Existence of stable standing waves for the nonlinear Schrödinger equation with attractive inverse-power potentials

<abstract><p>In this paper, we consider the following nonlinear Schrödinger equation with attractive inverse-power potentials</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ i\partial_t\psi+\Delta\psi+\gamma|x|^{-\sigma}\psi+|\psi|^\alpha\psi = 0, \; \; \; (t, x)\in\mathbb{R}\times\mathbb{R}^N, $\end{document} </tex-math></disp-formula></p> <p>where $ N\geq3 $, $ 0 &lt; \gamma &lt; \infty $, $ 0 &lt; \sigma &lt; 2 $ and $ \frac{4}{N} &lt; \alpha &lt; \frac{4}{N-2} $. By using the concentration compactness principle and considering a local minimization problem, we prove that there exists a $ \gamma_0 &gt; 0 $ sufficiently small such that $ 0 &lt; \gamma &lt; \gamma_0 $ and for any $ a\in(0, a_0) $, there exist stable standing waves for the problem in the $ L^2 $-supercritical case. Our results are complement to the result of Li-Zhao in ^{[<xref ref-type="bibr" rid="b23">23</xref>]}.</p></abstract>

Ground states for critical fractional Schr\”{o}dinger-Poisson systems with vanishing potentials

This paper deals with a class of fractional Schr\”{o}dinger-Poisson system \[\begin{cases}\displaystyle (-\Delta )^{s}u+V(x)u-K(x)\phi |u|^{2^*_s-3}u=a (x)f(u), &x \in \R^{3}\\ (-\Delta )^{s}\phi=K(x)|u|^{2^*_s-1}, &x \in \R^{3}\end{cases} \]with a critical nonlocal term and multiple competing potentials, which may decay and vanish at infinity, where $s \in (\frac{3}{4},1)$, $ 2^*_s = \frac{6}{3-2s}$ is the fractional critical exponent. The problem is set on the whole space and compactness issues have to be tackled. By employing the mountain pass theorem, concentration-compactness principle and approximation method, the existence of a positive ground state solution is obtained under appropriate assumptions imposed on $V, K, a$ and $f$.

Existence of groundstates for Choquard type equations with Hardy–Littlewood–Sobolev critical exponent

In this paper, we consider a class of Choquard equations with Hardy–Littlewood–Sobolev lower or upper critical exponent in the whole space $\mathbb{R}^{N}$ R N . We combine an argument of L. Jeanjean and H. Tanaka (see (Proc. Am. Math. Soc. 131:2399–2408, 2003) with a concentration–compactness argument, and then we obtain the existence of ground state solutions, which extends and complements the earlier results.

Concentration-Compactness Principle for Trudinger–Moser’s Inequalities on Riemannian Manifolds and Heisenberg Groups: A Completely Symmetrization-Free Argument

The concentration-compactness principle for the Trudinger–Moser-type inequality in the Euclidean space was established crucially relying on the Pólya–Szegő inequality which allows to adapt the symmetrization argument. As far as we know, the first concentration-compactness principle of Trudinger–Moser type in non-Euclidean settings, such as the Heisenberg (and more general stratified) groups where the Pólya–Szegő inequality fails, was found in [J. Li, G. Lu and M. Zhu, Concentration-compactness principle for Trudinger–Moser inequalities on Heisenberg groups and existence of ground state solutions, Calc. Var. Partial Differential Equations 57 2018, 3, Paper No. 84] by developing a nonsmooth truncation argument. In this paper, we establish the concentration-compactness principle of Trudinger–Moser type on any compact Riemannian manifolds as well as on the entire complete noncompact Riemannian manifolds with Ricci curvature lower bound. Our method is a symmetrization-free argument on Riemannian manifolds where the Pólya–Szegő inequality fails. This method also allows us to give a completely symmetrization-free argument on the entire Heisenberg (or stratified) groups which refines and improves a proof in the paper of Li, Lu and Zhu. Our results also show that the bounds for the suprema in the concentration-compactness principle on compact manifolds are continuous and monotone increasing with respect to the volume of the manifold.

Nonlinear perturbations of a periodic fractional Laplacian with supercritical growth

Our main goal is to explore the existence of positive solutions for a class of nonlinear fractional Schrödinger equation involving supercritical growth given by $$ (- \Delta)^{\alpha} u + V(x)u=p(u),\quad x\in \mathbb{R^N},\ N \geq 1. $$ We analyze two types of problems, with $V$ being periodic and asymptotically periodic; for this we use a variational method, a truncation argument and a concentration compactness principle.

On the existence of positive solutions for a nonlinear elliptic class of equations in R2 and R3

We study the existence of positive solutions for an elliptic equation in RN for N = 2, 3 which is related with the existence of standing (localized) waves and the existence of the ground state solutions for some physical model or systems in fluid mechanics to describe the evolution of weakly nonlinear water waves. We use a variational approach and the well-known principle of concentration-compactness due to P. Lions to obtain the existence of this type of solutions, even in the case that the nonlinear term g is a non-homogeneous function or an operator defined in H1(RN) with values in R.

Existence and Symmetry of Solutions for a Class of Fractional Schrödinger–Poisson Systems

In this paper, we investigate a class of Schrödinger–Poisson systems with critical growth. By the principle of concentration compactness and variational methods, we prove that the system has radially symmetric solutions, which improve the related results on this topic.