Weighted $\frac{N}{2}$-biharmonic equations with exponential growth nonlinearity without the Ambrosetti–Rabinowitz condition

Abstract

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.