arXiv subjects
Publications and source records attributed to Mousomi Bhakta.
The paper deals with existence and multiplicity of positive solutions to nonlocal equations with critical Hrardy-Sobolev nonlinearities and external terms. We establish the profile decomposition of the Palais-Smale sequences associated with the functional and existence of at least two positive solutions to the equation.
This paper deals with existence of a nontrivial positive solution to systems of equations involving nontrivial nonhomogeneous terms and critical or subcritical nonlinearities. Via a minimization argument we prove existence of a positive solution whose energy is negative provided that the nonhomogeneous terms are small enough in the dual norm.
This paper deals with existence and multiplicity of positive solutions to the following class of nonlocal equations with critical nonlinearity: \begin{equation} \tag{$\mathcal E$} (-Δ)^s u = a(x) |u|^{2^*_s-2}u+f(x)\;\;\text{in}\;\mathbb{R}^{N}, \quad u \in \dot{H}^s(\mathbb{R}^{N}), \end{equation} where $s \in (0,1)$, $N>2s$, $2_s^*:=\frac{2N}{N-2s}$, $0< a\in L^\infty(\mathbb{R}^{N})$ and $f$ is a nonnegative nontrivial functional in the dual space of $\dot{H}^s$. We prove existence of a positive solution whose energy is negative. Further, under the additional assumption that $a$ is a continuous function, $a(x)\geq 1$ in $\mathbb{R}^{N}$, $a(x)\to 1$ as $|x|\to\infty$ and $\|f\|_{\dot{H}^s(\mathbb{R}^{N})'}$ is small enough (but $f\not\equiv 0$), we establish existence of at least two positive solutions to ($\mathcal E$).
We study existence and multiplicity of positive solutions of the following class of nonlocal scalar field equations: \begin{equation} \tag{$\mathcal{P}$} \left\{\begin{aligned} (-Δ)^s u + u &= a(x) |u|^{p-1}u+f(x)\;\;\text{in}\;\mathbb{R}^{N}, u &\in H^{s}{(\mathbb{R}^{N})} \end{aligned} \right. \end{equation} where $s \in (0,1)$, $N>2s$, $1<p<2_s^*-1:=\frac{N+2s}{N-2s}$, $0< a\in L^\infty(\mathbb{R}^N)$ and $f\in H^{-s}(\mathbb{R}^N)$ is a nonnegative functional i.e., ${\langle}f,u{\rangle} \geq 0$ whenever $u$ is a nonnegative function in $H^s(\mathbb{R}^N)$. We prove existence of a positive solution when $f\equiv 0$ under certain asymptotic behavior on the function $a.$ Moreover, when $a(x)\geq 1$, $a(x)\to 1$ as $|x|\to\infty$ and $\|f\|_{H^{-s}(\mathbb{R}^N)}$ is small enough (but $f\not\equiv 0$), then we show that the above equation admits at least two positive solutions. Finally, we establish existence of three positive solutions to the above equation, under the condition that $a(x)\leq 1$ with $a(x)\to 1$ as $|x|\to\infty$ and $\|f\|_{H^{-s}(\mathbb{R}^N)}$ is small enough (but $f\not\equiv 0$).
Our main aim is to study Green function for the fractional Hardy operator $P:=(-Δ)^s -\fracθ{|x|^{2s}}$ in $\mathbb{R}^N$, where $0<θ<Λ_{N,s}$ and $Λ_{N,s}$ is the best constant in the fractional Hardy inequality. Using Green function, we also show that the integral representation of the weak solution holds.
We study positive solutions to the fractional Lane-Emden system \begin{equation*} \tag{S}\label{S} \left\{ \begin{aligned} (-Δ)^s u &= v^p+μ\quad &&\text{in } Ω\\ (-Δ)^s v &= u^q+ν\quad &&\text{in } Ω\\ u = v &= 0 \quad &&\text{in } Ω^c={\mathbb R}^N \setminus Ω, \end{aligned} \right. \end{equation*} where $Ω$ is a $C^2$ bounded domains in ${\mathbb R}^N$, $s\in(0,1)$, $N>2s$, $p>0$, $q>0$ and $μ,\, ν$ are positive measures in $Ω$. We prove the existence of the minimal positive solution of the above system under a smallness condition on the total mass of $μ$ and $ν$. Furthermore, if $p,q \in (1,\frac{N+s}{N-s})$ and $0 \leq μ,\, ν\in L^r(Ω)$ for some $r>\frac{N}{2s}$ then we show the existence of at least two positive solutions of the above system. We also discuss the regularity of the solutions.
In this paper we prove the existence of infinitely many nontrivial solutions for the class of $(p,\, q)$ fractional elliptic equations involving concave-critical nonlinearities in bounded domains in $\mathbb{R}^N$. Further, when the nonlinearity is of convex-critical type, we establish the multiplicity of nonnegative solutions using variational methods. In particular, we show the existence of at least $cat_Ω(Ω)$ nonnegative solutions.
We study the nonlocal scalar field equation with a vanishing parameter \[ \left\{\begin{array}{lll} (-Δ)^s u+εu &=|u|^{p-2}u -|u|^{q-2}u \quad\text{in}\quad\mathbb{R}^N \\ u >0, & u \in H^s(\mathbb{R}^N), \end{array} \right. \] where $s\in(0,1)$, $N>2s$, $q>p>2$ are fixed parameters and $ε>0$ is a vanishing parameter. For $ε>0$ small, we prove the existence of a ground state solution and show that any positive solution of above problem is a classical solution and radially symmetric and symmetric decreasing. We also obtain the decay rate of solution at infinity. Next, we study the asymptotic behavior of ground state solutions when $p$ is subcritical, supercritical or critical Sobolev exponent $2^*=\frac{2N}{N-2s}$. For $p<2^*$, the solution asymptotically coincides with unique positive ground state solution of $(-Δ)^s u+u=u^p$. On the other hand, for $p=2^*$ the asymptotic behaviour of the solutions is given by the unique positive solution of the nonlocal critical Emden-Fowler type equation. For $p>2^*$, the solution asymptotically coincides with a ground-state solution of $(-Δ)^s u=u^p-u^q$. Furthermore, using these asymptotic profile of solutions, we prove the \textit{local uniqueness} of solution in the case $p\leq 2^*$.
We study the fractional laplacian problem (-Δ)^s u &=& u^p -εu^q \quad\text{in }\quad Ω, u &\in& H^s(Ω)\cap L^{q+1}(Ω),u &>&0 \quad\text{in }\quad Ω, u&=&0 \quad\text{in}\quad \mathbb{R}^N\setminusΩ, where $s\in(0,1)$, $q>p\geq \frac{N+2s}{N-2s}$ and $ε>0$ is a parameter. Here $Ω\subseteq\mathbb{R}^N$ is a bounded star-shaped domain with smooth boundary and $N> 2 s$. We establish existence of a variational positive solution $u_ε$ and characterize the asymptotic behaviour of $u_ε$ as $ε\to 0$. When $p=\frac{N+2s}{N-2s}$, we describe how the solution $u_ε$ concentrates and blows up at a interior point of the domain. Furthermore, we prove the local uniqueness of solution of the above problem when $Ω$ is a convex symmetric domain of $\mathbb{R}^N$ with $N>4s$ and $p=\frac{N+2s}{N-2s}$.
In this paper we prove the existence of infinitely many nontrivial solutions of the following equations driven by a nonlocal integro-differential operator $L_K$ with concave-convex nonlinearities and homogeneous Dirichlet boundary conditions \begin{eqnarray*} \mathcal{L}_{K} u + μ\, |u|^{q-1}u + λ\,|u|^{p-1}u &=& 0 \quad\text{in}\quad Ω, \\[2mm] u&=&0 \quad\mbox{in}\quad\mathbb{R}^N\setminusΩ, \end{eqnarray*} where $Ω$ is a smooth bounded domain in $\mathbb{R}^N$, $N>2s$, $s\in(0, 1)$, $0 6s$, $λ=1$, we find $μ^*>0$ such that for any $μ\in(0,μ^*)$, there exists at least one sign changing solution.
In this article we study the existence of sign changing solution of the following p-fractional problem with concave-critical nonlinearities: \begin{eqnarray*} (-Δ)^s_pu &=& μ|u|^{q-1}u + |u|^{p^*_s-2}u \quad\mbox{in}\quad Ω, u&=&0\quad\mbox{in}\quad\mathbb{R}^N\setminusΩ, \end{eqnarray*} where $s\in(0,1)$ and $p\geq 2$ are fixed parameters, $0 ps$ .
We study the problem \begin{eqnarray*} (-Δ)^s u &=& u^p - u^q \quad\text{in }\quad \mathbb{R}^N, u &\in& \dot{H}^s(\mathbb{R}^N)\cap L^{q+1}(\mathbb{R}^N), u&>0& \quad\text{in}\quad\mathbb{R}^N, \end{eqnarray*} where $s\in(0,1)$ is a fixed parameter, $(-Δ)^s$ is the fractional laplacian in $\mathbb{R}^N$, $q>p\geq \frac{N+2s}{N-2s}$ and $N>2s$. For every $s\in(0,1)$, we establish regularity results of solutions of above equation (whenever solution exists) and we show that every solution is a classical solution. Next, we derive certain decay estimate of solutions and the gradient of solutions at infinity for all $s\in(0,1)$. Using those decay estimates, we prove Pohozaev type identity in $\mathbb{R}^N$ and show that the above problem does not have any solution when $p=\frac{N+2s}{N-2s}$. We also discuss radial symmetry and decreasing property of the solution and prove that when $p>\frac{N+2s}{N-2s}$, the above problem admits a solution. Moreover, if we consider the above equation in a bounded domain with Dirichlet boundary condition, we prove that it admits a solution for every $p\geq \frac{N+2s}{N-2s}$ and every solution is a classical solution.
We study the problem \begin{equation*} (I_ε) \left\{\begin{aligned} -Δu- \frac{μu}{|x|^2}&=u^p -εu^q \quad\text{in }\quad Ω, \\ u&>0 \quad\text{in }\quad Ω, \\ u &\in H^1_0(Ω)\cap L^{q+1}(Ω), \end{aligned} \right. \end{equation*} where $q>p\geq 2^*-1$, $ε>0$ is a parameter, $Ω\subseteq\mathbb{R}^N$ is a bounded domain with smooth boundary, $0\in Ω$, $N\geq 3$ and $0<μ<\barμ:=\big(\frac{N-2}{2}\big)^2$. We prove at $0$, any solution of $(I_ε)$ has the singularity of order $|x|^{-ν}$ when $q<\frac{2+ν}ν$ and of the order $|x|^{-\frac{2}{q-1}}$, when $q>\frac{2+ν}ν$, where $ν=\sqrt{\barμ}-\sqrt{\barμ-μ}$. Moreover, we show that when $q=\frac{2+ν}ν$ and $u$ is radial, $u\sim |x|^{-ν}|\log|x||^{-\fracν{2}}$. This gives the complete classification of singularity at $0$ in the supercritical case. We also obtain gradient estimate. Using the transformation $v=|x|^νu$, we reduce the problem $(I_ε)$ to $(J_ε)$ \begin{equation*} (J_ε) \left\{\begin{aligned} -div(|x|^{-2ν} \nabla v)&=|x|^{-(p+1)ν} v^p -ε|x|^{-(q+1)ν} v^q \quad\text{in }\quad Ω, \\ v&>0 \quad\text{in }\quad Ω, \\ v& \in H^1_0(Ω, |x|^{-2ν} )\cap L^{q+1}(Ω, |x|^{-(q+1)ν} ), \end{aligned} \right. \end{equation*} and then formulating a variational problem for $(J_ε)$, we establish the existence of a variational solution $v_ε$. Furthermore, we characterize the asymptotic behavior of $v_ε$ as $ε\to 0$ by variational arguments and when $p=2^*-1$, we show how the solution $v_ε$ blows-up at $0$. This is the first paper where the results have been established with super critical exponents for $μ>0$.
We study the existence/nonexistence of positive solution to the problem of the type: \begin{equation}\tag{$P_λ$} \begin{cases} Δ^2u-μa(x)u=f(u)+λb(x)\quad\textrm{in $Ω$,}\\ u>0 \quad\textrm{in $Ω$,}\\ u=0=Δu \quad\textrm{on $\partialΩ$,} \end{cases} \end{equation} where $Ω$ is a smooth bounded domain in $\mathbb R^N$, $N\geq 5$, $a, b, f$ are nonnegaive functions satisfying certain hypothesis which we will specify later. $μ,λ$ are positive constants. Under some suitable conditions on functions $a, b, f$ and the constant $μ$, we show that there exists $λ^*>0$ such that when $0<λ<λ^*$, ($P_λ$) admits a solution in $W^{2,2}(Ω)\cap W^{1,2}_0(Ω)$ and for $λ>λ^*$, it does not have any solution in $W^{2,2}(Ω)\cap W^{1,2}_0(Ω)$. Moreover as $λ\uparrowλ^*$, minimal positive solution of ($P_λ$) converges in $W^{2,2}(Ω)\cap W^{1,2}_0(Ω)$ to a solution of ($P_{λ^*}$). We also prove that there exists $\tildeλ^*<\infty$ such that $λ^*\leq\tildeλ^*$ and for $λ>\tildeλ^*$, the above problem ($P_λ$) does not have any solution even in the distributional sense/very weak sense and there is complete {\it blow-up}. Under an additional integrability condition on $b$, we establish the uniqueness of positive solution of ($P_{λ^*}$) in $W^{2,2}(Ω)\cap W^{1,2}_0(Ω)$.
We study the existence/nonexistence of positive solution of $$ {Δ^2u-μ\frac{u}{|x|^4}=\frac{|u|^{q_β-2}u}{|x|^β}\quad\textrm{in $Ω$,}} $$ when $Ω$ is a bounded domain and $N\geq 5$, $q_β=\frac{2(N-β)}{N-4}$, $0\leq β<4$ and $0\leqμ<\big(\frac{N(N-4)}{4}\big)^2$. We prove the nonexistence result when $Ω$ is an open subset of $\mathbf R^N$ which is star shaped with respect to the origin. We also study the existence of positive solution in $Ω$ when $Ω$ is a bounded domain with non trivial topology and $β=0$, $μ\in(0,μ_0)$, for certain $μ_0<\big(\frac{N(N-4)}{4}\big)^2$ and $N\geq 8$. Different behavior of PS sequences have been obtained depending on $β=0$ or $β>0$.
We study the existence and multiplicity of sign changing solutions of the following equation $ \begin{cases} -Δu = μ|u|^{2^{\star}-2}u+\frac{|u|^{2^{*}(t)-2}u}{|x|^t}+a(x)u \quad\text{in}\quad Ω, u=0 \quad\text{on}\quad\partialΩ, \end{cases} $where $Ω$ is a bounded domain in $R^N$, $0\in\partialΩ$, all the principal curvatures of $\partialΩ$ at $0$ are negative and $μ\geq 0, \ \ a>0, \ \ N\geq 7, \ \ 0<t<2, \ \ 2^{\star}=\frac{2N}{N-2}$ and $2^{\star}(t)=\frac{2(N-t)}{N-2}$.
We study the existence and non-existence of nontrivial weak solution of $$ {Δ^2u-μ\frac{u}{|x|^{4}} = \frac{|u|^{q_β-2}u}{|x|^β}+|u|^{q-2}u\quad\textrm{in ${\mathbb R}^N$,}} $$ where $N\geq 5$, $q_β=\frac{2(N-β)}{N-4}$, $0<β<4$, $1<q\leq 2^{**}$ and $μ<μ_1:=\big(\frac{N(N-4)}{4}\big)^2$. Using Pohozaev type of identity, we prove the non-existence result when $1<q< 2^{**}$. On the other hand when the equation has multiple critical nonlinearities i.e. $q=2^{**}$ and $-(N-2)^2\leqμ<μ_1$, we establish the existence of nontrivial solution using the Mountain-Pass theorem by Ambrosetti and Rabinowitz and the variational methods.
We study existence, multiplicity and qualitative properties of entire solutions for a noncompact problem related to p-biharmonic type equations with weights. More precisely, we deal with the following family of equations $$ Δ_{p}^2 u = λ|x|^{-2p}|u|^{p-2}u + |x|^{-β}|u|^{q-2} u\quad\text{in} \quad \mathbb R^N, $$ where $N> 2p$, $p>1$, $q>p$, $β= N - \frac{q}{p}(N-2p)$ and $λ\in\mathbb R$ is smaller than the Rellich constant.