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Consider the Hénon equation with the homogeneous Neumann boundary condition − Δ u + u = | x | α u p , u > 0 in Ω , ∂ u ∂ ν = 0 on ∂ Ω , where Ω ⊂ B ( 0 , 1 ) ⊂ R N , N ≥ 2 and ∂ Ω ∩ ∂ B ( 0 , 1 ) ≠ ∅ . We are concerned on the asymptotic behavior of ground state solutions as the parameter α → ∞ . As α → ∞ , the non-autonomous term | x | α is getting singular near | x | = 1 . The singular behavior of | x | α for large α > 0 forces the solution to blow up. Depending subtly on the ( N − 1 ) − dimensional measure | ∂ Ω ∩ ∂ B ( 0 , 1 ) | N − 1 and the nonlinear growth rate p , there are many different types of limiting profiles. To catch the asymptotic profiles, we take different types of renormalization depending on p and | ∂ Ω ∩ ∂ B ( 0 , 1 ) | N − 1 . In particular, the critical exponent 2 ⁎ = 2 ( N − 1 ) N − 2 for the Sobolev trace embedding plays a crucial role in the renormalization process. This is quite contrasted with the case of Dirichlet problems, where there is only one type of limiting profile for any p ∈ ( 1 , 2 ⁎ − 1 ) and a smooth domain Ω. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in standing waves with a vortex for the nonlinear Chern–Simons–Schrödinger equations (CSS for short). We study the existence and the nonexistence of standing waves when a constant λ > 0 , representing the strength of the interaction potential, varies. We prove every standing wave is trivial if λ ∈ ( 0 , 1 ) , every standing wave is gauge equivalent to a solution of the first order self-dual system of CSS if λ = 1 and for every positive integer N , there is a nontrivial standing wave with a vortex point of order N if λ > 1 . We also provide some classes of interaction potentials under which the nonexistence of standing waves and the existence of a standing wave with a vortex point of order N are proved. [ABSTRACT FROM AUTHOR]

In an earlier paper, [1] , the authors treated a family of Allen–Cahn model problems for which 0 and 1 are solutions and further solutions were found that are near 1 on a prescribed set, T + Ω , where T ⊂ Z n , and near 0 on ( Z n ∖ T ) + Ω . Here Ω ⊂ ( 0 , 1 ) n . In this paper, a more general class of potentials is treated for which the pair, { 0 , 1 } , is replaced by Z and the existence of a far richer structure of shadowing solutions, including unbounded ones, is established. [ABSTRACT FROM AUTHOR]

In this paper, we prove a partly clustering phenomenon for nonlinear Schrödinger systems with large mixed couplings of attractive and repulsive forces, which arise from the models in Bose-Einstein condensates and nonlinear optics. More precisely, we consider a system with three components where the interaction between the first two components and the third component is repulsive, and the interaction between the first two components is attractive. Recent studies [10–13] in this case show that for large interaction forces, the first two components are localized in a region with a small energy and the third component is close to a solution of a single equation. Especially, the results in the works [12,13] say that the region of localization for a (locally) least energy vector solution on a ball in the class of radially symmetric functions is the origin or the whole boundary depending on the space dimension 1 ≤ n ≤ 3. In this paper we construct a new type of solutions with a region of localization different from the origin or the whole boundary. In fact, we show that there exist radially symmetric positive vector solutions with clustering multi-bumps for the first two components near the maximum point of r n − 1 U 3 , where U is the limit of the third component and the maximum point is the only critical point different from the origin and the boundary. [ABSTRACT FROM AUTHOR]