We consider the p-adic Schro & uml;dinger equation i partial derivative tu = (D alpha+V(t))u with a time-dependent nonlo cal potential V defined on the field of p-adic numbers Qp. We construct a solution u in the space H = L2(Qp) by approximating it with a sequence of functions that solve analogous equations in finite-dimensional subspaces EN subset of H.
In this paper, we study the existence and L degrees degrees-regularity of solutions for a general class of strongly nonlinear elliptic problems associated with the differential inclusion gamma(u) + A(u) + (x, u, del u) f, where A is a Leray-Lions operator from W1,p 0 (S2, omega) into its dual W-1,p'(S2, omega & lowast;), gamma is a maximal monotone mapping with 0 is an element of gamma (0), is a nonlinear term satisfying only a growth condition, without any sign condition on s, and the right-hand side f is assumed to belong to L degrees degrees(S2).
In this paper, we classify Hopf real hypersurfaces in 2 & times; 2 and IHI2 & times; IHI2 for which the structure Jacobi operator and structure Lie operator are anticommutative.
In this paper, we first establish some maximum principles, including a simple maximum principle, the maximum principle for parabolic fractional p(x, & centerdot;)-Laplacians and the maximum principle for anti-symmetric functions. Then we consider a Hopf's lemma for parabolic fractional p(x, & centerdot;)-Laplacians, which is an extension of the results in Wang-Chen [Commun. Pure Appl. Anal. 21 (2022) 3055-3069]. In the end, applying the maximum principle for anti-symmetric functions and the method of moving planes, we obtain the symmetry and monotonicity of solutions to nonlinear parabolic fractional p(x, & centerdot;)-equations in exterior domains of a ball. With the hope of providing new perspectives and tools for subsequent research.
In this article, we introduce a new subclass of univalent functions. This class consists of all typically-real functions f such that the image of the unit disk under f is a union of arcs of ellipses with centers on the real axis, tangent to the imaginary axis and constant ratio of axes of the ellipses. We call these functions & micro;-ellipselike and denote their family by S-& micro;(c), where the parameter & micro; determines the ratio of axes. Moreover, for all z in the upper half of the unit disc the following analytic condition Im ( zf ' (z)/f(2)(z) ) < & micro; Re-2 (f (z)) Im(z f'(z))/|f(z)|(4) determines whether the function f belongs to the class S-& micro;(c). Some examples of functions from S-& micro;(c) are given.
