## Abstract

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

## Citation

Wei Li. Ping Yan. "Various Half-Eigenvalues of Scalar $p$-Laplacian with Indefinite Integrable Weights." Abstr. Appl. Anal. 2009 1 - 27, 2009. https://doi.org/10.1155/2009/109757

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