On a fourth-order elliptic equation involving the critical Sobolev exponent: the effect of the graph topology (Q1947403)

The author considers the following fourth-order elliptic equation involving the critical Sobolev exponent under Navier boundary conditions \[ \begin{cases} \Delta^2 u=Ku^{2^*-1}+\varepsilon u, \,\, u>0&\text{in } \Omega,\\ \Delta u=u=0 &\text{on } \partial \Omega,\end{cases}\tag{1} \] where \(\Omega \) is a bounded domain in \(\mathbb R^n\), \(n>8,\) with smooth boundary \(\partial\Omega\), \(K\) is a smooth positive function in \(\Omega\), \(\varepsilon \) is a small nonnegative number and \(2^*=\frac{2n}{n-4}\) is the critical exponent. The author first investigates the effect of the topological structure of the graph of the function \(K\) on the existence of the solution of (1) as \(\varepsilon \rightarrow 0.\) After that, solutions of (1), which blow up and concentrate around a boundary or interior local maximum point of \(K\) as \(\varepsilon \rightarrow 0,\) are constructed to estimate the number of such solutions. Finally, the nonexistence of solutions, which concentrate around a non degenerate critical point of \(K\), is also established.