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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]

Abstract: We consider the nonlinear elliptic problem where and with . It is known that as , the number of nonequivalent solutions of the above problem goes to ∞ when , . Here we prove the same phenomenon for any by finding -symmetric clustering bump solutions which concentrate near the set for large . [Copyright &y& Elsevier]

Abstract: We study standing waves for nonlinear Schrödinger equations with the gauge field. Some existence results of standing waves are established by applying variational methods to the functional which is obtained by representing the gauge field in terms of complex scalar field ϕ. We also show that there exists no standing wave for certain range of parameters by establishing a new inequality of Sobolev type. [Copyright &y& Elsevier]

Abstract: Let Ω be a bounded domain in with the boundary . We consider the following singularly perturbed nonlinear elliptic problem on Ω, where ν is the exterior normal to ∂Ω and the nonlinearity f is of subcritical growth. It has been known that under Berestycki and Lions conditions for and , there exists a solution of the problem which develops a spike layer near a local maximum point of the mean curvature H on ∂Ω for small . In this paper, we extend the previous result for and . [Copyright &y& Elsevier]

Abstract: Let Ω be a bounded domain in , , with the boundary . We consider the following singularly perturbed nonlinear elliptic problem on Ω where ν is an exterior normal to ∂Ω and a nonlinearity f of subcritical growth. Under rather strong conditions on f, it has been known that for small , there exists a solution of the above problem which exhibits a spike layer near local maximum points of the mean curvature H on ∂Ω as . In this paper, we obtain the same result under some conditions on f (Berestycki–Lions conditions), which we believe to be almost optimal. [Copyright &y& Elsevier]

Abstract: We consider the following singularly perturbed nonlinear elliptic problemwhere the nonlinearity f is of subcritical growth and is a bounded domain in . Under certain conditions of there exists a mountain pass solution for above problem and the solution exhibits a spike layer as . To see the location of the spike layer for small some additional assumptions, certain nondegeneracy for a limiting problem or monotonicity of have been assumed. In this paper, we characterize the location of the spike layer without such additional assumptions for . [Copyright &y& Elsevier]

Abstract: This paper is concerned with analyzing the limiting behavior of the least energy solutions for the Hénon equation. We study the problems in bounded domains with smooth or non–smooth boundaries, and in particular we examine the effect of the boundary on the limiting profile of the solutions. [Copyright &y& Elsevier]

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]