A note on the strong maximum principle for elliptic differential inequalities
The paper gives necessary and sufficient conditions for the validity of the strong maximum principle and the compact support principle for quasilinear elliptic differential inequalities in divergence form. More precisely, consider the differential inequalities \[ \text{ div} \big\{ A(|Du|)Du\big\}-f(u)\leq 0,\qquad u\geq 0,\tag{1} \] \[ \text{ div} \big\{ A(|Du|)Du\big\}-f(u)\geq 0,\qquad u\geq 0,\tag{2} \] in a (possibly unbounded) domain \(D\subset {\mathbb{R}}^n,\) \(n\geq 2,\) where \(A=A(t)\) and \(f(t)\) satisfy the following conditions: \(A\in C(0,\infty),\) \(\Omega(t)=tA(t)\) is strictly increasing in \((0,\infty)\) and \(\lim_{t\to 0}\Omega(t)=0,\) \(f\in C[0,\infty),\) \(f(0)=0\) and \(f\) is non-decreasing on some interval \([0,\delta),\) \(\delta>0.\) Set \(F(t)=\int_0^t f(s)ds,\) \(s>0\) and \[ H(t)=t\Omega(t)-\int_0^t \Omega(s)ds,\quad t\geq 0. \] By a solution of \((1)\) or \((2)\) one means a non-negative function \(u\in C^1(D)\) which satisfies \((1)\) or \((2)\) in a distribution sense. By definition, the strong maximum principle holds for \((1)\), if \(u(x_0)=0\) for some solution \(u\) of \((1)\) and some \(x_0\in D\) implies \(u\equiv 0\) in \(D\), while \((2)\) obeys the compact support principle, if \(\lim_{|x|\to\infty} u(x)=0\) for some solution \(u\) to \((2)\) in an exterior domain \(D\) yields that \(u\) has compact support in \(D.\) The main results of the paper are as follows: 1) The strong maximum principle holds for \((1)\) if and only if either \(f(s)\equiv 0\) for \(s\in [0,\mu),\) \(\mu>0,\) or \(f(s)>0\) for \(s\in(0,\delta)\) and \(\int_0^\delta {{ds}\over {H^{-1}(F(s))}}=\infty.\) 2) The compact support principle holds for \((2)\) if and only if \(f(s)>0\) for \(s\in(0,\delta)\) and \(\int_0^\delta {{ds}\over {H^{-1}(F(s))}}<\infty\).
elliptic differential inequalities in divergence form
scientific article; zbMATH DE number 1463866