arXiv subjects

Sushmita Rawat

Publications and source records attributed to Sushmita Rawat.

5 recordsLinked to original sources

arXiv2023

High energy solutions for $p$-Kirchhoff elliptic problems with Hardy-Littlewood-Sobolev nonlinearity

This article deals with the study of the following Kirchhoff-Choquard problem: \begin{equation*} \begin{array}{cc} \displaystyle M\left(\, \int\limits_{\mathbb{R}^N}|\nabla u|^p\right) (-Δ_p) u + V(x)|u|^{p-2}u = \left(\, \int\limits_{\mathbb{R}^N}\frac{F(u)(y)}{|x-y|^μ}\,dy \right) f(u), \;\;\text{in} \; \mathbb{R}^N, u > 0, \;\; \text{in} \; \mathbb{R}^N, \end{array} \end{equation*} where $M$ models Kirchhoff-type nonlinear term of the form $M(t) = a + bt^{θ-1}$, where $a, b > 0$ are given constants; $1<p<N$, $Δ_p = \text{div}(|\nabla u|^{p-2}\nabla u)$ is the $p$-Laplacian operator; potential $V \in C^2(\mathbb{R}^N)$; $f$ is monotonic function with suitable growth conditions. We obtain the existence of a positive high energy solution for $θ\in \left[1, \frac{2N-μ}{N-p}\right) $ via the Pohožaev manifold and linking theorem. Apart from this, we also studied the radial symmetry of solutions of the associated limit problem.

math.AP↗

arXiv2023

Critical exponent Neumann problem with Hardy-Littlewood-Sobolev nonlinearity

In this article, we study the Brezis-Nirenberg type problem of nonlinear Choquard equation with Neumann boundary condition \begin{equation*} \begin{aligned} -Δu &= λα(x)u + \left(\int\limits_Ω\frac{u(y)^{2^*_μ}}{|x-y|^μ}\;dy\right)u^{2^*_μ-1}, \;\;\text{in} \; Ω,\\ \frac{\partial u}{\partial ν} &= 0\;\; \text{on} \; \partialΩ, \end{aligned} \end{equation*} where $Ω$ is a bounded domain in $\mathbb{R}^N$ $(N\geq 4)$, $ν$ is the unit outer normal to $\partial Ω$ and $μ\in (0, N)$. According to the parameter $λ$, we prove necessary and sufficient conditions for the existence and non-existence of positive weak solutions to the problem. The proof is based on variational arguments.

math.AP↗

arXiv2022

Critical growth fractional Kirchhoff elliptic problems

This article is concerned with the existence and multiplicity of positive weak solutions for the following fractional Kirchhoff-Choquard problem: \begin{equation*} \begin{array}{cc} \displaystyle M\left( \|u\|^2\right) (-Δ)^s u = \dsλf(x)|u|^{q-2}u + \left( \int\limits_Ω \frac{|u(y)|^{2^{*}_{μ,s}}}{|x-y|^ μ}\, dy\right) |u|^{2^{*}_{μ,s}-2}u \;\text{in} \; Ω, u > 0\quad \text{in} \; Ω, \,\, u = 0\quad \text{in} \; \mathbb{R}^{N}\backslashΩ, \end{array} \end{equation*} where $Ω$ is open bounded domain of $\mathbb{R}^{N}$ with $C^2$ boundary, $N > 2s$ and $s \in (0,1)$, here $M$ models Kirchhoff-type coefficient of the form $M(t) = a + bt^{\te-1}$, where $a, b > 0$ are given constants. $(-Δ)^s$ is fractional Laplace operator, $λ> 0$ is a real parameter. We explore using the variational methods, the existence of solution for ${q} \in (1,2^*_s)$ and $\te \geq 1$. % and we also consider the case when $\te > 2^*_{μ,s}$ for $2< q < 2^*_{s}$. Here $2^*_s = \frac{2N}{N-2s}$ and $2^{*}_{μ,s} = \frac{2N-μ}{N-2s}$ is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality.

math.AP↗

arXiv2022

Existence of positive solutions for a class of quasilinear Schrödinger equations with critical Choquard nonlinearity

This article is concerned with the existence of positive weak solutions for the following quasilinear Schrödinger Choquard equation: \begin{equation*} \begin{array}{cc} \displaystyle -div(g^2(u)\nabla u) + g(u)g'(u)\nabla u + a(x) u = k(x, u) \;\text{in} \; \mathbb{R}^N, \end{array} \end{equation*} where $N \geq 3$, $\displaystyle k(x,u) := h(x,u) + (I_{\vartheta}*|u|^{α\cdot2^*_μ})|u|^{α\cdot2^*_μ-2}u$, $g : \mathbb{R} \to \mathbb{R}^+$ is a differentiable even function with $g(0) = 1$ and $g'(t) \geq 0$ for all $t \geq 0$; $h\in C( \mathbb{R}^N \times\mathbb{R}, \mathbb{R})$ and the potential $a \in C( \mathbb{R}^N, \mathbb{R})$. We establish the existence of a positive solution using the change of variable and variational methods under appropriate assumptions on $g$, $h$ and $a$.

math.AP↗

arXiv2021

Three solutions for a fractional elliptic problem with asymmetric critical Choquard nonlinearity

In this paper we study the existence and multiplicity of weak solutions for the following asymmetric nonlinear Choquard problem on fractional Laplacian: \begin{equation*} \begin{array}{rl} (-Δ)^s u &= \displaystyle-λ|u|^{q-2}u + au + b\left( \int\limits_Ω \frac{(u^{+}(y))^{2^{*}_{μ,s}}}{|x-y|^ μ}\, dy\right) (u^{+})^{2^{*}_{μ,s}-2}u \quad\text{in} \; Ω, u &= 0\quad \text{in} \; \mathbb{R}^{N}\backslashΩ, \end{array} \end{equation*} where $Ω$ is open bounded domain of $\mathbb{R}^{N}$ with $C^2$ boundary, $N > 2s$ and $s \in (0,1)$. Here $(-Δ)^s$ is the fractional Laplace operator, $λ> 0$ is a real parameter, $q \in (1, 2)$, $a > 0$ and $b> 0$ are given constants, and $2^{*}_{μ,s} = \frac{2N-μ}{N-2s}$ is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality and the notation $u^{+} = \max \{u, 0\}$. We prove that the above problem has at least three nontrivial solutions using the Mountain pass Lemma and Linking theorem.

math.AP↗