We study a semilinear elliptic equation with an asymptotic linear nonlinearity. Exact multiplicity of solutions are obtained under various conditions on the nonlinearity and the spectrum set. Our method combines a bifurcation approach and Leray–Schauder degree theory.

In this paper we study the existence of solutions for the initial value problem for semilinear functional differential equations of fractional order with state-dependent delay. The nonlinear alternative of Leray-Schauder type is the main tool in our analysis.

We establish interior Schauder estimates for kinetic equations with integro-differential diffusion. study of the form $f_t + v \cdot \nabla_x f = \mathcal L_v c$, where $\mathcal L_v$ is an diffusion operator order $2s$ acting in $v$-variable. Under suitable ellipticity and H\"older continuity conditions on kernel L_v$, we obtain a priori estimate $f$ properly scaled space.