A nonlinear boundary problem involving the \(p\)-bi-Laplacian operator (Q2575957)

First, the following new Sobolev's trace embedding \(W^{m,p}(\Omega)\hookrightarrow L^q(\partial\Omega)\) is shown: Theorem: Let \(\Omega\subset \mathbb R^N\), \(N > 2\), be a bounded domain of class \(C^m\) and \(1<p< +\infty\). A) For all \(u\in W^{m,p}(\Omega)\), \(mp < N\), the restriction of \(u\) to \(\partial\Omega\) (denoted also by \(u\)) belongs to \(L^q(\partial\Omega)\), for all \(q\in[1,p^*_m]\) where \(p^*_m =\frac{(N-1)p}{N-mp}\). B) For all \(u\in W^{m,p}(\Omega)\), \(mp\geq N\), the restriction of \(u\) to \(\partial\Omega\) (denoted also by \(u\)) belongs to \(L^q(\partial\Omega)\) for all \(q\in[1,+\infty)\). In case \(A: W^{m,p}(\Omega)\) is compactly embedding in \(L^q(\partial\Omega)\) for all \(q\in[1,p^*_m]\). In case \(B\) and \(mp > N\): \(W^{m,p}(\Omega)\) is compactly embedding in \(L^\infty(\partial\Omega)\cap C(\partial\Omega)\). Later this embedding is applied to the following problem: \[ \begin{aligned} \Delta^2_pu+|u|^{p-2}u & = 0\text{ in }\Omega,\\ -\frac{\partial}{\partial n}(|\Delta u|^{p-2}\Delta u)& =\lambda\rho(x)|u|^{p-2}u\text{ on }\partial\Omega,\quad u\in W^{2,p}_0(\Omega),\end{aligned}\tag{1} \] where \(\Omega\) is a bounded domain of class \(C^2\) in \(\mathbb R^N\), \(N\geq 2\), \(1<p<+\infty\), \(\lambda\in\mathbb R\) and \(\varphi\in L^r(\partial\Omega)\) with \(r=r(N,p)\) satisfying some condition. \(\Delta^2_p u:=\Delta(|\Delta u|^{p-2}\Delta u)\) is the operator of fourth order, so-called \(p\)-bi-Laplacian. Problem (1) is shown to have at least one nondecreasing sequence of positive eigenvalues \((\lambda_k)_{k\geq 1}\) by using the Ljusternik-Schnirelmann theory.