In this paper, we prove the existence of multiple weak solutions for a nonlinear nonlocal elliptic partial differential equation involving a singularity and a power nonlinearity, which is given as ${\left(-{\mathrm{\Delta }}_p\right)}^{s}u=\frac{\lambda }{u^{\gamma }}+u^q;u>0\text{\hspace{0.17em}in\hspace{0.17em}}\mathrm{\Omega }$ with zero Dirichlet boundary condition. Here, Ω is an open bounded domain in ${\mathbb{R}}^N$ with smooth boundary, N > ps, s ∈ (0, 1), λ > 0, 0 < γ < 1, 1 < p < ∞, and $p-1<q\le p_s^{*}=\frac{Np}{N-ps}$ . We employ variational techniques to show the existence of multiple positive weak solutions of the above problem. We also prove that for some α ∈ (0, s], the weak solution to the problem is in $C^{1,\alpha }\left(\overline{\mathrm{\Omega }}\right)$ .

The problem of non-isothermal fluid flow in and around a liquid drop has been studied. The temperature of the fluid is assumed to be non-constant, steady and hence is governed by the Laplace's equation. The thermal and hydrodynamic problems have been solved under nonisothermal boundary conditions assuming Stokes equations for the flow inside and outside the drop. The drag and torque on the droplet in the form of Faxen's laws are presented. The use of the drag formula has been demonstrated by few particular cases. Some important asymptotic limiting cases have been discussed.

The purpose of this article is to prove the existence of solution to a nonlinear nonlocal elliptic problem with a singularity and a discontinuous critical nonlinearity, which is given as ${(-\mathrm{\Delta })}_p^{s}u=$ $\mu g(x,u)+\frac{\lambda }{u^{\gamma }}+H(u-\alpha )u^{p_s^{*}-1}\text{in}\mathrm{\Omega },u>0\text{in}\mathrm{\Omega },$ with the zero Dirichlet boundary condition. Here, $\mathrm{\Omega }\subset {\mathbb{R}}^N$ is a bounded domain with Lipschitz boundary, s ∈ (0, 1), $2<p<\frac{N}{s}$ , γ ∈ (0, 1), λ, μ > 0, α ≥ 0 is real, $p_s^{*}=\frac{Np}{N-sp}$ is the fractional critical Sobolev exponent, and H is the Heaviside function, i.e., H(a) = 0 if a ≤ 0 and H(a) = 1 if a > 0. Under suitable assumptions on the function g, the existence of solution to the problem has been established. Furthermore, it will be shown that as α → 0⁺, the sequence of solutions of the problem for each such α converges to a solution of the problem for which α = 0.