We use variational methods to study the existence and multiplicity of solutions for following quasi-linear partial differential equation: \[ \left \{ \begin {matrix} {-\triangle _{p} u = \lambda |u|^{r-2}u + \mu \textstyle {\frac {|u|^{q-2}}{|x|^{s}}}u \quad \text {in $\Omega $}, {}} {\hphantom {-} u|_{\partial \Omega } 0, }\hfill \end {matrix}\right . \] where $\lambda$ $\mu$ are two positive parameters $\Omega$ is a smooth bounded domain in $\mathbf {R}^n$ containing $0$ its interior. The...

Relatively little is known about the ability of numerical methods for stochastic differential equations (SDEs) to reproduce almost sure and small‐moment stability. Here, we focus on these stability properties in limit as timestep tends zero. Our analysis motivated by an example exponentially surely stable nonlinear SDE which Euler–Maruyama (EM)method fails this behavior any nonzero timestep. We begin showing that EM correctly reproduces exponential sufficiently small timesteps scalar linear...

This paper focuses on stochastic partial differential equations (SPDEs) under two-time-scale formulation. Distinct from the work in existing literature, systems are driven by $\alpha$-stable processes with $\alpha \in(1,2)$. In addition, SPDEs either modulated a continuous-time Markov chain finite state space or have an addition fast jump component. The inclusion of is for needs treating random environment, whereas process enables consideration discontinuity sample paths processes. Assuming...

In this paper, by the weak convergence method, based on a variational representation for positive functionals of Poisson random measure and Brownian motion, we establish uniform large deviation principles (LDPs) class neutral stochastic differential equations driven jump processes. As byproduct, also obtain LDPs delay which, in particular, allow coefficients to be highly nonlinear with respect argument.

As the limit equations of mean-field particle systems perturbed by common environmental noise, McKean-Vlasov stochastic differential with noise have received a lot attention. Moreover, past dependence is an unavoidable natural phenomenon for dynamic in life sciences, economics, finance, automatic control, and other fields. Combining two aspects above, this paper delves into class nonlinear functional (MV-SFDEs) noise. The well-posedness MV-SFDEs first demonstrated through application Banach...

We consider stochastic different equations on $\mathbb R^d$ with coefficients depending the path and distribution for whole history. Under a local integrability condition time-spatial singular drift, well-posedness Lipschitz continuity in initial values are proved, which is new even independent case. Moreover, under monotone condition, asymptotic log-Harnack inequality established, extends corresponding result of [5] derived

In this paper, we are concerned with convergence rate of Euler–Maruyama scheme for stochastic differential equations Hölder–Dini continuous drifts. The key contributions as follows: (i) by means regularity non-degenerate Kolmogrov equation, investigate a class which allow the drifts to be Dini and unbounded; (ii) aid regularization properties degenerate discuss range where order $$\frac{2}{3}$$ respect first component merely Dini-continuous concerning second component.

This work is concerned with the approximate controllability of nonlinear fractional impulsive stochastic differential system under assumption that corresponding linear approximately controllable. Using calculus, analysis, and technique control theory, a new set sufficient conditions for obtained. The results in this paper are generalizations continuations recent on issue. An example given to illustrate efficiency main results.