We consider the scalar-field type nonlinear elliptic equation under the homogeneous Neumann condition on a spherical cap. We discuss all the non-trivial solutions which bifurcate from the trivial positive solution. We show that the bifurcating points of non-azimuthal solutions are close to each other when the cap covers almost whole the sphere.

We propose a systematic study of transformations of A-hypergeometric functions. Our approach is to apply changes of variables corresponding to automorphisms of toric rings, to Euler-type integral representations of A-hypergeometric functions. We show that all linear A-hypergeometric transformations arise from symmetries of the corresponding polytope. As an application of the techniques developed here, we show that the Appell function F₄ does not admit a certain kind of Euler-type integral representation.

The aim of this paper is to prove the existence of -bounded solution operator families for a resolvent problem on the upper half-space arising from a compressible fluid model of Korteweg type with free boundary condition. Such a compressible fluid model was derived by Dunn and Serrin (1985) and studied by Kotschote (2008) as a boundary value problem with non-slip boundary condition.

This paper deals with a two-species Keller-Segel-Stokes system with competitive kinetics under homogeneous Neumann boundary conditions in a bounded domain with smooth boundary. The main purpose of this paper is to obtain global existence and stabilization of classical solutions to the system in the 3-dimensional case under the smallness conditions for chemotactic interactions. To this end, this paper develops a maximal Sobolev regularity result for Stokes operator involving a time weighted function, which seems new in the existing literature (see Lemma 2.3 in this paper).

We consider higher order Schrödinger type operators with nonnegative potentials. We assume that the potential belongs to the reverse Hölder class which includes nonnegative polynomials. We establish estimates of the fundamental solution and show L^p boundedness of some Schrödinger type operators. We use pointwise estimates by the Hardy-Littlewood maximal operator to prove our results.