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In this paper, we prove the existence of a positive least energy vector solution and its asymptotic behavior for three-component nonlinear Schrödinger systems with mixed couplings (two repulsive and one attractive) forces and nonconstant potentials on the entire space when the interaction forces are large. When the system with mixed coupling forces has constant potentials, repulsive forces make a state more stable when the interaction between components are less; thus it loses a compactness due to translation segregating components. To get a compactness, we impose a potential wall at infinity; then, we can construct a least energy vector solution. A main interest in this work is its asymptotic behavior of the solution for large interaction forces; one component repelling other two components survives and the other two components diminish and concentrate at a point diverging to infinity as the interaction forces are getting larger and larger. The location of the concetration point, which we could characterize in terms of the limit of a surviving component, a repulsive force and potentials of diminishing components under the assumption of the nondegeneracy for the limit problem of the surviving component. [ABSTRACT FROM AUTHOR]

There have been numerous studies on Hardy's inequality on a bounded domain, which holds for functions vanishing on the boundary. On the other hand, the classical Legendre differential equation defined in an interval can be regarded as a Neumann version of the Hardy inequality with subcritical weight functions. In this paper we study a Neumann version of the Hardy inequality on a bounded C^2-domain in \mathbb {R}^n of the following form \begin{equation*} \int _\Omega d^{\beta }(x) |\nabla u(x) |^2 dx \ge C(\alpha,\beta) \int _\Omega \frac {|u(x)|^2}{d^{\alpha }(x)} dx \quad \text { with }\quad \int _\Omega \frac {u(x)}{d^{\alpha }(x)} dx=0, \end{equation*} where d(x) is the distance from x \in \Omega to the boundary \partial \Omega and \alpha,\beta \in \mathbb {R}. We classify all (\alpha,\beta) \in \mathbb {R}^2 for which C(\alpha,\beta) > 0. Then, we study whether an optimal constant C(\alpha,\beta) is attained or not. Our study on C(\alpha,\beta) for general (\alpha,\beta) \in \mathbb {R}^2 shows that the (classical) Hardy inequality can be regarded as a special case of the Neumann version. [ABSTRACT FROM AUTHOR]

This paper is concerned with asymptotic behavior of positive solutions for coupled Schrödinger equations with mixed interactions between components. We construct locally minimal energy solutions that show distinctively different limiting profile for simultaneously large attractive and repulsive couplings. The components of the solutions constructed exhibit partial synchronization and segregation. [ABSTRACT FROM AUTHOR]

In this paper we study the structure of positive vector solutions for nonlinear Schrödinger systems with 3 components when all interspecies interaction forces are positive and large while all intraspecies interaction forces are positive and fixed. We will show that the structure strongly depends on some relation of large interspecies interaction forces. [ABSTRACT FROM AUTHOR]

In this paper, we study Hardy's inequality in a limiting case: ∫ Ω | ∇ u | N d x ≥ C N (Ω) ∫ Ω | u (x) | N | x | N (log R | x |) N d x for functions u ∈ W 0 1 , N (Ω) , where Ω is a bounded domain in ℝ N with R = sup x ∈ Ω | x |. We study the attainability of the best constant C N (Ω) in several cases. We provide sufficient conditions that assure C N (Ω) > C N (B R) and C N (Ω) is attained, here B R is the N -dimensional ball with center the origin and radius R. Also, we provide an example of Ω ⊂ ℝ 2 such that C 2 (Ω) > C 2 (B R) = 1 / 4 and C 2 (Ω) is not attained. [ABSTRACT FROM AUTHOR]

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]

The paper is concerned with the asymptotic behavior of positive least energy vector solutions to nonlinear Schrödinger systems with mixed couplings which arise from models in Bose–Einstein condensates and nonlinear optics. We show that due to mixed attractive and repulsive interactions the least energy solutions exhibit new interesting component-wise pattern formations, including co-existence of partial synchronization and segregation. The novelty of our approach is the successful use of multiple scaling to carry out a refined asymptotic analysis of convergence to a multiply scaled limiting system. [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]

For $$N \le 3$$ and $$\beta > 0,$$ we consider the following singularly perturbed elliptic system There are an enormous number of results for localized solutions of singularly perturbed scalar problems using variational methods or finite dimensional reduction methods. However, there exist no general existence results of localized solutions for elliptic systems. We present some such results here. In the first, by a mini-max characterization for a limiting problem, for small $$\varepsilon > 0,$$ we show the existence of one bump solutions with a common concentration point of $$u_1,u_2$$ in a domain O when certain conditions for the limiting problem are satisfied. Typical examples of potentials $$W_1,W_2$$ satisfying the condition are the following: (1) $$W_1,W_2$$ have a common non-degenerate critical point in O which share the same stable, unstable directions; (2) for the outnormal n on $$\partial O$$ , $$\frac{\partial W_1}{\partial n} > 0, \frac{\partial W_2}{\partial n} > 0$$ or $$\frac{\partial W_1}{\partial n} < 0, \frac{\partial W_2}{\partial n} < 0$$ on $$\partial O;$$ (3) $$\max _{x \in O}W_i(x) >> \max _{x \in \partial O}W_i(x)$$ for $$i =1,2.$$ We also give some nonexistence results for some potentials $$W_1,W_2$$ , not satisfying these conditions, but each $$W_1,W_2$$ having a structurally stable critical point in O. [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]