21 research outputs found
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 is prescribed and 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 . 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
Positive ground state of coupled planar systems of nonlinear Schrödinger equations with critical exponential growth
In this paper, we prove the existence of a positive ground state solution to the following coupled system involving nonlinear Schrödinger equations: ââu + V1(x)u = f1(x, u) + λ(x)v, x â R2 ââv + V2(x)v = f2(x, v) + λ(x)u, x â R2 where λ, V1, V2 â C(R2 ,(0, +â)) and f1, f2 : R2 Ă R â R have critical exponential growth in the sense of TrudingerâMoser inequality. The potentials V1(x) and V2(x) satisfy a condition involving the coupling term λ(x), namely 0 < λ(x) †λ0 p V1(x)V2(x). We use non-Nehari manifold, Lionsâs concentration compactness and strong maximum principle to get a positive ground state solution. Moreover, by using a bootstrap regularity lifting argument and L q -estimates we get regularity and asymptotic behavior. Our results improve and extend the previous results
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
Existence of positive solutions to Kirchhoff type problems with zero mass
The existence of positive solutions depending on a nonnegative parameter lambda to Kirchhoff type problems with zero mass is proved by using variational method, and the new result does not require usual compactness conditions. A priori estimate and a Pohozaev type identity are used to obtain the bounded Palais-Smale sequences for constant coefficient nonlinearity, while a cut-off functional and Pohozaev type identity are utilized to obtain the bounded Palais-Smale sequences for the variable-coefficient case. (C) 2013 Elsevier Inc. All rights reserved
Existence of a positive solution to Kirchhoff type problems without compactness conditions
The existence of a positive solution to a Kirchhoff type problem on R-N is proved by using variational methods, and the new result does not require usual compactness conditions. A cut-off functional is utilized to obtain the bounded Palais-Smale sequences. (c) 2012 Elsevier Inc. All rights reserved
Standing waves in nonlinear Schrödinger equations
In the theory of nonlinear Schrödinger equations, it is expected that the solutions will either spread out because of the dispersive effect of the linear part of the equation or concentrate at one or several points because of nonlinear effects. In some remarkable cases, these behaviors counterbalance and special solutions that neither disperse nor focus appear, the so-called standing waves. For the physical applications as well as for the mathematical properties of the equation, a fundamental issue is the stability of waves with respect to perturbations. Our purpose in these notes is to present various methods developed to study the existence and stability of standing waves. We prove the existence of standing waves by using a variational approach. When stability holds, it is obtained by proving a coercivity property for a linearized operator. Another approach based on variational and compactness arguments is also presented. When instability holds, we show by a method combining a Virial identity and variational arguments that the standing waves are unstable by blow-up