arXiv subjects
Publications and source records attributed to Divya Goel.
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.
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.
In this article we are concern for the following Choquard equation \[ -Δu = λ|u|^{q-2}u +\left(\int_Ω\frac{|u(y)|^{2^*_μ}}{|x-y|^μ} dy \right)|u|^{2^*_μ-2} u \; \text{in}\; Ω,\quad u = 0 \; \text{ on } \partial Ω, \] where $Ω$ is an open bounded set with continuous boundary in $\mathbb{R}^N( N\geq 3)$, $2^*_μ=\frac{2N-μ}{N-2}$ and $q \in [2,2^*)$ where $2^*=\frac{2N}{N-2}$. Using Lusternik-Schnirelman theory, we associate the number of positive solutions of the above problem with the topology of $Ω$. Indeed, we prove if $λ< λ_1$ then problem has $\text{cat}_Ω(Ω)$ positive solutions whenever $q \in [2,2^*)$ and $N>3 $ or $4<q<6 $ and $N=3$.
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.
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.
In this paper, we study Mountain Pass Characterization of the second eigenvalue of the operator $-\De_p u -\De_{J,p}u$ and study shape optimization problems related to these eigenvalues.
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.