On a weighted elliptic equation of N-Kirchhoff type with double exponential growth

Abstract

Abstract In this work, we study the weighted Kirchhoff problem − g ∫ B σ ( x ) ∣ ∇ u ∣ N d x div ( σ ( x ) ∣ ∇ u ∣ N − 2 ∇ u ) = f ( x , u ) in B , u > 0 in B , u = 0 on ∂ B , \left\{\begin{array}{ll}-g\left(\mathop{\displaystyle \int }\limits_{B}\sigma \left(x)| \nabla u\hspace{-0.25em}{| }^{N}{\rm{d}}x\right){\rm{div}}\left(\sigma \left(x)| \nabla u\hspace{-0.25em}{| }^{N-2}\nabla u)=f\left(x,u)& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}B,\\ u\gt 0& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}B,\\ u=0& \hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}\partial B,\end{array}\right. where B B is the unit ball of R N {{\mathbb{R}}}^{N} , σ ( x ) = log e ∣ x ∣ N − 1 \sigma \left(x)={\left(\log \left(\frac{e}{| x| }\right)\right)}^{N-1} , the singular logarithm weight in the Trudinger-Moser embedding, and g g is a continuous positive function on R + {{\mathbb{R}}}^{+} . The nonlinearity is critical or subcritical growth in view of Trudinger-Moser inequalities. We first obtain the existence of a solution in the subcritical exponential growth case with positive energy by using minimax techniques combined with the Trudinger-Moser inequality. In the critical case, the associated energy does not satisfy the condition of compactness. We provide a new condition for growth, and we stress its importance to check the compactness level.