| Abstract: |
| This talk shall be on study of higher-order doubly critical elliptic problems with weights, driven by a polyharmonic double phase operator. Precisely, problems of the type
\begin{equation*}
\begin{cases}
\mathcal{L}^m_{p,q}(u) = f(x,u) ~&\text{in } \Omega,\
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 Caratheodary function, which has doubly critical growth, and $\mathcal{L}^m_{p,q}$ represents a polyharmonic double phase operator.
I shall present suitable Musielak--Orlic--Sobolev framework to study such problems, through application of variational methods. The existence of weak solutions can be derived by establishing a compactness result within these spaces and in addition, the nonexistence results can also be derived under appropriate assumptions by establishing a Pohozaev-type identity for higher-order derivatives. The overall approach is an extension of classical techniques to capture the intricate features of the double phase operator for higher-order derivatives, and shall address the difficulties arising from critical nonlinearities, while also overcoming the non closed feature of truncations in higher order Sobolev spaces. |
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