Searcharxiv⌕ Search

arXiv subjects

Prashanta Garain

Publications and source records attributed to Prashanta Garain.

At least 37 records · Page 2Linked to original sources
arXiv2024

Higher Hölder regularity for the fractional $p$-Laplace equation in the subquadratic case

We study the fractional $p$-Laplace equation $$ (-Δ_p)^s u = 0 $$ for $0<s<1$ and in the subquadratic case $1<p<2$. We provide Hölder estimates with an explicit Hölder exponent. The inhomogeneous equation is also treated and there the exponent obtained is almost sharp. Our results complement the previous results for the superquadratic case when $p\geq 2$. The arguments are based on a careful Moser-type iteration and a perturbation argument.

math.AP↗
arXiv2024

Higher Hölder regularity for a subquadratic nonlocal parabolic equation

In this paper, we are concerned with the Hölder regularity for solutions of the nonlocal evolutionary equation $$ \partial_t u+(-Δ_p)^s u = 0. $$ Here, $(-Δ_p)^s$ is the fractional $p$-Laplacian, $0<s<1$ and $1<p<2$. We establish Hölder regularity with explicit Hölder exponents. We also include the inhomogeneous equation with a bounded inhomogeneity. In some cases, the obtained Hölder exponents are almost sharp. Our results complement the previous results for the superquadratic case when $p\geq 2$.

math.AP↗
arXiv2023

On the regularity and existence of weak solutions for a class of degenerate singular elliptic problem

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 source function to establish regularity and existence results.

math.AP↗
arXiv2023

On a class of mixed local and nonlocal semilinear elliptic equation with singular nonlinearity

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 and approximation approach.

math.AP↗
arXiv2023

On the regularity theory for mixed anisotropic and nonlocal $p$-Laplace equations and its applications to singular problems

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 problems. More precisely, we obtain local boundedness of subsolutions, Harnack inequality for solutions and weak Harnack inequality for supersolutions.

math.AP↗
arXiv2023

Mixed local and nonlocal Dirichlet $(p,q)$-eigenvalue problem

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 $Ω\subset\mathbb{R}^N$ under the assumption that $1<p<\infty$ and $1<q<p^{*}$ where $p^{*}=\frac{N p}{N-p}$ if $1<p<N$ and $p^{*}=\infty$ if $p\geq N$.

math.AP↗
arXiv2022

Higher Hölder regularity for mixed local and nonlocal degenerate elliptic equations

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 gradient. In addition, we establish existence, uniqueness and local boundedness. The approach is based on an iteration in the spirit of Moser combined with an approximation method.

math.AP↗
arXiv2022

On regularity and existence of weak solutions to nonlinear Kolmogorov-Fokker-Planck type equations with rough coefficients

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 $A=A(ξ,X,Y,t):\R^m\times\R^m\times\R^m\times\R\to\R^m$ is assumed to be continuous with respect to $ξ$, and measurable with respect to $X,Y$ and $t$. $A=A(ξ,X,Y,t)$ is allowed to be nonlinear but with linear growth. We establish higher integrability and local boundedness of weak sub-solutions, weak Harnack and Harnack inequalities, and H{ö}lder continuity with quantitative estimates. In addition we establish existence and uniqueness of weak solutions to a Dirichlet problem in certain bounded $X$, $Y$ and $t$ dependent domains.

math.AP↗
arXiv2022

On an anisotropic double phase problem with singular and sign changing nonlinearity

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 critical case. The results in the critical case is even new in the classical $p$-Laplacian case.

math.AP↗
arXiv2021

On a class of weighted anisotropic $p$-Laplace equation with singular nonlinearity

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.

math.AP↗
arXiv2021

Weighted anisotropic Sobolev inequality with extremal and associated singular problems

For a given Finsler-Minkowski norm $\mathcal{F}$ in $\mathbb{R}^N$ and a bounded smooth domain $Ω\subset\mathbb{R}^N$ $\big(N\geq 2\big)$, we establish the following weighted anisotropic Sobolev inequality $$ S\left(\int_Ω|u|^q f\,dx\right)^\frac{1}{q}\leq\left(\int_Ω\mathcal{F}(\nabla u)^p w\,dx\right)^\frac{1}{p},\quad\forall\,u\in W_0^{1,p}(Ω,w)\leqno{\mathcal{(P)}} $$ where $W_0^{1,p}(Ω,w)$ is the weighted Sobolev space under a class of $p$-admissible weights $w$, where $f$ is some nonnegative integrable function in $Ω$. We discuss the case $0<q<1$ and observe that $$ μ(Ω):=\inf_{u\in W_{0}^{1,p}(Ω,w)}\Bigg\{\int_Ω\mathcal{F}(\nabla u)^p w\,dx:\int_Ω|u|^{q}f\,dx=1\Bigg\}\leqno{\mathcal{(Q)}} $$ is associated with singular weighted anisotropic $p$-Laplace equations. To this end, we also study existence and regularity properties of solutions for weighted anisotropic $p$-Laplace equations under the mixed and exponential singularities.

math.AP↗
arXiv2021

Singular subelliptic equations and Sobolev inequalities on nilpotent Lie groups

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 $(q,p)$-Sobolev inequality, $0<q<1$, $1<p<ν$.

math.AP↗
arXiv2021

On the regularity theory for mixed local and nonlocal quasilinear elliptic equations

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 Hölder continuity of weak solutions, Harnack inequality for weak solutions and weak Harnack inequality for weak supersolutions. We also discuss lower semicontinuity of weak supersolutions as well as upper semicontinuity of weak subsolutions. Our approach is purely analytic and it is based on the De Giorgi-Nash-Moser theory, the expansion of positivity and estimates involving a tail term. The main results apply to sign changing solutions and capture both local and nonlocal features of the equation.

math.AP↗
arXiv2021

On the regularity theory for mixed local and nonlocal quasilinear parabolic equations

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, lower semicontinuity of weak supersolutions as well as upper semicontinuity of weak subsolutions. We also discuss the pointwise behavior of the semicontinuous representatives. Our main results are valid for sign changing solutions. Our approach is purely analytic and is based on energy estimates and the De Giorgi theory.

math.AP↗
arXiv2021

On an anisotropic p-Laplace equation with variable singular exponent

In this article, we study the following anisotropic p-Laplacian equation with variable exponent given by \begin{equation*} (P)\left\{\begin{split} -Δ_{H,p}u&=\frac{\la f(x)}{u^{q(x)}}+g(u)\text{ in }Ω,\\ u&>0\text{ in }Ω,\,u=0\text{ on }\partialΩ, \end{split}\right. \end{equation*} under the assumption $Ω$ is a bounded smooth domain in $\mathbb{R}^N$ with $p,N\geq 2$, $\la>0$ and $0<q \in C(\bar \Om)$. For the purely singular case that is $g\equiv 0$, we proved existence and uniqueness of solution. We also demonstrate the existence of multiple solution to $(P)$ provided $f\equiv 1$ and $g(u)=u^r$ for $r\in (p-1,p^*-1)$.

math.AP↗
arXiv2021

Multiple solutions for a class of quasilinear problems with double criticality

We establish multiplicity results for the following class of quasilinear problems $$ \left\{ \begin{array}{l} -Δ_Φu=f(x,u) \quad \mbox{in} \quad Ω, \\ u=0 \quad \mbox{on} \quad \partial Ω, \end{array} \right. \leqno{(P)} $$ where $Δ_Φu=\text{div}(φ(x,|\nabla u|)\nabla u)$ for a generalized N-function $Φ(x,t)=\int_{0}^{|t|}φ(x,s)s\,ds$. We consider $Ω\subset\mathbb{R}^N$ to be a smooth bounded domain that contains two disjoint open regions $Ω_N$ and $Ω_p$ such that $\overline{Ω_N}\cap\overline{Ω_p}=\emptyset$. The main feature of the problem $(P)$ is that the operator $-Δ_Φ$ behaves like $-Δ_N$ on $Ω_N$ and $-Δ_p$ on $Ω_p$. We assume the nonlinearity $f:Ω\times\mathbb{R}\to\mathbb{R}$ of two different types, but both behaves like $e^{α|t|^\frac{N}{N-1}}$ on $Ω_N$ and $|t|^{p^*-2}t$ on $Ω_p$ as $|t|$ is large enough, for some $α>0$ and $p^*=\frac{Np}{N-p}$ being the critical Sobolev exponent for $1<p<N$. In this context, for one type of nonlinearity $f$, we provide multiplicity of solutions in a general smooth bounded domain and for another type of nonlinearity $f$, in an annular domain $Ω$, we establish existence of multiple solutions for the problem $(P)$ that are nonradial and rotationally nonequivalent.

math.AP↗