Related papers: Multiple normalized solutions for quasi-linear Sch…
In this paper we are concerned with the existence of normalized solutions for nonlinear Schr\"odinger equations on noncompact metric graphs with localized nonlinearities. In a $L^2$-supercritical regime, we obtain the existence of solutions…
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…
We consider the existence of solutions for nonlinear Schr\"odinger equations on noncompact metric graphs with localized nonlinearities. In an $L^2$-supercritical regime, we establish the existence of infinitely many solutions for any…
We consider the existence and multiplicity of solutions for a class of quasi-linear Schr\"{o}dinger equations which include the modified nonlinear Schr\"{o}dinger equations. A new perturbation approach is used to treat the sub-cubic…
We discuss the application of the Mountain Pass algorithm to the so-called quasi-linear Schrodinger equation, which is naturally associated with a class of nonsmooth functionals so that the classical algorithm is not directly applicable.
We study a class of gauged nonlinear Schr\"{o}dinger equations in the plane. We obtain existence of two nontrivial solutions via the Mountain-Pass theorem and Ekeland's variational principle. Moreover, we prove existence of infinitely many…
We obtain the existence of mountain pass solutions for quasi-linear equations without the typical assumptions which guarantee the boundedness of an arbitrary Palais-Smale sequence. This is done through a recent version of the monotonicity…
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 will utilize the dual method to construct multiple nonradial normalized solutions of the following quasilinear Schr\"{o}dinger equation: \begin{equation*} -\Delta u-\Delta(|u|^{2})u-\mu u=|u|^{p-2}u, \qquad in \quad…
We prove global existence and stability of solution to the mass-critical stochastic nonlinear Schr\"odinger equation in $d=1$ at $L^2$ regularity. Our construction starts with the existence of solution to the truncated subcritical problem.…
It is shown that a large subset of initial data with finite energy ($L^2$ norm)evolves nearly linearly in nonlinear Schr\" odinger equation with periodic boundary conditions. These new solutions are not perturbations of the known ones such…
We find a normalized solution $u=(u_1,\ldots,u_K)$ to the system of $K$ coupled nonlinear Schr\"odinger equations \begin{equation*} \left\{ \begin{array}{l} -\Delta u_i+ \lambda_i u_i = \sum_{j=1}^K\beta_{i,j}u_i|u_i|^{p/2-2}|u_j|^{p/2}…
We study normalized solutions for the nonlinear Schrodinger (NLS) equation with potential and Sobolev critical nonlinearity. By establishing suitable assumptions on the potential, together with new techniques, we find a mountain-pass type…
We study the existence and multiplicity of positive solutions in $H^1(\mathbb{R}^N)$, $N\ge3$, with prescribed $L^2$-norm, for the (stationary) nonlinear Schr\"odinger equation with Sobolev critical power nonlinearity. It is well known…
This work examines a quasilinear Schr\"odinger-Poisson system involving a critical nonlinearity, expressed as \[ -\Delta u + \phi u + \lambda u = |u|^{q-2} u + |u|^4 u, \quad x \in \Omega_r, \] \[ -\Delta \phi - \varepsilon^4 \Delta_4 \phi…
We obtain the existence, nonexistence and multiplicity of positive solutions with prescribed mass for nonlinear Schr\"{o}dinger equations in bounded domains via a global bifurcation approach. The nonlinearities in this paper can be mass…
In this paper, we study the fractional critical Schr\"{o}dinger-Poisson system \[\begin{cases} (-\Delta)^su +\lambda\phi u= \alpha u+\mu|u|^{q-2}u+|u|^{2^*_s-2}u,&~~ \mbox{in}~{\mathbb R}^3,\\ (-\Delta)^t\phi=u^2,&~~ \mbox{in}~{\mathbb…
We study the existence and nonexistence of normalized solutions $(u_a, \lambda_a)\in H^{1}(\mathbb{R}^N)\times \mathbb{R}$ to the nonlinear Schr\"{o}dinger equation with mixed nonlocal nonlinearities. This study can be viewed as a…
The multiplicity of positive weak solutions for a quasilinear Schr\"{o}dinger equations $-L_p u +(\lambda A(x)+1)|u|^{p-2}u= h(u)$ in $\mathbb{R}^N$ is established, where $L_p u\doteq \epsilon^{p}\Delta_p u +\epsilon^{p}\Delta_p (u^2)u$,…
Existence of two solutions to a parametric singular quasi-linear elliptic problem is proved. The equation is driven by the {\Phi}-Laplacian operator and the reaction term can be non-monotone. The main tools employed are a local minimum…