Related papers: A fractional p-Laplacian problem with multiple cri…

To our knowledge, this paper is the first attempt to consider the existence issue for fractional $p$-Laplacian equation: $(-\Delta)_p^s u= \lambda f(u),\; u> 0 ~\text{in}~\Omega;\; u=0\;\text{in}~ \mathbb{R}^N\setminus\Omega$, where $p>1$,…

The purpose of this paper is to study and classify singular solutions of the Poisson problem $$ %\begin{equation}\label{eq 0.1} \left \{ \begin{aligned} {\mathcal L}^s_\mu u = f \quad\ {\rm in}\ \, \Omega\setminus \{0\},\\ u =0 \quad\ {\rm…

It is establish regularity results for weak solutions of quasilinear elliptic problems driven by the well known $\Phi$-Laplacian operator given by \begin{equation*} \left\{\ \begin{array}{cl} \displaystyle-\Delta_\Phi u= g(x,u), &…

We study the boundary value problem $-{\rm div}(\log(1+ |\nabla u|^q)|\nabla u|^{p-2}\nabla u)=f(u)$ in $\Omega$, $u=0$ on $\partial\Omega$, where $\Omega$ is a bounded domain in $\RR^N$ with smooth boundary. We distinguish the cases where…

In this paper, we investigate the existence of multiple solutions to the following multi-critical elliptic problem \begin{equation}\label{eq:0.1} \left\{\begin{aligned} -\Delta u & =\lambda |u|^{p-2}u…

We investigate the fractional magnetic $p$-Laplacian operator in the physical dimension case $N=3$, with $0<s<1<p$ and $sp<3$. Our goal is twofold. First, we define and study suitable functional settings for such operator proving…

In this paper we study the following boundary value problem involving the weak p-Laplacian. \begin{equation*} \quad -M(\|u\|_{\mathcal{E}_p}^p)\Delta_p u = h(x,u) \; \text{in}\; \mathcal{S}\setminus\mathcal{S}_0; \quad u = 0 \; \mbox{on}\;…

This article gives an existence theory for weak solutions of second order non-elliptic linear Dirichlet problems of the form {eqnarray} \nabla'P(x)\nabla u +{\bf HR}u+{\bf S'G}u +Fu &=& f+{\bf T'g} \textrm{in}\Theta…

We prove the existence of infinitely many nonnegative solutions to the following nonlocal elliptic partial differential equation involving singularities \begin{align} (-\Delta)_{p(\cdot)}^{s}…

This paper deals with the following fractional Choquard equation $$\varepsilon^{2s}(-\Delta)^su +Vu=\varepsilon^{-\alpha}(I_\alpha*|u|^p)|u|^{p-2}u\ \ \ \mathrm{in}\ \mathbb{R}^N,$$ where $\varepsilon>0$ is a small parameter, $(-\Delta)^s$…

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…

In this paper we prove the existence of a positive solution of the nonlinear and nonlocal elliptic equation in $\mathbb{R}^n$ \[ (-\Delta)^s u =\varepsilon h u^q+u^{2_s^*-1} \] in the convex case $1\leq q<2_s^*-1$, where $…

In this article the problem to be studied is the following $$ (P) \left\{ \begin{array}{rcll} u_t+(-\D^s_{p}) u & = & f(x,t) & \text{ in } \O_{T}\equiv \Omega \times (0,T), \\ u & = & 0 & \text{ in }(\ren\setminus\O) \times (0,T), \\ u &…

In the present paper we investigate the following semilinear singular elliptic problem: \begin{equation*} (\rm P)\qquad \left \{\begin{array}{l} -\Delta u = \dfrac{p(x)}{u^{\alpha}}\quad \text{in} \Omega \\ u = 0\ \text{on} \Omega,\ u>0…

We study quasilinear degenerate singular elliptic equation of type -Delta_p u = \frac{u^{p^*(s)-1}}{|y|^t}$ in a smooth bounded domain \Omega in R^n=R^k \times R^{N-k}$, x=(y,z) in R^k \times R^{N-k}, 2 \leq k<N and N \geq 3, 1<p<2, 0\leq…

In this paper, we apply a self-similar transformation to convert the parabolic equation with a Sobolev-Hardy term \begin{align*} u_t-\Delta u= \frac{|u|^{q-2}u}{\left|x\right|^s} & \text { in } \mathbb{R}^N \times(0, \infty), \end{align*}…

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 work we study the existence of positive solution to the fractional quasilinear problem, $$ \left\{ \begin{array}{rcll} (-\Delta )^s u &=&\lambda \dfrac{u}{|x|^{2s}}+ |\nabla u|^{p}+ \mu f &\inn \Omega,\\ u&>&0 & \inn\Omega,\\ u&=&0…

This paper deals with multiplicity and bifurcation results for nonlinear problems driven by the fractional Laplace operator $(-\Delta)^s$ and involving a critical Sobolev term. In particular, we consider $$\begin{cases}…

In this paper, we deal with the following $(p,q)$-fractional problem $$ (-\Delta)^{s_{1}}_{p}u +(-\Delta)^{s_{2}}_{q}u=\lambda P(x)|u|^{k-2}u+\theta|u|^{p_{s_{1}}^{*}-2}u \, \mbox{ in }\, \Omega,\qquad u=0\, \mbox{ in }\, \mathbb{R}^{N}…