Non Local Weighted Fourth Order Equation in Dimension $4$ with Non-linear Exponential Growth

Abstract

In this work, we study the weighted Kirchhoff problem \[ \begin{cases} g\big( \int_{B} (w(x) |\Delta u|^{2}) \, dx \big) [\Delta (w(x) \Delta u)] = f(x,u) &\textrm{in $B$}, \\ u > 0 &\textrm{in $B$}, \\ u = \frac{\partial u}{\partial n} = 0 &\textrm{on $\partial B$}, \end{cases} \] where $B$ is the unit ball of $\mathbb{R}^{4}$, $w(x) = \big( \log \frac{e}{|x|} \big)^{\beta}$, the singular logarithm weight in Adam's embedding, $g$ is a continuous positive function on $\mathbb{R}^{+}$. The nonlinearities are critical growth in view of Adam's inequalities. We prove the existence of a positive ground state solution using mountain pass method combined with a concentration compactness result. The associated energy function does not satisfy the condition of compactness. We provide a new condition for growth and we stress its importance to check the min-max compactness level.