In this paper, we study the existence, nonexistence and mass concentration of L2-normalized solutions for nonlinear fractional Schrödinger equations. Comparingwith the Schrödinger equation, we encounter some new challenges due to the nonlocal nature of the fractional Laplacian. We first prove that the optimal embedding constant for the fractional Gagliardo–Nirenberg–Sobolev inequality can be expressed by exact form, which improves the results of [17, 18]. By doing this, we then establish the existence and nonexistence of L2-normalized solutions for this equation. Finally, under a certain type of trapping potentials, by using some delicate energy estimates we present a detailed analysis of the concentration behavior of L2-normalized solutions in the mass critical case.

We construct countably infinitely many non-radial singular solutions of the problem

of the form

where v(σ) depends only on σ ∈𝕊N−1. To this end we construct countably infinitely many solutions of

using ordinary differential equation techniques.

We are interested in non-negative non-trivial solutions of

where 1 < p and Ω is a bounded smooth domain in ℝN with 3 ⩽ N ⩽ 9. We show that given a non-negative integer M there is some large p(M,Ω) such that the only non-negative solution u, of Morse index at most M, is u = 0.

We investigate the decay for |x|→∞ of weak Sobolev-type solutions of semilinear nonlocal equations Pu = F(u). We consider the case when P = p(D) is an elliptic Fourier multiplier with polyhomogeneous symbol p(ξ), and we derive algebraic decay estimates in terms of weighted Sobolev norms. Our basic example is the celebrated Benjamin–Ono equation

for internal solitary waves of deep stratified fluids. Their profile presents algebraic decay, in strong contrast with the exponential decay for KdV shallow water waves.

The aim of this paper is to present a sign-changing solution for a class of radially symmetric asymptotically linear Schrödinger equations. The proof is variational and the Ekeland variational principle is employed as well as a deformation lemma combined with Miranda’s theorem.

In this paper we prove existence and qualitative properties of solutions for a nonlinear elliptic system arising from the coupling of the nonlinear Schrödinger equation with the Poisson equation. We use a contraction map approach together with estimates of the Bessel potential used to rewrite the system in an integral form.

We prove the existence of a family of slow-decay positive solutions of a supercritical elliptic equation with Hardy potential

and study the stability and oscillation properties of these solutions. We also show that if the equation on ℝN has a stable slow-decay positive solution, then for any smooth compact K ⊂ ℝN a family of the exterior Dirichlet problems in ℝN \ K admits a continuum of stable slow-decay infinite-energy solutions.