Sami Baraket

This paper studies a class of nonlocal ‐Laplacian equations defined on locally finite graphs, where the nonlinearity depends on an intrinsic operator applied to the solution. We develop a functional framework based on discrete Sobolev spaces and establish key properties of the associated operators. Under suitable growth and structural conditions, we prove the existence of weak solutions using pseudomonotone operator theory. Additionally, we derive a priori bounds for solutions and apply our main results to a concrete nonlocal problem involving convolution‐type operators. Our work extends some classical PDE techniques to the graph setting and provides a unified approach for analyzing nonlocal variational problems on discrete structures.

In this paper, we establish the existence of nontrivial solutions for a logarithmic weighted \frac{N}{2} -biharmonic problem within the unit ball B of \mathbb{R}^{N} , without imposing the Ambrosetti–Rabinowitz condition. The nonlinearity exhibits critical or subcritical growth in view of weighted Adams inequalities. Our proof relies on minimax techniques and the pass mountain theorem applied to Cerami sequences. In the critical case, the associated energy does not adhere to the compactness constraint. We introduce a novel growth condition and emphasise the importance of avoiding the compactness level.