| Abstract: |
| In this talk, based on joint work with U. Guarnotta, we study an eigenvalue problem for a double-phase operator. The lack of homogeneity of $\Delta_p^a + \Delta_q$ makes the notion of spectrum non-unique, allowing for different possible definitions, as already occurs for the $(p,q)$-Laplacian. We adopt a notion of spectrum inspired by the Fucik spectrum, following an approach previously introduced by Bobkov and Tanaka. A key point is that the structure of the spectrum depends strongly on the linear independence condition $\phi_p^{a} \neq k \phi_q$ for every $k \in \mathbb{R}$, leading to different spectral configurations depending on whether this condition holds. The problem exhibits several features, including the lack of homogeneity, unbalanced growth placing it in the Musielak-Orlicz framework, and the presence of a non-negative, non-trivial weight $a \in C^{0,1}(\Omega)$, which leads to a loss of local regularity. We discuss the region of parameters in which existence or non-existence of positive solutions occurs. The analysis relies on normalization arguments, the Nehari manifold, and truncation techniques, combined with Picone-type inequalities and a suitably tailored strong maximum principle. |
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