A note on the strong maximum principle and the compact support principle. (English) Zbl 1157.35018
Summary: We are concerned with the strong maximum principle (SMP) and the compact support principle (CSP) for non-negative solutions to quasilinear elliptic inequalities of the form
\[ \text{div}(A(|\nabla u|)\nabla u)+ G(|\nabla u|)- f(u)\leq 0\quad\text{in }\Omega, \]
and
\[ \text{div}(A(|\nabla u|)\nabla u)+ G(|\nabla u|)- f(u)\geq 0\quad\text{in }\mathbb R^N\setminus B_r(0), \]
respectively. We give new conditions on the data \((A,G,f)\) to obtain (SMP) and (CSP). When these conditions are particularized to the \(m\)-Laplacian and pure power nonlinearities we completely classify the data according to the validity of the (CSP) or the (SMP). In doing so we clarify the general situation and we consider a case not covered in the literature.
\[ \text{div}(A(|\nabla u|)\nabla u)+ G(|\nabla u|)- f(u)\leq 0\quad\text{in }\Omega, \]
and
\[ \text{div}(A(|\nabla u|)\nabla u)+ G(|\nabla u|)- f(u)\geq 0\quad\text{in }\mathbb R^N\setminus B_r(0), \]
respectively. We give new conditions on the data \((A,G,f)\) to obtain (SMP) and (CSP). When these conditions are particularized to the \(m\)-Laplacian and pure power nonlinearities we completely classify the data according to the validity of the (CSP) or the (SMP). In doing so we clarify the general situation and we consider a case not covered in the literature.
MSC:
| 35B50 | Maximum principles in context of PDEs |
| 35J60 | Nonlinear elliptic equations |
| 35R45 | Partial differential inequalities and systems of partial differential inequalities |
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