Singular potential Hamiltonian system

Abstract

We investigate multiple solutions for the Hamiltonian system with singular potential nonlinearity and periodic condition. We get a theorem which shows the existence of the nontrivial weak periodic solution for the Hamiltonian system with singular potential nonlinearity. We obtain this result by using the variational method, critical point theory for indefinite functional.

MSC:35Q70.

1 Introduction

Let D be an open subset in $R^{2n}$ with a compact complement $C=R^{2n}\mathrm{\setminus }D$, $n\ge 2$. Let $p\in R^n$, $q\in R^n$, $(p(t),q(t))\in C^2([0,2\pi ],D)$ and $G(t,(p(t),q(t)))\in C^2([0,2\pi ]\times D,R^1)$. In this paper we investigate the number of solutions $(p(t),q(t))\in C^2([0,2\pi ],D)$ with singular potential nonlinearity and periodic condition

$$\begin{array}{r}\dot{p(t)}=-G_q(t,p(t),q(t)),\\ \dot{q(t)}=G_p(t,p(t),q(t)),\\ p(0)=p(2\pi ),q(0)=q(2\pi ).\end{array}$$

(1.1)

Let us set $z=(p,q)$ and

$$J=\left(\begin{array}{cc}0& -I_n\\ I_n& 0\end{array}\right).$$

Then (1.1) can be rewritten as

$$\begin{array}{r}-J\dot{z}=G_z(t,z(t)),\\ z(0)=z(2\pi ),\end{array}$$

(1.2)

where $\dot{z}=\frac{dz}{dt}$. We assume that $G(t,z(t))\in C^2([0,2\pi ]\times D,R^1)$ satisfies the following conditions:

(G1) There exists $R_0>0$ such that

$$sup\{\left|G(t,z(t))\right|+{\parallel {grad}_{z}G(t,z(t))\parallel }_{R^{2n}}\mid (t,z)\in [0,2\pi ]\times \left(R^{2n}\mathrm{\setminus }{B}_{R_0}\right)\}<+\mathrm{\infty }.$$

(G2) There is a neighborhood Z of C in $R^{2n}$ such that

$$G(t,z)\ge \frac{A}{d^2(z,C)}\text{for }(t,z)\in [0,2\pi ]\times Z,$$

where $d(z,C)$ is the distance function to C and $A>0$ is a constant.

Several authors ([1–4], etc.) studied the Hamiltonian system with nonsingular potential nonlinearity. Jung and Choi [2, 3] considered (1.1) with nonsingular potential nonlinearity or jumping nonlinearity crossing one eigenvalue, or two eigenvalues, or several eigenvalues. Chang [1] proved that (1.1) has at least two nontrivial 2π-periodic weak solutions under some asymptotic nonlinearity. Jung and Choi [2] proved that (1.1) has at least m weak solutions, which are geometrically distinct and nonconstant under some jumping nonlinearity.

In this paper we are trying to find the weak solutions $z\in C^2([0,2\pi ],D)$ of system (1.2) such that

$${\int }_0^{2\pi }\dot{z}\cdot w-J{G}_z(t,z(t))\cdot Jwdt=0\text{for all }w\in E,$$

i.e.,

$${\int }_0^{2\pi }[(\dot{p}+G_q(t,z(t)))\cdot \psi -(\dot{q}-G_p(t,z(t)))\cdot \varphi ]dt=0\text{for all }\zeta =(\varphi ,\psi )\in E,$$

where E is introduced in Section 2.

Our main result is as follows.

Theorem 1.1 Assume that G satisfies conditions (G1)-(G2). Then system (1.1) has at least one 2π-periodic solution.

For the proof of Theorem 1.1, we introduce the perturbed operator $A_{ϵ}$, such that $A_{ϵ}^{-1}$ is a compact operator and the associated functional $I(z)$ corresponding to the operator $A_{ϵ}$, and approach the variational method, the critical point theory. In Section 2, we investigate the Fréchet differentiability of the associated functional $I(z)$ and recall the critical point theorem for indefinite functional. In Section 3, we show that the associated functional $I(z)$ satisfies the geometrical assumptions of the critical point theorem for indefinite functional and prove Theorem 1.1.

2 Variational method

Let $L^2([0,2\pi ],R^{2n})$ denote the set of 2n-tuples of the square integrable 2π-periodic functions and choose $z\in L^2([0,2\pi ],R^{2n})$. Then it has a Fourier expansion $z(t)={\sum }_{k=-\mathrm{\infty }}^{k=+\mathrm{\infty }}{a}_k{e}^{ikt}$, with $a_k=\frac{1}{2\pi }{\int }_0^{2\pi }z(t)e^{-ikt}dt\in C^{2n}$, $a_{-k}=\overline{a_k}$ and ${\sum }_{k\in Z}{|a_k|}^2<\mathrm{\infty }$. Let

$$A:z(t)\mapsto -J\dot{z}(t)$$

with the domain

$$\begin{array}{rcl}D(A)& =& \{z(t)\in H^1([0,2\pi ],R^{2n})\mid z(0)=z(2\pi )\}\\ =& \{z(t)\in L^2([0,2\pi ],R^{2n})|\sum _{k\in Z}{(ϵ+|k|)}^2{|a_k|}^2<+\mathrm{\infty }\},\end{array}$$

where ϵ is a positive small number. Then A is a self-adjoint operator. Let $\{M_{\lambda }\}$ be the spectral resolution of A, and let α be a positive number such that $\alpha \notin \sigma (A)$ and $[-\alpha ,\alpha ]$ contains only one element 0 of σA. Let

$$P_0={\int }_{-\alpha }^{\alpha }d{M}_{\lambda },P_{+}={\int }_{\alpha }^{+\mathrm{\infty }}d{M}_{\lambda },P_{-}={\int }_{-\mathrm{\infty }}^{-\alpha }d{M}_{\lambda }.$$

Let

$$L_0=P_0{L}^2([0,2\pi ],R^{2n}),L_{+}=P_{+}{L}^2([0,2\pi ],R^{2n}),L_{-}=P_{-}{L}^2([0,2\pi ],R^{2n}).$$

For each $u\in L^2([0,2\pi ],R^{2n})$, we have the composition

$$u=u_0+u_{+}+u_{-},$$

where $u_0\in L_0$, $u_{+}\in L_{+}$, $u_{-}\in L_{-}$. According to A, there exists a small number $ϵ>0$ such that $-ϵ\notin \sigma (A)$. Let us define the space E as follows:

$$E=D\left({|A|}^{\frac{1}{2}}\right)=\{z\in L^2([0,2\pi ],R^{2n})|\sum _{k\in Z}(ϵ+|k|){|a_k|}^2<\mathrm{\infty }\}$$

with the scalar product

$${(z,w)}_E=ϵ{(z,w)}_{L^2}+{({|A|}^{\frac{1}{2}}z,{|A|}^{\frac{1}{2}}w)}_{L^2}$$

and the norm

$$\parallel z\parallel ={(z,z)}_E^{\frac{1}{2}}={\left(\sum _{k\in Z}(ϵ+|k|){|a_k|}^2\right)}^{\frac{1}{2}}.$$

The space E endowed with this norm is a real Hilbert space continuously embedded in $L^2([0,2\pi ],R^{2n})$. The scalar product in $L^2$ naturally extends as the duality pairing between E and $E^{\prime }=W^{-\frac{1}{2},2}([0,2\pi ],R^{2n})$. We note that the operator ${(ϵ+|A|)}^{-1}$ is a compact linear operator from $L^2([0,2\pi ],R^{2n})$ to E such that

$${({(ϵ+|A|)}^{-1}w,z)}_E={\int }_0^{2\pi }(w(t),z(t))dt.$$

Let

$$A_{ϵ}=ϵI+A.$$

Let

$$E_0={|A_{ϵ}|}^{-\frac{1}{2}}{L}_0,E_{+}={|A_{ϵ}|}^{-\frac{1}{2}}{L}_{+},E_{-}={|A_{ϵ}|}^{-\frac{1}{2}}{L}_{-}.$$

Then $E=E_0\oplus E_{+}\oplus E_{-}$ and for $z\in E$, z has the decomposition $z=z_0+z_{+}+z_{-}\in E$, where

$$z_0={|A_{ϵ}|}^{-\frac{1}{2}}{u}_0,z_{+}={|A_{ϵ}|}^{-\frac{1}{2}}{u}_{+},z_{-}={|A_{ϵ}|}^{-\frac{1}{2}}{u}_{-}.$$

Thus we have

$${\parallel z_0\parallel }_{E_0}={\parallel u_0\parallel }_{L_0},{\parallel z_{+}\parallel }_{E_{+}}={\parallel u_{+}\parallel }_{L_{+}},{\parallel z_{-}\parallel }_{E_{-}}={\parallel u_{-}\parallel }_{L_{-}}$$

and that $E_0$, $E_{+}$, $E_{-}$ are isomorphic to $L_0$, $L_{+}$, $L_{-}$, respectively.

For the sake of simplicity, from now on we shall denote the subset of $C^2([0,2\pi ],D)$, satisfying the 2π-periodic condition, by $C^2(S^1,D)$.

Let us introduce an open set of the Hilbert space E as follows:

$$X=\{z\in E\mid z(t)\in D\subset R^{2n},t\in S^1\}.$$

Let

$$X_0=E_0\cap X,X_{+}=E_{+}\cap X,X_{-}=E_{-}\cap X.$$

Then $X=X_0\oplus X_{+}\oplus X_{-}$ and for $z\in X$, z has the decomposition $z=z_0+z_{+}+z_{-}\in X$, where

$$z_0\in X_0,z_{+}\in X_{+},z_{-}\in X_{-}.$$

The associated functional of (1.2) on X is as follows:

$$I(z)=\frac{1}{2}({\parallel A_{ϵ}^{\frac{1}{2}}{z}_{+}\parallel }_{L^2}^2+{\parallel A_{ϵ}^{\frac{1}{2}}{M}_{+}{z}_0\parallel }^2-{\parallel {(-A_{ϵ})}^{\frac{1}{2}}{z}_{-}\parallel }_{L^2}^2-{\parallel {(-A_{ϵ})}^{\frac{1}{2}}{M}_{-}{z}_0\parallel }^2)-{\psi }_{ϵ}(z),$$

(2.1)

where ${\psi }_{ϵ}(z)=\psi (z)+\frac{ϵ}{2}{\parallel z\parallel }_{L^2}^2$, $\psi (z)={\int }_0^{2\pi }G(t,z(t))dt$. Let

$$F(z)=G_z(t,z(t)).$$

By $G\in C^2$, $\psi (z)={\int }_0^{2\pi }G(t,z(t))\in C^2(S^1\times D,R^1)$. Let

$$F_{ϵ}(z)=ϵI+F(z)=ϵI+G_z(t,z(t)).$$

System (1.2) can be rewritten as

$$A_{ϵ}(z)=F_{ϵ}(z).$$

(2.2)

The Euler equation of the functional $I(z)$ is the system

$$u_{+}={|A_{ϵ}|}^{-\frac{1}{2}}{P}_{+}{F}_{ϵ}(z),$$

(2.3)

$$u_{-}=-{|A_{ϵ}|}^{-\frac{1}{2}}{P}_{-}{F}_{ϵ}(z),$$

(2.4)

$$M_{+}{u}_0={|A_{ϵ}|}^{-\frac{1}{2}}{M}_{+}{P}_0{F}_{ϵ}(z),M_{-}{u}_0=-{|A_{ϵ}|}^{-\frac{1}{2}}{M}_{-}{P}_0{F}_{ϵ}(z).$$

(2.5)

Thus $z=z_0+z_{+}+z_{-}$ is a solution of (2.2) if and only if $u=u_0+u_{+}+u_{-}$ is a critical point of I. System (2.3)-(2.5) is reduced to

$$A_{ϵ}{z}_{+}=P_{+}{F}_{ϵ}(z_0+z_{+}+z_{-})\text{or}{z}_{+}={(A_{ϵ})}^{-1}{P}_{+}{F}_{ϵ}(z_0+z_{+}+z_{-}),$$

(2.6)

$$A_{ϵ}{z}_{-}=P_{-}{F}_{ϵ}(z_0+z_{+}+z_{-})\text{or}{z}_{-}={(A_{ϵ})}^{-1}{P}_{-}{F}_{ϵ}(z_0+z_{+}+z_{-}),$$

(2.7)

$$A_{ϵ}{M}_{+}{z}_0=M_{+}{P}_0{F}_{ϵ}(z_0+z_{+}+z_{-}),A_{ϵ}{M}_{-}{z}_0=M_{-}{P}_0{F}_{ϵ}(z_0+z_{+}+z_{-}).$$

(2.8)

By the condition $F(z)\in C^2(E,E)$ and (G2),

$${\parallel F_{ϵ}(z)-F_{ϵ}(w)\parallel }_{L^2}\le (ϵ+\beta ){\parallel u-v\parallel }_H\mathrm{\forall }u,v\in E.$$

(2.9)

By (G3) and (G4), there exists $\gamma >min\{j_1-\alpha +ϵ,\beta -j_1+ϵ\}$ such that

$$\parallel A_{ϵ}^{-1}{|}_{L_{+}\oplus L_{-}}\parallel \le \frac{1}{\gamma }.$$

By the following lemma, the weak solutions of (1.2) coincide with the critical points of the functional $I(z)$.

Lemma 2.1 Assume that G satisfies conditions (G1)-(G2). Then $I(z)$ is continuous and Fréchet differentiable in X with the Fréchet derivative

$$DI(z)w={\int }_0^{2\pi }(A_{ϵ}z-F_{ϵ}(z))\cdot wdt\mathit{\text{for all}}w\in X.$$

Moreover, $DI\in C$. That is, $I\in C^1$.

Proof First we prove that $I(z)={\int }_0^{2\pi }[\frac{1}{2}{A}_{ϵ}z-G(t,z(t))-\frac{ϵ}{2}{z}^2]dt$ is continuous and Fréchet differentiable in X. For $z,w\in X$,

$$\begin{array}{c}|I(z+w)-I(z)|\hfill \\ =|{\int }_0^{2\pi }\frac{1}{2}{A}_{ϵ}(z+w)\cdot (z+w)-{\int }_0^{2\pi }[G(t,z+w)+\frac{ϵ}{2}{(z+w)}^2]\hfill \\ -{\int }_0^{2\pi }\frac{1}{2}{A}_{ϵ}(z)\cdot z+{\int }_0^{2\pi }[G(t,z)+\frac{ϵ}{2}{z}^2]|\hfill \\ =|{\int }_0^{2\pi }\frac{1}{2}[A_{ϵ}(z)\cdot w+A_{ϵ}(w)\cdot z+A_{ϵ}(w)\cdot w]\hfill \\ -{\int }_0^{2\pi }[G(t,z+w)-G(t,z)+\frac{ϵ}{2}(2z\cdot w+w^2)]|.\hfill \end{array}$$

We have

$$\begin{array}{c}\left|{\int }_0^{2\pi }[G(t,z+w)-G(t,z)]dt\right|\hfill \\ \le \left|{\int }_0^{2\pi }[G_z(t,z(t))\cdot w+O\left({\parallel w\parallel }_{R^{2n}}\right)]dt\right|=O\left({\parallel w\parallel }_{R^{2n}}\right).\hfill \end{array}$$

(2.10)

Thus we have

$$|I(z+w)-I(z)|=O\left({\parallel w\parallel }_{R^{2n}}\right).$$

Next we shall prove that $I(z)$ is Fréchet differentiable in X. For $z,w\in X$,

$$\begin{array}{c}|I(z+w)-I(z)-DI(z)w|\hfill \\ =|{\int }_0^{2\pi }\frac{1}{2}{A}_{ϵ}(z+w)\cdot (z+w)-{\int }_0^{2\pi }[G(t,z+w)+\frac{ϵ}{2}{(z+w)}^2]\hfill \\ -{\int }_0^{2\pi }\frac{1}{2}{A}_{ϵ}(z)\cdot z+{\int }_0^{2\pi }[G(t,z)+\frac{ϵ}{2}{z}^2]\hfill \\ -{\int }_0^{2\pi }{A}_{ϵ}(z)\cdot w+{\int }_0^{2\pi }[G_z(t,z)+ϵz\cdot w]|\hfill \\ =|{\int }_0^{2\pi }\frac{1}{2}[A_{ϵ}(w)\cdot z+A_{ϵ}(w)\cdot w]-{\int }_0^{2\pi }[G(t,z+w)-G(t,z)-G_z(t,z)+\frac{ϵ}{2}{w}^2]|.\hfill \end{array}$$

By (2.10), we have

$${\int }_0^{2\pi }[G(t,z+w)-G(t,z)-G_z(t,z)]dt=O\left({\parallel w\parallel }_{R^{2n}}\right).$$

Thus

$$|I(z+w)-I(z)-DI(z)w|=O\left({\parallel w\parallel }_{R^{2n}}\right).$$

 □

Lemma 2.2 Assume that G satisfies conditions (G1)-(G2). Let $\{z_k\}\subset X_{-}$ and $z_k\rightharpoonup z$ weakly in X with $z\in \partial X$. Then $I(z_k)\to -\mathrm{\infty }$.

Proof To prove the conclusion, it suffices to prove that

$${\int }_0^{2\pi }G(t,z_k(t))dt⟶+\mathrm{\infty }.$$

Since $G(t,z(t))$ is bounded from below, it suffices to prove that there is an interval $[a,b]\subset [0,2\pi ]$ such that

$${\int }_a^{b}G(t,z_k(t))dt⟶+\mathrm{\infty }.$$

By definition, $z\in \partial X$ means that there exists $t^{\ast }\in [0,2\pi ]$ such that $z(t^{\ast })\in \partial D$. By G(2), there exists a constant $B>0$ such that

$$G(x,t,U)\ge \frac{A}{d^2(z,C)}-B.$$

Thus we have

$${\int }_{t^{\ast }}^{t^{\ast }+\delta }G(t,z(t))dt\ge {\int }_{t^{\ast }}^{t^{\ast }+\delta }(\frac{A}{{\parallel z(t)-z(t^{\ast })\parallel }_{R^{2n}}^2}-B)dt$$

for all $\delta >0$. By Schwarz’s inequality, we have

$${\parallel z(t)-z_k(t)\parallel }_{R^{2n}}\le {|t-t^{\ast }|}^{\frac{1}{2}}{\left({\int }_0^{2\pi }{\parallel \dot{z}(t)\parallel }_{R^{2n}}^{2}dt\right)}^{\frac{1}{2}}.$$

Thus we have

$${\int }_{t^{\ast }}^{t^{\ast }+\delta }G(t,z(t))dt=+\mathrm{\infty }.$$

Since the embedding $H^1(S^1,R^{2n})\hookrightarrow C(S^1,R^{2n})$ is compact, we have

$$max\{{\parallel z(t)-z_k(t)\parallel }_{R^{2n}}^2\mid t\in S^1\}⟶0\text{as }k\to \mathrm{\infty }.$$

Thus by Fatou’s lemma, we have

$$\begin{array}{rcl}lim\hspace{0.17em}inf{\int }_{t^{\ast }}^{t^{\ast }+\delta }G(t,z_k(t))dt& \ge & {\int }_{t^{\ast }}^{t^{\ast }+\delta }lim\hspace{0.17em}infG(t,z_k(t))\\ =& {\int }_{t^{\ast }}^{t^{\ast }+\delta }G(t,z(t))dt=+\mathrm{\infty }.\end{array}$$

Thus

$$lim\hspace{0.17em}inf{\int }_{t^{\ast }}^{t^{\ast }+\delta }G(t,z_k(t))=+\mathrm{\infty }.$$

Thus if $\{z_k\}\subset X_{-}$,

$$\begin{array}{rcl}I(z_k)& =& {\int }_0^{2\pi }[\frac{1}{2}{A}_{ϵ}{z}_k(t)-{\psi }_{ϵ}(z_k)]dxdt\\ =& \frac{1}{2}({\parallel A_{ϵ}^{\frac{1}{2}}{(z_k)}_{+}\parallel }_{L^2}^2+{\parallel A_{ϵ}^{\frac{1}{2}}{M}_{+}{(z_k)}_0\parallel }^2-{\parallel {(-A_{ϵ})}^{\frac{1}{2}}{(z_k)}_{-}\parallel }_{L^2}^2\\ -{\parallel {(-A_{ϵ})}^{\frac{1}{2}}{M}_{-}{(z_k)}_0\parallel }^2)-{\psi }_{ϵ}(z_k)\\ \le & -{\parallel {(-A_{ϵ})}^{\frac{1}{2}}{(z_k)}_{-}\parallel }_{L^2}^2-{\psi }_{ϵ}(z_k)⟶-\mathrm{\infty },\end{array}$$

so we prove the lemma. □

Now, we recall the critical point theorem for indefinite functional (cf. [5]).

Let

$$\begin{array}{c}{B}_r=\{u\in X\mid \parallel u\parallel \le r\},\hfill \\ S_r=\{u\in X\mid \parallel u\parallel =r\}.\hfill \end{array}$$

Theorem 2.1 (Critical point theorem for indefinite functional)

Let X be a real Hilbert space with $X=X_1\oplus X_2$ and $X_2=X_1^{\perp }$. Suppose that $I\in C^1(X,R)$ satisfies (PS) and

• (I1) $I(u)=\frac{1}{2}(Lu,u)+bu$, where $Lu=L_1{P}_{1}u+L_2{P}_{2}u$ and $L_i:X_i\to X_i$ is bounded and self-adjoint, $i=1,2$,

• (I2) $b^{\prime }$ is compact, and

• (I3) there exists a subspace $\tilde{X}\subset X$ and sets $S\subset X$, $Q\subset \tilde{X}$ and constants $\alpha >\omega$ such that

  • (i) $S\subset X_1$ and $I{|}_S\ge \alpha$,

  • (ii) Q is bounded and $I{|}_{\partial Q}\le \omega$,

  • (iii) S and ∂Q link.

Then I possesses a critical value $c\ge \alpha$.

3 Proof of Theorem 1.1

We shall show that the functional $I(z)$ satisfies the geometric assumptions of the critical point theorem for indefinite functional.

Lemma 3.1 (Palais-Smale condition)

Assume that G satisfies conditions (G1) and (G2). Then there exists a constant ${\gamma }_0$ depending on the $C^1$ norm of the function G on $S^1\times (R^{2n}\mathrm{\setminus }{B}_{R_0})$ such that $I(z)$ satisfies the ${(P.S.)}_{\gamma }$ condition in X for $\gamma <{\gamma }_0$.

Proof We shall prove the lemma by contradiction. We suppose that there exists a sequence $\{z_k\}\subset X$ satisfying

$$I(z_k)\to \gamma$$

(3.1)

and

$$DI(z_k)=z_k-A_{ϵ}^{-1}(G_z(t,z_k)+ϵz_k)⟶\theta ,$$

(3.2)

where $A_{ϵ}^{-1}$ is a compact operator and $\theta =(0,\dots ,0)$. Since $G(t,z(t))$ is bounded from below, (3.1) implies that there exists a constant $C_1>0$ such that

$${\int }_0^{2\pi }{A}_{ϵ}{z}_k(t)dt\ge C_1.$$

(3.3)

We claim that $\{z_k\}$ has a convergent subsequence. It suffices to prove that the sequence $\{z_k\}$ is bounded in X. By contradiction, we suppose that up to a subsequence, $\{z_k\}$ satisfies ${\parallel z_k\parallel }_{R^{2n}}\to \mathrm{\infty }$. Then, for large k, we have

$${\parallel z_k\parallel }_{R^{2n}}\ge R_0.$$

(3.4)

It follows from (3.4) that

$$|{\int }_0^{2\pi }G(t,z_k)dt|\le 2\pi sup\{|G(t,z_k)|\mid (t,z_k)\in S^1\times \left(R^{2n}\mathrm{\setminus }{B}_{R_0}\right)\}.$$

(3.5)

By (3.3) and (3.5),

$$\begin{array}{rcl}I(z_k)& =& {\int }_0^{2\pi }[\frac{1}{2}{A}_{ϵ}(z_k)-G(t,z_k(t))-\frac{ϵ}{2}{z}_k^2]dt\\ \ge & \frac{C_1}{2}-2\pi sup\{|G(t,z_k)|\mid (t,z_k)\in S^1\times \left(R^{2n}\mathrm{\setminus }{B}_{R_0}\right)\}-\frac{ϵ}{2}{\parallel z_k\parallel }_{L^2}^2.\end{array}$$

Letting

$${\gamma }_0=\frac{C_1}{2}-2\pi sup\{|G(t,z_k)|\mid (t,z_k)\in S^1\times \left(R^{2n}\mathrm{\setminus }{B}_{R_0}\right)\}-\frac{ϵ}{2}{\parallel z_k\parallel }_{L^2}^2,$$

we have $I(z_k)\ge {\gamma }_0$, which is a contradiction. □

Let

$$Q=(\overline{B_r}\cap X^{-})\oplus \{\mathit{re}\mid e\in B_1\cap X^{+},0<r<R\}.$$

Lemma 3.2 Assume that G satisfies conditions (G1) and (G2). Then there exist sets $S_{\rho }\subset X^{+}$ with radius $\rho >0$, $Q\subset X$ and a constant $\alpha >0$ such that

• (i) $S_{\rho }\subset X^{+}$ and $I{|}_{S_{\rho }}\ge \alpha$,

• (ii) Q is bounded and $I{|}_{\partial Q}\le 0$,

• (iii) $S_{\rho }$ and ∂Q link.

Proof (i) Let us choose $z\in X^{+}\subset X$. Then $z(t)\in D$. By (G1), $G(t,z)$ is bounded above and there exists a constant $\tau >0$

$$\begin{array}{rcl}I(z)& =& \frac{1}{2}({\parallel A_{ϵ}^{\frac{1}{2}}{z}_{+}\parallel }_{L^2}^2+\parallel A_{ϵ}^{\frac{1}{2}}{M}_{+}{z}_0\parallel -{\parallel {(-A_{ϵ})}^{\frac{1}{2}}{z}_{-}\parallel }_{L^2}^2-{\parallel {(-A_{ϵ})}^{\frac{1}{2}}{M}_{-}{z}_0\parallel }^2)\\ -{\int }_0^{2\pi }G(t,z)dt-\frac{ϵ}{2}{\parallel z\parallel }_{L^2}^2\\ \ge & \frac{1}{2}{\parallel A_{ϵ}^{\frac{1}{2}}{z}_{+}\parallel }_{L^2}^2-\tau -\frac{ϵ}{2}{\parallel z\parallel }_{L^2}^2\end{array}$$

for $\tau >0$. Then there exist constants $\rho >0$ and $\alpha >0$ such that if $z\in S_{\rho }\cap X^{+}$, then $I(z)\ge \alpha$.

(ii) Let us choose $e\in B_1\cap X^{+}$. Let $z\in \overline{B_r}\cap X^{-}\oplus \{\mathit{re}\mid 0<r\}$. Then $z=w+y$, $w\in \overline{B_r}\cap X^{-}$, $y=\mathit{re}$. We note that

$$\text{If }w\in \overline{B_r}\cap X^{-},\text{ then }{\int }_0^{2\pi }{A}_{ϵ}(z)\cdot zdt=-{\parallel {(-A_{ϵ})}^{\frac{1}{2}}{z}_{-}\parallel }_{L^2}^2\le 0.$$

By (G2), $G(t,w+\mathit{re})$ is bounded from below. Thus, by Lemma 2.2, there exists a constant $\tau >0$ such that if $z=w+\mathit{re}$, then we have

$$\begin{array}{rcl}I(z)& =& \frac{1}{2}{r}^2-{\parallel {(-A_{ϵ})}^{\frac{1}{2}}{z}_{-}\parallel }_{L^2}^2-{\int }_0^{2\pi }G(t,w+\mathit{re})dt-\frac{ϵ}{2}{\parallel z\parallel }_{L^2}^2\\ \le & \frac{1}{2}{r}^2-{\parallel {(-A_{ϵ})}^{\frac{1}{2}}{z}_{-}\parallel }_{L^2}^2-{\int }_{\mathrm{\Omega }}\frac{\tau }{d^2(V+\mathit{re},C)}dxdt-\frac{ϵ}{2}{\parallel z\parallel }_{L^2}^2.\end{array}$$

We can choose a constant $R>r$ such that if $z=w+\mathit{re}\in Q=(\overline{B_r}\cap X^{-})\oplus \{\mathit{re}\mid e\in B_1\cap X^{+},0<r<R\}$, then $I(z)<0$. Thus we prove the lemma. □

Proof of Theorem 1.1 By Lemma 2.1, $I(z)$ is continuous and Fréchet differentiable in X and, moreover, $DI\in C$. By Lemma 2.2, if $\{z_k\}\subset X^{-}$ and $z_k\rightharpoonup z$ weakly in X with $z\in \partial X$, then $I(z_k)\to -\mathrm{\infty }$. By Lemma 3.1, $I(z)$ satisfies the ${(P.S.)}_{\gamma }$ condition for $\gamma <{\gamma }_0$. By Lemma 3.2, there exist sets $S_{\rho }\subset X^{+}$ with radius $\rho >0$, $Q\subset X$ and a constant $\alpha >0$ such that $I{|}_{S_{\rho }}\ge \alpha$, Q is bounded and $I{|}_{\partial Q}\le 0$, and $S_{\rho }$ and ∂Q link. By the critical point theorem, $I(z)$ possesses a critical value $c\ge \alpha$. Thus (1.1) has at least one nontrivial weak solution. Thus we prove Theorem 1.1. □