彭超權(quán),魏 盼

(中南民族大學(xué) 數(shù)學(xué)與統(tǒng)計(jì)學(xué)學(xué)院,武漢 430074)

一類帶漸近線性項(xiàng)的Schr?dinger方程非平凡解的存在性

彭超權(quán),魏 盼

(中南民族大學(xué) 數(shù)學(xué)與統(tǒng)計(jì)學(xué)學(xué)院,武漢 430074)

研究了如下Schr?dinger方程:-Δu+V(x)u+u=f(u),x∈N,其中N≥3,f(u)關(guān)于u在無(wú)窮遠(yuǎn)處漸近線性. 這類方程源于數(shù)學(xué)物理中的多種分支,在生物學(xué)的一些問(wèn)題中也有一定的體現(xiàn).利用山路定理,證明了在一定條件下該方程在H1(N)中非平凡解的存在性.

漸近線性;非平凡解;山路定理

1 主要問(wèn)題及結(jié)果

本文考慮如下一類Schr?dinger方程:

-Δu+V(x)u+u=f(u),x∈N,

(1)

在H1(N)中非平凡解的存在性,其中N≥3.

Schr?dinger方程(1)的研究受到了廣泛的關(guān)注,這類方程源于數(shù)學(xué)物理中的多種分支, 在生物學(xué)的一些問(wèn)題中也有一定的體現(xiàn)，例如文[1,2],另外文[3]在徑向?qū)ΨQ的Sobolev空間中討論了問(wèn)題(1)；文[4]得到了V(x)及f(t)關(guān)于變量具有周期性時(shí),f(t)關(guān)于t在無(wú)窮遠(yuǎn)處超線性或漸近線性時(shí)方程多解的存在性；文[5]考慮了V(x)變號(hào)且在無(wú)窮遠(yuǎn)處可能衰減到零的情形；文[6] 則討論了一類帶參數(shù)的漸近線性方程組的問(wèn)題,更多相關(guān)文章可見文[7-9].

本文擬討論V(x)在無(wú)窮遠(yuǎn)處可能衰減到零以及f(t)關(guān)于t在無(wú)窮遠(yuǎn)處漸近線性的情形,我們假設(shè)V(x)及f(t)滿足如下條件:

(i)V(x)連續(xù)且?a,A>0以及α∈(0,2)使得對(duì)?x∈N有:

(ii)f∈C(,), 當(dāng)s→0時(shí),f(s)s-1→0.

(iii) ?l∈(0,∞),使得當(dāng)s→+∞時(shí),f(s)s-1→l,并且對(duì)?s≠0有0≤f(s)s-1≤l.

我們的主要結(jié)果如下:

定理1 假設(shè)V(x)滿足條件(i)，并且條件(ii)～(v)成立,令l>μ*,其中：

則方程(1)在H1(N)中至少有一個(gè)非平凡解.

2 預(yù)備知識(shí)

令E={u∈H1(N,u|2+V(x)u2+u2)dx<+∞},它的內(nèi)積為:

u,vuv+V(x)uv+uv)dx,

為了證明定理1,我們將用到文[10]中提出的山路定理.

引理1 設(shè)E為Banach空間,E*是其相應(yīng)的對(duì)偶空間,I∈C1(E,)且滿足:

I{zn}→c≥β并且(1+‖zn‖)‖I′(zn)‖E*→0,

(2)

滿足式(2)的序列稱為Cerami序列(簡(jiǎn)稱(C)c序列).

引理2 假設(shè)函數(shù)f(t)滿足條件(ii)和(iii),則有:

(a) ?ρ,β>0,使得當(dāng)‖u‖=ρ時(shí),對(duì)?u∈E,有I(u)≥β>0.

(b) ?v∈E,使得當(dāng)‖v‖≥ρ,l>μ*時(shí),有I(v)<0.

證明 (a) 由條件(ii)和(iii)可知,對(duì)?ε>0,存在一個(gè)常數(shù)Cε>0,使得對(duì)?t∈有:

|F(t)|≤ε|t|2+Cε|t|p,

(3)

其中p∈(2,2*),則由式(3)和Sobolev不等式可知,對(duì)?u∈E有:

由此可知當(dāng)t→∞時(shí)有I(tφ)→-∞,從而問(wèn)題(b)得證.

3 定理1的證明

引理3 假設(shè)條件(i)～(v)成立,則泛函I的任何(C)c序列{un}在E中有界.

即

(4)

下證w不恒等于0. 反證法. 若w≡0,由條件(iii)、(v)可得,存在充分大的R>0以及η∈(0,1)使得:

(5)

(6)

則由式(4)～(6)可知1≤η+o(1),矛盾. 因此w不恒等于0.

矛盾. 因此{(lán)un}在E中有界.

定理1的證明 由引理1及引理2知,存在序列{un}∈E滿足:

I(un)→c≥β>0,
(1+‖un‖)I′(un)→0,

由引理3可知,{un}在E上有界,從而存在{un}的子列,不妨仍記為本身,使得當(dāng)n→∞時(shí),un?u在E中. 下證當(dāng)n→∞時(shí),‖un-u‖→0. 因?yàn)?

(7)

(8)

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Existence of Nontrivial Solutions to a Schr?dinger
Equation with Asymptotically Linear Terms

Peng Chaoquan,Wei Pan

(College of Mathematics and Statistics,South-Central University for Nationalities,Wuhan 430074,China)

In this paper, we study thd following Schr?dinger equation -Δu+V(x)u+u=f(u),x∈N,whereN≥3,f(u) is asymptotically linear. Equations of this form arise in various branches of mathematical physics and in some problems in biology as well. By using the Mountain Pass theorem and under certain conditions,we prove that the equation possess at infinity a nontrivial solution foruinH1(N).

asymptotically linear;nontrivial solution;Mountain Pass theorem

2015-10-13

彭超權(quán)(1979-),男,副教授,博士,研究方向：偏微分方程，E-mail：pcq1979@163.com

國(guó)家自然科學(xué)基金資助項(xiàng)目(61374085);中南民族大學(xué)研究生創(chuàng)新基金資助項(xiàng)目(2015sycxjj126)

O175.25

A

1672-4321(2015)04-0129-03