Weak Eigenfunctions for the Linearization of Extremal Elliptic Problems
Weak Eigenfunctions for the Linearization of Extremal Elliptic Problems
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DOI：
10.1006/jfan.1997.3171
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发表时间：
1998-06
影响因子：
1.7
通讯作者：
X. Cabré;Y. Martel
中科院分区：
文献类型：
--
作者：
X. Cabré;Y. Martel
考虑半线性椭圆问题[公式]，其中λ为非负参数，g为正的、非减的凸非线性项。存在一个参数值λ *，它是解存在性的极值。研究了半线性问题在参数λ = λ * 对应的极值弱解处的线性化。在某些情况下，该线性化问题具有离散的正H10谱.然而，我们证明了在L1（Ω）中总是存在一个正的弱本征函数，其特征值为零。当λ > λ * 时，零L1特征值与半线性问题解的不存在性是一致的.最后，当Ω是单位球且g（u）= eu或g（u）=（1+ u）p时，我们求出了极值问题线性化的所有弱特征函数和特征值.
Abstract We consider the semilinear elliptic problem[formula]where λ is a nonnegative parameter and g is a positive, nondecreasing, convex nonlinearity. There exists a value λ * of the parameter which is extremal in terms of existence of solution. We study the linearization of the semilinear problem at the extremal weak solution corresponding to the parameter λ = λ *. In some cases, this linearized problem has discrete and positive H 1 0 -spectrum. However, we prove that there always exists a positive weak eigenfunction in L 1 ( Ω ) with eigenvalue zero for this linearized problem. The zero L 1 -eigenvalue is coherent with the nonexistence of solutions of the semilinear problem for λ > λ *. Finally, we find all weak eigenfunctions and eigenvalues for the linearization of the extremal problem when Ω is the unit ball and g ( u )= e u or g ( u )=(1+ u ) p .