\(L^ n\) is sharp for the anti-maximum principle
This paper extends the following result of Clément and Peletier that reads as follows for the Dirichlet Laplacian on bounded smooth domains \(\Omega\subset\mathbb{R}^n\): for all \(f>0\) with \(f\in L^p(\Omega)\) and \(p>n\), there is \(\lambda_f>\lambda_1\), where \(\lambda_1\) is the first eigenvalue, such that one finds for \(\lambda\in(\lambda_1, \lambda_f)\) that the solution of \[ -\Delta u= \lambda u+f\quad\text{in }\Omega,\qquad u= 0\quad\text{on }\partial\Omega,\tag{\(*\)} \] satisfies \(u<0\). The author shows that the result above is no longer true for all \(f\in L^p(\Omega)\) with \(p\leq n\). In fact the following proposition is established: Let \(n\geq 2\). There exists \(f\in L^n(\Omega)\) with \(f>0\) such that, for all \(\lambda>\lambda_1\) and \(\lambda\) not an eigenvalue, the solution \(u_\lambda\) of \((*)\) changes sign.
scientific article; zbMATH DE number 1000134