Related papers: Normalized ground states for NLS equations with ma…

We study the existence of standing waves, of prescribed $L^2$-norm (the mass), for the nonlinear Schr\"{o}dinger equation with mixed power nonlinearities $$ i \partial_t \phi + \Delta \phi + \mu \phi |\phi|^{q-2} + \phi |\phi|^{2^* - 2} =…

In this paper we show the existence of ground-state solutions for the energy-critical NLS perturbed with subcritical terms when the space dimension $d\geq4$. However in dimension three, we show that when the perturbation is small enough,…

In this paper, we consider the existence, multiplicity and nonexistence of solutions for the following equation \begin{equation*} \begin{cases} \begin{aligned} &-\Delta u+\omega u=\mu u^{p-1}+u^{q-1},~ u>0 \quad &&\text { in } \Omega, \\…

This paper concerns the existence of positive ground state solutions for generalized quasilinear Schr\"odinger equations in $\mathbb{R}^N$ with critical growth which arise from plasma physics, as well as high-power ultrashort laser in…

In this paper, we study the existence and multiplicity of normalized solutions for the following $L^2$-supercritical Schr\"odinger equation on noncompact metric graph $\G=(\V,\E)$ with nonlinear point defects \begin{equation*} \begin{cases}…

In this paper, we are concerned with the following fractional $N/s$-Laplacian Choquard equation \begin{align*} \begin{cases} (-\Delta)^s_{N/s}u=\lambda |u|^{\frac{N}{s}-2}u +(I_\mu*F(u))f(u),\ \ \mbox{in}\ \mathbb{R}^N,…

This paper is devoted to the study of the following nonlocal equation: \begin{equation*} -\left(a+b\|\nabla u\|_{2}^{2(\theta-1)}\right) \Delta u =\lambda u+\alpha (I_{\mu}\ast|u|^{q})|u|^{q-2}u+(I_{\mu}\ast|u|^{p})|u|^{p-2}u \ \hbox{in} \…

We look for ground state solutions to the following nonlinear Schr\"{o}dinger equation $$-\Delta u + V(x)u = f(x,u)-\Gamma(x)|u|^{q-2}u\hbox{ on }\mathbb{R}^N,$$ where $V=V_{per}+V_{loc}\in L^{\infty}(\mathbb{R}^N)$ is the sum of a periodic…

In this paper, we study the existence and multiplicity of the normalized solutions to the following quasi-linear problem \begin{equation*} -\Delta u-\Delta(|u|^2)u+\lambda u=|u|^{p-2}u+\tau|u|^{q-2}u, \text{ in }\mathbb{R}^N,~ 1\leq N\leq4,…

We study the stationary nonlinear Schr\"odinger equation \begin{equation}-\Delta u+V(x)u+\lambda u=|u|^{q-2}u,\quad u \in H^1(\mathbb{R}^N), \quad N \geq 2\end{equation} where $V \in L^{\infty}(\mathbb{R}^N)$ is a radial potential. In the…

In the present paper, we study the normalized solutions with least energy to the following system: $$\begin{cases} -\Delta u+\lambda_1u=\mu_1 |u|^{p-2}u+\beta r_1|u|^{r_1-2}|v|^{r_2}u\quad &\hbox{in}\;\mathbb R^N,\\ -\Delta…

This paper is concerned with the existence of ground states for a class of Kirchhoff type equation with combined power nonlinearities \begin{equation*} -\left(a+b\int_{\mathbb{R}^{3}}|\nabla u(x)|^{2}\right) \Delta u =\lambda…

Normalized ground state solutions (NGSS) of Schrodinger equations (SE) have attracted the attention of many research groups during the last decades. This is essentially due to their relevance in many fields in physics and engineering, where…

We study the existence of radially symmetric solutions of the following nonlinear scalar field equations in ${\mathbb R}^N$ ($N\geq 2$): $$ (*)_m \left\{ \eqalign{ -&\Delta u = g(u) -\mu u \quad \hbox{in}\ {\mathbb R}^N, \cr &\|…

In this paper, we investigate the existence of normalized solutions for the following nonlinear Kirchhoff type problem \begin{equation*} \begin{cases} -(a+b\int_{\Omega}\vert\nabla u\vert^2dx)\Delta u+\lambda u=\vert u\vert^{p-2}u & \text{…

In this paper, we study the following class of nonlinear equations: $$ -\Delta u+V(x) u = \left[|x|^{-\mu}*(Q(x)F(u))\right]Q(x)f(u),\quad x\in\mathbb{R}^2, $$ where $V$ and $Q$ are continuous potentials, which can be unbounded or vanishing…

We study minimal mass blow-up solutions of the focusing $L^2$ critical nonlinear Schr\"odinger equation with inverse-square potential, \[ i\partial_t u + \Delta u + \frac{c}{|x|^2}u+|u|^{\frac{4}{N}}u = 0, \] with $N\geqslant 3$ and…

We consider the existence of normalized solutions to nonlinear Schr\"odinger equations on noncompact metric graphs in the $L^2$ supercritical regime. For sufficiently small prescribed mass ($L^2$ norm), we prove existence of positive…

In any dimension $N \geq 1$, for given mass $a>0$, we look to critical points of the energy functional $$ I(u) = \frac{1}{2}\int_{\mathbb{R}^N}|\nabla u|^2 dx + \int_{\mathbb{R}^N}u^2|\nabla u|^2 dx - \frac{1}{p}\int_{\mathbb{R}^N}|u|^p…

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…