A generalization of an extensible beam equation with critical growth in [formula omitted

Abstract

In this work we obtain an existence result for a generalized extensible beam equation with critical growth in RN of the type Δ2u−M(∫RN|∇u|2dx)Δu+u=λf(u)+|u|2∗∗−2uinRN, where N≥5 and λ>0. The functions M:[0,+∞)→R and f:R→R are continuous. Since there is a competition between the function M and the critical exponent given by 2∗∗=2NN−4, we need to make a truncation on function M. Using the size of λ, we show that each solution of auxiliary problem is a solution of original problem. Our approach is variational and uses minimax point critical theorems.