We establish a continuous embedding $W^{s(\cdot),2}(\Omega)\hookrightarrow L^{\alpha(\cdot)}(\Omega)$, where the variable exponent $\alpha(x)$ can be close to critical $2_{s}^*(x)=\frac{2N}{N-2\bar{s}(x)}$, with $\bar{s}(x)=s(x,x)$ for all $x\in\bar{\Omega}$. Subsequently, this is used prove multiplicity of solutions nonlocal degenerate Kirchhoff problems singular exponent. Moreover, we also ob...

In this work, we are concerned with the terminal value problem for time fractional equation (in sense of Conformable derivative) a nonlocal term Kirchhoff type$ \partial_t^\alpha u = K\Big(\|\nabla u\|_{L^2(\mathcal{D})}\Big)\Delta + f(x,t), \quad (x,t) \in (0,T)\times \mathcal{D} $subject to final data which is blurred by random Gaussian white noise. The principal goal article recover solution...

Abstract In this paper, the existence of two solutions for superlinear fourth-order impulsive elastic beam equations is obtained. We get theorems via variational methods and corresponding two-critical-point theorems. Combining with Newton-iterative method, an example presented to illustrate value obtained