Ratan K. Giri | Searcharxiv

arXiv2020

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

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} (-Δ)^su&= \frac{1}{u^γ}+λu^{2_s^*-1}+μ~\text{in}~Ω, u&>0~\text{in}~Ω, u&= 0~\text{in}~\mathbb{R}^N\setminusΩ. \end{split} \end{equation} Here, $Ω$ is a bounded domain of $\mathbb{R}^N$, $s\in (0,1)$, $2s<N$, $λ,γ\in (0,1)$, $2_s^*=\frac{2N}{N-2s}$ is the fractional critical Sobolev exponent and $μ$ is a nonnegative bounded Radon measure in $Ω$.

math.AP↗

arXiv2016

A problem involving the $p$-Laplacian operator

Using a variational technique we guarantee the existence of a solution to the \emph{resonant Lane-Emden} problem $-Δ_p u=λ|u|^{q-2}u$, $u|_{\partialΩ}=0$ if and only if a solution to $-Δ_p u=λ|u|^{q-2}u+f$, $u|_{\partialΩ}=0$, $f\in L^{p'}(Ω)$ ($p'$ being the conjugate of $p$), exists for $q\in (1,p)\bigcup (p,p^{*})$ under a certain condition for both the cases, i.e., $1<q<p<p^{*}$ and $1< p < q < p^{*}$ - the sub-linear and the super-linear cases.

math.AP↗