Prashanta Garain

• PhD
• Assistant Professor at Indian Institute of Science Education and Research, Berhampur

This paper proves the existence and regularity of weak solutions for a class of mixed local–nonlocal problems with singular nonlinearities. We examine both the purely singular problem and perturbed singular problems. A central contribution of this work is the inclusion of a variable singular exponent in the context of measure-valued data. Another n...

This article proves the existence and regularity of weak solutions for a class of mixed local-nonlocal problems with singular nonlinearities. We examine both the purely singular problem and perturbed singular problems. A central contribution of this work is the inclusion of a variable singular exponent in the context of measure-valued data. Another...

In this paper, we will study Neumann ( p , q )-eigenvalue problem for the weighted p -Laplace operator in outward Hölder cuspidal domains. The suggested method is based on the composition operators on weighted Sobolev spaces.

In this article, we establish variational characterization of solutions of a mixed local and nonlocal singular semilinear elliptic problem. As an application of this fact, we deduce decomposition property. Further, using moving plane method, we prove symmetry of solutions of a mixed local and nonlocal singular perturbed problem, where the decomposi...

This article is twofold. In the first part of the article, we consider Brezis-Oswald problem involving the mixed anisotropic and nonlocal $p$-Laplace operator and establish existence, uniqueness, boundedness and strong maximum principle. Further, for some mixed anisotropic and nonlocal $p$-Laplace type equations, we obtain Sturmian comparison theor...

In the article we study the Neumann (p, q)-eigenvalue problems in bounded Hölder γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document}-singular domai...

We study the fractional p-Laplace equation (-Δp)su=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} (-\Delta _p)^s u = 0 \end{aligned}$$\end{document}fo...

We establish mixed anisotropic and nonlocal Sobolev type inequalities with an extremal. We show that the extremal function is unique up to a multiplicative constant that is associated with the corresponding mixed anisotropic and nonlocal singular partial differential equation. We prove that such a mixed Sobolev type inequality is necessary and suff...

We establish existence results for a class of mixed anisotropic and nonlocal p -Laplace equations with singular nonlinearities. We consider both constant and variable singular exponents. Our argument is based on an approximation method. To this end, we also discuss the necessary regularity properties of weak solutions of the associated non-singular...

In this article, we consider a class of degenerate singular problems. The degeneracy is captured by the presence of a class of p-admissible weights, which may vanish or blow up near the origin. Further, the singularity is allowed to vary inside the domain. We provide sufficient conditions on the weight function, on the singular exponent and the sou...

We consider a class of singular weighted anisotropic p-Laplace equations. We provide sufficient condition on the weight function that may vanish or blow up near the origin to ensure the existence of at least one weak solution in the purely singular case and at least two different weak solutions in the purturbed singular case.

In this article, we consider a combination of local and nonlocal Laplace equation with singular nonlinearities. For such mixed problems, we establish the existence of at least one weak solution for a parameter-dependent singular nonlinearity and existence of multiple solutions for perturbed singular nonlinearity. Our argument is based on the variat...

We consider the spectral problem for the mixed local and nonlocal p-Laplace operator. We discuss the existence and regularity of eigenfunction of the associated Dirichlet (p, q)-eigenvalue problem in a bounded domain Ω ⊂ ℝN under the assumption that 1 < p < ∞ and 1 < q < p∗ where p∗ = Np/(N − p) if 1 < p < N and p∗ = ∞ if p ⩾ N.

This article consists of study of anisotropic double phase problems with singular term and sign changing subcritical as well as critical nonlinearity. Seeking the help of well known Nehari manifold technique, we establish existence of at least two opposite sign energy solutions in the subcritical case and one negative energy solution in the critica...

We establish existence results for a class of mixed anisotropic and nonlocal $p$-Laplace equation with singular nonlinearities. We consider both constant and variable singular exponents. Our argument is based on an approximation method. To this end, we also discuss the necessary regularity properties of weak solutions of the associated non-singular...

In this article, we consider the spectral problem for the mixed local and nonlocal $p$-Laplace operator. We discuss the existence and regularity of eigenfunction of the associated Dirichlet $(p,q)$-eigenvalue problem in a bounded domain $\Omega\subset\mathbb{R}^N$ under the assumption that $1<p<\infty$ and $1<q<p^{*}$ where $p^{*}=\frac{N p}{N-p}$...

In the article we study the Neumann $(p,q)$-eigenvalue problems in bounded H\"older $\gamma$-singular domains $\Omega_{\gamma}\subset \mathbb{R}^n$. In the case $1<p<\infty$ and $1<q<p^{*}_{\gamma}$ we prove solvability of this eigenvalue problem and existence of the minimizer of the associated variational problem. In addition, we establish some re...

We consider equations involving a combination of local and nonlocal degenerate p-Laplace operators. The main contribution of the paper is almost Lipschitz regularity for the homogeneous equation and Hölder continuity with an explicit Hölder exponent in the general case. For certain parameters, our results also imply Hölder continuity of the gradien...

In this article, we consider a combination of local and nonlocal Laplace equation with singular nonlinearities. For such mixed problems, we establish existence of at least one weak solution for a parameter dependent singular nonlinearity and existence of multiple solution for purturbed singular nonlinearity. Our argument is based on the variational...

We establish multiplicity results for the following class of quasilinear problems P\begin{equation*} \left\{ \begin{array}{@{}l} -\Delta_{\Phi}u=f(x,u) \quad \mbox{in} \quad \Omega, \\ u=0 \quad \mbox{on} \quad \partial \Omega, \end{array} \right. \end{equation*} where $\Delta _{\Phi }u=\text {div}(\varphi (x,|\nabla u|)\nabla u)$ for a generalized...

In this article, we consider mixed local and nonlocal Sobolev (q,p)-inequalities with extremal in the case 0<q<1<p<∞. We prove that the extremal of such inequalities is unique up to a multiplicative constant that is associated with a singular elliptic problem involving the mixed local and nonlocal p-Laplace operator. Moreover, it is proved that the...

In this article we provide existence, uniqueness and regularity results of a degenerate singular elliptic boundary value problem whose prototype is given by −div(w(x)|∇u|p−2∇u)=f(x)uδinΩ,u>0inΩ,u=0on∂Ω,\begin{equation*}\hskip7pc \left\{ \def\eqcellsep{&}\begin{array}{l} -{\operatorname{div}}\big (w(x)|\nabla u|^{p-2}\nabla u\big )=\dfrac{f(x)}{u^\d...

We discuss pointwise behavior of weak supersolutions for a class of doubly nonlinear parabolic fractional [Formula: see text]-Laplace equations which includes the fractional parabolic [Formula: see text]-Laplace equation and the fractional porous medium equation. More precisely, we show that weak supersolutions have lower semicontinuous representat...

For a bounded smooth domain \(\Omega \subset {\mathbb {R}}^N\) with \(N\ge 2\), we establish a weighted and an anisotropic version of Sobolev inequality related to the embedding \(W_{0}^{1,p}(\Omega )\hookrightarrow L^q(\Omega )\) for \(1<p<\infty \) and \(2\le p<\infty \) respectively. Our main emphasize is the case of \(0<q<1\) and we deal with a...

We consider equations involving a combination of local and nonlocal degenerate $p$-Laplace operators. The main contribution of the paper is almost Lipschitz regularity for the homogeneous equation and H\"older continuity with an explicit H\"older exponent in the general case. For certain parameters, our results also imply H\"older continuity of the...

We consider nonlinear Kolmogorov-Fokker-Planck type equations of the form \begin{equation}\label{abeqn} (\partial_t+X\cdot\nabla_Y)u=\nabla_X\cdot(A(\nabla_X u,X,Y,t)). \end{equation} The function $A=A(\xi,X,Y,t):\R^m\times\R^m\times\R^m\times\R\to\R^m$ is assumed to be continuous with respect to $\xi$, and measurable with respect to $X,Y$ and $t$....

In this article we study singular subelliptic p-Laplace equations and best constants in Sobolev inequalities on Carnot groups. We prove solvability of these subelliptic p-Laplace equations and existence of the minimizer of the corresponding variational problem. It leads to existence of the best constant in the corresponding (q, p)-Sobolev inequalit...

This article consists of study of anisotropic double phase problems with singular term and sign changing subcritical as well as critical nonlinearity. Seeking the help of well known Nehari manifold technique, we establish existence of at least two opposite sign energy solutions in the subcritical case and one negative energy solution in the critica...

We consider nonlinear Kolmogorov-Fokker-Planck type equations of the form \begin{document}$ \begin{equation*} (\partial_t+X\cdot\nabla_Y)u = \nabla_X\cdot(A(\nabla_X u, X, Y, t)). \end{equation*} $\end{document} The function $ A = A(\xi, X, Y, t): \mathbb R^m\times \mathbb R^m\times \mathbb R^m\times \mathbb R\to \mathbb R^m $ is assumed to be cont...

We consider a class of singular weighted anisotropic $p$-Laplace equations. We provide sufficient condition on the weight function that may vanish or blow up near the origin to ensure the existence of at least one weak solution in the purely singular case and at least two different weak solutions in the purturbed singular case.

This paper is concerned with the qualitative analysis of solutions to the following class of quasilinear problems $$\begin{aligned} \left\{ \begin{array}{ll} -\Delta _{\Phi }u=f(x,u) &{}\quad \text {in } \Omega ,\\ u=0 &{}\quad \text {on }\partial \Omega , \end{array} \right. \end{aligned}$$ (P) where \(\Delta _{\Phi }u=\mathrm{div}\,(\varphi (x,|\...

We establish a local boundedness estimate for weak subsolutions to a doubly nonlinear parabolic fractional p-Laplace equation. Our argument relies on energy estimates and a parabolic nonlocal version of De Giorgi’s method. Furthermore, by means of a new algebraic inequality, we show that positive weak supersolutions satisfy a reverse Hölder inequal...

In this article we study singular subelliptic $p$-Laplace equations and best constants in Sobolev inequalities on nilpotent Lie groups. We prove solvability of these subelliptic $p$-Laplace equations and existence of the minimizer of the corresponding variational problem. It leads to existence of the best constant in the corresponding $(q,p)$-Sobol...

We establish existence and uniqueness of weak solutions for a class of weighted singular problems. The main feature of the article is that our results hold for a wide class of $p$-admissible weights, that may vanish or blow up near the origin and simultaneously the singular exponent is allowed to vary inside the domain. Moreover, our results are va...

In the article we study the Dirichlet $(p,q)$-eigenvalue problem for subelliptic non-commutative operators on nilpotent Lie groups. We prove solvability of this eigenvalue problem and existence of the minimizer of the corresponding variational problem.

We consider mixed local and nonlocal quasilinear parabolic equations of p-Laplace type and discuss several regularity properties of weak solutions for such equations. More precisely, we establish local boundeness of weak subsolutions, local Hölder continuity of weak solutions, lower semicontinuity of weak supersolutions as well as upper semicontinu...

We consider mixed local and nonlocal quasilinear parabolic equations of p-Laplace type and discuss several regularity properties of weak solutions for such equations. More precisely, we establish local boundeness of weak subsolutions, local H\"older continuity of weak solutions, lower semicontinuity of weak supersolutions as well as upper semiconti...

For a given Finsler-Minkowski norm F in R N and a bounded smooth domain Ω ⊂ R N N ≥ 2 , we establish the following weighted Finsler Sobolev inequality (P) S Ω

We establish multiplicity results for the following class of quasilinear problems (P) −∆Φu = f (x, u) in Ω, u = 0 on ∂Ω, where ∆Φu = div(ϕ(x, |∇u|)∇u) for a generalized N-function Φ(x, t) = |t| 0 ϕ(x, s)s ds. We consider Ω ⊂ R N to be a smooth bounded domain that contains two disjoint open regions ΩN and Ωp such that ΩN ∩ Ωp = ∅. The main feature o...

In this article, we consider mixed local and nonlocal Sobolev $(q,p)$-inequalities with extremal in the case $0<q<1<p<\infty$. We prove that the extremal of such inequalities is unique up to a multiplicative constant that is associated with a singular elliptic problem involving the mixed local and nonlocal $p$-Laplace operator. Moreover, it is prov...

This article proves a weak Harnack inequality with a tail term for sign changing supersolutions of a mixed local and nonlocal parabolic equation. Our argument is purely analytic. It is based on energy estimates and the Moser iteration technique. Instead of the parabolic John-Nirenberg lemma, we adopt a lemma of Bombieri to the mixed local and nonlo...

In this article, we study the following anisotropic p-Laplacian equation with variable exponent given by \begin{equation*} (P)\left\{\begin{split} -\Delta_{H,p}u&=\frac{\la f(x)}{u^{q(x)}}+g(u)\text{ in }\Omega,\\ u&>0\text{ in }\Omega,\,u=0\text{ on }\partial\Omega, \end{split}\right. \end{equation*} under the assumption $\Omega$ is a bounded smoo...

We consider a combination of local and nonlocal $p$-Laplace equations and discuss several regularity properties of weak solutions. More precisely, we establish local boundedness of weak subsolutions, local H\"older continuity of weak solutions, Harnack inequality for weak solutions and weak Harnack inequality for weak supersolutions. We also discus...

We discuss pointwise behavior of weak supersolutions for a class of doubly nonlinear parabolic fractional $p$-Laplace equations which includes the fractional parabolic $p$-Laplace equation and the fractional porous medium equation. More precisely, we show that weak supersolutions have lower semicontinuous representative. We also prove that the semi...

By analysing the structure of the fibering map associated with the Euler functional and the Nehari manifold technique, we prove that the following quasilinear singular Schrödinger equation, −∆u − 1 2 ∆(u 2)u = λ u −q + u p in Ω, u > 0 in Ω, u = 0 on ∂ Ω admits at least two weak solutions in X := {u ∈ H 1 0 (Ω) : u 2 ∈ H 1 0 (Ω)} for λ > 0 small, pr...

In this article we consider a variational problem related to a quasilinear singular problem and obtain a nonexistence result in a metric measure space with a doubling measure and a Poincaré inequality. Our method is purely variational and to the best of our knowledge, this is the first work concerning singular problems in a general metric setting.

We establish a local boundedness estimate for weak subsolutions to a doubly nonlinear parabolic fractional $p$-Laplace equation. Our argument relies on energy estimates and a parabolic nonlocal version of De Giorgi's method. Furthermore, by means of a new algebraic inequality, we show that positive weak supersolutions satisfy a reverse H\"older ine...

In this article we consider a variational problem related to a quasilinear singular problem and obtain a nonexistence result in a metric measure space with a doubling measure and a Poincar\'e inequality. Our method is purely variational and to the best of our knowledge, this is the first work concerning singular problems in a general metric setting...

Let p i ≥ 2 and consider the following anisotropic p-Laplace equation − ∑ i = 1 N ∂ ∂ x i ∂ u ∂ x i p i − 2 ∂ u ∂ x i = g ( x ) f ( u ) , u > 0 i n Ω . Under suitable hypothesis on the weight function g we present an existence result for f ( u ) = e 1 u in a bounded smooth domain Ω and nonexistence results for f ( u ) = − e 1 u or − ( u − δ + u − γ...

This article was unintentionally published twice in this journal, by the same authors

This article deals with the existence of the following quasilinear degenerate singular elliptic equation (P λ) −div(w(x)|∇u| p−2 ∇u) = g λ (u), u > 0 in Ω, u = 0 on ∂Ω where Ω ⊂ R n is a smooth bounded domain, n ≥ 3, λ > 0, p > 1 and w is a Muckenhoupt weight. Using variational techniques, for g λ (u) = λf (u)u −q , under certain assumptions on f ,...

For an open, bounded domain \(\Omega \) in \({\mathbb {R}}^N\) which is strictly convex with smooth boundary, we show that there exists a \(\Lambda >0\) such that for \(0<\lambda <{\Lambda } \), the quasilinear singular problem $$\begin{aligned} -\Delta _pu= & {} \lambda u^{-\delta }+u^q\,\,\text{ in }\,\,\Omega \\ u= & {} 0\,\,\text{ on }\,\,\part...

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 =\cfrac{\lambda}{u^{\del}}+u^q\,\,\mbox{in}\,\,\Om\\ &u=0\,\,\mbox{on}\,\,\partial\Om;\, \,\,u>0\,\,\mbox{in}\,\,\Om \end{eqnarray*} admits atl...

In this article, we prove that the following singular quasilinear Schrödinger equation, −∆u − 1 2 ∆(u 2)u = λu −q + u p in Ω, u > 0 in Ω, u = 0 on ∂Ω, admits at least two positive weak solutions for λ > 0 small, provided Ω is a bounded smooth domain in R N (≥3) , 0 < q < 1 and 4 < p + 1 < 4N N −2. We analyse the structure of fibering map on the ass...

In this paper we present some non existence results concerning the stable solutions to the equation $$\operatorname{div}(w(x)|\nabla u|^{p-2}\nabla u)=g(x)f(u)\;\;\mbox{in}\;\;\mathbb{R}^N;\;\;p\geq 2$$ when $f(u)$ is either $u^{-\delta}+u^{-\gamma}$, $\delta,\gamma>0$ or $\exp(\frac{1}{u})$ and for a suitable class of weight functions $w,g$.

This article deals with the existence of the following quasilinear degenerate singular elliptic equation \begin{equation*} (P_\la)\left\{ \begin{split} -\text{div}(w(x)|\nabla u|^{p-2}\nabla u) &= g_{\la}(u),\;u>0\; \text{in}\; \Om, u&=0 \; \text{on}\; \partial \Om \end{split}\right. \end{equation*} where $ \Om \subset \mb R^n$ is a smooth bounded...

Journal ref: Complex Variables and Elliptic Equations

Let $p_i\geq 2$ and consider the following anisotropic $p$-Laplace equation $$ -\sum_{i=1}^{N}\frac{\partial}{\partial x_i}\Big(\Big|\frac{\partial u}{\partial x_i}\Big|^{p_i-2}\frac{\partial u}{\partial x_i}\Big)=g(x)f(u),\,\,u>0\text{ in }\Omega. $$ Under suitable hypothesis on the weight function $g$ we present an existence result for $f(u)=e^\f...

For any bounded smooth domain $\Omega$ of $\mathbb{R}^N$ with $N\geq 2$, we provide existence, uniqueness and regularity results for weak solutions to the degenerated singular problem \begin{gather*} \begin{cases} -\operatorname{div}(\mathcal A(x,\nabla u))=\frac{f}{u^\delta}\,\,\text{ in }\,\,\Omega, u>0\text{ in }\Omega,\,\, u = 0 \text{ on } \pa...

In this paper, we prove some qualitative properties for the positive solutions to some degenerate elliptic equation given by \[-\operatorname{div}(w|\nabla u|^{p-2}\nabla u)=f(x,u);\;\;w\in \mathcal{A}_p\] on smooth domain and for varying nonlinearity $f$.

In this paper, we prove some qualitative properties for the positive solutions to some degenerate elliptic equation given by \[-\operatorname{div}(w|\nabla u|^{p-2}\nabla u)=f(x,u);\;\;w\in \mathcal{A}_p\] on smooth domain and for varying nonlinearity $f$.

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 $C^{1,\zeta}_{loc}$ solution in $\mathbb{R}^N$ where $f(x)$ is either $-x^{-\delta}$ or $e^x$ for any $0<\zeta<1$.

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 =\cfrac{\lambda}{u^{\del}}+u^q\,\,\mbox{in}\,\,\Om\\ &u=0\,\,\mbox{on}\,\,\partial\Om;\, \,\,u>0\,\,\mbox{in}\,\,\Om \end{eqnarray*} admits atl...

The Finsler p-Laplacian is the class of nonlinear differential operators given by ΔH,pu≔div(H(∇u)p−1∇ηH(∇u)) where p>1, H:Rn→[0,∞) is a convex function which is in C1(Rn∖{0}) and is positively homogeneous of degree 1. In this article we provide a comparison principle, weighted Poincare Inequality, Liouville Theorem and Hardy type inequality for the...