Classical solutions up to the boundary to some higher order semilinear Dirichlet problems
The authors consider a domain \(\Omega \subset \mathbb{R}^{n}\), \(n\geq 2m\), with boundary \(\partial \Omega \in C^{2m,\gamma }\), \(f\in C^{0,\gamma }( \overline{\Omega })\), \(g\in C^{0,\gamma }(\overline{\Omega }\times \mathbb{R} )\), and the semilinear elliptic problem \((-\Delta )^{m}u(x)+g(x,u(x))=f(x)\) posed in \(\Omega \), with the homogeneous Dirichlet boundary conditions \(( \frac{\partial }{\partial \nu })^{k}u(x)=0\), on \(\partial \Omega \), for \( k=0,\ldots ,m-1\). \(g\) further satisfies the sign condition \(tg(x,t)\geq 0\) for all \(x\in \Omega \) and \(t\in \mathbb{R}\), and one growth condition among three ones. The main result of the paper proves the existence of a classical solution \(u\in C^{2m,\gamma }(\overline{\Omega })\cap C_{0}^{m-1}(\overline{ \Omega })\), with \[C_{0}^{m-1}(\overline{\Omega })=\left\{u\in C^{m-1}(\overline{ \Omega }): \left(\frac{\partial }{\partial \nu }\right)^{k}u(x)=0\text{ on }\partial \Omega,\text{ for }k=0,\ldots ,m-1\right\}.\] For the proof, the authors first quote from their preprint [``Note on a sign-dependent regularity for the polyharmonic Dirichlet problem'', Preprint, \url{arXiv:2009.09747}] regularity results for the linear polyharmonic problem. They then consider the semilinear problem obtained when replacing \(g\) by \(g_{M}\) defined as \(g_{M}(x,t)=\min \{g(x,t),M\}\) for \(t\geq 0\) and \(g_{M}(x,t)=\max \{g(x,t),-M\}\) for \(t<0\). They prove the existence of a weak solution to the corresponding semilinear problem, proving the existence of a minimizing sequence for the associated energy functional and using compactness arguments. They prove uniform bounds on this solution in the \(W_{0}^{m,2}(\Omega )\)-norm and on \( g_{M}(\cdot ,u_{M})u_{M}\) in the \(L^{1}(\Omega )\)-norm. They observe that the weak solution is in fact a classical one. Coming back to the original problem, the authors distinguish the cases \(n\in \lbrack 2m,6m]\) and \(n>6m\) in the case of a one-sided growth condition on \(g\). They use the previously proved result, Sobolev embeddings and the Agmon-Douglis-Nirenberg results.
higher order semilinear elliptic problem
scientific article; zbMATH DE number 7335756
scientific article