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

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