Antimaximum principle for quasilinear problems. (English) Zbl 1241.35059

Arcoya, David; Rossi, Julio D.

Adv. Differ. Equ. 9, No. 9-10, 1185-1200 (2004).

Summary: In this paper, for a bounded, smooth domain \(\Omega\in\mathbb R^N\), \(h\in L^r(\Omega)\), \(r>N/2\), \(g\in L^s(\partial\Omega)\), \(s>N-1\), we prove the maximum and antimaximum principle for the quasilinear boundary value problem \[ -\text{div}(A(u)\nabla u)+u=h\quad\text{in}\;\Omega,\quad A(u)\nabla u\cdot\eta=\lambda f(u)+g\quad \text{on}\;\partial\Omega, \] where \(A\) is elliptic and bounded and \(f\) is asymptotically linear. The sharpness of this result (\(r>N/2\) and \(s>N-1\)) is discussed for the linear boundary-value problem \[ -\Delta u+u=h\quad\text{in}\;\Omega,\quad \partial u/\partial\eta=\lambda u+g\quad \text{on}\;\partial\Omega. \]

MSC:

35J60	Nonlinear elliptic equations
35P30	Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
35B50	Maximum principles in context of PDEs