Related papers: Peak Solutions for the fractional Nirenberg proble…

In this article, we study the following fractional $p$-Laplacian equation with critical growth singular nonlinearity \begin{equation*} \quad (-\De_{p})^s u = \la u^{-q} + u^{\alpha}, u>0 \; \text{in}\; \Om,\quad u = 0 \; \mbox{in}\; \mb R^n…

We study the existence of solutions of the following nonlinear Schr\"odinger equation $$ -\Delta u+V(x)u-\frac{(N-2)^2}{4|x|^2}u=f(x,u) $$ where $V:\mathbb{R}^N\to\mathbb{R}$ and $f:\mathbb{R}^N\times \mathbb{R}\to \mathbb{R}$ are periodic…

We get multiplicity of normalized solutions for the fractional Schr\"{o}dinger equation $$ (-\Delta)^su+V(\varepsilon x)u=\lambda u+h(\varepsilon x)f(u)\quad \mbox{in $\mathbb{R}^N$}, \qquad\int_{\mathbb{R}^N}|u|^2dx=a, $$ where…

We prove that the fractional Yamabe equation $\mathcal L_\gamma u=|u|^\frac{4\gamma}{Q-2\gamma}u$ on the Heisenberg group $\mathbb H^n$ has $[\frac{n+1}{2}]$ sequences of nodal (sign-changing) weak solutions whose elements have mutually…

Let $(M,g)$ be a $n-$dimensional compact Riemannian manifold without boundary and $\Gamma$ be a non degenerate closed geodesic of $(M,g)$. We prove that the supercritical problem $$-\Delta_gu+h u=u^{\frac{n+1}{n-3}\pm\epsilon},\ u>0,\…

For $1<p<\infty$, we consider the following problem $$ -\Delta_p u=f(u),\quad u>0\text{ in }\Omega,\quad\partial_\nu u=0\text{ on }\partial\Omega, $$ where $\Omega\subset\mathbb R^N$ is either a ball or an annulus. The nonlinearity $f$ is…

We study a fractional analogue of a plasma problem arising from physics. Specifically, for a fixed bounded domain $\Omega$ we study solutions to the eigenfunction equation \[ (- \Delta)^s u = \lambda(u- \gamma)_+ \] with $u \equiv 0$ on…

This paper studies the nonlinear stochastic partial differential equation of fractional orders both in space and time variables: \[ \left(\partial^\beta+\frac{\nu}{2}(-\Delta)^{\alpha/2}\right)u(t,x) =…

We study the existence of solutions to the fractional elliptic equation (E1) $(-\Delta)^\alpha u+\epsilon g(|\nabla u|)=\nu $ in a bounded regular domain $\Omega$ of $\R^N (N\ge2)$, subject to the condition (E2) $u=0$ in $\Omega^c$, where…

We consider the nonlinear heat equation $u_t-\Delta u =|u|^p+b |\nabla u|^q$ in $(0,\infty)\times \R^n$, where $n\geq 1$, $p>1$, $q\geq 1$ and $b>0$. First, we focus our attention on positive solutions and obtain an optimal Fujita-type…

In this paper, we provide two-sided estimates and uniform asymptotics for the solution of $d$-dimensional critical fractal Burgers equation $u_t-\Delta^{\alpha/2}u+b\cdot \nabla\left(u|u|^q\right)=0$, $\alpha\in(1,2)$, $b\in\mathbb R^d$ for…

We introduce a new minimax principle to prove the existence of multi-peak solutions to the Neumann problem of the $p$-Laplace equation $$ -\varepsilon^p \Delta_p u = u^{q-1} - u^{p-1} \ \ \text{in}\ \Omega,$$ where $\Om$ is a bounded domain…

A generalized Neumann solution for the two-phase fractional Lam\'e--Clapeyron--Stefan problem for a semi--infinite material with constant initial temperature and a particular heat flux condition at the fixed face is obtained, when a…

We consider a class of nonlinear fractional equations having the Caputo fractional derivative of the time variable $t$, the fractional order of the self-adjoint positive definite unbounded operator in a Hilbert space and a singular…

We study the existence/nonexistence and qualitative properties of the positive solutions to the problem \begin{align*} (-\Delta)^s u -\theta\frac{u}{|x|^{2s}}&=u^p - u^q \quad\text{in }\,\, \mathbb{R}^N,\quad u > 0 \quad\text{in }\,\,…

In this paper we study the Neumann problem\begin{equation*}\begin{cases}-\Delta u+u=u^p \& \text{ in }B\_1 \\u \textgreater{} 0, \& \text{ in }B\_1 \\\partial\_\nu u=0 \& \text{ on } \partial B\_1,\end{cases}\end{equation*}and we show the…

We consider an optimal rearrangement maximization problem involving the fractional Laplace operator $(-\Delta)^s$, $0<s<1$, and the Gagliardo-Nirenberg seminorm $[u]_s$. We prove the existence of a maximizer, analyze its properties and show…

We study the following nonlinear and nonlocal elliptic equation in~$\R^n$ $$ (-\Delta)^s u = \epsilon\,h\,u^q + u^p \ {\mbox{ in }}\R^n, $$ where~$s\in(0,1)$, $n>2s$, $\epsilon>0$ is a small parameter, $p=\frac{n+2s}{n-2s}$, $q\in(0,1)$,…

In this paper, we derive sufficient conditions ensuring the existence of a weak solution $u$ for a tempered fractional Euler-Lagrange equations $$ \frac{\partial L}{\partial x}(u,{^C}\mathbb{D}_{a^+}^{\alpha, \sigma} u, t) +…

This article establishes the existence and multiplicity of normalized solutions to the weighted nonlinear Schr\"odinger-type equation governed by the Caffarelli-Kohn-Nirenberg operator, $$ -\text{div}(|x|^{-2a}\nabla u)=\lambda…