Related papers: Kirchhoff equations with Hardy-Littlewood-Sobolev …

In this paper, we consider the following Kirchhoff type equation $$ -\left(a+ b\int_{\R^3}|\nabla u|^2\right)\triangle {u}+V(x)u=f(u),\,\,x\in\R^3, $$ where $a,b>0$ and $f\in C(\R,\R)$, and the potential $V\in C^1(\R^3,\R)$ is positive,…

On a closed Riemannian manifold $(M^n ,g)$, we consider the Yamabe-type equation $-\Delta_g u + \lambda u = \lambda |u|^{q-1}u$, where $\lambda \in \mathbb{R}_{+}$ and $q>1$. We assume that $M$ admits a proper isoparametric function $f$…

This paper deals with the existence of ground states for degenerative ($a=0$) and non-degenerative ($a>0$) double weighted critical Kirchhoff equation \begin{eqnarray*} \left\{ \begin{array}{ll} \displaystyle-\left(a+b\int_B |\nabla…

In this paper we study a class of critical Choquard equations with a symmetric potential, i.e. we consider the equation $$-\Delta u +V(|x|) u =\left(|x|^{-\mu}* |u|^{2^\star_\mu}\right)|u|^{2^\star_\mu-2}u,\quad\mbox{in}\quad\mathbb R^N$$…

In this paper, we study the multiplicity and concentration of the positive solutions to the following critical Kirchhoff type problem: \begin{equation*} -\left(\varepsilon^2 a+\varepsilon b\int_{\R^3}|\nabla u|^2\mathrm{d} x\right)\Delta u…

We study the existence and multiplicity of positive solutions with prescribed $L^2$-norm for the Sobolev critical Schr\"odinger equation on a bounded domain $\Omega\subset\mathbb{R}^N$, $N\ge3$: \[ -\Delta U = \lambda U + U^{2^{*}-1},\qquad…

In this paper, we consider the following problem $$ -\Delta u -\zeta \frac{u}{|x|^{2}} = \sum_{i=1}^{k} \left( \int_{\mathbb{R}^{N}} \frac{|u|^{2^{*}_{\alpha_{i}}}}{|x-y|^{\alpha_{i}}} \mathrm{d}y \right) |u|^{2^{*}_{\alpha_{i}}-2}u +…

In this paper we study the normalized solutions of the following critical growth Choquard equation with mixed local and non-local operators: \begin{equation*} \begin{array}{rcl} -\Delta u +(-\Delta)^s u & = & \lambda u +\mu |u|^{p-2}u…

We consider the following Choquard equation $$ -\Delta_\gamma u + u = \left(d(z)^{-\mu} \ast |u|^p\right)|u|^{p-2}u, \text{ in } \mathbb{R}^N, $$ where $\Delta_\gamma$ is the Grushin operator. For a suitable range of the parameter $p$ we…

We investigate the degenerate fractional Schr\"{o}dinger-Kirchhoff-Poisson equation in $\mathbb{R}^3$ with critical nonlinearity and electromagnetic fields $\varepsilon^{2s} M([u]_{s,A}^2)(-\Delta)_{A}^su + V(x)u + \phi u = k(x)|u|^{r-2}u +…

In this paper, we study the multiplicity of nonnegative solutions for mixed local and non-local problem involving critical nonlinearity with sign changing weight. Using Nehari manifold method and fibering map analysis, we have shown…

Using the Mountain--Pass Theorem of Ambrosetti and Rabinowitz we prove that $-\Delta_p u-\mu|x|^{-p}{u^{p-1}}=|x|^{-s}{u^{\crits-1}}+u^{\crit-1}$ admits a positive weak solution in $\rn$ of class $\dunp\cap C^1(\rn\setminus\{0\})$, whenever…

The paper deals with existence and multiplicity of solutions of the fractional Schr\"{o}dinger--Kirchhoff equation involving an external magnetic potential. As a consequence, the results can be applied to the special case \begin{equation*}…

We are interested in finding prescribed $L^2$-norm solutions to inhomogeneous nonlinear Schr\"{o}dinger (INLS) equations. For $N\ge 3$ we treat the equation with combined Hardy-Sobolev power-type nonlinearities $$ -\Delta u+\lambda…

In this paper, we first establish the uniqueness and non-degeneracy of positive solutions to the fractional Kirchhoff problem \begin{equation*}…

We investigate normalized solutions with prescribed $L^2$-norm for the upper critical fractional Choquard equation \[(-\Delta)^s u+V(\varepsilon x)u=\lambda…

Given $\mu > 0$, we study the elliptic problem: \begin{align*} \text{ find } (u,\lambda) \in H_0^1(\Omega) \times \mathbb{R} \text{ such that } -\Delta u + \lambda u = |u|^{p-2}u \text{ in } \Omega \text{ and } \int_\Omega|u|^2dx = \mu,…

We prove the existence of multiple solutions for the following sixth-order $p(x)$-Kirchhoff-type problem: $-M(\int_\Omega \frac{1}{p(x)}|\nabla \Delta u|^{p(x)}dx)\Delta^3_{p(x)} u = \lambda f(x)|u|^{q(x)-2}u + g(x)|u|^{r(x)-2}u + h(x) \ \…

In this paper, we study the Brezis-Nirenberg type problem for Choquard equations in $\mathbb{R}^N$ \begin{equation*} -\Delta u+u=(I_{\alpha}\ast|u|^{p})|u|^{p-2}u+\lambda|u|^{q-2}u \quad \mathrm{in}\ \mathbb{R}^N, \end{equation*} where…

We establish the existence of a positive ground state solution for a Kirchhoff problem in $\mathbb{R}^2$ involving critical exponential growth, that is, the nonlinearity behaves like $\exp(\alpha_{0}s^{2})$ as $|s| \to \infty$, for some…