Existence, multiplicity and bifurcation for critical polyharmonic equations (Q2721856)

Let \(\Omega\) be a bounded domain in \(\mathbb{R}^N\) with \(N\geq 3\) and \(\partial \Omega\) be sufficiently smooth. The authors deal with the semilinear elliptic equation of higher order involving critical Sobolev exponents NEWLINE\[NEWLINE\begin{cases} (-\Delta_x)^m u=|u|^{p-1} u+f(x,u),\quad & x\in\Omega\\ \left.{ \partial^j u\over\partial \nu^j}\right |_{\partial \Omega}=0,\quad & j=0,1, \dots,m-1 \end{cases}\tag{1}NEWLINE\]NEWLINE where, and throughout this paper, \(\vec\nu\) is the unit outward normal to \(\partial\Omega\), \(2\leq 2m<n\), \(p={n+2m\over n-2m}\), \(f(x,0)=0\) and \(f(x,u)\) is a lower-order perturbation of \(|u|^p\) in the sense \(\lim_{|u|\to+\infty} f(x,u)/ |u|^p=0\) uniformly for \(x\in\Omega\). By using the abstract critical point theorem, the multiple results and bifurcation for (1) are obtained.