关于一些椭圆型方程及方程组的解的存在性的研究
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摘要
本文主要研究半线性椭圆型方程及方程组,带电磁位势的非线性Schrodinger方程及Schrodinger-Pooisson方程组的解的存在性.
本文共分七章：
在第一章中,我们概述本文所研究问题的背景及国内外研究现状,并简要介绍本文的主要工作及相关的预备知识和一些记号.
在第二章中,我们研究下述非线性椭圆型方程的非平凡解的存在性.其中Ω是RN中的有界区域,a∈LN/2(Ω),N≥3.f∈C0(Ω×R1,R1)在t=0处是超线性且在t=∞是次临界增长.在某些给定条件下,(S1)具有所谓的环绕几何结构.在不假设Ambrosetti-Rabinowitz条件下,我们证明了(S1)至少有一个非平凡解.我们的结果推广了Miyagaki和Souto文献[103]中的结果,他们考虑a(x)=0的情况,此时(S1)具有山路几何结构.在第三章中,我们应用环绕定理和集中紧原理证明下述半线性椭圆型方程组至少存在一对正解(u,v)∈H1(RN)×H1(RN).这里关于f,g∈C0(RN×R1)的主要假设是：f(x,t)和g(x,t)在t=0处是超线性,在t=+∞是次临界增长,同时f和g满足某种单调性条件.这里我们不假设f或g满足通常的Ambrosetti-Rabinowitz条件.我们将文献[103]中的主要结果从单个方程推广到了方程组.
在第四章中,我们进一步研究半线性椭圆型方程组(S2).此处f(x,t)和g(x,t)满足一组与第三章不同的条件.应用关于无穷维的强不定的泛函的临界点理论,我们证明了(S2)有一个正的基态解.此外,如果f(x,t)和g(x,t)关于x是周期函数,关于t是奇函数,则(S2)有无穷多个几何不同的解.我们的结果改进了文献[88,89]中的结果.
在第五章中,我们研究下述带电磁位势的非线性Schrodinger方程其中A(r)=(A1(r),A2(r),…,AN(r))为向量,Aj(r)(j=1,2,…,N)是R+上的实函数,V(r)是R+上的正函数,当N≥3时,1＜p＜N+2/N-2；当N=1,2时1 在第六章中,我们研究下述带电磁位势的非线性Schrodinger方程其中2＜p＜2N/N-2,如果N≥3；2＜p＜+∞,如果N=1,2.ε>0是参数,a(x)是RN上的正的连续函数,A(x)=(A1(x),A2(x),…,AN(x))是向量,Aj(x)(j= 1,2,...,N)是RN上的实函数.我们证明了在某些给定的条件下对任意的正整数m,存在ε(m)>0使得,对任意的0<ε<ε(m),(E2)有一个m-峰的复值解.因此,当ε→0,(E2)有越来越多的多峰复值解.我们的结果推广了Liu和Lin文献[96]中的结果,他们考虑(E2)中A(x)三0的情况.
在第七章中,我们研究下述非线性Schrodinger-Poisson方程组
其中K(x)是R3中的正的连续函数,lim K(x)=0,2＜p＜6,∈＞0 |x|→∞为参数.对任意的正整数m,我们证明了存在ε(m)>0使得在某些给定的条件下对任意的0＜∈＜∈(m),(SP)有一个正的m-峰解.因此当ε→0时,(SP)有越来越多的多峰解.我们的结果推广了文献[96]中关于单个非线性Schrodinger方程的结果.
In this paper, we mainly study the existence of solutions for some semilinear elliptic equations and systems, some nonlinear Schrodinger equations with electro-magnetic fields and nonlinear Schrodinger-Poisson system.
The thesis consists of seven chapters:
In Chapter One, we summarize the background of the related problems and state the main results of the present thesis. We also give some preliminary results and notations used in the whole thesis.
In Chapter Two, we study the existence of a nontrivial solution to the following nonlinear elliptic problem: whereΩis a bounded domain of RN and a∈LN/2(Ω),N≥3,f∈C0(CΩ×(Q×R1,R1) is superlinear at t=0 and subcritical at t=∞. Under suitable conditions, the equation (S1) possesses the so-called linking geometric structure. We prove that the equation (S1) has at least one nontrivial solution without assuming the Ambrosetti-Rabinowitz condition. Our main result extends a recent result of Miyagaki and Souto given in [103] for the equation (S1) with a(x)=0 and possessing the mountain-pass geometric structure.
In Chapter Three, we prove the existence of at least one positive solution pair (u,v) E Hl(RN)×(RN) to the following semilinear elliptic system
by using a linking theorem and the concentration-compactness principle. The main conditions we imposed on the nonnegative functionsf, g∈C0(RN×R1) are that, f(x,t) and g(x,t) are superlinear at t=0 as well as at t=+∞, that f and g are subcritical in t and satisfy a kind of monotonic conditions. We mention that we do not assume that f or g satisfies the Ambrosetti-Rabinowitz condition as usual. We generalize the results in [103] from a single equation to the system.
In Chapter Four, we study the following semilinear elliptic system (S2) further. Here f(x,t) and g(x,t) satisfy some different conditions from Chapter Three. By using critical point theory of strongly indefinite functionals, we obtain a positive ground state solution for (S2). Moreover, if f(x,t) and g(x,t) are periodic in x and odd in t, then (S2) has infinitely many geometrically distinct solutions. We generalize the results in [88,89].
In Chapter Five, we study the nonlinear Schrodinger equation with electromag-netic fields
where the vector A(r)=(A1(r), A2(r),…, AN(r)) is such that Aj(r)(j=1,2,…, N) is a real function on R+ and V(r) is a positive function on R+,1＜p＜N+2/N-2 if N≥3 and 1＜p＜+∞ifN=1,2. We prove that the equation (E1) has infinitely many non-radial complex-valued solutions under conditions (H1) and (H2). Our main re-sult extends a result of Wei and Yan given in [134] for the equation (S1) with A(y)= 0.
In Chapter Six, we are concerned with the existence of multi-bump solutions for a nonlinear Schrodinger equations with electromagnetic fields where 2＜p＜2N/N-2 if N≥3 and 2＜p＜+∞if N= 1,2 and∈＞0 is a parameter. a(x) is a positive continuous function on RN, and A(x) = (A1(x), A2(x),..., AN(x)) is such that Aj(x)(j= 1,2,..., N) is a real function on RN. We prove under some suitable conditions that for any positive integer m, there exists∈(m)> 0 such that, for 0＜∈＜∈(m), the problem (E2) has an m-bump complex-valued solution. As a result, when∈→0, the equation has more and more multi-bump complex-valued solutions. Our main result extends a result of Liu and Lin given in [96] for the equation (S2) with A(x)≡0.
In Chapter Seven, we study the following nonlinear Schrodinger-Poisson system
where K(x) is positive and continuous function in R3, and lim K(x)=0,2＜p＜6 and (?)＞0 is a parameter. For any positive integer m, we prove that there exists (?)(m)＞0 such that, for 0＜(?)＜(?)(m), equation (SP) has an m-bump positive solution under some suitable conditions. As a consequence, equation (SP) has more and more multi-bump positive solutions as (?)→0. We generalize the result in [96] from a single nonlinear Schrodinger equation to the nonlinear Schrodinger-Poisson system.
本文共分七章：
在第一章中,我们概述本文所研究问题的背景及国内外研究现状,并简要介绍本文的主要工作及相关的预备知识和一些记号.
在第二章中,我们研究下述非线性椭圆型方程的非平凡解的存在性.其中Ω是RN中的有界区域,a∈LN/2(Ω),N≥3.f∈C0(Ω×R1,R1)在t=0处是超线性且在t=∞是次临界增长.在某些给定条件下,(S1)具有所谓的环绕几何结构.在不假设Ambrosetti-Rabinowitz条件下,我们证明了(S1)至少有一个非平凡解.我们的结果推广了Miyagaki和Souto文献[103]中的结果,他们考虑a(x)=0的情况,此时(S1)具有山路几何结构.在第三章中,我们应用环绕定理和集中紧原理证明下述半线性椭圆型方程组至少存在一对正解(u,v)∈H1(RN)×H1(RN).这里关于f,g∈C0(RN×R1)的主要假设是：f(x,t)和g(x,t)在t=0处是超线性,在t=+∞是次临界增长,同时f和g满足某种单调性条件.这里我们不假设f或g满足通常的Ambrosetti-Rabinowitz条件.我们将文献[103]中的主要结果从单个方程推广到了方程组.
在第四章中,我们进一步研究半线性椭圆型方程组(S2).此处f(x,t)和g(x,t)满足一组与第三章不同的条件.应用关于无穷维的强不定的泛函的临界点理论,我们证明了(S2)有一个正的基态解.此外,如果f(x,t)和g(x,t)关于x是周期函数,关于t是奇函数,则(S2)有无穷多个几何不同的解.我们的结果改进了文献[88,89]中的结果.
在第五章中,我们研究下述带电磁位势的非线性Schrodinger方程其中A(r)=(A1(r),A2(r),…,AN(r))为向量,Aj(r)(j=1,2,…,N)是R+上的实函数,V(r)是R+上的正函数,当N≥3时,1＜p＜N+2/N-2；当N=1,2时1
在第七章中,我们研究下述非线性Schrodinger-Poisson方程组
其中K(x)是R3中的正的连续函数,lim K(x)=0,2＜p＜6,∈＞0 |x|→∞为参数.对任意的正整数m,我们证明了存在ε(m)>0使得在某些给定的条件下对任意的0＜∈＜∈(m),(SP)有一个正的m-峰解.因此当ε→0时,(SP)有越来越多的多峰解.我们的结果推广了文献[96]中关于单个非线性Schrodinger方程的结果.
In this paper, we mainly study the existence of solutions for some semilinear elliptic equations and systems, some nonlinear Schrodinger equations with electro-magnetic fields and nonlinear Schrodinger-Poisson system.
The thesis consists of seven chapters:
In Chapter One, we summarize the background of the related problems and state the main results of the present thesis. We also give some preliminary results and notations used in the whole thesis.
In Chapter Two, we study the existence of a nontrivial solution to the following nonlinear elliptic problem: whereΩis a bounded domain of RN and a∈LN/2(Ω),N≥3,f∈C0(CΩ×(Q×R1,R1) is superlinear at t=0 and subcritical at t=∞. Under suitable conditions, the equation (S1) possesses the so-called linking geometric structure. We prove that the equation (S1) has at least one nontrivial solution without assuming the Ambrosetti-Rabinowitz condition. Our main result extends a recent result of Miyagaki and Souto given in [103] for the equation (S1) with a(x)=0 and possessing the mountain-pass geometric structure.
In Chapter Three, we prove the existence of at least one positive solution pair (u,v) E Hl(RN)×(RN) to the following semilinear elliptic system
by using a linking theorem and the concentration-compactness principle. The main conditions we imposed on the nonnegative functionsf, g∈C0(RN×R1) are that, f(x,t) and g(x,t) are superlinear at t=0 as well as at t=+∞, that f and g are subcritical in t and satisfy a kind of monotonic conditions. We mention that we do not assume that f or g satisfies the Ambrosetti-Rabinowitz condition as usual. We generalize the results in [103] from a single equation to the system.
In Chapter Four, we study the following semilinear elliptic system (S2) further. Here f(x,t) and g(x,t) satisfy some different conditions from Chapter Three. By using critical point theory of strongly indefinite functionals, we obtain a positive ground state solution for (S2). Moreover, if f(x,t) and g(x,t) are periodic in x and odd in t, then (S2) has infinitely many geometrically distinct solutions. We generalize the results in [88,89].
In Chapter Five, we study the nonlinear Schrodinger equation with electromag-netic fields
where the vector A(r)=(A1(r), A2(r),…, AN(r)) is such that Aj(r)(j=1,2,…, N) is a real function on R+ and V(r) is a positive function on R+,1＜p＜N+2/N-2 if N≥3 and 1＜p＜+∞ifN=1,2. We prove that the equation (E1) has infinitely many non-radial complex-valued solutions under conditions (H1) and (H2). Our main re-sult extends a result of Wei and Yan given in [134] for the equation (S1) with A(y)= 0.
In Chapter Six, we are concerned with the existence of multi-bump solutions for a nonlinear Schrodinger equations with electromagnetic fields where 2＜p＜2N/N-2 if N≥3 and 2＜p＜+∞if N= 1,2 and∈＞0 is a parameter. a(x) is a positive continuous function on RN, and A(x) = (A1(x), A2(x),..., AN(x)) is such that Aj(x)(j= 1,2,..., N) is a real function on RN. We prove under some suitable conditions that for any positive integer m, there exists∈(m)> 0 such that, for 0＜∈＜∈(m), the problem (E2) has an m-bump complex-valued solution. As a result, when∈→0, the equation has more and more multi-bump complex-valued solutions. Our main result extends a result of Liu and Lin given in [96] for the equation (S2) with A(x)≡0.
In Chapter Seven, we study the following nonlinear Schrodinger-Poisson system
where K(x) is positive and continuous function in R3, and lim K(x)=0,2＜p＜6 and (?)＞0 is a parameter. For any positive integer m, we prove that there exists (?)(m)＞0 such that, for 0＜(?)＜(?)(m), equation (SP) has an m-bump positive solution under some suitable conditions. As a consequence, equation (SP) has more and more multi-bump positive solutions as (?)→0. We generalize the result in [96] from a single nonlinear Schrodinger equation to the nonlinear Schrodinger-Poisson system.
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