Existence of a solution for a m-biharmonic Kirchhoff type equation

In this thesis, we proved the existence of a ground-state solution for the following $m$-biharmonic Kirchhoff-type equation. \begin{equation} (P_\lambda)\left\{\begin{array}{l} \displaystyle \Delta_{m}^{2} u(x)-\left[ a \left( \int_{\Omega}\mid\nabla u(x)\mid^{m} d x \right)^{m-1}+b\right] \Delta_{m} u(x)+ c\mid u(x)\mid^{m-2}u(x) \\ =f(x)\mid u(x)\mid^{-\gamma}-\lambda\mid u(x)\mid^{p-2} u(x), \\ \Delta u(x)=u(x)=0, \end{array}\right. \end{equation} where $\Omega \subseteq \mathbb{R}^{N}$ is a smooth bounded domain, $N \geq 3, \quad \gamma \in] 0,1[, \quad a, b, c \in] 0,+\infty[, $\\$\lambda \in] 0,+\infty[, \quad f \in \Lp^{q}(\Omega),\quad f(x) \geq 0, \text{ for almost everywhere } x \in \Omega,\quad p \in] 0, m^{* *}[,\quad q=\dfrac{m^{**}}{m^{**}+\gamma-1},$\\ $m^{**}=\left\{\begin{array}{lll} \dfrac{m N}{N-2 m}, & \text { lf } 1<m<\frac{N}{2}; \\ +\infty, & \text { if } m \geq\frac{N}{2}. \end{array}\right.$\\ Here $m^{** }$ is the critical Sobolev exponent for the embedding $$ \W^{2, m}(\Omega) \subseteq \Lp^{m^{**}}(\Omega). $$ Tahri and Yasid (2021) used variational methods to show the existence and uniqueness of a weak solution of a singular biharmonic problem of Kirchhoff type involving critical Sobolev exponent, $$\Delta^2 u - (a\int_\Omega |\nabla u|^2 dx + b) \Delta u + cu = f(x) |u|^{-\gamma} - \lambda |u|^{p-2} u,$$ working on a smooth bounded domain $\Omega \subseteq R^N$ with Dirichlet conditions on the boundary and some adequate conditions for the data. We extend their results whenever the biharmonic operator is replaced by the $m$-biharmonic operator, $$\Delta_m^2 u = \Delta (| \Delta u|^{m-2} \Delta u).$$ This involves adequate use and application of advanced tools of Nonlinear Analysis.

• Auteur principal: Calle Cárdenas, Christian Andrés (author)
• Format: bachelorThesis
• Langue: eng
• Publié: 2023
• Sujets: Cálculo de variaciones; Sobolev; Ecuación m-biarmónica tipo Kirchhoff; m-biharmonic Kirchhoff type equation; Calculus of variations
• Accès en ligne: http://repositorio.yachaytech.edu.ec/handle/123456789/620