Abstract multiplicity results for \((p, q)\)-Laplace equations with two parameters (Q6645849)

The paper considers the Dirichlet problem for the equation \[-\Delta_p u -\Delta_q u = \alpha |u|^{ p-2} u + \beta |u| ^{q-2} u\] in a bounded smooth domain, with \(1 < q < p\) and \(\alpha,\beta\in\mathbb{R}\). The aim is to obtain multiplicity of solutions (either positive or sign changing) for suitable combination of the parameters \(\alpha,\beta\).\N\NSolutions, either with positive or with negative energy, are obtained via variational methods and in particular using a Nehari manifold analysis and Ljusternik-Schnirelmann theory.\N\NThe conditions on \(\alpha,\beta\) are related to the variational eigenvalues of \(-\Delta_p\) and \(-\Delta_q\), and to the following critical parameters \[ \beta_*(\alpha)=\inf \left\{\frac{\|\nabla u\|_q^q}{\|u\|_q^q}: u \in W_0^{1, p} \backslash\{0\} \text { and } \|\nabla u\|_p^p\leq \alpha\|u) \|_p^p\right\}. \]