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K. Sreenadh

Publications and source records attributed to K. Sreenadh.

66 records · Page 4Linked to original sources
arXiv2017

On doubly nonlocal $p$-fractional coupled elliptic system

\noi We study the following nonlinear system with perturbations involving p-fractional Laplacian \begin{equation*} (P)\left\{ \begin{split} (-\De)^s_p u+ a_1(x)u|u|^{p-2} &= α(|x|^{-μ}*|u|^q)|u|^{q-2}u+ β(|x|^{-μ}*|v|^q)|u|^{q-2}u+ f_1(x)\; \text{in}\; \mb R^n,\\ (-\De)^s_p v+ a_2(x)v|v|^{p-2} &= γ(|x|^{-μ}*|v|^q)|v|^{q-2}v+ β(|x|^{-μ}*|u|^q)|v|^{q-2}v+ f_2(x)\; \text{in}\; \mb R^n, \end{split} \right. \end{equation*} where $n>sp$, $0 0$, $0< a_i \in C^1(\mb R^n, \mb R)$, $i=1,2$ and $f_1,f_2: \mb R^n \to \mb R$ are perturbations. We show existence of atleast two nontrivial solutions for $(P)$ using Nehari manifold and minimax methods.

math.AP↗
arXiv2016

Fractional Choquard Equation with Critical Nonlinearities

In this article, we study the Brezis-Nirenberg type problem of nonlinear Choquard equation involving a fractional Laplacian \[ (-\De)^s u = \left( \int_{\Om}\frac{|u|^{2^*_{μ,s}}}{|x-y|^μ}\mathrm{d}y \right)|u|^{2^*_{μ,s}-2}u +\la u \; \text{in } \Om,\] where $\Om $ is a bounded domain in $\mathbb R^n$ with Lipschitz boundary, $\la $ is a real parameter, $s \in (0,1)$, $n >2s$ and $2^*_{μ,s}= (2n-μ)/(n-2s)$ is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality. We obtain some existence, multiplicity, regularity and nonexistence results for solution of the above equation using variational methods.

math.AP↗
arXiv2016

Elliptic Problems in $\mathbb{R}^N$ with Critical and Singular Discontinuous Nonlinearities

Let $Ω$ be a bounded domain in $\mathbb R^{N}$, $N\geq3$ with smooth boundary, $a>0, λ>0$ and $0<δ<3$ be real numbers. Define $2^*:=\displaystyle\frac{2N}{N-2}$ and the characteristic function of a set $A$ by $χ_A$. We consider the following critical problem with singular and discontinuous nonlinearity: \begin{eqnarray*} (P_\la^a)~~~~ \qquad \Biggl\{\begin{array}{rl} -Δu &= λ\left(u^{2^*-1}+ \displaystyle χ_{\{u 0~~\text{in} ~~Ω, \\ u & = 0 ~\text{on}~ \partial Ω. \end{array} \end{eqnarray*} \noindent We study the existence and the global multiplicity of solutions to the above problem.

math.AP↗
arXiv2016

On Dirichlet problem for fractional $p$-Laplacian with singular nonlinearity

In this article, we study the following fractional $p$-Laplacian equation with critical growth singular nonlinearity \begin{equation*} \quad (-\De_{p})^s u = \la u^{-q} + u^α, 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 > sp, s \in (0,1), \la >0, 0 < q \leq 1 $ and $α\le p^*_s-1$. We use variational methods to show the existence and multiplicity of positive solutions of above problem with respect to parameter $\la$.

math.AP↗
arXiv2016

Polyharmonic Kirchhoff type equations with singular exponential nonlinearities

\noi In this article, we study the existence of non-negative solutions of the following polyharmonic Kirchhoff type problem with critical singular exponential nolinearity $$ \quad \left\{ \begin{array}{lr} \quad -M\left(\displaystyle\int_Ω|\nabla^m u|^{\frac{n}{m}}dx\right)Δ_{\frac{n}{m}}^{m} u = \frac{f(x,u)}{|x|^α} \; \text{in}\; \Om{,} \quad \quad u = \nabla u=\cdot\cdot\cdot= {\nabla}^{m-1} u=0 \quad \text{on} \quad \partial \Om{,} \end{array} \right. $$ where $\Om\subset \mb R^n$ is a bounded domain with smooth boundary, $n\geq 2m\geq 2$ and $f(x,u)$ behaves like $e^{|u|^{\frac{n}{n-m}}}$ as $|u|\ra\infty$. Using mountain pass structure and {the} concentration compactness principle, we show the existence of a nontrivial solution. %{OR}\\ In the later part of the paper, we also discuss the above problem with convex-concave type sign changing nonlinearity. Using {the} Nehari manifold technique, we show the existence and multiplicity of non-negative solutions. \medskip

math.AP↗
arXiv2016

Critical growth fractional elliptic problem with singular nonlinearities

In this article, we study the following fractional Laplacian equation with critical growth and singular nonlinearity $$\quad (-Δ)^s u = λa(x) u^{-q} + u^{2^*_s-1}, \quad u>0 \; \text{in}\; Ω,\quad u = 0 \; \mbox{in}\; \mathbb{R}^n \setminusΩ,$$ where $Ω$ is a bounded domain in $\mathbb{R}^n$ with smooth boundary $\partial Ω$, $n > 2s,\; s \in (0,1),\; λ>0,\; 0 < q \leq 1 $, $θ\leq a(x) \in L^\infty(Ω)$, for some $θ>0$ and $2^*_s=\frac{2n}{n-2s}$. We use variational methods to show the existence and multiplicity of positive solutions of the above problem with respect to the parameter $λ$.

math.AP↗
arXiv2015

Existence and multiplicity results for fractional $p$-Kirchhoff equation with sign changing nonlinearities

In this paper, we show the existence and multiplicity of nontrivial, non-negative solutions of the fractional $p$-Kirchhoff problem \begin{equation*} \begin{array}{rllll} M\left(\displaystyle\int_{\mathbb{R}^{2n}}\frac{|u(x)-u(y)|^p}{\left|x-y\right|^{n+ps}}dx\,dy\right)(-Δ)^{s}_p u &=λf(x)|u|^{q-2}u+ g(x)\left|u\right|^{r-2}u\, \text{in} Ω,\\ u&=0 \;\mbox{in} \mathbb{R}^{n}\setminus Ω, \end{array} \end{equation*} where $(-Δ)^{s}_p$ is the fractional $p$-Laplace operator, $Ω$ is a bounded domain in $\mathbb{R}^n$ with smooth boundary, $f \in L^{\frac{r}{r-q}}(Ω)$ and $g\in L^\infty(Ω)$ are sign changing, $M$ is continuous function, $ps<n<2ps$ and $1<q<p<r\leq p_s^*=\frac{np}{n-ps}$.

math.AP↗
arXiv2015

On The Fučik Spectrum Of Non-Local Elliptic Operators

In this article, we study the Fučik spectrum of fractional Laplace operator which is defined as the set of all $(\al,\ba)\in \mb R^2$ such that \begin{equation*} \quad \left. \begin{array}{lr} \quad (-\De)^s u = \al u^{+} - \ba u^{-} \; \text{in}\; \Om \quad \quad \quad \quad u = 0 \; \mbox{in}\; \mb R^n \setminus\Om.\\ \end{array} \quad \right\} \end{equation*} has a non-trivial solution $u$, where $\Om$ is a bounded domain in $\mb R^n$ with Lipschitz boundary, $n>2s$, $s\in(0,1)$. The existence of a first nontrivial curve $\mc C$ of this spectrum, some properties of this curve $\mc C$, e.g. Lipschitz continuous, strictly decreasing and asymptotic behavior are studied in this article. A variational characterization of second eigenvalue of the fractional eigenvalue problem is also obtained. At the end, we study a nonresonance problem with respect to Fučik spectrum.

math.FA↗
arXiv2015

$n$Kirchhoff type equations with exponential nonlinearities

In this article, we study the existence of non-negative solutions of the class of non-local problem of $n$-Kirchhoff type $$ \left\{ \begin{array}{lr} \quad - m(\int_Ω|\nabla u|^n)Δ_n u = f(x,u) \; \text{in}\; Ω,\quad u =0\quad\text{on} \quad \partial Ω, \end{array} \right.$$ where $Ω\subset \mathbf{R}^n$ is a bounded domain with smooth boundary, $n\geq 2$ and $f$ behaves like $e^{|u|^{\frac{n}{n-1}}}$ as $|u|\to\infty$. Moreover, by minimization on the suitable subset of the Nehari manifold, we study the existence and multiplicity of solutions, when $f(x,t)$ is concave near $t=0$ and convex as $t\rightarrow \infty.$

math.AP↗
arXiv2014

Existence of multiple solutions of $p$-fractional Laplace operator with sign-changing weight function

In this article, we study the following $p$-fractional Laplacian equation \begin{equation*} (P_{\la}) \left\{ \begin{array}{lr} - 2\int_{\mb R^n}\frac{|u(y)-u(x)|^{p-2}(u(y)-u(x))}{|x-y|^{n+p\al}} dy = \la |u(x)|^{p-2}u(x) + b(x)|u(x)|^{\ba-2}u(x)\; \text{in}\; \Om \quad \quad\quad\quad \quad\quad\quad\quad\quad \quad u = 0 \; \mbox{in}\; \mb R^n \setminus\Om,\quad u\in W^{\al,p}(\mb R^n).\\ \end{array} \quad \right. \end{equation*} where $\Om$ is a bounded domain in $\mb R^n$ with smooth boundary, $n> p\al$, $p\geq 2$, $\al\in(0,1)$, $\la>0$ and $b:\Om\subset\mb R^n \ra \mb R$ is a sign-changing continuous function. We show the existence and multiplicity of non-negative solutions of $(P_{\la})$ with respect to the parameter $\la$, which changes according to whether $1<\ba<p$ or $p< \ba< p^{*}=\frac{np}{n-p\al}$ respectively. We discuss both the cases separately. Non-existence results are also obtained.

math.AP↗
arXiv2013

A Nehari manifold for non-local elliptic operator with concave-convex non-linearities and sign-changing weight function

In this article, we study the existence and multiplicity of non-negative solutions of following $p$-fractional equation: $$ \quad \left\{\begin{array}{lr}\ds \quad - 2\int_{\mb R^n}\frac{|u(y)-u(x)|^{p-2}(u(y)-u(x))}{|x-y|^{n+p\al}} dxdy = \la h(x)|u|^{q-1}u+ b(x)|u|^{r-1} u \; \text{in}\; \Om \quad \quad \quad \quad u \geq 0 \; \mbox{in}\; \Om,\quad u\in W^{\al,p}(\mb R^n), \quad \quad\quad \quad\quad u =0\quad\quad \text{on} \quad \mb R^n\setminus \Om \end{array} \right. $$ where $\Om$ is a bounded domain in $\mb R^n$, $p\geq 2$, $n> p\al$, $\al\in(0,1)$, $0< q 0$ and $h$, $b$ are sign changing smooth functions. We show the existence of solutions by minimization on the suitable subset of Nehari manifold using the fibering maps. We find that there exists $\la_0$ such that for $\la\in (0,\la_0)$, it has at least two solutions.

math.AP↗