On the Principal Eigenvalue of the $(p,A)$-Laplacian with Sign-Changing Morrey Potentials: Complete Continuity, Optimization, and Differentiability

Abstract

We establish a unified framework to study the continuity and complete continuity of the principal eigenvalue functional $λ_1(V,m)$ associated with the following $(p,A)$-Laplacian operator involving both sign-changing potential $V$, and positive weight $m$ in a Morrey space: \begin{align*} -Δ_{p,A}(u) + V|u|^{p-2}u &= λm|u|^{p-2}u \quad \text{in},, Ω, u &= 0 \quad\quad\quad \quad\quad \text{on},, \partialΩ, \end{align*} where $1<p<N$, $λ\in\mathbb{R}$ is a parameter, and $Ω\subset \mathbb{R}^N (N \geq 2)$ is a bounded domain with boundary $\partial Ω$. As a consequence, we investigate the existence and uniqueness of optimal principal eigenvalues $λ_1(V,m)$ over a weakly compact subset of the Morrey space. Finally, we derive a formula for the derivative of $λ_1(V,m)$ with respect to regular perturbations of potential $V$, and positive weight $m$ within a closed subspace of Morrey space. Our results develop analytic tools to overcome the lack of separability and reflexivity in Morrey spaces to investigate this eigenvalue problem. In particular, the study is new even for the linear case $p=2$. We also emphasize that the tools established are applicable for a wide class of PDEs with coefficients in Morrey spaces.