A class of fourth-order elliptic equations with concave and convex nonlinearities in RN

In this article, we study the multiplicity of solutions for a class of fourth-order elliptic equations with concave and convex nonlinearities in RN. Under the appropriate assumption, we prove that there are at least two solutions for the equation by Nehari manifold and Ekeland variational principle, one of which is the ground state solution

Direct Aerobic Carbonylation of C(sp2)-H and C(sp3)-H Bonds through Ni/Cu Synergistic Catalysis with DMF as the CO Source

The direct carbonylation of aromatic sp2 and unactivated sp3 C–H bonds of amides was achieved via nickel/copper catalysis under atmospheric O2 with the assistance of a bidentate directing group. The sp2 C–H functionalization showed high regioselectivity and good functional group compatibility. The sp3 C–H functionalization showed high site-selectivity by favoring the C–H bonds of α-methyl groups over those of the α-methylene, β- or γ-methyl groups. Moreover, this reaction showed a predominant preference for functionalizing the α-methyl over α-phenyl group. Mechanistic studies revealed that nickel/copper synergistic catalysis is involved in this process

Step-like contrast structure of singularly perturbed optimal control problem

In this paper, the existence of step-like contrast structure for a class of singularly perturbed optimal control problem is shown by the contrast structure theory. By means of direct scheme of boundary function method, we construct the uniformly valid asymptotic solution for the singularly perturbed optimal control problem. Finally, an example is presented to show the result

The Relationship between Virtual Reality Technology and Anxiety State of Parturient Women with Labor Pain

As per the authors' request, the sequence of the affiliations have been adjusted accordingly