Existence and Symmetry of Multi-bump Solutions for Nonlinear Schrödinger Equations

The Method of Moments for Nonlinear Schrödinger Equations: Theory and Applications

“…(7.14) we obtain specific nonlinear terms for which the quadratic phase approximation allows us to write closed equations for the moments. Depending on the parameter δ = (1 + a F ) 2 + 4a J there exist three families of solutions 17) where…”

“…For this reason, most studies about the dynamics of this type of equation are exclusively numerical. The rigorous studies carried out to date concentrate on (i) properties of stationary solutions [17], (ii) particular results on the existence of solutions [18,19], and (iii) asymptotic properties [13,14].…”

Multiscale-bump standing waves with a critical frequency for nonlinear Schrödinger equations

“…Their method, based on an interesting Lyapunov-Schmidt finite dimensional reduction, was extended by Oh in [28,29] to include a similar result in higher dimensions, provided 1 < p < N +2 N −2 . Other existence results for positive solutions of problem (1.2) under the condition inf x∈R N W (x) > E can be found in [1,2,3,10,12,13,14,15,16,17,20,22,24,25,27,31,32].…”

Multiplicity of asymmetric solutions for nonlinear elliptic problems

“…Among other conditions, V (x) and K(x) are assumed to be at least continuous in all the previous work except in [5], where one positive solution was obtained for = 1 when V (x) ≡ 1 and K(x) satisfies certain assumptions. In [28], Z.-Q. Wang has considered the following problem:…”

“…By applying the concentration compactness principle to deal with a local minimization problem in a suitable subspace of H 1 (R N ), Z.-Q. Wang proved in [28] that (1.3) has a solution which is G-invariant for λ large and concentrates on {x | |x| = 1} as λ goes to ∞. For problem (1.2) with V (x) a positive constant, if K(x) satisfies the same conditions of symmetry as V (x) in (1.3) and K(x) achieves its global maximum (say K 0 ) on {x | |x| = 1} and K(x) < K 0 for 1 − σ < |x| < 1 + σ, where σ ∈ (0, 1), then Wang's method in [28] can be applied to obtain a similar result as in [28] for (1.2) when ε is small enough.…”