abstract 104

Bulletin of Computational Applied Mathematics (Bull CompAMa) 

104

m-Biharmonic Kirchhoff-type equation with singularity and critical exponent (Research Paper)

Juan Ricardo Mayorga Zambrano, Christian Calle-Cárdenas, Josué Castillo-Jaramillo.

We consider the m-biharmonic Kirchhoff-type integro-differential equation

\Delta_{m}^{2} u-\left[ a \left( \int_{\Omega}\mid\nabla u(y)\mid^{m} d y \right)^{m-1}+b\right]  \Delta_{m} u \\

\qquad  + c | u|^{m-2}u = f(x) | u|^{-\gamma}-\lambda| u|^{p-2} u, \qquad x\in \Omega, \\

 u = \Delta u = 0, \qquad \qquad\qquad\qquad\qquad\quad\qquad \:\:\:\:\: x\in \partial \Omega,

 (Pm)

where $\Omega \subseteq \mathbb{R}^\mathrm{N}$ is a smooth bounded domain,  $N\geq 3$, $\Delta_m^2$ is the $m$-biharmonic operator, $\gamma\in ]0,1[$, $a, b, c, \lambda\in ]0,+\infty[$, $p\in]1-\gamma, m^{**}]$, $f\in \mathrm{L}^q(\Omega)$ is a.e. positive, $q=m^{**}/(m^{**}+\gamma-1)$ and $m^{**} = mN/(N-2m)$ if $1<m<N/2$, and $m^{**}=+\infty$ if $m\geq N/2$. Even though the energy functional associated to (Pm) presents a singularity and, as a consequence, is not  Fréchet differentiable, it's proved by a direct method - including the case of Sobolev critical exponent - that (Pm) has a positive ground-state solution.

Keywords: Kirchhoff-type equation; m-biharmonic operator; integro-differential equation; critical Sobolev exponent.

Cite this paper:

Mayorga-Zambrano J.R., Calle-Cárdenas C., Castillo-Jaramillo J.

m-Biharmonic Kirchhoff-type equation with singularity and critical exponent,

Bull. Comput. Appl. Math. (Bull CompAMa),

Vol. 12, No.2 pp.129-146, 2024.