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We investigate the existence of normalized solutions for the following nonlinear fractional Choquard equation: $$ (-\Delta)^s u+V(\epsilon x)u=\lambda u+\left(I_\alpha *|u|^q\right)|u|^{q-2} u+\left(I_\alpha *|u|^p\right)|u|^{p-2} u, \quad…
We prove the existence of positive solutions for the supercritical nonlinear fractional Schr\"odinger equation $(-\Delta)^s u+V(x)u-u^p=0$ in $\mathbb R^n$, with $u(x)\to 0$ as $|x|\to +\infty$, where $p>\frac{n+2s}{n-2s}$ for $s\in (0,1),…
In this paper, we study a nonlocal elliptic problem with the fractional Laplacian on $R^n$. We show that the problem has infinite positive solutions in $C^\tau(R^n)\bigcap H^\alpha_{loc}(R^n)$. Moreover each of these solutions tends to some…
In this paper, first we study carefully the positive solutions to $\Delta u+\lambda_{1}u\ln u +\lambda_{2}u^{b+1}=0$ defined on a complete noncompact Riemannian manifold $(M, g)$ with $Ric(g)\geq -Kg$, which can be regarded as…
In this paper we study the existence and multiplicity of weak solutions for the following asymmetric nonlinear Choquard problem on fractional Laplacian: \begin{equation*} \begin{array}{rl} (-\Delta)^s u &= \displaystyle-\lambda|u|^{q-2}u +…
We obtain an improved version of a recent result concerning the existence of nonnegative nonradial solutions $u\in D^{1,2}(\mathbb{R}^{N})\cap L^{2}(\mathbb{R}^{N},\left| x\right| ^{-\alpha }dx)$ to the equation \[ -\triangle…
In this note we complete a previous study, where we got existence results for the quasilinear elliptic equation \begin{equation*} -\Delta w+ V\left( \left| x\right| \right) w - w \left( \Delta w^2 \right)= K(|x|) g(w) \quad \text{in…
We consider the singular elliptic problem of the form \[ -\Delta u + V(x)u = \mathcal{B}(x)|u|^{2^*-2}u + \frac{\mathcal{A}(x)}{|u|^{2^*}u}, \qquad u\in H^1(M), \] where the coefficients are allowed to have low regularity. Under natural…
In this article, we prove the existence of solutions to a nonlinear nonlocal elliptic problem with a singualrity and a discontinuous critical nonlinearity which is given as follows. \begin{align} \begin{split}\label{main_prob}…
This paper deals with the existence of positive radial solutions to the iterative system of nonlinear elliptic equations of the form $$ \begin{aligned} \triangle{\mathtt{u}_{{\dot{\iota}} }}-\frac{(\mathtt{N}-2)^2r_0^{2\mathtt{N}-2}}{\vert…
We consider NLS on $\T^2$ with multiplicative spatial white noise and nonlinearity between cubic and quartic. We prove global existence, uniqueness and convergence almost surely of solutions to a family of properly regularized and…
We prove the existence and $C^{1,\alpha}$ regularity of solutions to nonlocal fully nonlinear elliptic equations with gradient constraints. We do not assume any regularity about the constraints; so the constraints need not be $C^1$ or…
In this paper we consider the existence of solution for the following class of fractional elliptic problem \begin{equation}\label{00} \left\{\begin{aligned} (-\Delta)^su + u &= Q(x) |u|^{p-1}u\;\;\mbox{in}\;\;\R^N \setminus \Omega\\…
This paper investigates the existence of infinitely many positive solutions for the logarithmic scalar field equation \begin{equation} \tag{$P$} \label{equ1} -\Delta u+ V(x) u= u\log u^2, \quad u\in H^1(\mathbb{R}^N), \end{equation} and its…
We study the existence of solutions of the following nonlinear Schr\"odinger equation $$ -\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…
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 discuss recent advances in the theory of quasilinear equations of the type $ -\Delta_{p} u = \sigma u^{q} \; \; \text{in} \;\; \mathbb{R}^n, $ in the case $0<q< p-1$, where $\sigma$ is a nonnegative measurable function, or measure, for…
This paper explores the existence of solutions to a class of nonlinear elliptic equations involving a mixed local-nonlocal operator of the form $-\Delta_{\mathbb{B}^N} + (-\Delta_{\mathbb{B}^N})^s$, with $0 < s < 1$, set in the hyperbolic…
We study the semilinear elliptic problem \[ -\Delta u = Q_{\Omega} |u|^{p-2}u \quad \text{in } \mathbb{R}^N, \] where \( Q_{\Omega} = \chi_{\Omega} - \chi_{\mathbb{R}^N \setminus \Omega} \) for a bounded smooth domain \( \Omega \subset…
We study the existence of positive solutions to the quasilinear elliptic problem -\epsilon \Delta u+V(x)u-\epsilon k(\Del(|u|^{2}))u=g(u), \quad u>0, x \in R^N, where g has superlinear growth at infinity without any restriction from above…