On the Fredholm alternative for the p-Laplacian at the first eigenvalue
On the Fredholm alternative for the p-Laplacian at the first eigenvalue
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关于第一特征值处 p-拉普拉斯算子的 Fredholm 替代方案
DOI：
10.1512/iumj.2002.51.2156
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发表时间：
2002
影响因子：
1.1
通讯作者：
P. Takáč
中科院分区：
文献类型：
--
作者：
P. Takáč
研究了退化拟线性Dirichlet边值问题（P）-Δ pu = λ 1的弱解u ∈ W1，p0（Ω）的存在性|u| p-2 u + f（x）in Ω; u = 0 on Ω.设1 < p < ∞，p <$= 2，Ω是RN中的有界区域，f ∈ L∞（Q）是给定函数，λ 1表示正p-Laplacian -Ap的第一（最小）特征值，其中Δ p u <$div（|拉乌|p-2职等）。本征值λ 1是简单的，设φ 1表示与λ 1相关的本征函数。当f满足正交条件<$Ω fφ 1dx = 0时，证明了问题（P）解的存在性，在这种情况下，当p <$= 2且f <$0在Ω中时，解集在C1（Ω）中有界.在我们的证明中，一个二阶泰勒公式在其积分形式（接近φ 1）中起着关键作用，它包含某些Gâteaux导数和一个半正定二次型。该二次形式补偿了对应于问题（P）的能量泛函中的不连续性的缺乏。当与众所周知的正则性结果相结合时，它的正半定性给出了先验估计，由此可以得出存在性和有界性（p = 2）。
We investigate the existence of a weak solution u ∈ W 1,p 0 (Ω) to the degenerate quasi-linear Dirichlet boundary value problem (P) -Δ p u = λ 1 |u| p-2 u + f(x) in Ω; u = 0 on ∂Ω. It is assumed that 1 < p < ∞, p ¬= 2, Ω is a bounded domain in R N , f ∈ L∞ (Q) is a given function, and the number λ 1 stands for the first (smallest) eigenvalue of the positive p-Laplacian -Ap, where Δ p u ≡ div(|⊇u| p-2 ⊇u). The eigenvalue λ 1 being simple, let φ 1 denote the eigenfunction associated with λ 1 . We show the existence of a solution for problem (P) when f satisfies the orthogonality condition ⊃ Ω fφ 1 dx = 0, in which case the set of solutions is bounded in C 1 (Ω) provided p ¬= 2 and f ≢ 0 in Ω. A key role in our proofs is played by a second-order Taylor formula in its integral form (near φ 1 ) which contains certain Gâteaux derivatives and a positive semidefinite quadratic form. This quadratic form compensates for lack of coercivity in the energy functional corresponding to problem (P). When combined with well-known regularity results, its positive semidefiniteness renders a priori estimates from which existence and boundedness (for p ¬= 2) follow.