In this article, we mainly study the existence of normalized ground state solutions for Choquard-Kirchhoff type equations with Sobolev critical exponents, -( a+b ∫ _ℝ^3|∇ u|^2dx) Δ u=λ u + |u|^4u + μ (I_α *|u|^q)|u|^q-2u in ℝ^3. We fix their mass, that is, ∫ _ℝ^3|u|^2dx=c^2 and assume that a>0, b>0, c>0, μ∈ℝ, (3+α)/30 , under appropriate assumptions about α , we obtain a local minimizer and mountain pass type solutions on their respective intervals. When μ <0 , we show that there is no ground state for the question. We briefly discuss the asymptotic behavior and the behavior at critical points.
In mathematical optimization theory, the linear complementarity problem plays a vital role in many areas such as bimatrix game theory, market equilibrium, computational complexity, and many more. The nature of the solution of the linear complementarity problem can be discussed with the help of the matrix A involved in the problem. Sign reversing is a property of matrices along with a given vector. An n × n matrix A reverses the sign of a vector x in an n-dimensional real vector space if it satisfies x_i(Ax)_i≤ 0 for all index i. The concept of sign reversing is a useful tool to identify and characterize certain matrix classes in linear complementarity problems. The sign reversing set of a matrix A is defined as {x: x_i(Ax)_i ≤ 0, for all i} . In this paper, we have defined the concept of sign reversing set for operators on separable Hilbert spaces relative to a given orthonormal basis and we have observed that an operator can have different sign reversing sets relative to different orthonormal bases. With the help of sign reversing property, we generalized the sufficient matrix classes to operators in infinite dimensional Hilbert spaces.
The present paper aims to investigate super-biderivations and 2-local super-derivations on the n-th super Schrödinger algebra 𝒮_n , which is viewed as a semidirect product of the Lie superalgebra 𝔬𝔰𝔭(1|2) and the (3n+1) -dimensional Heisenberg Lie superalgebra. We first give a description of all super-derivations of 𝒮_n . We also show that any derivation of the n-th Schrödinger algebra (𝒮_n)_0̅ is isomorphic to the restriction of an even super-derivation on 𝒮_n to (𝒮_n)_0̅ . Then we prove that all super-biderivations (without skew-supersymmetric conditions) on 𝒮_n are inner. Based on this result, we also characterize the linear super-commuting maps and commutative post-Lie superalgebra structures on 𝒮_n . Finally, we show that every 2-local super-derivation on 𝒮_n is a super-derivation.
In this paper we introduce the concepts of α -neutralized Bowen entropy ( α >0 ) and neutralized Bowen entropy dimension for subsets, defined via neutralized Bowen open balls. We then establish a number of variational principles for these quantities. Finally, by introducing a new measure‑theoretic version of α -neutralized entropy for subsets, we prove several inverse variational principles for α -neutralized Bowen entropy and for neutralized Bowen entropy dimension.
