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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 } \Omega,\\[6pt]
u=\nabla u=\cdots\nabla^{m-1} u=0
&\text{on }{\partial\Omega}, \end{cases} \end{equation} where $\Omega \subset \mathbb{R}^N$ with $N \geq 2$ is a smooth bounded domain with Lipschitz boundary $\partial\Omega$, $m \in \mathbb{N}$, $1 < p < q < \frac{N}{m}$ with $(N-1)q\leq Np$, the nonlinear term $f\colon\Omega\times\mathbb{R}\to \mathbb{R}$ is a Carathéodory function, which has doubly critical growth, and $\mathcal{L}^m_{p,q}$ 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.

Submission history

From: Ashutosh Dixit [view email]
[v1] Tue, 23 Jun 2026 17:34:58 UTC (38 KB)