A generalization of an extensible beam equation with critical growth in RN

Abstract

In this work we obtain an existence result for a generalized extensible beam equation with critical growth in $\mathbb{R}^{N}$ of the type $$\Delta^{2}u-M\biggl(\dis\int_{\mathbb{R}^{N}}|\nabla u|^{2} \ dx\ \biggl)\Delta u +u = \lambda f(u)+ |u|^{2^{**}-2}u \ \mbox{in} \ \ \mathbb{R}^{N}, $$ where $N\geq 5$ and $\lambda>0$. The functions $M:[0,+\infty)\rightarrow \mathbb{R}$ and $f:\mathbb{R}\rightarrow \mathbb{R}$ are continuous. Since there is a competition between the function $M$ and the critical exponent given by $2^{**}=\frac{2N}{N-4}$, we need to make a truncation on function $M$. Using the size of $\lambda$, we show that each solution of auxiliary problem is a solution of original problem. Our approach is variational and uses minimax point critical theorems