This article concerns about the existence of a positive SOLA (Solutions Obtained as Limits of Approximations) for the following singular critical Choquard problem involving fractional power of Laplacian and a critical Hardy potential. 0.1
 $$\begin{aligned} \begin{aligned} (-\Delta )^su-\alpha \frac{u}{|x|^{2s}}&=\lambda u+ u^{-\gamma }+\beta \left( \int _{\Omega }\frac{u^{2_b^*}(y)}{|x-y|^b}dy\right) u^{2_b^*-1}+\mu ~\text {in}~\Omega ,\\ u&>0~\text {in}~\Omega ,\\ u&= 0~\text {in}~\mathbb {R}^N{\setminus }\Omega . \end{aligned} \end{aligned}$$
 Here, $$\Omega $$
 is a bounded domain of $$\mathbb {R}^N$$
 , $$s\in (0,1)$$
 , $$\alpha ,\lambda $$
 and $$\beta $$
 are positive real parameters, $$N>2s$$
 , $$\gamma \in (0,1)$$
 , $$0<b<\min \{N,4s\}$$
 , $$2_b^*=\frac{2N-b}{N-2s}$$
 is the critical exponent in the sense of Hardy–Littlewood–Sobolev inequality and $$\mu $$
 is a bounded Radon measure in $$\Omega $$
 .

A critical fractional choquard problem involving a singular nonlinearity and a radon measure

We prove the existence of a positive {\it SOLA (Solutions Obtained as Limits of Approximations)} to the following PDE involving fractional power of Laplacian \begin{equation} 
\begin{split} 
(-\Delta)^su&= \frac{1}{u^\gamma}+\lambda u^{2_s^*-1}+\mu ~\text{in}~\Omega, u&>0~\text{in}~\Omega, u&= 0~\text{in}~\mathbb{R}^N\setminus\Omega. 
\end{split} 
\end{equation} Here, $\Omega$ is a bounded domain of $\mathbb{R}^N$, $s\in (0,1)$, $2s<N$, $\lambda,\gamma\in (0,1)$, $2_s^*=\frac{2N}{N-2s}$ is the fractional critical Sobolev exponent and $\mu$ is a nonnegative bounded Radon measure in $\Omega$.

Existence of positive solutions for a singular elliptic problem with critical exponent and measure data

In this work, we determine the general terms of t-cobalancers, t-cobalancing numbers and Lucas t-cobalancing numbers by solving the Pell equation 2x − y = 2t − 1 for some fixed integer t ≥ 1.

/pdf/t-cobalancing-numbers-and-t-cobalancers-50y51khtj0.pdf

t-cobalancing numbers and t-cobalancers

In this paper, we study the effect of Hardy potential on the existence or non-existence of solutions to a fractional Laplacian problem involving a singular nonlinearity. Also, we mention a stability result.

Nonlocal Lazer-McKenna type problem perturbed by the Hardy's potential and its parabolic equivalence

In this paper, we study the existence of a positive solution to the elliptic problem: (−Δ)su=u−q+f(x)h(u)+μin Ω,u>0in Ω,u=0in (RN∖Ω). Here Ω⊂RN (N>2s) is an open bounded domain with smooth boundary...

Existence of a unique positive entropy solution to a singular fractional Laplacian

The aim of this paper is to prove the existence of solution for a partial differential equation involving a singularity with a general nonnegative, Radon measure μ as its nonhomogenous term which is given as −Δu = f(x)h(u) + μ in Ω, u = 0 on ∂Ω, u > 0 on Ω, where Ω is a bounded domain of R N , f is a nonnegative function over Ω.

Elliptic Partial Differential Equation Involving a Singularity and a Radon Measure

In this article, we prove the existence of solutions to a nonlinear nonlocal elliptic problem with a singualrity and a discontinuous critical nonlinearity which is given as follows. 
\begin{align} 
\begin{split}\label{main_prob} 
(-\Delta)_p^su&=\mu g(x,u)+\frac{\lambda}{u^\gamma}+H(u-\alpha)u^{p_s^*-1},~\text{in}~\Omega 
u&>0,~\text{in}~\Omega, 
u&=0,~\text{in}~\mathbb{R}^N\setminus\Omega, 
\end{split} 
\end{align} 
where $\Omega\subset\mathbb{R}^N$ is a bounded domain with Lipschitz boundary, $s\in (0,1)$, $2 0$, $\alpha\geq 0$ is real, $H$ is the Heaviside function, i.e. $H(a)=0$ if $a\leq 0$, $H(a)=1$ if $a>0$ and $p_s^*=\frac{Np}{N-sp}$ is the fractional critical Sobolev exponent. 
Under suitable assumptions on the function $g$, we prove the existence of solution to the problem. Furthermore, we show that as $\alpha\rightarrow0^+$, the sequence of solutions of $\eqref{main_prob}$ for each such $\alpha$ converges to a solution of the problem for which $\alpha=0$.

A singular elliptic problem involving fractional $p$-Laplacian and a discontinuous critical nonlinearity

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 (−Δ)psu= μg(x,u)+λuγ+H(u−α)ups*−1inΩ,u>0inΩ, with the zero Dirichlet boundary condition. Here, Ω⊂RN is a bounded domain with Lipschitz boundary, s ∈ (0, 1), 2 0, α ≥ 0 is real, ps*=NpN−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.

A singular elliptic problem involving fractional p-Laplacian and a discontinuous critical nonlinearity

Almost balancing numbers are defined from a Diophantine equation slightly different from the defining equation for balancing numbers. There are two types of almost balancing numbers and are respectively the ceiling and floor functions of square roots of two types of almost square triangular numbers. These numbers are very closely associated with balancing, Lucas balancing, Pell and associated Pell numbers.

Akasmika Panda

Papers

Elliptic Partial Differential Equation Involving a Singularity and a Radon Measure

A singular elliptic problem involving fractional $p$-Laplacian and a discontinuous critical nonlinearity

A singular elliptic problem involving fractional p-Laplacian and a discontinuous critical nonlinearity

Almost Balancing Numbers

A critical fractional choquard problem involving a singular nonlinearity and a radon measure