arXiv subjects
Publications and source records attributed to Kaushik Bal.
We provide sufficient conditions on $w\in L^1_{loc}(\mathbb{R}^N)$ such that the weighted $p$-Laplace equation $$-\operatorname{div}\big(w(x)|\nabla u|^{p-2}\nabla u\big)=f(u)\;\;\mbox{in}\;\;\mathbb{R}^N$$ does not admit any stable $C^{1,ζ}_{loc}$ solution in $\mathbb{R}^N$ where $f(x)$ is either $-x^{-δ}$ or $e^x$ for any $0<ζ<1$.
For an open, bounded domain $\Om$ in $\mathbb{R}^N$ which is strictly convex with $C^2$ boundary, we show that there exists a $\land>0$ such that the singular quasilinear problem \begin{eqnarray*} &-\delp u =\cfracλ{u^{\del}}+u^q\,\,\mbox{in}\,\,\Om\\ &u=0\,\,\mbox{on}\,\,\partial\Om;\, \,\,u>0\,\,\mbox{in}\,\,\Om \end{eqnarray*} admits atleast two solution $ u$ and $v$ in $W^{1,p}_{loc}(\Om)\cap L^{\infty}(\Om)$ for any $\del>0$ and $0<\lam<\land$ provided $1 0$.
In this article we are interested in addressing the question of existence and uniqueness of positive solution to a quasilinear elliptic equation involving p-laplacian in Heisenberg Group. The idea is to prove the uniqueness by using Diaz-Saa Inequality in Heisenberg Group which we obtain via a generalized version of Picone's Identity.
We investigate the following quasilinear parabolic and singular equation, {equation} \tag{\rm P$_t$} \{{aligned} & u_t-Δ_p u =\frac{1}{u^δ}+f(x,u)\;\text{in}\,(0,T)\timesΩ, & u =0\,\text{on} \;(0,T)\times\partialΩ,\quad u>0 \text{in}\, (0,T)\timesΩ, &u(0,x) =u_0(x)\;\text{in}Ω, {aligned}. {equation} % where $Ω$ is an open bounded domain with smooth boundary in $\R^{\rm N}$, $1 < p< \infty$, $0<δ$ and $T>0$. We assume that $(x,s)\inΩ\times\R^+\to f(x,s)$ is a bounded below Caratheodory function, locally Lipschitz with respect to $s$ uniformly in $x\inΩ$ and asymptotically sub-homogeneous, i.e. % {equation} \label{sublineargrowth} 0 \leq\displaystyle\lim_{t\to +\infty}\frac{f(x,t)}{t^{p-1}}=α_f<λ_1(Ω), {equation} % (where $λ_1(Ω)$ is the first eigenvalue of $-Δ_p$ in $Ω$ with homogeneous Dirichlet boundary conditions) and $u_0\in L^\infty(Ω)\cap W^{1,p}_0(Ω)$, satisfying a cone condition defined below. Then, for any $δ\in (0,2+\frac{1}{p-1})$, we prove the existence and the uniqueness of a weak solution $u \in{\bf V}(Q_T)$ to $({\rm P_t})$. Furthermore, $u\in C([0,T], W^{1,p}_0(Ω))$ and the restriction $δ<2+\frac{1}{p-1}$ is sharp. Finally, in the last section we analyse the case $p=2$. Using the interpolation spaces theory and the semigroup theory, we prove the existence and the uniqueness of weak solutions to $({\rm P}_t)$ for any $δ>0$ in $C([0,T], L^2(Ω))\cap L^\infty(Q_T)$ and under suitable assumptions on the initial data we give additional regularity results. Finally, we describe their asymptotic behaviour in $L^\infty(Ω)\cap H^1_0(Ω)$ when $δ<3$.