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Konijeti Sreenadh

Publications and source records attributed to Konijeti Sreenadh.

15 recordsLinked to original sources
arXiv2025

Elliptic Problems Involving Mixed Local-Nonlocal Operator in the Hyperbolic Space

This paper explores the existence of solutions to a class of nonlinear elliptic equations involving a mixed local-nonlocal operator of the form $-Δ_{\mathbb{B}^N} + (-Δ_{\mathbb{B}^N})^s$, with $0 < s < 1$, set in the hyperbolic space $\mathbb{B}^N$. By employing variational methods, we address both subcritical and critical nonlinearities, establishing the existence of weak solutions under appropriate conditions.

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arXiv2025

Global Compactness Result for a Brézis-Nirenberg-Type Problem Involving Mixed Local Nonlocal Operator

This paper investigates the profile decomposition of Palais-Smale sequences associated with a Brezis-Nirenberg type problem involving a combination of mixed local nonlocal operators, given by \begin{equation*} \left\{\begin{aligned} &-Δu + (-Δ)^s u - λu = |u|^{2^*-2}u \;\;\mbox{ in } Ω, &\quad u=0\,\mbox{ in }\mathbb{R}^N\setminus Ω. \end{aligned} \right. \end{equation*} where $Ω\subseteq \mathbb{R}^{N}$ is a smooth bounded domain with $N \geq 3$, $s\in (0,1),\,λ\in\mathbb{R}$ is a real parameter and $2^* = \frac{2N}{N - 2} $ denotes the critical Sobolev exponent. As an application of the derived global compactness result, we further study the existence of positive solution of the corresponding Coron-type problem (C. R. Acad. Sci. Paris Sér I Math, 299(7):209-212, 1984) when $λ=0$.

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arXiv2024

Existence and regularity results for space-time fractional integro-differential equation of Kirchhoff type with memory

This paper analyses a Kirchhoff type quasilinear space-time fractional integro-differential equation with memory $(\mathcal{K}^{s}_α)$. Various a priori bounds are derived in different norms on the solution of the considered equation. Utilizing these a priori bounds, existence and uniqueness of the weak solution to the proposed model are proved. Furthermore, regularity results on the solution of $(\mathcal{K}^{s}_α)$ are established. The contribution made in this work provides a framework for further investigation of such types of partial integro-differential equation $(\mathcal{K}^{s}_α)$.

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arXiv2023

An elliptic problem involving critical Choquard and singular discontinuous nonlinearity

The present article investigates the existence, multiplicity and regularity of weak solutions of problems involving a combination of critical Hartree type nonlinearity along with singular and discontinuous nonlinearity. By applying variational methods and using the notion of generalized gradients for Lipschitz continuous functional, we obtain the existence and the multiplicity of weak solutions for some suitable range of $λ$ and $γ$. Finally by studying the $L^\infty$-estimates and boundary behavior of weak solutions, we prove their Hölder and Sobolev regularity.

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arXiv2023

A Linearized L1-Galerkin FEM for Non-smooth Solutions of Kirchhoff type Quasilinear Time-fractional Integro-differential Equation

In this article, we study the semi discrete and fully discrete formulations for a Kirchhoff type quasilinear integro-differential equation involving time-fractional derivative of order $α\in (0,1) $. For the semi discrete formulation of the equation under consideration, we discretize the space domain using a conforming FEM and keep the time variable continuous. We modify the standard Ritz-Volterra projection operator to carry out error analysis for the semi discrete formulation of the considered equation. In general, solutions of the time-fractional partial differential equations (PDEs) have a weak singularity near time $t=0$. Taking this singularity into account, we develop a new linearized fully discrete numerical scheme for the considered equation on a graded mesh in time. We derive a priori bounds on the solution of this fully discrete numerical scheme using a new weighted $H^{1}(Ω)$ norm. We prove that the developed numerical scheme has an accuracy rate of $O(P^{-1}+N^{-(2-α)})$ in $L^{\infty}(0,T;L^{2}(Ω))$ as well as in $L^{\infty}(0,T;H^{1}_{0}(Ω))$, where $P$ and $N$ are degrees of freedom in the space and time directions respectively. The robustness and efficiency of the proposed numerical scheme are demonstrated by some numerical examples.

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arXiv2022

Choquard equation involving mixed local and nonlocal operators

In this article, we study an elliptic problem involving an operator of mixed order with both local and nonlocal aspects and in the presence of critical nonlinearity of Hartree type. To this end, we first investigate the corresponding Hardy-Littlewood-Sobolev inequality and detect the optimal constant. Using variational methods and a Pohožaev identity we then show the existence and nonexistence results for the corresponding subcritical perturbation problem.

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arXiv2022

Finite Element Analysis of Time Fractional Integro-differential Equations of Kirchhoff type for Non-homogeneous Materials

In this paper, we study a time-fractional initial-boundary value problem of Kirchhoff type involving memory term for non-homogeneous materials. The energy argument is applied to derive the a priori bounds on the solution of the considered problem. Consequently, we prove the existence and uniqueness of the weak solution to the problem under consideration. We keep the time variable continuous and discretize the space domain using a conforming FEM to obtain the semi discrete formulation of the problem. The semi discrete error analysis is carried out by modifying the standard Ritz-Volterra projection operator. To obtain the numerical solution to the problem efficiently, we develop a new linearized L1 Galerkin FEM. This numerical scheme is shown to have a convergence rate of $O(h+k^{2-α})$, where $α~ (0<α<1)$ is the fractional derivative exponent, $h$ and $k$ are the discretization parameters in the space and time directions respectively. Further, this convergence rate is improved in the time direction by proposing a novel linearized L2-1$_σ$ Galerkin FEM. We prove that this numerical scheme has an accuracy rate of $O(h+k^{2})$. Finally, a numerical experiment is conducted to validate our theoretical claims.

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arXiv2022

A Choquard type equation with a singular absorption nonlinearity in two dimension

In this article, we show the existence of a nonnegative solution to the singular problem $(\mc P_\la)$ posed in a bounded domain $Ω$ in $\mb R^2$ (see below). We achieve this by approximating the singular function $u^{-β}\log(u)$ by a function $l_\e(u)$ which pointwisely converges to $-u^β\log(u)$ as $\e \ra 0$. Using variational techniques, the perturbed equation $-\De u+l_\e(u)=\ds\la \left(\I{\Om}\frac{F(u(y))}{|x-y|^μ}dy\right)f(u(x))$ is shown to have a solution $u_\e \in H_0^{1}(\Om)$ when the parameter $\la >0$ is small enough. Letting $\e \ra 0$ and proving a pointwise gradient estimate, we show that the solution $u_\e$ converges to a nontrivial nonnegative solution of the original problem $(\mc P_\la)$.

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arXiv2018

n-Kirchhoff Choquard equations with exponenetial nonlinearity

This article deals with the study of the following Kirchhoff equation with exponential nonlinearity of Choquard type (see $(KC)$ below). We use the variational method in the light of Moser-Trudinger inequality to show the existence of weak solutions to $(KC)$. Moreover, analyzing the fibering maps and minimizing the energy functional over suitable subsets of the Nehari manifold, we prove existence and multiplicity of weak solutions to convex-concave problem $(\mathcal{P}_{\la,M})$ below.

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arXiv2018

Existence of three positive solutions for a nonlocal singular dirichlet boundary problem

In this article, we prove the existence of at least three positive solutions for the following nonlocal singular problem \begin{equation*} (P_\la)\left\{ \begin{split} (-\De)^su &= \la\frac{f(u)}{u^q}, \; \; u>0 \;\; \text{in}\;\; \Om,\\ u &= 0\;\; \text{in}\;\; \mb R^n \setminus \Om \end{split} \right. \end{equation*} where $(-\De)^s$ denotes the fractional Laplace operator for $s\in (0,1)$, $n>2s$, $q \in (0,1)$, $\la>0$ and $\Om$ is smooth bounded domain in $\mb R^n$. Here $f :[0,\infty) \to [0,\infty)$ is a continuous nondecreasing map satisfying $\lim\limits_{u\to \infty}\frac{f(u)}{u^{q+1}}=0$. We show that under certain additional assumptions on $f$, $(P_\la)$ possesses at least three distinct solutions for a certain range of $\la$. We use the method of sub-supersolutions and a critical point theorem by Amann \cite{amann} to prove our results. Moreover, we prove a new existence result for a suitable infinite semipositone nonlocal problem which played a crucial role to obtain our main result and is of independent interest. \medskip

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arXiv2016

Positive solutions for nonlinear Choquard equation with singular nonlinearity

In this article, we study the following nonlinear Choquard equation with singular nonlinearity \begin{equation*} \quad -\De u = \la u^{-q} + \left( \int_{\Om}\frac{|u|^{2^*_μ}}{|x-y|^μ}\mathrm{d}y \right)|u|^{2^*_μ-2}u, \quad u>0 \; \text{in}\; \Om,\quad u = 0 \; \mbox{on}\; \partial\Om, \end{equation*} where $\Om$ is a bounded domain in $\mb{R}^n$ with smooth boundary $\partial \Om$, $n > 2,\; \la >0,\; 0 < q < 1, \; 0<μ<n$ and $2^*_μ=\frac{2n-μ}{n-2}$. Using variational approach and structure of associated Nehari manifold, we show the existence and multiplicity of positive weak solutions of the above problem, if $\la$ is less than some positive constant. We also study the regularity of these weak solutions.

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arXiv2015

Critical growth fractional systems with exponential nonlinearity

We study the existence of positive solutions for the system of fractional elliptic equations of the type, \begin{equation*} \begin{array}{rl} (-Δ)^{\frac{1}{2}} u &=\frac{p}{p+q}λf(x)|u|^{p-2}u|v|^q + h_1(u,v) e^{u^2+v^2},\;\textrm{in}\; (-1, 1),\\ (-Δ)^{\frac{1}{2}} v &=\frac{q}{p+q}λf(x)|u|^p|v|^{q-2}v + h_2(u,v) e^{u^2+v^2},\;\textrm{in}\; (-1, 1), u,v&>0 \;\textrm{in } \; (-1,1), u&=v=0 \; \text{in} \; \mathbb R\setminus (-1,1). \end{array} \end{equation*} where {$1 2}$. Here $(-Δ)^{\frac{1}{2}}$ is the fractional Laplacian operator. We show the existence of multiple solutions for suitable range of $λ$ by analyzing the fibering maps and the corresponding Nehari manifold. We also study the existence of positive solutions for a superlinear system with critical growth exponential nonlinearity.

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arXiv2015

Critical exponent for half-Laplacian in the whole space

We study the existence of {weak} solutions for fractional elliptic equations of the type, \begin{equation*} (-Δ)^{\frac{1}{2}} u+ V(x) u= h(u), u> 0 \;\textrm{in} \;\mathbb R, \end{equation*} %where $1 2,\;1<β\leq2\;, λ>0, K(x)>0, f$ is continuous and sign changing. where $h$ is a real valued function that behaves like $e^{u^2}$ as $u\rightarrow \infty$ and $V(x)$ is a positive, continuous unbounded function. Here $(-Δ)^{\frac{1}{2}}$ is the fractional Laplacian operator. We show the existence of mountain-pass solution when the nonlinearity is superlinear near $t=0$. We also study the corresponding critical exponent problem for the Kirchhoff equation \[ m\left(\int_{\mathbb R}|(-Δ)^{\frac{1}{2}}u|^2 dx+ \int_\mb R u^2 V(x)dx\right)\left((-Δ)^{\frac{1}{2}} u+ V(x) u\right)= f(u)\;\, \text{in}\, \mathbb R \] where $f(u)$ behaves like $e^{u^2}$ as $u\rightarrow \infty$ and $f(u)\sim u^3$ as $u\rightarrow 0$.

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