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K. Sreenadh

Publications and source records attributed to K. Sreenadh.

At least 55 records · Page 3Linked to original sources
arXiv2020

A qualitative study of (p,q) Singular parabolic equations: local existence, Sobolev regularity and asymptotic behaviour

The purpose of the article is to study the existence, regularity, stabilization and blow up results of weak solution to the following parabolic $(p,q)$-singular equation: \begin{equation*} (P_t)\; \left\{\begin{array}{rllll} u_t-Δ_{p}u -Δ_{q}u & = \vth \; u^{-\de}+ f(x,u), \; u>0 \text{ in } \Om\times (0,T), \\ u&=0 \quad \text{ on } \pa\Om\times (0,T), u(x,0)&= u_0(x) \; \text{ in }\Om, \end{array} \right. \end{equation*} where $\Om$ is a bounded domain in $\mathbb{R}^N$ with $C^2$ boundary $\pa\Om$, $1 0$, $N\ge 2$ and $\vth>0$ is a parameter. Moreover, we assume that $f:\Om\times [0,\infty) \to \mb R$ is a bounded below Carathéodory function, locally Lipschitz with respect to the second variable uniformly in $x\in\Om$ and $u_0\in L^\infty(\Om)\cap W^{1,p}_0(\Om)$. We distinguish the cases as $q$-subhomogeneous and $q$-superhomogeneous depending on the growth of $f$ (hereafter we will drop the term $q$). In the subhomogeneous case, we prove the existence and uniqueness of the weak solution to problem $(P_t)$ for $\de<2+1/(p-1)$. For this, we first study the stationary problems corresponding to $(P_t)$ by using the method of sub and super solutions and subsequently employing implicit Euler method, we obtain the existence of a solution to $(P_t)$. Furthermore, in this case, we prove the stabilization result, that is, the solution $u(t)$ of $(P_t)$ converges to $u_\infty$, the unique solution to the stationary problem, in $L^\infty(\Om)$ as $t\ra\infty$. For the superhomogeneous case, we prove the local existence theorem by taking help of nonlinear semigroup theory. Subsequently, we prove finite time blow up of solution to problem $(P_t)$ for small parameter $\vartheta>0$ in the case $\de\leq 1$ and for all $\vth>0$ in the case $\de>1$.

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arXiv2020

Singular elliptic problems with unbalanced growth and critical exponent

In this article, we study the existence and multiplicity of solutions of the following $(p,q)$-Laplace equation with singular nonlinearity: \begin{equation*} \left\{\begin{array}{rllll} -Δ_{p}u-\baΔ_{q}u & = \la u^{-\de}+ u^{r-1}, \ u>0, \ \text{ in } \Om \\ u&=0 \quad \text{ on } \pa\Om, \end{array} \right. \end{equation*} where $\Om$ is a bounded domain in $\mathbb{R}^n$ with smooth boundary, $1< q< p p$ and $\la,\, \ba>0$ are parameters. We prove existence, multiplicity and regularity of weak solutions of $(P_\la)$ for suitable range of $\la$. We also prove the global existence result for problem $(P_\la)$.

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arXiv2020

Singular doubly nonlocal elliptic problems with Choquard type critical growth nonlinearities

The theory of elliptic equations involving singular nonlinearities is well studied topic but the interaction of singular type nonlinearity with nonlocal nonlinearity in elliptic problems has not been investigated so far. In this article, we study the very singular and doubly nonlocal singular problem $(P_λ)$(See below). Firstly, we establish a very weak comparison principle and the optimal Sobolev regularity. Next using the critical point theory of non-smooth analysis and the geometry of the energy functional, we establish the global multiplicity of positive weak solutions.

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arXiv2020

Unbalanced $(p,2)$-fractional problems with critical growth

We study the existence, multiplicity and regularity results of non-negative solutions of following doubly nonlocal problem: $$ (P_\la) \left\{ \begin{array}{lr}\ds \quad (-Δ)^{s_1}u+\ba (-Δ)^{s_2}_{p}u = \la a(x)|u|^{q-2}u+ \left(\int_{\Om}\frac{|u(y)|^r}{|x-y|^μ}~dy\right)|u|^{r-2} u \quad \text{in}\; \Om, \quad \quad\quad \quad u =0\quad \text{in} \quad \mb R^n\setminus \Om, \end{array} \right. $$ where $\Om\subset\mb R^n$ is a bounded domain with $C^2$ boundary $\pa\Om$, $0 2 s_1$, $1< q 0$ and $a\in L^{\frac{d}{d-q}}(\Om)$, for some $q<d<2^{*}_{s_1}:=\frac{2n}{n-2s_1}$, is a sign changing function. We prove that each nonnegative weak solution of $(P_\la)$ is bounded. Furthermore, we obtain some existence and multiplicity results using Nehari manifold method.

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arXiv2019

Polyharmonic Kirchhoff problems involving exponential non-linearity of Choquard type with singular weights

In this work, we study the higher order Kirchhoff type Choquard equation $(KC)$ involving a critical exponential non-linearity and singular weights. We prove the existence of solution to $(KC)$ using Mountain pass Lemma in light of Moser-Trudinger and singular Adams-Moser inequalities. In the second part of the paper, using the Nehari manifold technique and minimization over its suitable subsets, we prove the existence of at least two solutions to the Kirchhoff type Choquard equation $(\mathcal{P_{\la,M}})$ involving convex-concave type non-linearity.

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arXiv2019

Regularity results on a class of doubly nonlocal problems

The purpose of this article is twofold. First, an issue of regularity of weak solution to the problem $(P)$ (See below) is addressed. Secondly, we investigate the question of $H^s$ versus $C^0$- weighted minimizers of the functional associated to problem $(P)$ and then give applications to existence and multiplicity results.

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arXiv2019

Brezis-Nirenberg type result for Kohn Laplacian with critical Choquard Nonlinearity

In this article, we are study the following Dirichlet problem with Choquard type non linearity \[ -Δ_{\mathbb{H}} u = a u+ \left(\int_Ω\frac{|u(η)|^{Q^*_λ}}{|η^{-1}ξ|^λ}dη\right)|u|^{Q^*_λ-2}u \; \text{in}\; Ω,\quad u = 0 \; \text{ on } \partial Ω, \] where $Ω$ is a smooth bounded subset of the Heisenberg group $\mathbb{H}^N, N\in \mathbb N$ with $C^2$ boundary and $Δ_{\mathbb{H}}$ is the Kohn Laplacian on the Heisenberg group $\mathbb{H}^N$. Here, $Q^*_λ=\frac{2Q-λ}{Q-2},\; Q= 2N+2$ and $a$ is a positive real parameter. We derive the Brezis-Nirenberg type result for the above problem. Moreover, we also prove the regularity of solutions and nonexistence of solutions depending on the range of $a$.

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arXiv2019

Coron problem for nonlocal equations invloving Choquard nonlinearity

We study the problem \[ -\De u = \left(\int_{\Om}\frac{|u(y)|^{2^*_μ}}{|x-y|^μ}dy\right)|u|^{2^*_μ-2}u, \; \text{in}\; \Om,\quad u = 0 \; \text{ on } \pa \Om , \] where $\Om$ is a smooth bounded domain in $\mathbb{R}^N( N\geq 3)$, $2^*_μ=\frac{2N-μ}{N-2}$. we prove the existence of a positive solution of the above problem in an annular type domain when the inner hole is sufficiently small.

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arXiv2019

Critical growth elliptic problems involving Hardy-Littlewood-Sobolev critical exponent in non-contractible domains

The paper is concerned with the existence and multiplicity of positive solutions of the nonhomogeneous Choquard equation over an annular type bounded domain. Precisely, we consider the following equation \[ -\De u = \left(\int_{\Om}\frac{|u(y)|^{2^*_μ}}{|x-y|^μ}dy\right)|u|^{2^*_μ-2}u+f \; \text{in}\; \Om,\quad u = 0 \; \text{ on } \pa \Om , \] where $\Om$ is a smooth bounded annular domain in $\mathbb{R}^N( N\geq 3)$, $2^*_μ=\frac{2N-μ}{N-2}$, $f \in L^{\infty}(\Om)$ and $f \geq 0$. We prove the existence of four positive solutions of the above problem using the Lusternik-Schnirelmann theory and varitaional methods, when the inner hole of the annulus is sufficiently small.

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arXiv2019

Regularity and multiplicity results for fractional $(p,q)$-Laplacian equations

This article deals with the study of the following nonlinear doubly nonlocal equation: \begin{equation*} (-Δ)^{s_1}_{p}u+\ba(-Δ)^{s_2}_{q}u = \la a(x)|u|^{δ-2}u+ b(x)|u|^{r-2} u,\; \text{ in }\; \Om, \; u=0 \text{ on } \mathbb{R}^n\setminus \Om, \end{equation*} where $\Om$ is a bounded domain in $\mathbb{R}^n$ with smooth boundary, $1< \de \le q\leq p p s_1$ and $\la, \ba>0$ are parameters. Here $a\in L^{\frac{r}{r-\de}}(\Om)$ and $b\in L^{\infty}(\Om)$ are sign changing functions. We prove the $L^\infty$ estimates, weak Harnack inequality and Interior Hölder regularity of the weak solutions of the above problem in the subcritical case $(r<p_{s_1}^*).$ Also, by 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 above convex-concave problem. In case of $\de=q$, we show the existence of solution.

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arXiv2019

Kirchhoff equations with Hardy-Littlewood-Sobolev critical nonlinearity

We consider the following Kirchhoff - Choquard equation \[ -M(\|\na u\|_{L^2}^{2})\De u = \la f(x)|u|^{q-2}u+ \left(\int_{\Om}\frac{|u(y)|^{2^*_μ}}{|x-y|^μ}dy\right)|u|^{2^*_μ-2}u \; \text{in}\; \Om,\quad u = 0 \; \text{ on } \pa \Om , \] where $\Om$ is a bounded domain in $\mathbb{R}^N( N\geq 3)$ with $C^2$ boundary, $2^*_μ=\frac{2N-μ}{N-2}$, $1<q\leq 2$, and $f$ is a continuous real valued sign changing function. When $1<q< 2$, using the method of Nehari manifold and Concentration-compactness Lemma, we prove the existence and multiplicity of positive solutions of the above problem. We also prove the existence of a positive solution when $q=2$ using the Mountain Pass Lemma.

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arXiv2018

Critical growth elliptic problems with Choquard type nonlinearity:A survey

This article deals with a survey of recent developments and results on Choquard equations where we focus on the existence and multiplicity of solutions of the partial differential equations which involve the nonlinearity of convolution type. Because of its nature, these equations are categorized under the nonlocal problems. We give a brief survey on the work already done in this regard following which we illustrate the problems we have addressed. Seeking the help of variational methods and asymptotic estimates, we prove our main results.

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arXiv2018

A Global multiplicity result for a very singular critical nonlocal equation

In this article, we show the global multiplicity result for the following nonlocal singular problem \begin{equation*} (P_\la):\;\quad (-\De)^s u = u^{-q} + \la u^{{2^*_s}-1}, \quad u>0 \; \text{in}\; \Om,\quad u = 0 \; \mbox{in}\; \mb R^n \setminus\Om, \end{equation*} where $\Om$ is a bounded domain in $\mb{R}^n$ with smooth boundary $\partial \Om$, $n > 2s,\; s \in (0,1),\; \la >0,\; q>0$ satisfies $q(2s-1)<(2s+1)$ and $2^*_s=\frac{2n}{n-2s}$. Employing the variational method, we show the existence of at least two distinct weak positive solutions for $(P_\la)$ in $X_0$ when $\la \in (0,\La)$ and no solution when $\la>\La$, where $\La>0$ is appropriately chosen. We also prove a result of independent interest that any weak solution to $(P_λ)$ is in $C^α(\R^n)$ with $α=α(s,q)\in (0,1)$. The asymptotic behaviour of weak solutions reveals that this result is sharp.

math.AP↗
arXiv2018

On Concentration of least energy solutions for magnetic critical Choquard equations

In the present paper, we consider the following magnetic nonlinear Choquard equation $$ \left\{ \begin{array}{ll} & (-i \nabla+A(x))^2u + μg(x)u = λu + (|x|^{-α} * |u|^{2^*_α})|u|^{2^*_α-2}u ,\; u>0 \;\text{in} \; \mathbb{ R}^n , & u \in H^1(\mathbb{R}^n, \mathbb{ C}) \end{array} \right\}.$$ where $n \geq 4$, $2^*_α= \frac{2n-α}{n-2}$, $λ>0$, $μ\in \mathbb{ R}$ is a parameter, $α\in (0,n)$, $A(x): \mathbb{R}^n \rightarrow \mathbb{ R}^n$ is a magnetic vector potential and $g(x)$ is a real valued potential function on $\mathbb{R}^n$. Using variational methods, we establish the existence of least energy solution under some suitable conditions. Moreover, the concentration behavior of solutions is also studied as $μ\rightarrow +\infty$.

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arXiv2017

Doubly nonlocal system with Hardy-Littlewood-Sobolev critical nonlinearity

This article concerns about the existence and multiplicity of weak solutions for the following nonlinear doubly nonlocal problem with critical nonlinearity in the sense of Hardy-Littlewood-Sobolev inequality \begin{equation*} \left\{ \begin{split} (-Δ)^su &= λ|u|^{q-2}u + \left(\int_Ω\frac{|v(y)|^{2^*_μ}}{|x-y|^μ}~\mathrm{d}y\right) |u|^{2^*_μ-2}u\; \text{in}\; Ω (-Δ)^sv &= δ|v|^{q-2}v + \left(\int_{\Om}\frac{|u(y)|^{2^*_μ}}{|x-y|^μ}~\mathrm{d}y \right) |v|^{2^*_μ-2}v \; \text{in}\; Ω u &=v=0\; \text{in}\; \mb R^n\setminusΩ, \end{split} \right. \end{equation*} where $Ω$ is a smooth bounded domain in $\mb R^n$, $n >2s$, $s \in (0,1)$, $(-Δ)^s$ is the well known fractional Laplacian, $μ\in (0,n)$, $2^*_μ= \displaystyle\frac{2n-μ}{n-2s}$ is the upper critical exponent in the Hardy-Littlewood-Sobolev inequality, $1 0$ are real parameters. We study the fibering maps corresponding to the functional associated with $(P_{λ,δ})$ and show that minimization over suitable subsets of Nehari manifold renders the existence of atleast two non trivial solutions of $(P_{\la,δ})$ for suitable range of $\la$ and $δ$.

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arXiv2017

On the first curve of Fučik Spectrum Of $p$-fractional Laplacian Operator with nonlocal normal boundary conditions

In this article, we study the Fučik spectrum of the $p$-fractional Laplace operator with nonlocal normal derivative conditions which is defined as the set of all $(a,b)\in \mb R^2$ such that $$ \mc (F_p)\left\{ \begin{array}{lr} Λ_{n,p}(1-\al)(-Δ)_{p}^{\al} u + |u|^{p-2}u = \frac{χ_{Ω_\e}}{\e} (a (u^{+})^{p-1} - b (u^{-})^{p-1}) \;\quad \text{in}\; Ω,\quad \\ \mc{N}_{\al,p} u = 0 \; \quad \mbox{in}\; \mb R^n \setminus \overlineΩ, \end{array} \right. $$ has a non-trivial solution $u$, where $Ω$ is a bounded domain in $\mb R^n$ with Lipschitz boundary, $p \geq 2$, $n>p \al $, $\e, \al \in(0,1)$ and $Ω{_\e}:=\{x \in Ω: d(x,\pa Ω)\leq \e \}$. We showed existence of the first non-trivial curve $\mc C$ of this spectrum which is used to obtain the variational characterization of a second eigenvalue of the problem $\mc (F_p)$. We also discuss some properties of this curve $\mc C$, e.g. Lipschitz continuous, strictly decreasing and asymptotic behaviour and nonresonance with respect to the Fučik spectrum.

math.AP↗
arXiv2017

Existence and stabilization results for a singular parabolic equation involving the fractional Laplacian

In this article, we study the following parabolic equation involving the fractional Laplacian with singular nonlinearity \begin{equation*} \quad (P_{t}^s) \left\{ \begin{split} \quad u_t + (-Δ)^s u &= u^{-q} + f(x,u), \;u >0\; \text{in}\; (0,T) \times Ω, u &= 0 \; \mbox{in}\; (0,T) \times (\mb R^n \setminusΩ), \quad \quad \quad \quad u(0,x)&=u_0(x) \; \mbox{in} \; {\mb R^n}, \end{split} \quad \right. \end{equation*} where $Ω$ is a bounded domain in $\mb{R}^n$ with smooth boundary $\partial Ω$, $n> 2s, \;s \in (0,1)$, $q>0$, ${q(2s-1)<(2s+1)}$, $u_0 \in L^\infty(Ω)\cap X_0(Ω)$ and $T>0$. We suppose that the map $(x,y)\in Ω\times \mb R^+ \mapsto f(x,y)$ is a bounded below Carathéodary function, locally Lipschitz with respect to second variable and uniformly for $x \in Ω$ it satisfies \begin{equation}\label{cond_on_f} { \limsup_{y \to +\infty} \frac{f(x,y)}{y}<λ_1^s(Ω)}, \end{equation} where $\la_1^s(Ω)$ is the first eigenvalue of $(-Δ)^s$ in $Ω$ with homogeneous Dirichlet boundary condition in $\mathbb{R}^n \setminus Ω$. We prove the existence and uniqueness of weak solution to $(P_t^s)$ on assuming $u_0$ satisfies an appropriate cone condition. We use the semi-discretization in time with implicit Euler method and study the stationary problem to prove our results. We also show additional regularity on the solution of $(P_t^s)$ when we regularize our initial function $u_0$.

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