arXiv subjects
Publications and source records attributed to D. Choudhuri.
The article is about an elliptic problem defined on a {\it stratified Lie group}. Both sub- and superlinear cases are considered whose solutions are guaranteed to exist in light of the interplay between the nonlinearities and the weak $L^1$ datum. The existence of infinitely many solutions is proved for suitable values of $\lambda,p,q$ by using the Symmetric Mountain Pass Theorem.
We prove the existence of infinitely many solutions to an elliptic problem by borrowing the techniques from algebraic topology. The solution(s) thus obtained will also be proved to be bounded.
We prove the existence of ground state solution to the following problem. \begin{align*} (-Δ)^{s}u+u&=λ|u|^{-γ-1}u+P(x)|u|^{p-1}u,~\text{in}~\mathbb{R}^N\setminusΩ\\ N_su(x)&=0,~\text{in}~Ω\end{align*} where $N\geq2$, $λ>0$, $0<s,γ<1$, $p\in(1,2_s^*-1)$ with $2_s^*=\frac{2N}{N-2s}$. % $0<s^-=\underset{(x,y)\inΩ\timesΩ}{\inf}\{s(x,y)\}\leq s(x,y)\leq s^+=\underset{(x,y)\inΩ\timesΩ}{\sup}\{s(x,y)\}<1$, $0<γ^-=\underset{x\inΩ}{\inf}\{γ(x)\}\leq γ(x)\leq γ^+=\underset{x\inΩ}{\sup}\{γ(x)\}<1$, $1-γ^-<1<p^-=\underset{x\inΩ}{\inf}\{p(x)\}\leq p(x)\leq p^+=\underset{x\inΩ}{\sup}\{p(x)\}<2_{s^-}^*=\underset{x\inΩ}{\inf}\{2_s^*(x)\}$ with $2_s^*(x)=\frac{2N}{N-2\tilde{s}(s)}$ where $\tilde{s}(x)=s(x,x)$. Moreover, $Ω\subset\mathbb{R}^N$ is a smooth bounded domain, $(-Δ)^s$ denotes the $s$-fractional Laplacian and finally $N_s$ denotes the nonlocal operator that describes the Neumann boundary condition which is given as follows. \begin{align*} N_{s}u(x)&=C_{N,s}\int_{\mathbb{R}^N\setminusΩ}\frac{u(x)-u(y)}{|x-y|^{N+2s}}dy,~x\inΩ. \end{align*} We further establish the existence of infinitely many bounded solutions to the problem.
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 $s(x,y)p(x,y) < N \;\forall\;(x,y)\in\overlineΩ\times\overlineΩ$.
In this paper, we prove the existence of multiple nontrivial solutions of the following equation. \begin{align*} \begin{split} -Δ_{p}u & = \fracλ{u^γ}+g(u)+μ~\mbox{in}\,\,Ω, u & = 0\,\, \mbox{on}\,\, \partialΩ, u&>0 \,\,\mbox{in}\,\,Ω, \end{split} \end{align*} where $Ω\subset \mathbb{R}^N$ is a smooth bounded domain with $N \geq 3$, $1 < p-1 < q$ , $ λ>0$, $γ>0$, $g$ satisfies certain conditions, $μ\geq 0$ is a bounded Radon measure.
In this article, we will prove the existence of infinitely many positive weak solutions to the following nonlocal elliptic PDE. \begin{align} (-Δ)^s u&= \fracλ{u^γ}+ f(x,u)~\text{in}~Ω,\nonumber u&=0~\text{in}~\mathbb{R}^N\setminusΩ,\nonumber \end{align} where $Ω$ is an open bounded domain in $\mathbb{R}^N$ with Lipschitz boundary, $N>2s$, $s\in (0,1)$, $γ\in (0,1)$. We will employ variational techniques to show the existence of infinitely many weak solutions of the above problem.
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 \begin{eqnarray} -Δu&=& f(x)h(u)+μ~\text{in}~Ω,\nonumber u&=&0~\text{on}~\partialΩ,\nonumber u&>& 0~\text{on}~Ω\nonumber, \end{eqnarray} where $Ω$ is a bounded domain of $\mathbb{R}^N$, $f$ is a nonnegative function over $Ω$.
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.
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.
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 $p(x)$.
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 $q$.
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.
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.