No abstract

A generalized nonlinear Picone identity for the p-biharmonic operator is established in this paper. As applications, a Sturmian comparison principle to the p-biharmonic equation with singular term, a Liouville's theorem to the p-biharmonic system, and a generalized Hardy-Rellich type inequality are obtained. MSC: Primary 26D10; secondary 26D15

In this paper, we derive an anisotropic Picone identity for the anisotropic Laplacian, which contains some known Picone identities. As applications, a Sturmian comparison principle to the anisotropic elliptic equation and an anisotropic Hardy type inequality are shown.

<abstract><p>In this paper, we concern with the predator-prey system with generalist predator and density-dependent prey-taxis in two-dimensional bounded domains. We derive the existence of classical solutions with uniform-in-time bound and global stability for steady states under suitable conditions through the Lyapunov functionals. In addition, by linear instability analysis and numerical simulations, we conclude that the prey density-dependent motility function can trigger the periodic pattern formation when it is monotone increasing.</p></abstract>

In this paper, we consider anisotropic elliptic problems on a bounded smooth domain, nonexistence of nontrivial weak solutions are obtained by two different simplified methods. We point out the results of nonexistence of nontrivial weak solutions not only depend on the size and shape of domain, but also the constants p 1 and p N in anisotropic elliptic problems.