论文标题:一类带权函数的拟线性椭圆方程
A Class of Quasilinear Elliptic Equations with Weights
论文作者 刘祥清
论文导师 黄毅生,论文学位 硕士,论文专业 基础数学
论文单位 云南师范大学,点击次数 201,论文页数 31页File Size716k
2004-05-01论文网 http://www.lw23.com/lunwen_1616502/ 椭圆型方程;p-Laplacian; Fu(?)ik谱; 变形山路引理; 鞍点定理; 共振; 非共振; Landesman-Lazer条件; (PS)条件; 变分方法
Elliptic equation, p-Laplacian, Fucik spectrum, a variant of the Mountain-pass Lemma, the saddle point theorem, resonance, nonresonance, the Landesman-Lazer condition, (PS) condition, variational methods.
本文研究下面的拟线性椭圆方程其中 是有界光滑区域，为p-Laplacian，权重函数a(x)＞0 a．e．于Ω，且a(x)∈L~r(Ω)(r≥N/P)．当f(x，u)=0时，方程(A1．1)就转化为下列方程我们把方程(A1．2)的Fuik谱定义为集合∑_p={(α，β)∈R~2，方程(A1．2)有非平凡解}。在方程(A1．2)中，当α=β时，上述的Fuik谱为R~2上形如(λ，λ)的点的集合，其中λ为方程的特征值。由文献[9]可知，方程(A1．3)存在一列特征值{λ_n}：0＜λ_1＜λ_2≤…≤λ_k…，并且λ_k→∞(k→∞)。 本文首先利用文献[10]的方法及[16]中的一个变形山路引理，讨论了方程(A1．2)的第一条非平凡Fuik谱曲线c的存在性及其性质；其次分(λ，μ)∑_p和(λ，μ)∈∑_p两种情况利用鞍点定理及其它变分方法讨论了方程(A1．1)在f(x，u)满足一定条件下的可解性。 本文的结果归纳如下： 定理A1．1 方程(A1．2)的第一条非平凡Puik谱曲线c存在，并且c是连续的、严格递减的。 定理A1．2 设f(x，u)满足下列条件 关于x一致成立； 关于x一致成立；且 其中(α，β)∈C。则方程(A1．1)有一非平凡解。 定理A1．3 假设存在常数δ＞0，使得λ∈(λ_1-δ，λ_1+δ)，μ=λ_1或者λ=λ_1，β∈(λ_1-δ，λ_1+δ)。若f(x，u)满足 且并且下面的条件成立 或云南师范大学硕士学位论文其中F(士oo)=lim suPF(t)，互(士oo)之一)士。c=liminfF(t)、 t一)士以二蛋关‘。(·，“一。(‘，‘并0(尹一1)夕(o)亡=0.了lseZ、...、 l一 孟乙 F那么方程(Al.l)至少有一弱解.定理Al.4若林，川任Cl，a(对满足条件 (a)存在常数凡>0，a(x)>k，且。(x)〔厂(卿. f(x，u)满足定理Al.3中的(f)，并且下面的条件成立(”一1，五h·d一卫‘十·，五·‘d一了(一)旅一dx，(，一‘，五h·d·>歹(一，五二dx一卫(+加)五一dx，丫?〔E(。，口)\{0}(Al·6)丫v任E(a，口)\{o}，(AI·7)其中C‘(l全2)为任意柞平凡Fu己ik谱曲线，E(。，口)为方程(Al·2)对应于(a，尽)任c‘c名，的解的集合，了(士oo)、互(士oo)的定义和定理Al.3中的一样.则方程(Al.l)至少有一非平凡解.
In this paper, we consider the quasilinear elliptic equationwhere ft is a bounded domain in RN(N 3) with smooth boundary 3ft, pu = div (| u|p-2 u) is the p-Laplacian, 1 < p < N, u = max{ u,0}, u = u+ - u-, the weight function a(x) > 0 a.e. in 0 and a(x) Lr( )(r N/p). If f(x,u) = 0, then Equation (A1.1) turns to the equationThe Fu ik spectrum of the p-Laplacian on W01,p( ) is defined as the set p of those ( , ) R2 such that Equation (A1.2) has a nontrivial solution. If = in (A1.2), then the Pucik spectrum Ep is the subset of R1 with the form of (A, A) ? M2, where A is the eigenvalue of problemBy [9], there exists a sequence of eigenvalues of (A1.3) {An} : 0 < 1 < 2 < < k , and k - (k - ).In our paper, first, we study the existence of the first nontrivial curve C in p and it"s some properties by using methods of [10] and a variant of the Mountain-pass Lemma given by [16]. Then under suitable conditions for f(x,u), we consider the solvability of (Al.l) in the cases of ( , ) p and ( , ) p by using the saddle point theorem and other variational methods.Our results can be included in the following:Theorem A1.1 The first nontrivial Fu ik spectrum curve C in p exists, moreover C is continuous and strictly decreasing.Theorem A1.2 If f(x,u) satisfies the following hypotheses: Then Equation (Al.l) has a nontrivial solution.Theorem A1.3 Let 6 > 0 be such that If f(x, u) satisfiesAnd suppose thatThen Equation (A1.1) has at least one weak solution.Theorem A1.4 If( , ) Cl. If a(x) satisfies(a) There exists a constant k > 0, such that a(x) > k, and a(x) Lr( ), f(x,u) satisfies (/) of the theorem A1.3, and assume thatwhere C[(l 2) are nontrivial Fucik spectrum curves, E( ) is the set of nontrivial solutions of (A1.2) with (, ) Cl p; F( ) and F( ) are defined as in Theorem Al.3. Then Equation (Al.l) has at least one nontrivial solution.
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