Related papers: Normalized solutions to Kirchhoff type equations w…

\noindent We are concerned with positive normalized solutions $(u,\lambda)\in H^1(\mathbb{R}^2)\times\mathbb{R}$ to the following semi-linear Schr\"{o}dinger equations $$ -\Delta u+\lambda u=f(u), \quad\text{in}~\mathbb{R}^2, $$ satisfying…

This paper is concerned with the following focusing biharmonic Schr\"{o}dinger equation with mixed dispersion and Sobolev critical growth: $$ \begin{cases} {\Delta}^2u-\Delta u-\lambda u-\mu|u|^{p-2}u-|u|^{4^*-2}u=0\ \ \mbox{in}\…

We consider the following Kirchhoff - Choquard equation \[ -M(\|\na u\|_{L^2}^{2})\De u = \la f(x)|u|^{q-2}u+ \left(\int_{\Om}\frac{|u(y)|^{2^*_{\mu}}}{|x-y|^{\mu}}dy\right)|u|^{2^*_{\mu}-2}u \; \text{in}\; \Om,\quad u = 0 \; \text{ on }…

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…

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…

We study asymptotic behavior of positive ground state solutions of the nonlinear Kirchhoff equation $$ -\Big(a+b\int_{\mathbb R^N}|\nabla u|^2\Big)\Delta u+ \lambda u= u^{q-1}+ u^{p-1} \quad {\rm in} \ \mathbb R^N, $$ as $\lambda\to 0$ and…

This paper is devoted to the study of the following autonomous Kirchhoff-type equation $$-M\left(\int_{\mathbb{R}^N}|\nabla{u}|^2\right)\Delta{u}= f(u),~~~~u\in H^1(\mathbb{R}^N),$$ where $M$ is a continuous non-degenerate function and…

We propose a simple minimization method to show the existence of least energy solutions to the normalized problem \begin{cases} -\Delta u + \lambda u = g(u) \quad \mathrm{in} \ \mathbb{R}^N, \ N \geq 3, \\ u \in H^1(\mathbb{R}^N), \\…

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…

This paper is concerned with the existence of sign-changing solutions to non local Kirchhoff type problems of the form \begin{equation}\label{s}\tag{S} -\Big(a+b\int_\Omega|\nabla u|^2dx\Big)\Delta u=f(x,u)\, \text{ in }\Omega,\quad\quad…

We are concerned with solutions of the following quasilinear Schr\"odinger equations \begin{eqnarray*} -{\mathrm{div}}\left(\varphi^{2}(u) \nabla u\right)+\varphi(u) \varphi^{\prime}(u)|\nabla u|^{2}+\lambda u=f(u), \quad x \in…

This paper investigates the existence of normalized solutions for the following Chern-Simons-Schr\"odinger equation: \begin{equation*} \left\{ \begin{array}{ll} -\Delta u+\lambda u+\left(\frac{h^{2}(\vert x\vert)}{\vert…

In this paper, we study a type of p-Kirchhoff equation $$ -\left( a+b\int_{\mathbb{R} ^3}{\left| \nabla u \right|^pdx} \right) \varDelta _pu=\lambda \left| u \right|^{p-2}u+\left| u \right|^{q-2}u, x \in \mathbb{R}^3 $$ with the prescribed…

We study normalized solutions $(\mu,u)\in \mathbb{R} \times H^1(\mathbb{R}^N)$ to nonlinear Schr\"odinger equations $$ -\Delta u + \mu u = g(u)\quad \hbox{in}\ \mathbb{R}^N, \qquad \frac{1}{2}\int_{\mathbb{R}^N} u^2 dx = m, $$ where $N\geq…

We investigate a class of Kirchhoff type equations involving a combination of linear and superlinear terms as follows: \begin{equation*} -\left( a\int_{\mathbb{R}^{N}}|\nabla u|^{2}dx+1\right) \Delta u+\mu V(x)u=\lambda…

In this paper, we consider the existence, multiplicity and the asymptotic behavior of prescribed mass solutions to the following nonlinear Kirchhoff equation with mixed nonlinearities: -(a+b\int|\nabla u|^2\mathrm{d}x)\Delta u+V(x)u+\lambda…

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…

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…

We consider the following autonomous Kirchhoff-type equation \begin{equation*} -\left(a+b\int_{\mathbb{R}^N}|\nabla{u}|^2\right)\Delta u= f(u),~~~~u\in H^1(\mathbb{R}^N), \end{equation*} where $a\geq0,b>0$ are constants and $N\geq1$. Under…

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…