## Abstract and Applied Analysis

### Various Half-Eigenvalues of Scalar $p$-Laplacian with Indefinite Integrable Weights

#### Abstract

Consider the half-eigenvalue problem ${({\phi}_{p}({x}^{\prime }))}^{\prime }+\lambda a(t){\phi}_{p}({x}_{+})-\lambda b(t){\phi }_{p}({x}_{-})=0$ a.e. $t\in [0,1]$, where $1\lt p \lt \infty$, ${\phi }_{p}(x)={|x|}^{p-2}x$, ${x}_{\pm }(\cdot\,\!)=\max \{\pm x(\cdot\,\!),0\}$ for $x\in {\mathcal{C}}^{0}:=C([0,1],\mathbb{R})$, and $a(t)$ and $b(t)$ are indefinite integrable weights in the Lebesgue space ${\mathcal{L}}^{\gamma }:={L}^{\gamma }([0,1],\mathbb{R}),1\leq \gamma \leq \infty$. We characterize the spectra structure under periodic, antiperiodic, Dirichlet, and Neumann boundary conditions, respectively. Furthermore, all these half-eigenvalues are continuous in $(a,b)\in {({\mathcal{L}}^{\gamma },{w}_{\gamma})}^{2}$, where ${w}_{\gamma }$ denotes the weak topology in ${\mathcal{L}}^{\gamma }$ space. The Dirichlet and the Neumann half-eigenvalues are continuously Fréchet differentiable in $(a,b)\in {({\mathcal{L}}^{\gamma },{\Vert \cdot\,\!\Vert}_{\gamma })}^{2}$, where ${\Vert \cdot\,\!\Vert }_{\gamma }$ is the ${L}^{\gamma }$ norm of ${\mathcal{L}}^{\gamma }$.

#### Article information

Source
Abstr. Appl. Anal., Volume 2009 (2009), Article ID 109757, 27 pages.

Dates
First available in Project Euclid: 16 March 2010

https://projecteuclid.org/euclid.aaa/1268745594

Digital Object Identifier
doi:10.1155/2009/109757

Mathematical Reviews number (MathSciNet)
MR2539910

Zentralblatt MATH identifier
1188.34018

#### Citation

Li, Wei; Yan, Ping. Various Half-Eigenvalues of Scalar $p$ -Laplacian with Indefinite Integrable Weights. Abstr. Appl. Anal. 2009 (2009), Article ID 109757, 27 pages. doi:10.1155/2009/109757. https://projecteuclid.org/euclid.aaa/1268745594

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