Related papers: Existence and Uniqueness for the Fractional Gelfan…

We study finite Morse index solutions to the non-local Gelfand-Liouville problem $$ (-\Delta)^su=e^u\quad\mathrm{in}\quad \mathbb{R}^n,$$ for every $s\in(0,1)$ and $n>2s$. Precisely, we prove non-existence of finite Morse index solutions…

This work establishes the existence and uniqueness of solutions to the fractional diffusion equation $$\frac{\partial^\alpha u}{\partial t^{\alpha}} + K(-\Delta)^{\beta} u - \nabla \cdot (\nabla V u) = f$$ on a $d$-dimensional torus,…

We study the problem $-\Delta u=\lambda u-u^{-1}$ with a Neumann boundary condition; the peculiarity being the presence of the singular term $-u^{-1}$. We point out that the minus sign in front of the negative power of $u$ is particularly…

In this paper we study the problem -\Delta u =\left(\frac{2+\alpha}{2}\right)^2\abs{x}^{\alpha}f(\lambda,u), & \hbox{in}B_1 \\ u > 0, & \hbox{in}B_1 u = 0, & \hbox{on} \partial B_1 where $B_1$ is the unit ball of $\R^2$, $f$ is a smooth…

In this paper, we study the existence and uniqueness of positive solutions for the following nonlinear fractional elliptic equation: \begin{eqnarray*} (-\Delta)^\alpha u=\lambda a(x)u-b(x)u^p&{\rm in}\,\,\R^N, \end{eqnarray*} where $…

We study the existence of nonnegative solutions to the following nonlocal elliptic problem involving singularity \begin{align} \mathfrak{M}\left(\int_{Q}\frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}dxdy\right)(-\Delta)_{p}^{s}…

In this paper, we are concerned with stable solutions to the fractional elliptic equation $$ (-\Delta)^s u=e^u\mbox{ in }\mathbb R^{N}, $$ where $(-\Delta)^s$ is the fractional Laplacian with $0<s<1$. We establish the nonexistence of stable…

This paper concerns the existence of a nontrivial solution for the following problem \begin{equation} \left\{\begin{aligned} -\Delta u + V(x)u & \in \partial_u F(x,u)\;\;\mbox{a.e. in}\;\;\mathbb{R}^{N},\nonumber u \in…

In this paper, we establish the uniqueness of positive solutions to the following fractional nonlinear elliptic equation with harmonic potential \begin{align*} (-\Delta)^s u+ \left(\omega+|x|^2\right) u=|u|^{p-2}u \quad \mbox{in}\,\, \R^n,…

We study the existence and multiplicity of solutions for a class of fractional Schr\"{o}dinger-Kirchhoff type equations with the Trudinger-Moser nonlinearity. More precisely, we consider \begin{gather*} \begin{cases}…

We study the asymptotic behavior as $p\to\infty$ of the Gelfand problem \[ -\Delta_{p} u=\lambda\,e^{u}\ \textrm{in}\ \Omega\subset\mathbb{R}^n,\quad u=0 \ \textrm{on}\ \partial\Omega. \] Under an appropriate rescaling on $u$ and $\lambda$,…

In this paper we are concerned with the construction of periodic solutions of the nonlocal problem $(-\Delta)^s u= f(u)$ in $\mathbb{R}$, where $(-\Delta)^s$ stands for the $s$-Laplacian, $s\in (0,1)$. We introduce a suitable framework…

We study the existence of {weak} solutions for fractional elliptic equations of the type, \begin{equation*} (-\Delta)^{\frac{1}{2}} u+ V(x) u= h(u), u> 0 \;\textrm{in} \;\mathbb R, \end{equation*} %where $1<q<2,\;p>2,\;1<\beta\leq2\;,…

In this paper, we study real solutions of the nonlinear Helmholtz equation $$ - \Delta u - k^2 u = f(x,u),\qquad x\in \R^N $$ satisfying the asymptotic conditions $$ u(x)=O(|x|^{\frac{1-N}{2}}) \quad \text{and} \quad \frac{\partial^2…

We investigate stable solutions of elliptic equations of the type \begin{equation*} \left \{ \begin{aligned} (-\Delta)^s u&=\lambda f(u) \qquad {\mbox{ in $B_1 \subset \R^{n}$}} \\ u&= 0 \qquad{\mbox{ on $\partial B_1$,}}\end{aligned}\right…

In this paper, we establish Liouville type theorems for stable solutions on the whole space $\mathbb R^N$ to the fractional elliptic equation $$(-\Delta)^su=f(u)$$ where the nonlinearity is nondecreasing and convex. We also obtain a…

We consider the following elliptic system with fractional laplacian $$ -(-\Delta)^su=uv^2,\ \ -(-\Delta)^sv=vu^2,\ \ u,v>0 \ \mbox{on}\ \R^n,$$ where $s\in(0,1)$ and $(-\Delta)^s$ is the $s$-Lapalcian. We first prove that all positive…

The purpose of this paper is to study the weak solutions of the fractional elliptic problem \begin{equation}\label{000} \begin{array}{lll} (-\Delta)^\alpha u+\epsilon g(u)=k\frac{\partial^\alpha\nu}{\partial \vec{n}^\alpha}\quad &{\rm…

This work is devoted to the study of the existence of at least one weak solution to nonlocal equations involving a general integro-differential operator of fractional type. As a special case, we derive an existence theorem for the…

We study the existence of solutions to the problem $$ (-\Delta)^{\frac{n}{2}}u = Qe^{nu}\quad\text{in }\mathbb{R}^n, \quad V := \int_{\mathbb{R}^n}e^{nu}dx < \infty,$$ where $Q=(n-1)!$ or $Q=-(n-1)!$. Extending the works of Wei-Ye and…