arXiv subjects

Sanjiban Santra

Publications and source records attributed to Sanjiban Santra.

5 recordsLinked to original sources

arXiv2017

Positive solutions to a fractional equation with singular nonlinearity

In this paper, we study the positive solutions to the following singular and non local elliptic problem posed in a bounded and smooth domain $Ω\subset \R^N$, $N> 2s$: % \begin{eqnarray*} (P_λ)\left\{\begin{array}{lll} &(-Δ)^s u=λ(K(x)u^{-δ}+f(u))\mbox{ in }Ω&u>0 \mbox{ in }Ω& u\equiv\, 0\mbox{ in }\R^N\backslashΩ. \end{array}\right. \end{eqnarray*} % Here $0 0$, $λ>0$ and $f\,:\, \R^+\to\R^+$ is a positive $C^2$ function. $K\,:\, Ω\to \R^+$ is a Hölder continuous function in $Ω$ which behave as ${\rm dist}(x,\partialΩ)^{-β}$ near the boundary with $0\leq β<2s$. First, for any $δ>0$ and for $λ>$ small enough, we prove the existence of solutions to $(P_λ)$. Next, for a suitable range of values of $δ$, we show the existence of an unbounded connected branch of solutions to $(P_λ)$ emanating from the trivial solution at $λ=0$. For a certain class of nonlinearities $f$, we derive a global multiplicity result that extends results proved in \cite{peral-al}. To establish the results, we prove new properties which are of independent interest and deal with the behavior and Hölder regularity of solutions to $(P_λ)$.

math.AP↗

arXiv2017

Profile of solutions for nonlocal equations with critical and supercritical nonlinearities

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 $p=\frac{N+2s}{N-2s}$.

math.AP↗

arXiv2016

On singular equations with critical and supercritical exponents

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 $μ>0$.

math.AP↗

arXiv2011

New entire positive solution for the nonlinear Schrodinger equation: Coexistence of fronts and bumps

In this paper we construct a new kind of positive solutions of $$\De u-u+u^{p}=0 \text{on} \R^2$$ when $p> 2.$ These solutions $\displaystyle{u(x,z)\sim \om(x-f(z))+ \sum_{i=1}^{\infty}\om_{0}((x, z)-ξ_i\vec{e}_{1})}$ as $L\rightarrow +\infty$ where $\om$ is a unique positive homoclinic solution of $\om"-\om+\om^{p}=0$ in $\R$ ; $\om_{0}$ is the two dimensional positive solution and $\vec{e}_{1}= (1, 0)$ and $ξ_{j}$ are points such that $ξ_{j}= jL+ \mathcal{O}(1)$ for all $j\geq 1.$ This represents a first result on the {\em coexistence} of fronts and bumps. Geometrically, our new solutions correspond to {\em triunduloid} in the theory of CMC surface.

math.AP↗

arXiv2009

Homoclinic solutions for fourth order traveling wave equations

We consider homoclinic solutions of fourth order equations $$ u^{""} + β^2 u^{"} + V_u (u)=0 {in} \R ,$$ where $V(u)$ is either the suspension bridge type $V(u)=e^u-1-u$ or Swift-Hohenberg type $ V(u)= {1/4}(u^2-1)^2$. For the suspension bridge type equation, we prove existence of a homoclinic solution for {\em all} $ β\in (0, β_*)$ where $ β_{*}= 0.7427...$. For the Swift-Hohenberg type equation, we prove existence of a homoclinic solution for each $β\in (0, β_{*})$, where $β_{*}=0.9342...$. This partially solves a conjecture of Chen--McKenna \cite{YCM1}.

math.AP↗