Normalized ground states of the nonlinear Schr\"{o}dinger equation with at least mass critical growth

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), \\ \int_{\mathbb{R}^N} |u|^2 \, dx = \rho > 0, \end{cases} where $\rho$ is prescribed and $(\lambda, u) \in \mathbb{R} \times H^1 (\mathbb{R}^N)$ is to be determined. The new approach based on the direct minimization of the energy functional on the linear combination of Nehari and Pohozaev constraints is demonstrated, which allows to provide general growth assumptions imposed on $g$. We cover the most known physical examples and nonlinearities with growth considered in the literature so far as well as we admit the mass critical growth at $0$

Existence and concentration of ground state solutions for a critical nonlocal Schr\"odinger equation in R2\R^2

We study the following singularly perturbed nonlocal Schr\"{o}dinger equation -\vr^2\Delta u +V(x)u =\vr^{\mu-2}\Big[\frac{1}{|x|^{\mu}}\ast F(u)\Big]f(u) \quad \mbox{in} \quad \R^2, where $V(x)$ is a continuous real function on $\R^2$, $F(s)$ is the primitive of $f(s)$, $0<\mu<2$ and \vr is a positive parameter. Assuming that the nonlinearity $f(s)$ has critical exponential growth in the sense of Trudinger-Moser, we establish the existence and concentration of solutions by variational methods.Comment: 3

On the logarithmic Schrodinger equation

In the framework of the nonsmooth critical point theory for lower semi-continuous functionals, we propose a direct variational approach to investigate the existence of infinitely many weak solutions for a class of semi-linear elliptic equations with logarithmic nonlinearity arising in physically relevant situations. Furthermore, we prove that there exists a unique positive solution which is radially symmetric and nondegenerate.Comment: 10 page

Multiplicity and concentration results for local and fractional NLS equations with critical growth

Goal of this paper is to study the following singularly perturbed nonlinear Schrödinger equation: eps^2s (-Delta)^s v + V(x)v = f(v), x in R^N, where s is in (0,1), N is greater or equal to 2, V in C(R^N,R) is a positive potential and f is assumed critical and satisfying general Berestycki-Lions type conditions. When eps is greater than 0 is small, we obtain existence and multiplicity of semiclassical solutions, relating the number of solutions to the cup-length of a set of local minima of V; in particular, we improve the result in [37]. Furthermore, these solutions are proved to concentrate in the potential well, exhibiting a polynomial decay. Finally, we prove the previous results also in the limiting local setting s = 1 and N greater or equal to 3, with an exponential decay of the solutions

Infinitely many solutions for a gauged nonlinear Schrödinger equation with a perturbation

In this paper, we use the Fountain theorem under the Cerami condition to study the gauged nonlinear Schrödinger equation with a perturbation in R2. Under some appropriate conditions, we obtain the existence of infinitely many high energy solutions for the equation

Multiplicity and concentration results for some nonlinear Schr\"odinger equations with the fractional pp-Laplacian

We consider a class of parametric Schr\"odinger equations driven by the fractional $p$-Laplacian operator and involving continuous positive potentials and nonlinearities with subcritical or critical growth. By using variational methods and Ljusternik-Schnirelmann theory, we study the existence, multiplicity and concentration of positive solutions for small values of the parameter