arXiv subjects
Publications and source records attributed to Gaurav Dwivedi.
In this article, we prove the existence of at least one positive solution for the mixed local-nonlocal semipositone problem \begin{equation*} \left\{ \begin{aligned} -Δ_p u+ (-Δ)^s_p u &= λf(u) && \text{in } Ω, u &= 0 && \text{in } \mathbb{R}^N \setminus Ω, \end{aligned} \right. \end{equation*} using mountain pass arguments, comparison principles and regularity principles.
This paper aims to establish the existence of a weak solution for the following problem: \begin{equation*} (-Δ)^{s}_{\mathcal{H}}u(x) +V(x)h(x,x,|u|)u(x)=\left(\int_{\mathbb{R}^{N}}\dfrac{K(y)F(u(y))}{|x-y|^λ}dy \right) K(x)f(u(x)) \ \hbox{in} \ \mathbb{R}^{N}, \end{equation*} where $N\geq 1$, $s\in(0,1), λ\in(0,N), \mathcal{H}(x,y,t)=\int_{0}^{|t|} h(x,y,r)r\ dr,$ $ h:\mathbb{R}^{N}\times\mathbb{R}^{N}\times [0,\infty)\rightarrow[0,\infty)$ is a generalized $N$-function and $(-Δ)^{s}_{\mathcal{H}}$ is a generalized fractional Laplace operator. The functions $V,K:\mathbb{R}^{N}\rightarrow (0,\infty)$, non-linear function $f:\mathbb{R}\rightarrow \mathbb{R}$ are continuous and $ F(t)=\int_{0}^{t}f(r)dr.$ First, we introduce the homogeneous fractional Musielak-Sobolev space and investigate their properties. After that, we pose the given problem in that space. To establish our existence results, we prove and use the suitable version of Hardy-Littlewood-Sobolev inequality for Lebesque Musielak spaces together with variational technique based on the mountain pass theorem. We also prove the existence of a ground state solution by the method of Nehari manifold.
In this paper, we show the existence of a weak solution for a fractional sub-Laplace equation involving a term with the critical Sobolev exponent, namely, \begin{align*} (-Δ_\mathbb{H})^su - λu &= |u|^{Q^*_s -2}u \text{ in } Ω,\\ u &= 0 \text{ in } \mathbb{H}^N \setminus Ω, \end{align*} where $Ω\subseteq \mathbb{H}^N$ is bounded and has continuous boundary, $(-Δ_\mathbb{H})^s$ is the horizontal fractional Laplacian, $s \in (0,1), λ> 0,$ and $Q^*_s=\frac{2Q}{Q-2s}$ is the Sobolev critical exponent. This problem is motivated by the celebrated Brezis-Nirenberg problem \cite{brezis1983positive}.
In this paper, we obtain the existence of weak solutions to the Choquard-Kirchhoff type critical multiphase problem: \begin{equation*} \left\{\begin{array}{cc} &-M(φ_{\h}(\lvert{\nabla u}\rvert))div(\lvert{\nabla u}\rvert^{p(x)-2}\nabla u+a_1(x)\lvert{\nabla u}\rvert^{q(x)-2}\nabla u+a_2(x)\lvert{\nabla u}\rvert^{r(x)-2}\nabla u) & =λg(x)\lvert{u}\rvert^{γ(x)-2}u+θB(x,u)+κ\left(\int_{\q}\frac{F(y,u(y))}{\lvert{x-y}\rvert^{d(x,y)}}\, dy\right) f(x,u) \ \text{in} \ Ω, & u=0 \ \text{on} \ {\partial Ω}. \end{array}\right. \end{equation*} The term $B(x,u)$ on the right-hand side generalizes the critical growth. We obtain existence and multiplicity results by establishing certain embedding results and concentration compactness principle along with the Hardy-Littlewood-Sobolev type inequality for the Musielak Orlicz Sobolev space $ W^{1,\mathcal{T}}(\q)$.
This paper aims to establish the existence of a weak solution for the non-local problem: \begin{equation*} \left\{\begin{array}{ll} -a\left(\int_Ω\mathcal{H}(x,|\nabla u|)dx \right) Δ_{\mathcal{H}}u &=f(x,u) \ \ \hbox{in} \ \ Ω, \ \ \ \\ \hspace{3.3cm} u &= 0 \ \ \hbox{on} \ \ \partial Ω, \end{array}\right. \end{equation*} where $Ω\subseteq \mathbb{R}^{N},\, N\geq 2$ is a bounded and smooth domain containing two open and connected subsets $Ω_p$ and $Ω_N$ such that $ \barΩ_{p}\cap\barΩ_{N}=\emptyset$ and $Δ_{\mathcal{H}}u=\hbox{div}( h(x,|\nabla u|)\nabla u)$ is the $\mathcal{H}$-Laplace operator. We assume that $Δ_{\mathcal{H}}$ reduces to $ Δ_{p(x)}$ in $Ω_{p}$ and to $ Δ_{N}$ in $Ω_{N},$ the non-linear function $f:Ω\times\mathbb{R}\rightarrow \mathbb{R}$ act as $|t|^{p^{\ast}(x)-2}t$ on $Ω_{p}$ and as $e^{α|t|^{N/(N-1)}}$ on $Ω_{N}$ for sufficiently large $|t|$. To establish our existence results in a Musielak-Sobolev space, we use a variational technique based on the mountain pass theorem.
We prove the existence of a weak solution to the problem \begin{equation*} \begin{split} -Δ_{p}u+V(x)|u|^{p-2}u & =f(u,|\nabla u|^{p-2}\nabla u), \ \ \ \\ u(x) & >0\ \ \forall x\in\mathbb{R}^{N}, \end{split} \end{equation*} where $Δ_{p}u=\hbox{div}(|\nabla u|^{p-2}\nabla u)$ is the $p$-Laplace operator, $1<p<N$ and the nonlinearity $f:\mathbb{R}\times\mathbb{R}^{N}\rightarrow\mathbb{R}$ is continuous and it depends on gradient of the solution. We use an iterative technique based on the Mountain pass theorem to prove our existence result.
In this note, we establish the existence of a positive solution and its stability to the following problem $$Δ_{\mathbb{H}^n}^2u=a(ξ)u-f(ξ,u)\text{ in }Ω, \,\,\, u|_{\partialΩ} = 0 =\left.Δ_{\mathbb{H}^n} u|_{\partialΩ},$$ on Heisenberg group.
The goal of this paper is to establish singular Adams type inequality for biharmonic operator on Heisenberg group. As an application, we establish the existence of a solution to \begin{equation*} Δ_{\mathbb{H}^n}^2 u=\frac{f(ξ,u)}{ρ(ξ)^a}\,\,\text{ in }Ω,\,\, u|_{\partialΩ}=0=\left.\frac{\partial u}{\partial ν}\right|_{\partialΩ}, \end{equation*} where $0\in Ω\subseteq \mathbb{H}^4$ is a smooth bounded domain, $0\leq a<Q,\,(Q=10).$ The special feature of this problem is that it contains an exponential nonlinearity and singular potential.
In this paper, we consider the bifurcation problem for fractional Laplace equation \begin{eqnarray*} \begin{array}{ll} (-Δ)^{s} u = λu + f(λ,\,x,\,u)& \mbox{in }Ω, u = 0 &\mbox{in }\mathbb{R}^n\backslash Ω, \end{array} \end{eqnarray*} where $Ω\subset \mathbb{R}^n,\,n> 2s (0<s<1)$ is an open bounded subset with smooth boundary, $(-Δ)^{s}$ stands for the fractional Laplacian. We show that a continuum of solutions bifurcates out from the principal eigenvalue $λ_1$ of the eigenvalue problem \begin{eqnarray*} \begin{gathered} (-Δ)^{s} v = λv\,\,\,\mbox{in}\,\,Ω, v = 0 \,\,\,\,\mbox{in}\,\,\,\,\mathbb{R}^n \backslashΩ, \end{gathered} \end{eqnarray*} and, conversely.
In this article we prove the nonlinear analogue of Picone's identity for $p-$biharmonic operator. As an application of our result we show that the Morse index of the zero solution to a $p-$biharmonic boundary value problem is $0$. We also prove a Hardy type inequality and Sturmian comparison principle. We also show the strict monotonicity of the principle eigenvalue and linear relationship between the solutions of a system of singular $p$-biharmonic system.