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

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.

Existence Solutions for a Weighted Biharmonic Equation with Critical Exponential Growth

“…Furthermore, problem (1.4), involving a potential, has been studied by Baraket and Jaidane [6]. Moreover, we mention that Abid et al [1] have proved the existence of a positive ground state solution for a weighted second-order elliptic problem of Kirchhoff type, with nonlinearities having a double exponential growth at infinity, using minimax techniques combined with Trudinger-Moser inequality. This notion of critical exponential growth was then extended to higher order Sobolev spaces by Adams' [2].…”

Anisotropic Moser–Trudinger-Type Inequality with Logarithmic Weight

“…They obtained the existence of solutions via variational methods based on a new Moser-Trudinger-type inequality for the Heisenberg group H n . Moreover, in [16], the authors also focus on a Kirchhoff-type problem and establish the existence of a radial solution in the subcritical growth case by the Moser-Trudinger inequality and minimax method.…”

Ground states of weighted 4D biharmonic equations with exponential growth

“…The authors proved that there is a non-trivial solution to this problem using mountain pass Theorem. Also, Abid et al [16] investigated the weighted second-order elliptic problem of Kirchhoff type, which is defined as follows:…”

“…The authors proved that there is a non‐trivial solution to this problem using mountain pass Theorem. Also, Abid et al [16] investigated the weighted second‐order elliptic problem of Kirchhoff type, which is defined as follows: $$$$ \left\{\begin{array}{rcll}-g\left({\int}_B\sigma (x){\left|\nabla u\right|}^N\kern0.1em dx\right)\operatorname{div}\left(\sigma (x){\left|\nabla u\right|}^{N-2}\nabla u\right)& =f\left(x,u\right)& \mathrm{in}\kern0.4em & B,\\ {}u& >0& \mathrm{in}\kern0.4em & B,\\ {}u& =0& \mathrm{on}\kern0.4em & \partial B,\end{array}\right. $$$$ where $$$ N\ge 2,\rho (x)={\left(\log \left(e/|x|\right)\right)}^{N-1} $$$, and the function $$$ f\left(x,t\right) $$$ is continuous in $$$ B\times \mathrm{\mathbb{R}} $$$ and behaves like …”