## Hopf’s lemma and anti-maximum principle in general domains.(English)Zbl 0831.35114

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 H. Berestycki, L. Nirenberg and 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.
Reviewer: G.Porru (Cagliari)

### MSC:

 35P05 General topics in linear spectral theory for PDEs 35B40 Asymptotic behavior of solutions to PDEs 35J15 Second-order elliptic equations

Zbl 0806.35129
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