Related papers: Critical growth fractional elliptic problem with s…

In this paper, we consider the fractional elliptic equation \begin{align*} \left\{\begin{aligned} &(-\Delta)^s u-\mu\frac{u}{|x|^{2s}} = \frac{|u|^{2_s^\ast (\alpha)-2}u}{|x|^{\alpha}} + f(x,u), && \mbox{in} \ \Omega,\\ &u=0, && \mbox{in} \…

We prove the existence of a positive {\it SOLA (Solutions Obtained as Limits of Approximations)} to the following PDE involving fractional power of Laplacian \begin{equation} \begin{split} (-\Delta)^su&= \frac{1}{u^\gamma}+\lambda…

We consider a Br\'ezis-Nirenberg type critical growth $p$-Laplacian problem involving a parameter $\mu > 0$ in a smooth bounded domain $\Omega$. We prove the existence of multiple nontrivial solutions if either $\mu$ or the volume of…

In this paper, we study the positive solutions to the following singular and non local elliptic problem posed in a bounded and smooth domain $\Omega\subset \R^N$, $N> 2s$: % \begin{eqnarray*} (P_\lambda)\left\{\begin{array}{lll}…

We study the existence and multiplicity of positive solutions for a family of fractional Kirchhoff equations with critical nonlinearity of the form \begin{equation*}…

In this article, we study the following nonlinear doubly nonlocal problem involving the fractional Laplacian in the sense of Hardy-Littlewood-Sobolev inequality \begin{equation*} \left\{\begin{aligned} (-\Delta)^s u & =…

In this work, the following fractional Laplacian problem with pure critical nonlinearity is considered \begin{equation*} \left\{ \begin{array}{ll} (-\Delta)^{s} u=|u|^{\frac{4s}{N-2s}}u, &\mbox{in}\ \mathbb{R}^N, \\ u\in…

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}…

We establish the existence of solutions to the following semilinear Neumann problem for fractional Laplacian and critical exponent: \begin{align*}\left\{\begin{array}{l l} { (-\Delta)^{s}u+ \lambda u= \abs{u}^{p-1}u } & \text{in $ \Omega,$…

We consider the boundary value problem $-\Delta_p u = \lambda c(x) |u|^{p-2}u + \mu(x) |\grad u|^p + h(x)$, $u \in W^{1,p}_0(\Omega) \cap L^{\infty}(\Omega)$, where $\Omega \subset \mathbb R^N$, $N \geq 2$, is a bounded domain with smooth…

In this paper we establish the existence and multiplicity of nontrivial solutions to the following problem \begin{align*} \begin{split} (-\Delta)^{\frac{1}{2}}u+u+(\ln|\cdot|*|u|^2)&=f(u)+\mu|u|^{-\gamma-1}u,~\text{in}~\mathbb{R},…

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}…

We study the following fractional elliptic equations of the type, \begin{equation*} (-\Delta)^{\frac12}_A u = \lambda u+f(|u|)u ,\;\textrm{in } \;(-1, 1),\; u=0\;\textrm{in } \;\mathbb R\setminus (-1, 1), \end{equation*} where $\lambda$ is…

In this paper, we prove that there exists at most one positive radial weak solution to the following quasilinear elliptic equation with singular critical growth \[ \begin{cases} -\Delta_{p}u-{\displaystyle…

In this paper, we study the following critical system with fractional Laplacian: \begin{equation*} \begin{cases} (-\Delta)^{s}u= \mu_{1}|u|^{2^{\ast}-2}u+\frac{\alpha\gamma}{2^{\ast}}|u|^{\alpha-2}u|v|^{\beta} \ \ \ \text{in} \ \…

In this paper we study the multiplicity of normalized solutions to the following nonlinear Schr\"{o}dinger equation with critical growth \begin{align*} \left\{ \begin{aligned} &-\Delta u=\lambda u+\mu |u|^{q-2}u+f(u), \quad \quad \hbox{in…

In this paper we consider the existence of solution for the following class of fractional elliptic problem \begin{equation}\label{00} \left\{\begin{aligned} (-\Delta)^su + u &= Q(x) |u|^{p-1}u\;\;\mbox{in}\;\;\R^N \setminus \Omega\\…

In this work, we establish the multiplicity of positive solutions for the following critical fractional Choquard equation with a perturbation on the star-shaped bounded domain $$ \left\{ \begin{array}{lr} (-\Delta)^s u = \lambda u…

In this article, we study the following fractional-Laplacian system with singular nonlinearity \begin{equation*} (P_{\lambda,\mu}) \left\{ \begin{array}{lr} (-\Delta)^s u = \lambda f(x) u^{-q}+ \frac{\alpha}{\alpha+\beta}b(x) u^{\alpha-1}…

In this article, we study the Brezis-Nirenberg type problem of nonlinear Choquard equation involving a fractional Laplacian \[ (-\De)^s u = \left( \int_{\Om}\frac{|u|^{2^*_{\mu,s}}}{|x-y|^{\mu}}\mathrm{d}y \right)|u|^{2^*_{\mu,s}-2}u +\la u…