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

Abstract

In this work, we study the weighted Kirchhoff problem −g∫Bσ(x)∣∇u∣Ndxdiv(σ(x)∣∇u∣N−2∇u)=f(x,u)inB,u>0inB,u=0on∂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 BB is the unit ball of RN{{\mathbb{R}}}^{N}, σ(x)=loge∣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 gg 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.

Keywords

• nonlocal kirchhoff equation
• trudinger-moser’s inequality
• critical double exponential nonlinearities
• mountain pass theorem
• 35j20
• 35j25
• 35j60