Related papers: Normalized solutions to a class of Kirchhoff type …

In this paper, we consider the following singularly perturbed Kirchhoff equation \begin{equation*} -(\varepsilon^2a+\varepsilon b\int_{\mathbb{R}^3}|\nabla u|^2dx)\Delta u+V(x)u=P(x)|u|^{p-2}u+Q(x)|u|^4u,\quad x\in\mathbb{R}^3,…

In this paper, we deal with the existence and concentration of normalized solutions to the supercritical nonlinear Schr\"{o}dinger equation \begin{equation*} \left\{ \begin{array}{l} -\Delta u + V(x) u = \mu_q u + a|u|^q u \quad {\rm…

The aim of this work is the study of the existence of normalized solutions to the nonlinear Schr\"odinger equation with nonlocal nonlinearities: \begin{equation}\nonumber \left\{\begin{aligned} &-\Delta u =\lambda…

The purpose of this paper is to study the indefinite Kirchhoff type problem: \begin{equation*} \left\{ \begin{array}{ll} M\left( \int_{\mathbb{R}^{N}}(|\nabla u|^{2}+u^{2})dx\right) \left[ -\Delta u+u\right] =f(x,u) & \text{in…

We investigate the differentiability of minimal average energy associated to the functionals $S_\ep (u) = \int_{\mathbb{R}^d} (1/2)|\nabla u|^2 + \ep V(x,u)\, dx$, using numerical and perturbative methods. We use the Sobolev gradient…

In this paper, we consider the existence of solutions of the following Kirchhoff-type problem \[ \left\{ \begin{array} [c]{ll} -\left(a+b\int_{\mathbb{R}^3}|\nabla u|^2dx\right)\Delta u+ V(x)u=f(x,u),~{\rm{in}}~ \mathbb{R}^{3},\\ u\in…

Consider the following Kirchhoff type problem $$ \left\{\aligned -\bigg(a+b\int_{\mathbb{B}_R}|\nabla u|^2dx\bigg)\Delta u&= \lambda u^{q-1} + \mu u^{p-1}, &\quad \text{in}\mathbb{B}_R, \\ u&>0,&\quad\text{in}\mathbb{B}_R,\\…

In this paper we study the existence and regularity results of normalized solutions to the following quasilinear elliptic Choquard equation with critical Sobolev exponent and mixed diffusion type operators: \begin{equation*}…

This paper concerns the existence of normalized solutions to a class of $(2,q)$-Laplacian equations in all the possible cases according to the value of $p$ with respect to the critical exponent $2(1+2/N)$. In the $L^2$-subcritical case, we…

In this paper, we study the mass-constrained fractional Choquard equation \( (-\Delta)^s u = \lambda u + \alpha (I_\mu * |u|^{\frac{2N-\mu}{N}})|u|^{\frac{2N-\mu}{N}-2}u + (I_\mu * |u|^p)|u|^{p-2}u \) in \( \mathbb{R}^N \), under the…

We study the Choquard equation with a local perturbation \begin{equation*} -\Delta u=\lambda u+(I_\alpha\ast|u|^p)|u|^{p-2}u+\mu|u|^{q-2}u,\ x\in \mathbb{R}^{N} \end{equation*} having prescribed mass \begin{equation*}…

In this paper, we show the existence and multiplicity of solutions for the following fourth-order Kirchhoff type elliptic equations \begin{eqnarray*} \Delta^{2}u-M(\|\nabla u\|_{2}^{2})\Delta u+V(x)u=f(x,u),\ \ \ \ \ x\in \mathbb{R}^{N},…

In this paper, we study normalized ground states for the following critical fractional NLS equation with prescribed mass: \begin{equation*} \begin{cases} (-\Delta)^{s}u=\lambda u +\mu|u|^{q-2}u+|u|^{2_{s}^{\ast}-2}u,&x\in\mathbb{R}^{N},…

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

This paper is concerned with the following focusing biharmonic HLS upper critical Hartree equation with a local perturbation $$ \begin{cases} {\Delta}^2u-\lambda u-\mu|u|^{p-2}u-(I_\alpha*|u|^{4^*_\alpha})|u|^{4^*_\alpha-2}u=0\ \ \mbox{in}\…

The paper addresses an open problem raised in [Bartsch, Molle, Rizzi, Verzini: Normalized solutions of mass supercritical Schr\"odinger equations with potential, Comm. Part. Diff. Equ. 46 (2021), 1729-1756] on the existence of normalized…

In this paper, we study the existence of normalized ground state solutions for the following biharmonic Choquard system \begin{align*} \begin{split} \left\{ \begin{array}{ll} \Delta^2u=\lambda_1 u+(I_\mu*F(u,v))F_u (u,v), \quad\mbox{in}\ \…

We are concerned with qualitative properties of positive solutions to the following coupled Sobolev critical Schr\"odinger equations $$ \begin{cases} -\Delta u+\lambda_1 u=\mu_1|u|^{2^*-2}u+\nu\alpha |u|^{\alpha-2}|v|^{\beta}u ~\hbox{in}~…

In this paper, we consider the following nonlinear Kirchhoff type problem: \[ \left\{\begin{array}{lcl}-\left(a+b\displaystyle\int_{\mathbb{R}^3}|\nabla u|^2\right)\Delta u+V(x)u=f(u), & \textrm{in}\,\,\mathbb{R}^3,\\ u\in…

In this paper, we consider the existence of positive solutions with prescribed $L^2$-norm for the following nonlinear Schr\"{o}dinger equation involving potential and Sobolev critical exponent \begin{equation*} \begin{cases} -\Delta…