Existence and bifurcation of positive solutions to a Kirchhoff type equationare considered by using topological degree argument and variational method. Here f is a continuous function which is asymptotically linear at zero and is asymptotically 3-linear at infinity. The new results fill in a gap of recent research about the Kirchhoff type equation in bounded domain, and in our results the nonlinearity may be resonant near zero or infinity.

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In this paper we study the multiplicity of nontrivial solutions of semilinear elliptic boundary value problems which may be double resonance near infinity between two consecutive eigenvalues of − with zero Dirichlet boundary data. The methods we use here are Morse theory, minimax methods and bifurcation theory.

In this paper, we study the existence and multiplicity of solutions of the operator equation Kfu = u in the real Hilbert space L 2 (G). Under certain conditions on the linear operator K, we establish the conditions on f which are able to guarantee that the operator equation has at least one solution, a unique solution, and infinitely many solutions, respectively. The monotone operator principle and the critical point theory are employed to discuss this problem, respectively. In argument, quadratic root operator K 1/2 and its properties play an important role. As an application, we investigate the existence and multiplicity of solutions to fourthorder boundary value problems for ordinary differential equations with two parameters, and give some new existence results of solutions.

Microglia are brain resident cells that function as brain phagocytic macrophages. The inflammatory responses of microglia induced by pathologic insults are key regulators in the progression of various neurological disorders. Currently, little is known about how these responses are regulated intrinsically. Here, it is observed that LPS-activated microglia exhibit distinct N6-methyladenosine (m6A) methylation patterns that are positively correlated with the expression patterns of corresponding mRNAs. High-throughput analyses and molecular studies both identified Igf2bp1 as the most significantly regulated m6A modifiers in activated microglia. Perturbation of function approaches further indicated Igf2bp1 as a key mediator for LPS-induced m6A modification and microglial activation presumably via enhancing the m6A methylation and stability of Gbp11 and Cp mRNAs. Thus, our study provides a possible mechanism for the m6A methylation-mediated microglia regulation and identifies Igf2bp1 as a potential target for modulating the inflammatory responses of microglia.