arXiv subjects
Publications and source records attributed to Debajyoti Choudhuri.
We study an elliptic nonlocal problem driven by a source term which is a Radon measure. We prove the existence of a weak solution by a weak convergence method. In the process, we define a new function space which we name the Walker space. The question about the existence of infinitely many nontrivial solutions leads to an interesting conjecture.
In this paper, we study an elliptic problem involving a $p$-Laplacian operator and a potential well which is driven by a critical and singular nonlinearity. Under the limiting case of a parameter blowing up to $\infty$ yields solutions to a different problem where the effect of the potential well becomes negligible.
This study is devoted to proving the existence of weak solutions for a nonlinear elliptic problem with Neumann-type boundary data. The problem is driven by a discontinuous power nonlinearity and a nonsmooth prescribed data. Additionally, we aim to derive an estimate that proves the well-posedness of the problem. This estimate serves as an evidence for the uniqueness of the existing solution when the boundary term is ``smooth".
In this paper, we investigate the existence of nontrivial weak solutions for the Prandtl-Batchelor type free boundary value elliptic problem driven by a power nonlinearity. The algebraic topology approach will be used to establish the existence of solutions of approximate problem, while variational techniques will be used to determine the existence of major problem solutions. In the process several classical results are improved.
We study the existence of nonnegative solutions to the following nonlocal elliptic problem involving singularity \begin{align} \mathfrak{M}\left(\int_{Q}\frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}dxdy\right)(-Δ)_{p}^{s} u&=\fracλ{|u|^{γ-1}u}+|u|^{p_s^*-2}u~\text{in}~Ω,\nonumber u&>0~\text{in}~Ω,\nonumber u&=0~\text{in}~\mathbb{R}^N\setminusΩ,\nonumber \end{align} where $Ω\subset\mathbb{R}^N$, is a bounded domain with Lipschitz boundary, $λ>0$, $N>ps$, $0<s,γ<1$, $(-Δ)_{p}^{s}$ is the fractional $p$-Laplacian operator for $1<p<\infty$ and $p_s^*=\frac{Np}{N-ps}$ is the critical Sobolev exponent. We employ a {\it cut-off} argument to obtain the existence of $k$ (being an arbitrarily large integer) solutions. Furthermore, by using the Moser iteration technique, we prove a uniform $L^{\infty}(Ω)$ bound for the solutions. The novelty of this work lies in proving the existence of small energy solutions by using the symmetric mountain pass theorem in spite of the presence of a critical nonlinear term which, of course, is super-linear.
We consider a multiphase spectral problem on a stratified Lie group. We prove the existence of an eigenfunction of $(2,q)$-eigenvalue problem on a bounded domain. Furthermore, we also establish a Pohozaev-like identity corresponding to the problem on the Heisenberg group.
We will study a free boundary value problem driven by a source term which is quite {\it irregular}. In the process, we will establish a monotonicity result, and regularity of the solution.
We establish the existence of at least two solutions of the {\it Prandtl-Batchelor} like elliptic problem driven by a power nonlinearity and a singular term. The associated energy functional is nondifferentiable and hence the usual variational techniques do not work. We shall use a novel approach in tackling the associated energy functional by a sequence of $C^1$ functionals and a {\it cutoff function}. Our main tools are fundamental elliptic regularity theory and the mountain pass theorem.
Using variational methods, we establish the existence of infinitely many solutions to an elliptic problem driven by a Choquard term and a singular nonlinearity. We further show that if the problem has a positive solution, then it is bounded a.e. in the domain $Ω$ and is Hölder continuous.
We prove the existence of solutions for the following critical Choquard type problem with a variable-order fractional Laplacian and a variable singular exponent \begin{align*} \begin{split} a(-Δ)^{s(\cdot)}u+b(-Δ)u&=λ|u|^{-γ(x)-1}u+\left(\int_Ω\frac{F(y,u(y))}{|x-y|^{μ(x,y)}}dy\right)f(x,u) & +ηH(u-α)|u|^{r(x)-2}u,~\text{in}~Ω, u&=0,~\text{in}~\mathbb{R}^N\setminusΩ. \end{split} \end{align*} where $a(-Δ)^{s(\cdot)}+b(-Δ)$ is a mixed operator with variable order $s(\cdot):\mathbb{R}^{2N}\rightarrow (0,1)$, $a, b\geq 0$ with $a+b>0$, $H$ is the Heaviside function (i.e., $H(t)=0$ if $t\leq0$, $H(t) = 1$ if $t>0),$ $Ω\subset\mathbb{R}^N$ is a bounded domain, $N\geq 2$, $λ>0$, $0<γ^{-}=\underset{x\in\barΩ}{\inf}\{γ(x)\}\leqγ(x)\leqγ^+=\underset{x\in\barΩ}{\sup}\{γ(x)\}<1$, $μ$ is a continuous variable parameter, and $F$ is the primitive function of a suitable $f$. The variable exponent $r(x)$ can be equal to the critical exponent $2_{s}^*(x)=\frac{2N}{N-2\bar{s}(x)}$ with $\bar{s}(x)=s(x,x)$ for some $x\in\barΩ,$ and $η$ is a positive parameter. We also show that as $α\rightarrow 0^+$, the corresponding solution converges to a solution for the above problem with $α=0$.
In this paper we establish existence of a solution to a semilinear equation with free boundary conditions on stratified Lie groups. In the process, a monotonicity condition is proved, which is quintessential in establishing the regularity of the solution.
The short note here is to give a few heuristic arguments on the weird looking fractional Laplacian operator. This is certainly going to expand the vision of a reader who is looking to develope a taste for research in this direction.
We establish a continuous embedding $W^{s(\cdot),2}(Ω)\hookrightarrow L^{α(\cdot)}(Ω)$, where the variable exponent $α(x)$ can be close to the critical exponent $2_{s}^*(x)=\frac{2N}{N-2\bar{s}(x)}$, with $\bar{s}(x)=s(x,x)$ for all $x\in\barΩ$. Subsequently, this continuous embedding is used to prove the multiplicity of solutions for critical nonlocal degenerate Kirchhoff problems with a variable singular exponent. Moreover, we also obtain the uniform $L^{\infty}$-estimate of these infinite solutions by a bootstrap argument.
In this paper we establish the existence and multiplicity of nontrivial solutions to the following problem \begin{align*} \begin{split} (-Δ)^{\frac{1}{2}}u+u+(\ln|\cdot|*|u|^2)&=f(u)+μ|u|^{-γ-1}u,~\text{in}~\mathbb{R}, \end{split} \end{align*} where $μ>0$, $(*)$ is the convolution operation between two functions, $0<γ<1$, $f$ is a function with a certain type of growth. We prove the existence of a nontrivial solution at a certain mountain pass level and another ground state solution when the nonlinearity $f$ is of exponential critical growth.
In this article, we show the existence of a unique entropy solution to the following problem: \begin{equation} \begin{split} (-Δ)_{p,α}^su&= f(x)h(u)+g(x) ~\text{in}~Ω,\\ u&>0~\text{in}~Ω,\\ u&= 0~\text{in}~\mathbb{R}^N\setminusΩ,\nonumber \end{split} \end{equation} where the domain $Ω\subset \mathbb{R}^N$ is bounded and contains the origin, $ α\in[0,\frac{N-ps}{2})$, $s\in (0,1)$, $2-\frac{s}{N} 1$ and $h$ is a general singular function with singularity at 0. Further, the fractional $p$-Laplacian with weight $α$ is given by $$(-Δ)_{p,α}^su(x)=\text{P. V.}\int_{\mathbb{R}^N}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{N+ps}}\frac{dy}{|x|^α|y|^α},~\forall x\in \mathbb{R}^N.$$
In this article we develop a concentration compactness type principle in a variable exponent setup. As an application of this principle we discuss a problem involving fractional `{\it $(p(x),p^+)$-Laplacian}' and power nonlinearities with exponents $(p^+)^*$, $p_s^*(x)$ with the assumption that the critical set $\{x\inΩ:p_s^*(x)=(p^+)^*\}$ is nonempty.
The purpose of this paper is to study nonlinear singular parabolic equations with $p(x)$- Laplacian. Precisely, we consider the following problem and discuss the existence of a non-negative weak solution. \begin{align*} \frac{\partial u}{\partial t}-Δ_{p(x)}u&=λu^{q(x)-1} + u^{-δ(x)}g+ f&&\text{in}~Q_T, u&= 0&&\text{on}~Σ_T, u(0,\cdot)&=u_0(\cdot)&&\text{in}~Ω\nonumber. \end{align*} Here $Q_T=Ω\times(0,T)$, $Σ_T=\partialΩ\times(0,T)$, $Ω$ is a bounded domain in $\mathbb{R}^N$ ($N\geq 2$) with Lipschitz continuous boundary $\partialΩ$, $λ\in(0,\infty)$, $f\in L^1(Q_T)$, $g\in L^\infty(Ω)$, $u_0\in L^r(Ω)$ with $r\geq 2$, $δ:\overlineΩ\rightarrow(0,\infty)$ is continuous, and $p,q\in C(\overlineΩ)$ with $\underset{x\in\overlineΩ}{\max}~p(x)<N$, $q(\cdot)<p^*(\cdot)$. The article is distinguished into two cases according to the choice of $f$ with different range of parameters $p(\cdot)$, $q(\cdot)$.
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} (-Δ)_p^su&=μg(x,u)+\fracλ{u^γ}+H(u-α)u^{p_s^*-1},~\text{in}~Ω u&>0,~\text{in}~Ω, u&=0,~\text{in}~\mathbb{R}^N\setminusΩ, \end{split} \end{align} where $Ω\subset\mathbb{R}^N$ is a bounded domain with Lipschitz boundary, $s\in (0,1)$, $2 0$, $α\geq 0$ 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 $α\rightarrow0^+$, the sequence of solutions of $\eqref{main_prob}$ for each such $α$ converges to a solution of the problem for which $α=0$.