Nonexistence results of entire solutions for superlinear elliptic inequalities
The paper is concerned with nonexistence results of classical entire solutions of the differential inequality: \(\Delta u\geq p(x)f(u)\), where \(\Delta\) is the Laplace operator, \(p(x)\) is a positive continuous function in \(\mathbb{R}^ N(N\geq 2)\), and \(f(u)\) is a positive continuous function which is defined either in \(\mathbb{R}_ +\) or in \(\mathbb{R}\). Since the case \(p(x)=1\) has been investigated in full detail, here \(p(x)\) is allowed to decay to zero as \(| x|\to\infty\). First \(f\) is supposed to be convex, and nonexistence criteria are derived through the analysis of an ordinary differential equation satisfied by the spherical mean of a possible entire solution of the given inequality. A special result for \(N\geq 3\) is the following. If \(\liminf_{| x|\to\infty}| x|^ 2p(x)>0\) and if \(\int^ \infty_ 1G(z)dz<\infty\), with \(G(z)^{-2}=\int^ z_ 0f(s)ds\) then the above inequality has no positive entire solutions when \(\text{dom} f=\mathbb{R}_ +\), and it has no entire solutions when \(\text{dom} f=\mathbb{R}\). Moreover \(f\) is supposed to be (instead of convex) Lipschitz continuous and strictly increasing, and a nonexisting criterion is developed by using a comparison method based on the maximum principle.
scientific article; zbMATH DE number 60582
scientific article