arXiv subjects

Amita Soni

Publications and source records attributed to Amita Soni.

6 recordsLinked to original sources

arXiv2019

Existence and multiplicity of solutions to a nonlocal elliptic PDE with variable exponent in a Nehari manifold using the Banach fixed point theorem

In this paper we study the existence and multiplicity of two distinct nontrivial weak solutions of the following equation in Nehari manifold. We have also proved that these solutions are in $L^{\infty}(Ω)$. \begin{align*} \begin{split} -Δ_{p(x,y)}^{s(x,y)}u &= β|u|^{α(x)-2}u+λf(x,u)\,\,\mbox{in}\,\,Ω,\\ u &= 0\,\, \mbox{in}\,\, \mathbb{R}^{N}\setminusΩ\end{split} \end{align*} Here, $λ, β> 0$ are parameters and $f(x,u)$ is a general nonlinear term satisfying certain conditions. The domain $Ω\subset\mathbb{R}^N (N\geq 2)$ is smooth and bounded. The relation between the exponents are assumed in the order $2 < α^{-}\leqα(x)\leqα^{+} < p^{-}\leq p(x,y)\leq p^{+} < q^{+} < r^{+} < r^{+2} < p_{s}^{*}(x)$. Also, $α(x)\leq p(x,x)\;\forall\;x\in\overlineΩ$ and $s(x,y)p(x,y) < N \;\forall\;(x,y)\in\overlineΩ\times\overlineΩ$.

math.AP↗

arXiv2019

Existence of solution for a system involving fractional Laplacians and a Radon measure

An existence of a nontrivial solution in some `weaker' sense of the following system of equations \begin{align*} (-Δ)^{s}u+l(x)ϕu+w(x)|u|^{k-1}u&=μ~\text{in}~Ω\nonumber\\ (-Δ)^{s}ϕ&= l(x)u^2~\text{in}~Ω\nonumber\\ u=ϕ&=0 ~\text{in}~\mathbb{R}^N\setminusΩ\end{align*} has been proved. Here $s \in (0,1)$, $l,w$ are bounded nonnegative functions in $Ω$, $μ$ is a Radon measure and $k > 1$ belongs to a certain range.

math.AP↗

arXiv2018

Three nontrivial solutions of a nonlocal problem involving critical exponent

In this paper we will prove the existence of three nontrivial weak solutions of the following problem involving a nonlinear integro-differential operator and a term with critical exponent. \begin{align*} \begin{split} -\mathscr{L}_Φu & = |u|^{{p_{s}^{\ast}}-2}u+λf(x,u)\,\,\mbox{in}\,\,Ω,\\ u & = 0\,\, \mbox{in}\,\, \mathbb{R}^N\setminus Ω, \end{split} \end{align*} Here $q\in(p, p_s^*)$, where $p_s^*$ is the fractional Sobolev conjugate of $p$ and $-\mathscr{L}_Φ$ represents a general nonlocal integro-differential operator of order $s\in(0,1)$. This operator is possibly degenerate and covers the case of fractional $p$-Laplacian operator.

math.AP↗

arXiv2018

Existence of solution of the $p(x)$-Laplacian problem involving critical exponent and Radon measure

In this paper we are proving the existence of a nontrivial solution of the ${p}(x)$- Laplacian equation with Dirichlet boundary condition. We will use the variational method and concentration compactness principle involving positive radon measure $μ$. \begin{align*} \begin{split} -Δ_{p(x)}u & = |u|^{q(x)-2}u+f(x,u)+μ\,\,\mbox{in}\,\,Ω,\\ u & = 0\,\, \mbox{on}\,\, \partialΩ, \end{split} \end{align*} where $Ω\subset \mathbb{R}^N$ is a smooth bounded domain, $μ> 0$ and $1 < p^{-}:=\underset{x\in Ω}{\text{inf}}\;p(x) \leq p^{+}:= \underset{x\in Ω}{\text{sup}}\;p(x) < q^{-}:=\underset{x\in Ω}{\text{inf}}\;q(x)\leq q(x) \leq p^{\ast}(x) < N$. The function $f$ satisfies certain conditions. Here, $q^{\prime}(x)=\frac{q(x)}{q(x)-1}$ is the conjugate of $q(x)$ and $p^{\ast}(x)=\frac{Np(x)}{N-p(x)}$ is the Sobolev conjugate of $p(x)$.

math.AP↗

arXiv2018

Existence of multiple solutions to an elliptic problem with measure data

In this paper we prove the existence of multiple nontrivial solutions of the following equation. \begin{align*} \begin{split} -Δ_{p}u & = λ|u|^{q-2}u+f(x,u)+μ\,\,\mbox{in}\,\,Ω, u & = 0\,\, \mbox{on}\,\, \partialΩ; \end{split} \end{align*} where $Ω\subset \mathbb{R}^N$ is a smooth bounded domain with $N \geq 3$, $1 < q^{\prime} < q < p-1; \; λ,\;$ and $f$ satisfies certain conditions, $μ>0$ is a Radon measure, $q^{\prime}=\frac{q}{q-1}$ is the conjugate of $q$.

math.AP↗

arXiv2017

Problem involving nonlocal operator

The aim of this paper is to deal with the elliptic pdes involving a nonlinear integrodifferential operator, which are possibly degenerate and covers the case of fractional $p$-Laplacian operator. We prove the existence of a solution in the weak sense to the problem \begin{align*} \begin{split} -\mathscr{L}_Φu & = λ|u|^{q-2}u\,\,\mbox{in}\,\,Ω,\\ u & = 0\,\, \mbox{in}\,\, \mathbb{R}^N\setminus Ω\end{split} \end{align*} if and only if a weak solution to \begin{align*} \begin{split} -\mathscr{L}_Φu & = λ|u|^{q-2}u +f,\,\,\,f\in L^{p'}(Ω),\\ u & = 0\,\, \mbox{on}\,\, \mathbb{R}^N\setminus Ω\end{split} \end{align*} ($p'$ being the conjugate of $p$), exists in a weak sense, for $q\in(p, p_s^*)$ under certain condition on $λ$, where $-\mathscr{L}_Φ$ is a general nonlocal integrodifferential operator of order $s\in(0,1)$ and $p_s^*$ is the fractional Sobolev conjugate of $p$. We further prove the existence of a measure $μ^{*}$ corresponding to which a weak solution exists to the problem \begin{align*} \begin{split} -\mathscr{L}_Φu & = λ|u|^{q-2}u +μ^*\,\,\,\mbox{in}\,\, Ω,\\ u & = 0\,\,\, \mbox{in}\,\,\mathbb{R}^N\setminus Ω\end{split} \end{align*} depending upon the capacity.

math.AP↗