Hopf's lemma and anti-maximum principle in general domains
Let \(L : = M + c(x)\), where \(M : = a_{ij}\partial_{ij} + b_i \partial_i\) is a uniformly elliptic operator in a bounded domain \(\Omega \subset \mathbb{R}^n\). For \(\lambda \in\mathbb{R}\), the Dirichlet problem \[ (L + \lambda) u = f \text{ in } \Omega, \quad u = u_0 \text{ on } \partial \Omega \tag{*} \] is investigated. The function \(f\) is assumed to be in \(L^n (\Omega)\) and the solution \(u\) is in \(W^{1,n}_{\text{loc}} (\Omega)\). The boundary condition \(u = u_0\) on \(\partial \Omega\) is understood in a sense introduced by \textit{H. Berestycki, L. Nirenberg} and \textit{S. R. S. Varadhan} in [Commun. Pure Appl. Math. 47, No. 1, 47-92 (1994; Zbl 0806.35129)]. The author considers the map \(T : L^n (\Omega) \to L^n (\Omega)\) that associates to a function \(f \in L^n (\Omega)\) the solution \(u\) of the Dirichlet problem \[ (L - k) u = f \text{ in } \Omega, \quad u = u_0 \text{ on } \partial \Omega, \] where \(k\) is a constant such that \(c(x) - k < 0\). The main results are the following. (1) \(T\) is compact and there exists \(\mu_0\) positive eigenvalue of \(T\) and \(T^*\) to which correspond respectively the positive eigenfunctions \(\varphi \in L^n (\Omega)\) and \(\psi \in L^{n/(n - 1)} (\Omega)\). (2) The spectrum of \(L\) contains only isolated eigenvalues with no finite accumulation point. (3) If \(\lambda_1\) is the first eigenvalue of \(L\) and if \(f\) satisfies suitable conditions then there exists \(\delta\) such that if \(0 < \lambda - \lambda_1 \leq \delta\) then a positive solution of \((*)\) exists. (4) Suppose the origin belongs to the boundary of \(\Omega\) and that an interior cone condition holds. Let \(a_{ij} (0) = \delta_{ij}\) and let \(w\) be a negative smooth solution of \(Lw \geq 0\) such that \(w(0) = 0\). Then, there exist positive constants \(\delta\) and \(\alpha\) such that \(w(x) + \delta |x |^\alpha \leq 0\) in a suitable cone with vertex at the origin.
scientific article; zbMATH DE number 784948