We consider the existence of solutions of the following weighted problem: { L : = - d i v ( ρ ( x ) | ∇ u | N - 2 ∇ u ) + ξ ( x ) | u | N - 2 u = f ( x , u ) i n B u > 0 i n B u = 0 o n ∂ B , \left\{ {\matrix{{L: = - div\left( {\rho \left( x \right){{\left| {\nabla u} \right|}^{N - 2}}\nabla u} \right) + \xi \left( x \right){{\left| u \right|}^{N - 2}}} \hfill & {u = f\left( {x,u} \right)} \hfill & {in} \hfill & B \hfill \cr {} \hfill & {u > 0} \hfill & {in} \hfill & B \hfill \cr {} \hfill & {u = 0} \hfill & {on} \hfill & {\partial B,} \hfill \cr } } \right. where B is the unit ball of ℝ ^N , N #62; 2, ρ ( x ) = ( log e | x | ) N - 1 \rho \left( x \right) = {\left( {\log {e \over {\left| x \right|}}} \right)^{N - 1}} the singular logarithm weight with the limiting exponent N − 1 in the Trudinger-Moser embedding, and ξ( x ) is a positif continuous potential. The nonlinearities are critical or subcritical growth in view of Trudinger-Moser inequalities of double exponential type. We prove the existence of positive solution by using Mountain Pass theorem. In the critical case, the function of Euler Lagrange does not fulfil the requirements of Palais-Smale conditions at all levels. We dodge this problem by using adapted test functions to identify this level of compactness.

In this work, we are concerned with the existence of a ground state solution for a Kirchhoff weighted problem under boundary Dirichlet condition in the unit ball of $\mathbb{R}^{4}$. The nonlinearities have critical growth in view of Adams' inequalities. To prove the existence result, we use Pass Mountain Theorem. The main difficulty is the loss of compactness due to the critical exponential growth of the nonlinear term $f$. 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.

In this work, we study the weighted Kirchhoff problem {g(?B(?(x)|?u|2 + V(x)u2)dx) [?div(?(x)?u) + V(x)u] = f (x, u) in B u > 0 in B u = 0 on ?B, where B is the unit ball in R2, ?(x) = log e |x| , the singular logarithm weight in the Trudinger-Moser embedding, g is a continuous positive function on R+ and the potential V is a continuous positve function. The nonlinearities are critical or subcritical growth in view of Trudinger-Moser inequalities. We prove the existence of non-trivial solutions via the critical point theory. In the critical case, 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.

"This work is concerned with the existence of a positive ground state solution for the following non local weighted problem \begin{equation*} \displaystyle \left\{ \begin{array}{rclll} L_{(\sigma,V)}u &= & \displaystyle f(x,u)& \mbox{in} \ B \\ u &>&0 &\mbox{in }B\\ u&=&0 &\mbox{on } \partial B, \end{array} \right. \end{equation*} where $$L_{(\sigma,V)}u:=g(\int_{B}(\sigma(x)|\nabla u|^{N}+V(x)|u|^{N})dx)\big[-\textmd{div} (\sigma(x)|\nabla u|^{N-2} \nabla u)+V(x)|u|^{N-2}u\big],$$ B is the unit ball of $\mathbb{R}^{N}$, $ N>2$, $\sigma(x)=\Big(\log(\frac{e}{|x|})\Big)^{\beta(N-1)}$, $\beta \in[0,1)$ the singular logarithm weight , $V(x)$ is a positif continuous potential.The Kirchhoff function $g$ is positive and continuous on $(0,+\infty)$. The nonlinearities are critical or subcritical growth in view of Trudinger-Moser inequalities of exponential type. We prove the existence of a positive ground state solution by using Mountain Pass theorem . In the critical case, the Euler-Lagrange function loses compactness except for a certain level. We dodge this problem by using adapted test functions to identify this level of compactness."