On doubly critical polyharmonic double phase problems: Existence and non-existence of solutions

Abstract

In this article, we investigate the existence and nonexistence of weak solutions to higher-order doubly critical elliptic problems with weights, driven by a polyharmonic double phase operator. More precisely, we deal with the following problem \begin{equation} \begin{cases} \mathcal{L}^m_{p,q}(u) = f(x,u) ~&\text{in } Ω,\\[6pt] u=\nabla u=\cdots\nabla^{m-1} u=0 &\text{on }{\partialΩ}, \end{cases} \end{equation} where $\hat{I}\text{c◯}\subset {\mathbb{R}}^N$ with $N\ge 2$ is a smooth bounded domain with Lipschitz boundary $\partial \hat{I}\text{c◯}$, $m\in \mathbb{N}$, $1<p<q<\frac{N}{m}$ with $(N-1)q\le Np$, the nonlinear term $f:\hat{I}\text{c◯}\times \mathbb{R}\to \mathbb{R}$ is a Carathéodory function, which has doubly critical growth, and ${\mathcal{L}}_{p,q}^m$ represents a polyharmonic double phase operator. By establishing new compactness results within a suitable Musielak--Orlicz--Sobolev framework and applying variational methods, we prove the existence of nontrivial weak solutions. In addition, we derive nonexistence results under appropriate assumptions by establishing a Pohozaev-type identity for higher--order derivatives. Our approach extends classical techniques to capture the intricate features of the double-phase operator for higher--order derivatives, and addresses the difficulties arising from critical nonlinearities, in particular extending the results of [F. Colasuonno, K. Perera, J. Differ. Equ., 422 (2025), 426--488] in a polyharmonic double phase setup overcoming the non-closedness of truncations in higher-order Sobolev spaces.