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

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