy}{|x-y|}\hfill \\ \hfill & & \le g^0[{\int }_a^b{\int }_a^b\left[{\mathsf{\Sigma }}_{i=1}^{n}G(\frac{|u_i(x)-u_i(y)|}{{|x-y|}^s})\right]\frac{dxdy}{|x-y|}.\hfill \end{array}$$

Sequence $\left\{{\int }_a^b{\int }_a^b\left[{\mathsf{\Sigma }}_{i=1}^{n}g(\frac{|u_{k_i}(x)-u_{k_i}(y)|}{{|x-y|}^s})\frac{u_{k_i}(x)-u_{k_i}(y)}{|u_{k_i}(x)-u_{k_i}(y)|}\frac{u_{k_i}(x)-u_{k_i}(y)}{{|x-y|}^s}\right]\frac{dxdy}{|x-y|}\right\}$ is bounded and converges to

${\int }_a^b{\int }_a^b\left[{\mathsf{\Sigma }}_{i=1}^{n}g(\frac{|u_i(x)-u_i(y)|}{{|x-y|}^s})\frac{u_i(x)-u_i(y)}{|u_i(x)-u_i(y)|}\frac{u_i(x)-u_i(y)}{{|x-y|}^s}\right]\frac{dxdy}{|x-y|}$. By (6), we have

$$\begin{array}{ccc}& & {\int }_a^b{\int }_a^b\left[{\mathsf{\Sigma }}_{i=1}^{n}g(\frac{|u_{k_i}(x)-u_{k_i}(y)|}{{|x-y|}^s})\frac{u_{k_i}(x)-u_{k_i}(y)}{|u_{k_i}(x)-u_{k_i}(y)|}\frac{u_{k_i}(x)-u_{k_i}(y)}{{|x-y|}^s}\right]\frac{dxdy}{|x-y|}\hfill \\ \hfill & & \ge g_0[{\int }_a^b{\int }_a^b\left[{\mathsf{\Sigma }}_{i=1}^{n}G(\frac{|u_{k_i}(x)-u_{k_i}(y)|}{{|x-y|}^s})\right]\frac{dxdy}{|x-y|}.\hfill \end{array}$$

Sequence $\left\{{\int }_a^b{\int }_a^b\left[{\mathsf{\Sigma }}_{i=1}^{n}G(\frac{|u_{k_i}(x)-u_{k_i}(y)|}{{|x-y|}^s})\right]\frac{dxdy}{|x-y|}\right\}$ is bounded and converges to

${\int }_a^b{\int }_a^b\left[{\mathsf{\Sigma }}_{i=1}^{n}G(\frac{|u_i(x)-u_i(y)|}{{|x-y|}^s})\right]\frac{dxdy}{|x-y|}$. Thus the sequence $\left\{u_k\right\}$ is bounded and converges weakly to u in $W^s{L}_G(S^1,R^n)$. Since the embedding $W^s{L}_G(S^1,R^n)\hookrightarrow L_G(S^1,R^n)$ is continuous and compact, $\left\{u_k\right\}$ converges strongly to u in $W^s{L}_G(S^1,R^n)$. □

Lemma 11.

If $u_k=(u_{k_1},\cdots ,u_{k_n})$, $u=(u_1,\cdots ,u_n)\in W^s{L}_G(S^1,R^n)$, $k=1,2,\dots$, then the following statement are equivalent to each other

(i) ${lim}_{k\to \infty }{\parallel u_k-u\parallel }_{s,G}=0$, $k=1,2$,

(ii) ${lim}_{k\to \infty }{\int }_a^b[{\mathsf{\Sigma }}_{i=1}^n(G(u_{k_i}(x)-u_i(x))]dx=0$ and

${lim}_{k\to \infty }{[u_k-u]}_{s,G}=0$,

(iii) $u_k\to u$ in measure in $W^s{L}_G(S^1,R^n)$ and

${lim}_{k\to \infty }{\int }_a^b\left[{\mathsf{\Sigma }}_{i=1}^{n}G(u_{k_i}(x))\right]dx={\int }_a^b\left[{\mathsf{\Sigma }}_{i=1}^{n}G(u_i(x))\right]dx$.

Proof. 

By the definition of ${\parallel ·\parallel }_{s,G}$, (i)⇔(ii) holds. We shall show that (i) implies (iii). We assume that (i) holds. Then

$$\begin{array}{ccc}& & {\int }_a^b[({\mathsf{\Sigma }}_{i=1}^{n}G(u_{k_i})(x))-({\mathsf{\Sigma }}_{i=1}^{n}G(u_i)(x))]dx\hfill \\ \hfill & & \le {\int }_a^b\left[{\mathsf{\Sigma }}_{i=1}^n(g(u_i+\lambda (u_{k_i}-u_i))(u_{k_i}-u_i))\right]dx\hfill \\ \hfill & & \le 2[{\mathsf{\Sigma }}_{i=1}^n(\parallel g(u_i+\lambda (u_{k_i}-u_i)){\parallel }_{G^{-1}}\parallel u_{k_i}-u_i{\parallel }_G)]\hfill \\ \hfill & & \le C({\mathsf{\Sigma }}_{i=1}^n(\parallel g(u_i+\lambda (u_{k_i}-u_i)){\parallel }_{G^{-1}}\parallel u_{k_i}-u_i{\parallel }_{s,G}))\hfill \\ \hfill & & \to 0\hfill \end{array}$$

for $0<\lambda <1$ and some $C>0$. It follows that (iii) holds. Assume that (iii) holds. Since

${lim}_{k\to \infty }{\int }_a^b\left[{\mathsf{\Sigma }}_{i=1}^{n}G(u_{k_i}(x))\right]dx={\int }_a^b\left[{\mathsf{\Sigma }}_{i=1}^{n}G(u_i(x))\right]dx$, $\left\{u_k\right\}$ converges weakly to u in $L_G(S^1,$$R^n)$. By assumption (iii), $u_k\to u$ in measure in $W^s{L}_G(S^1,R^n)$. It follows that $\left\{u_k\right\}$ is bounded in $W^s{L}_G(S^1,R^n)\subset L_G(S^1,R^n)$. By Lemma 9, the embedding $W^s{L}_G(S^1,R^n)\hookrightarrow L_G(S^1,R^n)$ is continuous and compact. Thus $u_k\to u$ strongly in $W^s{L}_G(S^1,R^n)$. Thus (i) holds. □

3. Variational Results

Let us set an open set of the Hilbert space $W^s{L}_G(S^1,R^n)$ as

$${\mathsf{\Lambda }}_{s,G}D=\{u\in W^s{L}_G(S^1,R^n)|{\int }_a^b{\int }_a^b\left[{\mathsf{\Sigma }}_{i=1}^{n}G(\frac{|u_i(x)-u_i(y)|}{{|x-y|}^s})\right]\frac{dxdy}{|x-y|}<\infty ,$$

$$\forall u(x)=(u_1(x),\cdots ,u_n(x))\in D=R^n\backslash C\subset R^n,\forall x\in S^1,\theta \in C,\theta =(0,\cdots ,0)\}$$

the loop space on D. We note that ${\mathrm{grad}}_u(\frac{m(x)}{{\left|u(x)\right|}^{\gamma (x)}})$ is not singular on $W^s{L}_G(S^1,D)$ because D is away from 0.

Let us define the functional $J_{s,G}:\mathsf{\Lambda }D\subset W^s{L}_G(S^1,R^n)\to R$ by

$$J_{s,G}(u)=Q_{s,G}(u)+{\int }_a^b\alpha (x){\left|u(x)\right|}^{p(x)}dx+{\int }_a^{b}m(x)\frac{1}{{\left|u\right|}^{\gamma (x)}}dx,$$

where

$$Q_{s,G}(u)={\int }_a^b{\int }_a^b\left[{\mathsf{\Sigma }}_{i=1}^{n}G(\frac{|u_i(x)-u_1(y)|}{{|x-y|}^s})\right]\frac{dxdy}{|x-y|}$$

for $u=(u_1,\cdots ,u_n)\in \mathsf{\Lambda }D$. Then the functional $Q_{s,G}$ is of class $C^1(W^s{L}_G(S^1,R^n),R)$ with

$$\begin{array}{ccc}& & <Q_{s,G}^{\prime }(u),v>=<{(-\Delta )}_g^{s}u,v>\hfill \\ \hfill & & ={\int }_a^b{\int }_a^b\left[{\mathsf{\Sigma }}_{i=1}^{n}g(\frac{|u_i(x)-u_i(y)|}{{|x-y|}^s})\frac{u_i(x)-u_i(y)}{|u_i(x)-u_i(y)|}\frac{v_i(x)-v_i(y)}{{|x-y|}^s}\right]\frac{dxdy}{|x-y|}\hfill \end{array}$$

(12)

$\forall u=(u_1,\cdots ,u_n)\in \mathsf{\Lambda }D$, which is proved in Proposition 3.3 in [89].

Lemma 12.

$J_{s,G}(u)$ is continuous and $C^1$ on ${\mathsf{\Lambda }}_{s,G}D$ with $Frchet$ derivative

$$\begin{array}{ccc}\hfill & & D{J}_{s,G}(u)v\hfill \\ \hfill & & ={\int }_a^b{\int }_a^b\left[{\mathsf{\Sigma }}_{i=1}^{n}g(\frac{|u_i(x)-u_i(y)|}{{|x-y|}^s})\frac{u_i(x)-u_i(y)}{|u_i(x)-u_i(y)|}\frac{v_i(x)-v_i(y)}{{|x-y|}^s}\right]\frac{dxdy}{|x-y|}\hfill \\ \hfill & & +{\int }_a^b[({grad}_u{\alpha (x)\left|u(x)\right|}^{p(x)}))·v(x)+({grad}_u(\frac{m(x)}{{\left|u(x)\right|}^{\gamma (x)}})·v(x)]dx,\hfill \end{array}$$

(13)

$\forall v=(v_1,\cdots ,v_n)\in {\mathsf{\Lambda }}_{s,G}D$. Moreover $D{J}_{s,G}\in C$. That is, $J_{s,G}\in C^1$.

Proof. 

By Proposition 3.3 in [89],

$$Q_{s,G}(u)={\int }_a^b{\int }_a^b\left[{\mathsf{\Sigma }}_{i=1}^{n}G(\frac{|u_i(x)-u_i(y)|}{{|x-y|}^s})\right]\frac{dxdy}{|x-y|}$$

is $C^1$ with $Frchet$ derivative

$$\begin{array}{ccc}\hfill <Q_{s,G}^{\prime }(u),v>& =& <{(-\Delta )}_g^{s}u,v>\hfill \\ \hfill & =& {\int }_a^b{\int }_a^b\left[{\mathsf{\Sigma }}_{i=1}^{n}g(\frac{|u_i(x)-u_i(y)|}{{|x-y|}^s})\frac{u_i(x)-u_i(y)}{|u_i(x)-u_i(y)|}\frac{v_i(x)-v_i(y)}{{|x-y|}^s}\right]\frac{dxdy}{|x-y|}.\hfill \end{array}$$

It suffices to show that ${\int }_a^b\alpha (x){\left|u(x)\right|}^{p(x)}dx+{\int }_a^b\frac{m(x)}{{\left|u(x)\right|}^{\gamma (x)}}dx$ is $C^1$ with $Frchet$ derivative ${\int }_a^b({\mathrm{grad}}_u{\alpha (x)\left|u(x)\right|}^{p(x)})·v(x)+{\int }_a^b({\mathrm{grad}}_u(\frac{m(x)}{{\left|u(x)\right|}^{\gamma (x)}})·v(x)dx\forall v\in {\mathsf{\Lambda }}_{s,G}D$. Let us set

$$\mathsf{\Psi }(u)={\mathsf{\Psi }}_1(u)+{\mathsf{\Psi }}_2(u),$$

where

$${\mathsf{\Psi }}_1(u)={\int }_a^b\alpha (x){\left|u(x)\right|}^{p(x)}dx,{\mathsf{\Psi }}_2(u)={\int }_a^b\frac{m(x)}{{\left|u(x)\right|}^{\gamma (x)}}dx.$$

We shall prove that $\mathsf{\Psi }(u)$ is continuous. For $v,w\in {\mathsf{\Lambda }}_{s,G}D$, we have

$$\begin{array}{ccc}\hfill & & |{\mathsf{\Psi }}_1(v+w)-{\mathsf{\Psi }}_1(v)|\hfill \\ \hfill & & \le {|\alpha (x)|}_{\infty }|{\int }_a^b[({\mathrm{grad}}_v|v(x)+{\tau }_1{w(x)|}^{p(x)}{)·w(x)+O(\parallel w\parallel }_H)]dx|={O(\parallel w\parallel }_H)\hfill \end{array}$$

(14)

for some $0<{\tau }_1<1$. and

$$\begin{array}{ccc}\hfill & & |{\mathsf{\Psi }}_2(v+w)-{\mathsf{\Psi }}_2(v)|\hfill \\ \hfill & & \le {|m(x)|}_{\infty }|{\int }_a^b[({\mathrm{grad}}_v\frac{1}{|v(x)+{\tau }_2{w(x)|}^{\gamma (x)}})·w(x)+{O(\parallel w\parallel }_H)]dx|={O(\parallel w\parallel }_H)\hfill \end{array}$$

(15)

for some $0<{\tau }_2<1$. Thus we have

$|\mathsf{\Psi }(v+w)-\mathsf{\Psi }(v)|\le |{\mathsf{\Psi }}_1(v+w)-{\mathsf{\Psi }}_1(v)|+|{\mathsf{\Psi }}_2(v+w)-{\mathsf{\Psi }}_2{(v)|=O(\parallel w\parallel }_H).$ Next we shall prove that $\mathsf{\Psi }(u)$ is $C^1$ in ${\mathsf{\Lambda }}_{s,G}D$. Let $v,w\in {\mathsf{\Lambda }}_{s,G}D$. By (14) and (15), we have

$$\begin{array}{c}\hfill |{\mathsf{\Psi }}_1(v+w)-{\mathsf{\Psi }}_1(v)-D{\mathsf{\Psi }}_1(v+{\tau }_{1}w){·w|=0(\parallel w\parallel }_H).\end{array}$$

for some ${\tau }_1>0$ and

$$\begin{array}{c}\hfill |{\mathsf{\Psi }}_1(v+w)-{\mathsf{\Psi }}_2(v)-D{\mathsf{\Psi }}_1(v+{\tau }_{2}w){·w|=0(\parallel w\parallel }_H).\end{array}$$

for some ${\tau }_2>0$. Thus $\mathsf{\Psi }(u)$ is $C^1$. □

Now we shall investigate the boundary behaviour of $J_{s,G}$

Lemma 13.

Let $\{u_k=(u_{k_1},\cdots ,u_{k_n})\}\subset {\mathsf{\Lambda }}_{s,G}D$ and $u_k\rightharpoonup u$ weakly in ${\mathsf{\Lambda }}_{s,G}D$ with $u=(u_1\cdots ,u_n)\in \partial {\mathsf{\Lambda }}_{s,G}D$. Then $J(u_k)\to +\infty$.

Proof. 

Since ${\int }_a^b{\int }_a^b\left[{\mathsf{\Sigma }}_{i=1}^{n}G(\frac{|u_{k_i}(x)-u_{k_i}(y)|}{{|x-y|}^s})\right]\frac{dxdy}{|x-y|}+{\int }_a^b\alpha (x){\left|u_k(x)\right|}^{p(x)}dx>0$, it suffices to show that

$${\int }_a^b\frac{m(x)}{|u_k{(x)|}^{\gamma (x)}}dx\to +\infty .$$

Since $\frac{m(x)}{|u_k{(x)|}^{\gamma (x)}}>0$,

$${\int }_{a_0}^{b_0}\frac{m(x)}{|u_k{(x)|}^{\gamma (x)}}dx\to +\infty .$$

(16)

for some subinterval $(a_0,b_0)$ of $(a,b)$. Since $u\in \partial {\mathsf{\Lambda }}_{s,G}D$, there exists $x_0\in (a,b)$ and a neighbourhood $N_{ϵ}(x_0)=\{x\in (a,b)\left|\right|x-x_0|<ϵ\}$ of $x_0$ such that $u(x_0)\in \partial D$ and for any $x\in N_{ϵ}(x_0)$,

$$\frac{m(x)}{{\left|u(x)\right|}^{\gamma (x)}}\ge \frac{c_1}{|u(x)-u(x_0)|}-c_2$$

for some $c_1>0$ and $c_2>0$. Thus we have

$${\int }_{N_{ϵ}(x_0)}\frac{m(x)}{{\left|u(x)\right|}^{\gamma (x)}}dx\ge {\int }_{N_{ϵ}(x_0)}[\frac{c_1}{|u(x)-u(x_0)|}-c_2]dx$$

(17)

$\forall ϵ>0$. Thus

$$\begin{array}{ccc}\hfill & & |u(x)-u(x_0)|\le \parallel u(x){\parallel }_{W^s{L}_G(S^1,R^n)}\frac{|x-x_0|}{b-a}\hfill \\ \hfill & & \le \frac{|x-x_0|}{b-a}{\left({\int }_a^b{\int }_a^b\left[{\mathsf{\Sigma }}_{i=1}^{n}G(\frac{|u_i(x)-u_i(y)|}{{|x-y|}^s})\right]\frac{dxdy}{|x-y|}]\right)}^{\frac{1}{g_0}}\hfill \\ \hfill & & <ϵ{\left({\int }_a^b{\int }_a^b\left[{\mathsf{\Sigma }}_{i=1}^{n}G(\frac{|u_i(x)-u_i(y)|}{{|x-y|}^s})\right]\frac{dxdy}{|x-y|}]\right)}^{\frac{1}{g_0}}.\hfill \end{array}$$

(18)

From (17) and (18),

$$\begin{array}{ccc}& & {\int }_{N_{ϵ}(x_0)}\frac{m(x)}{{\left|u(x)\right|}^{\gamma (x)}}dx\hfill \\ \hfill & & \ge {\int }_{N_{ϵ}(x_0)}[\frac{c_1}{ϵ{\left({\int }_a^b{\int }_a^b\left[{\mathsf{\Sigma }}_{i=1}^{n}G(\frac{|u_i(x)-u_i(y)|}{{|x-y|}^s})\right]\frac{dxdy}{|x-y|}]\right)}^{\frac{1}{g_0}}}-c_2]dx.\hfill \end{array}$$

Then we have

$${\int }_{N_{ϵ}(x_0)}\frac{m(x)}{{\left|u(x)\right|}^{\gamma (x)}}dx\to \infty .$$

 □

Now, we shall prove that $J_{s,G}(u)$ satisfies ${(P.S.)}_{{\tau }_{s,G}}$ condition for any ${\tau }_{s,G}>{{\tau }_{s,G}}^0$ in ${\mathsf{\Lambda }}_{s,G}D$.

Lemma 14.

Let ${(u_k)}_k$ be a sequence such that $J_{s,G}(u_k)\to {\tau }_{s,G}$ and $D{J}_{s,G}(u_k)\to \theta$ for some ${\tau }_{s,G}>0$. Then if $\parallel u_k{\parallel }_H\to \infty$, then $J_{s,G}(u_k)\le {{\tau }_{s,G}}^0$ in ${\mathsf{\Lambda }}_{s,G}D$ for some constant ${{\tau }_{s,G}}^0>0$.

Proof. 

Let $u_k=(u_{k_1},\cdots ,u_{k_n})\in {\mathsf{\Lambda }}_{s,G}D\subset H$. Assume that $\parallel u_k{\parallel }_H\to \infty$. Since $D{J}_{s,G}(u_k)={(-\Delta )}_g^s{u}_k+{\mathrm{grad}}_u(\alpha (x)|u_k(x){|}^{p(x)})+{\mathrm{grad}}_u(m(x)\frac{1}{|u_k{|}^{\gamma (x)}})\to \theta$, we have, for $u_k\in {\mathsf{\Lambda }}_{s,G}D$,

$$\begin{array}{ccc}\hfill {\parallel u\parallel }_H& \ge & D{J}_{s,G}(u_k)·u_k\hfill \\ \hfill & =& {\int }_a^b{(-\Delta )}_g^s{u}_k·u_k(x)dx+{\int }_a^b({\mathrm{grad}}_u(\alpha (x)|u_k{(x)|}^{p(x)}))·u_k(x)dx\hfill \\ \hfill & & +{\int }_a^b({\mathrm{grad}}_u(m(x)\frac{1}{|u_k{|}^{\gamma (x)}}))·u_k(x)dx.\hfill \end{array}$$

(19)

By (6), we have

$$\begin{array}{ccc}\hfill & & D{J}_{s,G}(u_k)·u_k\hfill \\ \hfill & & ={\int }_a^b{\int }_a^b\left[{\mathsf{\Sigma }}_{i=1}^{n}g(\frac{|u_{k_i}(x)-u_{k_i}(y)|}{{|x-y|}^s})\frac{u_{k_i}(x)-u_{k_i}(y)}{|u_{k_i}(x)-u_{k_i}(y)|}\frac{u_{k_i}(x)-u_{k_i}(y)}{{|x-y|}^s}\right]\frac{dxdy}{|x-y|}\hfill \\ \hfill & & +{\int }_a^b[({\mathrm{grad}}_u(\alpha (x)|u_k{(x)|}^{p(x)}))·u_k(x)+({\mathrm{grad}}_u(m(x)\frac{1}{|u_k{|}^{\gamma (x)}}))·u_k(x)]dx\hfill \\ \hfill & & \ge g_0[{\int }_a^b{\int }_a^b\left[{\mathsf{\Sigma }}_{i=1}^{n}G(\frac{|u_{k_i}(x)-u_{k_i}(y)|}{{|x-y|}^s})\right]\frac{dxdy}{|x-y|}\hfill \\ \hfill & & +{\int }_a^b({\mathrm{grad}}_u(\alpha (x)|u_k{(x)|}^{p(x)}))·u_k(x)dx+{\int }_a^b({\mathrm{grad}}_u(m(x)\frac{1}{|u_k{|}^{\gamma (x)}}))·u_k(x)dx.\hfill \end{array}$$

(20)

for large k. By Lemma 7, (20) becomes that

$$\begin{array}{ccc}\hfill \parallel u_k{\parallel }_H& \ge & D{J}_{s,G}(u_k)·u_k\hfill \\ \hfill & \ge & g_0[{\int }_a^b{\int }_a^b\left[{\mathsf{\Sigma }}_{i=1}^{n}G(\frac{|u_{k_i}(x)-u_{k_i}(y)|}{{|x-y|}^s})\right]\frac{dxdy}{|x-y|}\hfill \\ \hfill & & +{\int }_a^b({\mathrm{grad}}_u(\alpha (x)|u_k{(x)|}^{p(x)}))·u_k(x)dx+{\int }_a^b({\mathrm{grad}}_u(m(x)\frac{1}{|u_k{|}^{\gamma (x)}}))·u_k(x)dx\hfill \\ \hfill & \ge & g_0\parallel u_k{\parallel }_H^{g_0}+{\int }_a^b({\mathrm{grad}}_u(\alpha (x)|u_k{(x)|}^{p(x)}))·u_k(x)dx\hfill \\ \hfill & & +{\int }_a^b({\mathrm{grad}}_u(m(x)\frac{1}{|u_k{|}^{\gamma (x)}}))·u_k(x)dx\hfill \end{array}$$

(21)

and

$$\parallel u_k{\parallel }_H\ge g_0\parallel u_k{\parallel }_H^{g_0}+{\int }_a^b({\mathrm{grad}}_u(\alpha (x)|u_k{(x)|}^{p(x)}))·u_k(x)dx+{\int }_a^b({\mathrm{grad}}_u(m(x)\frac{1}{|u_k{|}^{\gamma (x)}}))·u_k(x)dx.$$

(22)

By (22) and $H\ddot{o}lder$ inequality in ${\mathsf{\Lambda }}_{s,G}D\subset W^s{L}_G(S^1,R^n)$, we have

$$\begin{array}{ccc}\hfill & & g_0\parallel u_k{\parallel }_H^{g_0}-{\parallel u_k\parallel }_H\hfill \\ \hfill & \le & -{\int }_a^b({\mathrm{grad}}_u(\alpha (x)|u_k{(x)|}^{p(x)}))·u_k(x)dx-{\int }_a^b({\mathrm{grad}}_u(m(x)\frac{1}{|u_k{|}^{\gamma (x)}}))·u_k(x)dx\hfill \\ \hfill & \le & 2\parallel {\mathrm{grad}}_u(\alpha (x)|u_k{(x)|}^{p(x)}{)\parallel }_{L_{G^{*}}}\parallel u_k{\parallel }_{L_G}+2\parallel {\mathrm{grad}}_u(m(x)\frac{1}{|u_k{|}^{\gamma (x)}}){\parallel }_{L_{G^{*}}}{\parallel u_k\parallel }_{L_G}\hfill \\ \hfill & \le & 2\parallel {\mathrm{grad}}_u(\alpha (x)|u_k{(x)|}^{p(x)}{)\parallel }_{L_{G^{*}}}\parallel u_k{\parallel }_H+2\parallel {\mathrm{grad}}_u(m(x)\frac{1}{|u_k{|}^{\gamma (x)}}){\parallel }_{L_{G^{*}}}{\parallel u_k\parallel }_H.\hfill \end{array}$$

(23)

From (23),

$$g_0\parallel u_k{\parallel }_H^{g_0-1}\le 1+2\parallel {\mathrm{grad}}_u(\alpha (x)|u_k{(x)|}^{p(x)}{)\parallel }_{L_{G^{*}}}+2{\parallel {grad}_u(m(x)\frac{1}{|u_k{|}^{\gamma (x)}})\parallel }_{L_{G^{*}}}.$$

(24)

$\parallel u_k{\parallel }_H\to \infty$ implies that there exists a constant $R_0>0$ such that $\parallel u_k{\parallel }_H\ge R_0$ for large k. Thus we have

$${\int }_a^b\frac{m(x)}{|u_k{|}^{\gamma (x)}}\le {(b-a)\left|m(x)\right|}_{\infty }\sup \{\frac{1}{|u_k{|}^{\gamma (x)}}|x\in (a,b),u_k\in H\backslash B_{R_0}\}.$$

(25)

By Lemma 9, the embedding

$$W^s{L}_G(S^1,R^n)\hookrightarrow L^{r(x)}(S^1,R^n)$$

is continuous and compact for all $1\le r(x)<g_0^{*}$. Since $p(x)<g_0^{*}$, there exists a positive constant C such that

$$\parallel u_k{\parallel }_{L^{p(x)}}\le C{\parallel u_k\parallel }_H.$$

(26)

By (24)–(26), we have

$$\begin{array}{ccc}\hfill J_{s,G}(u_k)& =& {\int }_a^b{\int }_a^b\left[{\mathsf{\Sigma }}_{i=1}^{n}G(\frac{|u_{k_i}(x)-u_{k_i}(y)|}{{|x-y|}^s})\right]\frac{dxdy}{|x-y|}\hfill \\ \hfill & & +{\int }_a^b\alpha (x){\left|u_k(x)\right|}^{p(x)}dx+{\int }_a^{b}m(x)\frac{1}{|u_k{|}^{\gamma (x)}}dx\hfill \\ \hfill & \le & \parallel u_k{\parallel }^{g^0}+{\left|\alpha (x)\right|}_{\infty }\parallel u_k{(x)\parallel }_{L^{p(x)}(a,b),R^n}^{p(x)}+(b-a){\left|m(x)\right|}_{\infty }\sup \left\{\frac{1}{|u_k{(x)|}^{\gamma (x)}}\right\}\hfill \\ \hfill & \le & \parallel u_k{\parallel }^{g^0}+{C\left|\alpha (x)\right|}_{\infty }\parallel u_k{(x)\parallel }_H^{p(x)}+(b-a){\left|m(x)\right|}_{\infty }\sup \left\{\frac{1}{|u_k{(x)|}^{\gamma (x)}}\right\}\hfill \\ \hfill & \le & [\frac{1}{g_0}(1+2\parallel {\mathrm{grad}}_u(\alpha (x)|u_k{(x)|}^{p(x)}{)\parallel }_{L_{G^{*}}}+2{\parallel {\mathrm{grad}}_u(m(x)\frac{1}{|u_k{|}^{\gamma (x)}}{\parallel }_{L_{G^{*}}})]}^{\frac{g^0}{g_0-1}}\hfill \\ \hfill & & +{C\left|\alpha (x)\right|}_{\infty }([\frac{1}{g_0}(1+2\parallel {\mathrm{grad}}_u(\alpha (x)|u_k{(x)|}^{p(x)}{)\parallel }_{L_{G^{*}}}\hfill \\ \hfill & & +2\parallel {\mathrm{grad}}_u(m(x)\frac{1}{|u_k{|}^{\gamma (x)}}{\parallel }_{L_{G^{*}}}){]}^{\frac{p_{+}}{g_0-1}})+(b-a){\left|m(x)\right|}_{\infty }\sup \{\frac{1}{|u_k{(x)|}^{\gamma (x)}}\}.\hfill \end{array}$$

If we set,

$$\begin{array}{ccc}\hfill {{\tau }_{s,G}}^0& =& [\frac{1}{g_0}(1+2\parallel {\mathrm{grad}}_u(\alpha (x)|u_k{(x)|}^{p(x)}{)\parallel }_{L_{G^{*}}}+2\parallel {\mathrm{grad}}_u(m(x)\frac{1}{|u_k{|}^{\gamma (x)}}){{\parallel }_{L_{G^{*}}}]}^{\frac{g^0}{g_0-1}}\hfill \\ \hfill & & +{C\left|\alpha (x)\right|}_{\infty }([\frac{1}{g_0}(1+2\parallel {\mathrm{grad}}_u(\alpha (x)|u_k{(x)|}^{p(x)}{)\parallel }_{L_{G^{*}}}\hfill \\ \hfill & & +2\parallel {\mathrm{grad}}_u(m(x)\frac{1}{|u_k{|}^{\gamma (x)}}){\parallel }_{L_{G^{*}}}{]}^{\frac{p_{+}}{g_0-1}})+(b-a){\left|m(x)\right|}_{\infty }sup\{\frac{1}{|u_k{(x)|}^{\gamma (x)}}\},\hfill \end{array}$$

we have $J_{s,G}(u_k)\le {{\tau }_{s,G}}^0$. □

Lemma 15.

$J_{s,G}(u)$ satisfies the ${(P.S.)}_{{\tau }_{s,G}}$ condition for any ${\tau }_{s,G}>{{\tau }_{s,G}}^0$ in ${\mathsf{\Lambda }}_{s,G}D$.

Proof. 

Let ${\tau }_{s,G}>0$ and ${(u_k)}_k\subset {\mathsf{\Lambda }}_{s,G}D$ be a sequence such that $J_{s,G}(u_k)\to {\tau }_{s,G}$ and $D{J}_{s,G}(u_k)\to \theta$, $\theta =(0,\cdots ,0)$ in ${\mathsf{\Lambda }}_{s,G}D$. First we shall show that $\left\{u_k\right\}$ is bounded in ${\mathsf{\Lambda }}_{s,G}D$. Suppose that $\parallel u_k{\parallel }_H\to \infty$ as $k\to \infty$. By Lemma 14, $J_{s,G}(u_k)\le {{\tau }_{s,G}}^0$, where ${{\tau }_{s,G}}^0$ is introduced in the proof of Lemma 14. This leads to a contradiction because $J_{s,G}(u_k)\to {\tau }_{s,G}>{{\tau }_{s,G}}^0$. Thus $\left\{u_k\right\}$ is bounded in ${\mathsf{\Lambda }}_{s,G}D$. Up to a subsequence, $\left\{u_k\right\}$ converges weakly to some $u\in {\mathsf{\Lambda }}_{s,G}D$. We claim that $\left\{u_k\right\}$ converges strongly to $u\in {\mathsf{\Lambda }}_{s,G}D$. Since $D{J}_{s,G}(u_k)\to \theta$, we have

$$u_k(x)+{({(-\Delta )}_g^s)}^{-1}({\mathrm{grad}}_u(\alpha (x)|u_k{(x)|}^{p(x)})+{\mathrm{grad}}_u(m(x)\frac{1}{|u_k{|}^{\gamma (x)}}))\to \theta$$

(27)

and we also have

$$D{J}_{s,G}(u_k(x))·u_k(x)={(-\Delta )}_g^s{u}_k(x)·u_k(x)+({\mathrm{grad}}_u(\alpha (x)|u_k{(x)|}^{p(x)}))·u_k(x)$$

$$+({\mathrm{grad}}_u(m(x)\frac{1}{|u_k{|}^{\gamma (x)}}))·u_k(x)⟶0.$$

(28)

The second and third part of the right-hand side of (28) satisfy

$${\int }_a^b[{\mathrm{grad}}_u(\alpha (x)|u_k{(x)|}^{p(x)}·u_k(x)]dx\le p_{+}{\left|\alpha (x)\right|}_{\infty }{\parallel u_k(x)\parallel }_{L^{p(x)}(S^1,R^n)}^{p_{+}}$$

and

$${\int }_a^b({\mathrm{grad}}_u(m(x)\frac{1}{|u_k{|}^{\gamma (x)}})·u_{k}dx\le {\left|m(x)\right|}_{\infty }{\gamma }_{+}{\int }_a^b\frac{1}{|u_k{(x)|}^{\gamma (x)}}dx.$$

Continuous and compact embedding $W^s{L}_G(S^1,R^n)\hookrightarrow L^{p(x)}(S^1,R^n)$ for $1<p(x)<g_0^{*}$ implies that $\parallel u_k{\parallel }_{L^{p(x)}(S^1,R^n)}\le C{\parallel u_k\parallel }_H$ for some constant $C>0$. Thus if $\left\{u_k\right\}$ is bounded in ${\mathsf{\Lambda }}_{s,G}D\subset W^s{L}_G(S^1,R^n)$, then $\left\{u_k\right\}$ is bounded in $L^{p(x)}(S^1,R^n)$. It follows that ${\int }_a^b[{\mathrm{grad}}_u(\alpha (x)|u_k(x){|}^{p(x)}·u_k(x)]dx$ and ${\int }_a^b({\mathrm{grad}}_u(m(x)\frac{1}{|u_k{|}^{\gamma (x)}})·u_{k}dx$ are bounded in $L^{p(x)}(S^1,R^n)$. By (28), ${(-\Delta )}_g^s{u}_k(x)·u_k(x)$ is bounded. Since $\left\{u_k\right\}$ is bounded in ${\mathsf{\Lambda }}_{s,G}D$,

$${(-\Delta )}_g^s{u}_k(x)\mathrm{is}\mathrm{bounded}\mathrm{in}{L}^{p(x)}(S^1,R^n).$$

Lemma 9 leads that $\left\{u_k\right\}$ converges to u strongly in ${\mathsf{\Lambda }}_{s,G}D$. □

4. Proofs of Theorems 1 and 2

Let us set the level set

$${J_{s,G}}_{{\tau }_{s,G}}=\{u\in {\mathsf{\Lambda }}_{s,G}D|J_{s,G}(u)\le {\tau }_{s,G}\}.$$

Lemma 16.

$H_{b_i}({\mathsf{\Lambda }}_{s,G}D)\ne 0$ for some strictly increasing sequence $b_i$.

Proof. 

Let ${\overline{C}}_{ϵ}(\theta )$ be a neighbourhood $\theta$ with radius $ϵ>0$ and choose $R>0$ such that ${\overline{C}}_{ϵ}(\theta )\subset \mathrm{int}(B_R(\theta ))$. Then we have

$$R^n\backslash B_R(\theta )\subset (R^n\backslash \left\{\theta \right\})$$

and $R^n\backslash B_R(\theta )$ is a deformation retract of $R^n\backslash \left\{\theta \right\}$. Then ${\mathsf{\Lambda }}_{s,G}(R^n\backslash B_R(\theta ))$ is a deformation retract of ${\mathsf{\Lambda }}_{s,G}D$. Thus we have

$$H_{*}({\mathsf{\Lambda }}_{s,G}D)\cong H_{*}({\mathsf{\Lambda }}_{s,G}(R^n\backslash B_R(\theta )))\oplus H_{*}({\mathsf{\Lambda }}_{s,G}D,{\mathsf{\Lambda }}_{s,G}(R^n\backslash B_R(\theta )))$$

$$\cong H_{*}({\mathsf{\Lambda }}_{s,G}(S^{n-1})\oplus H_{*}({\mathsf{\Lambda }}_{s,G}D,{\mathsf{\Lambda }}_{s,G}(S^{n-1})).$$

In [111], the $Poincar$ series of ${\mathsf{\Lambda }}_{s,G}(S^{n-1})$ is written as

$$P_t({\mathsf{\Lambda }}_{s,G}(S^{n-1}))=(1+t^n)+\frac{t^{n-1}}{1-t^{2(n-1)}}(1+t^n)(1+t^{n-1}).$$

 □

Lemma 17.

The level set ${J_{s,G}}_{{\tau }_{s,G}}$ is deformed into ${\mathsf{\Sigma }}_{s,G}$ for some finite dimensional singular complex ${\mathsf{\Sigma }}_{s,G}={\mathsf{\Sigma }}_{{\tau }_{s,G}}$.

Proof. 

Let $u\in {J_{s,G}}_{{\tau }_{s,G}}$. Then $u=(u_1,\cdots ,u_n)\in {\mathsf{\Lambda }}_{s,G}D$ and

$$J_{s,G}(u)={\int }_a^b{\int }_a^b\left[{\mathsf{\Sigma }}_{i=1}^{n}G(\frac{|u_i(x)-u_i(y)|}{{|x-y|}^s})\right]\frac{dxdy}{|x-y|}$$

$$+{\int }_a^b\alpha (x){\left|u(x)\right|}^{p(x)}dx+{\int }_a^{b}m(x)\frac{1}{{\left|u\right|}^{\gamma (x)}}dx\le {\tau }_{s,G}.$$

(29)

If $(x,u(x))\in [a,b]\times (R^n\backslash B_{R_0}(\theta ))$, then

$$0<\frac{m(x)}{{\left|u(x)\right|}^{\gamma (x)}}<+\infty ,\mathrm{and}\left|{\mathrm{grad}}_u(\frac{m(x)}{{\left|u(x)\right|}^{\gamma (x)}})\right|<+\infty$$

(30)

for some constant $R_0>0$. We also note that if $u(x)\in B_{R_0}\backslash {\overline{C}}_{ϵ}(\theta )$, then

$$(b-a)inf\left\{\alpha (x)\right\}{ϵ}^{p_{-}}+(b-a)\frac{inf\{m(x)\}}{R_0^{{\gamma }_{+}}}<{\int }_a^b\alpha (x){\left|u(x)\right|}^{p(x)}dx+{\int }_a^b\frac{m(x)}{{\left|u(x)\right|}^{\gamma (x)}}dx<{\tau }_{s,G}.$$

(31)

(29)–(31) imply that

$${\int }_a^b{\int }_a^b[{\mathsf{\Sigma }}_{i=1}^{n}G(\frac{|u_i(x)-u_i(y)|}{{|x-y|}^s}]\frac{dxdy}{|x-y|}<{{\tau }_{s,G}}^1$$

(32)

for some constant ${{\tau }_{s,G}}^1>0$. By Lemma 13, there exists ${{ϵ}_{s,G}}^0={ϵ}_{s,G}({\tau }_{s,G},{{\tau }_{s,G}}^1)$ such that

$$d(u,{\overline{C}}_{{ϵ}_{s,G}}(\theta ))\ge {{ϵ}_{s,G}}^0\forall u\in {J_{s,G}}_{{\tau }_{s,G}},\forall x\in [a,b].$$

Let us choose an integer $N_{s,G}=N_{{{\tau }_{s,G}}^1}>\frac{{({{\tau }_{s,G}}^1)}^{\frac{1}{g_0}}}{{{ϵ}_{s,G}}^0}$ and set

$${x_{s,G}}^i=\frac{(b-a)i}{N_{s,G}}i=1,2,\cdots ,N_{s,G}$$

and

$${\overline{u}}_{s,G}(x)=(1-\frac{1}{b-a}{N}_{s,G}(x-{x_{s,G}}^{i-1}))u({x_{s,G}}^{i-1})+\frac{1}{b-a}{N}_{s,G}(x-{x_{s.G}}^{i-1})u({x_{s,G}}^i)$$

$\forall x\in [{x_{s,G}}^{i-1},{x_{s,G}}^i],i=1,2,\dots ,N_{s,G},$ $\forall u\in {J_{s,G}}_{{\tau }_{s,G}}.$ Let

$${\mathsf{\Sigma }}_{s,G}=\left\{{\overline{u}}_{s,G}(x)\right|u\in {J_{s,G}}_{{\tau }_{s,G}}\}.$$

Mapping $D\times D\times \cdots {\times }_{N_{s,G}}D\ni {\overline{u}}_{s,G}\mapsto (u({x_{s,G}}^1),u({x_{s,G}}^2),\dots ,u({x_{s,G}}^{N_{s,G}}))$ is a homeomorphism. We shall show that ${\mathsf{\Sigma }}_{s,G}\subset {\mathsf{\Lambda }}_{s,G}D$. In fact, for $\forall u\in {J_{s,G}}_{{\tau }_{s,G}}$ and ${x_{s,G}}^i>{x_{s,G}}^{i-1}$, by (32), we have

$$\begin{array}{ccc}& & |u({x_{s,G}}^i)-u({x_{s,G}}^{i-1})|\le \parallel {\overline{u}}_{s,G}(x){\parallel }_{W^s{L}_G(S^1,R^n)}\frac{|{x_{s,G}}^i-{x_{s,G}}^{i-1}|}{b-a}\hfill \\ \hfill & & \le \frac{|{x_{s,G}}^i-{x_{s,G}}^{i-1}|}{b-a}{\left({\int }_a^b{\int }_a^b\left[{\mathsf{\Sigma }}_{i=1}^{n}G(\frac{|u_i(x)-u_i(y)|}{{|x-y|}^s})\right]\frac{dxdy}{|x-y|}]\right)}^{\frac{1}{g_0}}\hfill \\ \hfill & & <\frac{{ϵ}_{s,G}}{b-a}{({{\tau }_{s,G}}^1)}^{\frac{1}{g_0}}.\hfill \end{array}$$

Thus

$$\begin{array}{ccc}\hfill d({\overline{u}}_{s,G}(x),{\overline{C}}_{{ϵ}_{s,G}}(\theta ))& \ge & d(u({x_{s,G}}^i),{\overline{C}}_{{ϵ}_{s,G}}(\theta ))\hfill \\ \hfill & & -\left(1-\frac{1}{b-a}{N}_{s,G}(x-{x_{s,G}}^{i-1})\right)|u({x_{s,G}}^i)-u({x_{s,G}}^{i-1})|\hfill \\ \hfill & \ge & {{ϵ}_{s,G}}^0-{(N_{s,G})}^{-1}{({{\tau }_{s,G}}^1)}^{\frac{1}{g_0}}>0\hfill \end{array}$$

$\forall x\in [{x_{s,G}}^{i-1},{x_{s,G}}^i]$. We shall show that there exists ${\eta }_{s,G}\in C([0,1]\times {J_{s,G}}_{{\tau }_{s,G}},{\mathsf{\Lambda }}_{s,G}D)$ such that ${\eta }_{s,G}(0,·)=\mathrm{id}$, and ${\eta }_{s,G}(1,{J_{s,G}}_{{\tau }_{s,G}})={\mathsf{\Sigma }}_{s,G}$. In fact, Let us choose $u(x)\in {\mathsf{\Lambda }}_{s,G}D$ and let us define ${\eta }_{s,G}$ as follows:

$${\eta }_{s,G}(y,u)(x)=\left\{\begin{array}{c}u(x)\mathrm{for}x\ge (b-a)y,\hfill \\ (1-\frac{x-{x_{s,G}}^{i-1}}{(b-a)y-{x_{s,G}}^{i-1}})u({x_{s,G}}^{i-1})+\frac{x-{x_{s,G}}^{i-1}}{(b-a)y-{x_{s,G}}^{i-1}}u((b-a)y)\hfill \\ \mathrm{for}{x_{s,G}}^{i-1}<x<(b-a)y,\hfill \\ {\overline{u}}_{s,G}(x)\mathrm{for}x\le {x_{s,G}}^{i-1}.\hfill \end{array}\right.$$

Then ${\eta }_{s,G}(0,·)=\mathrm{id}$, and ${\eta }_{s,G}(1,{J_{s,G}}_{{\tau }_{s,G}})={\mathsf{\Sigma }}_{s,G}$. Thus we prove that ${J_{s,G}}_{{\tau }_{s,G}}$ is deformed into ${\mathsf{\Sigma }}_{s,G}$ in the loop space ${\mathsf{\Lambda }}_{s,G}D$. □

Proof of Theorem 1.

We claim that for each $q>n{N}_{s,G}$, where $N_{s,G}={N_{s,G}}_{{{\tau }_{s,G}}^0}$ is defined in Lemma 17, let us set

$${\tau }_{s,G}=\underset{z\in \beta }{inf}\underset{u\in \left[z\right]}{max}{J}_{s,G}(u),$$

where $\beta \in H_q({\mathsf{\Lambda }}_{s,G}D)$ is nontrivial. We claim that ${\tau }_{s,G}>{{\tau }_{s,G}}^0$ and ${\tau }_{s,G}$ is a critical value of $J_{s,G}$. In fact, suppose that ${\tau }_{s,G}\le {{\tau }_{s,G}}^0$ and ${\tau }_{s,G}$ is not a critical value of $J_{s,G}$. Then $\left[u\right]\in {J_{s,G}}_{{{\tau }_{s,G}}^0+1}$ for some $\left[u\right]\in \beta$ . By Lemma 17, there exists a deformation ${\eta }_{s,G}:[0,1]\times {J_{s,G}}_{{{\tau }_{s,G}}^0+1}\to {\mathsf{\Lambda }}_{s,G}D$ such that ${\eta }_{s,G}(1,{J_{s,G}}_{{{\tau }_{s,G}}^0+1})\subset {{\mathsf{\Sigma }}_{s,G}}_{{{\tau }_{s,G}}^0+1}$ with $dim{{\mathsf{\Sigma }}_{s,G}}_{{{\tau }_{s,G}}^0+1}\le n{N_{s,G}}_{{{\tau }_{s,G}}^0}.$ This implies that ${\eta }_{s,G}(1,\left[u\right])\subset {{\mathsf{\Sigma }}_{s,G}}_{{{\tau }_{s,G}}^0+1}$. However, ${\eta }_{s,G}(1,\left[u\right])\in \beta$ and $\beta \in H_q({\mathsf{\Lambda }}_{s,G}D)$ with $q>n{N_{s,G}}_{{{\tau }_{s,G}}^0}$, which is absurd. □

Proof of Theorem 2.

We suppose that $J_{s,G}(u)$ has only finitely many critical points $u_1$, $u_2$, …, $u_{m_{s,G}}$ such that we can obtain $J_{s,G}(u_k)>{{\tau }_{s,G}}^0$, $1\le k\le m_{s,G}$. Let us set

$$K=\{u_1,u_2,\dots ,u_{m_{s,G}}\}.$$

We note that $dim\ker (D^2{J}_{s,G}(u_k))\le 2n$, for all k. Letting

$$b_{s,G}^{*}>\max \{n{N_{s,G}}_{{{\tau }_{s,G}}^0},\mathrm{ind}(J_{s,G},u_k)+\dim \ker (D^2{J}_{s,G}(u_k))|1\le k\le m_{s,G}\}.$$

and

$$d_{s,G}>\max \{{{\tau }_{s,G}}^0,J_{s,G}(u_k)|1\le k\le m_s,G\},$$

we have

$$C_b(J_{s,G},u_k)=0\forall b_{s,G}\ge b_{s,G}^{*},k=1,2,\dots ,m_{s,G}$$

and

$$H_{*}({\mathsf{\Lambda }}_{s,G}D,{J_{s,G}}_{{{\tau }_{s,G}}^0})=H_{*}({J_{s,G}}_{d_{s,G}},{J_{s,G}}_{{{\tau }_{s,G}}^0}).$$

Hence

$$H_b({\mathsf{\Lambda }}_{s,G}D,{J_{s,G}}_{{{\tau }_{s,G}}^0})=0\forall b_{s,G}>b_{s,G}^{*}.$$

Since

$$i_{*}:H_b({\mathsf{\Lambda }}_{s,G}D)⟶H_b({\mathsf{\Lambda }}_{s,G}D,{J_{s,G}}_{{{\tau }_{s,G}}^0})\mathrm{is}\mathrm{injective}\mathrm{for}{b}_{s,G}\ge b_{s,G}^{*}$$

we have

$$H_b({\mathsf{\Lambda }}_{s,G}D)=0\mathrm{for}{b}_{s,G}\ge b_{s,G}^{*},$$

which is absurd. □