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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]

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 study the existence of multi-bump positive solutions of the following nonlinear elliptic problem: Here $$1

We give a simple proof of the fact that for a large class of quasilinear elliptic equations and systems the solutions that minimize the corresponding energy in the set of all solutions are radially symmetric. We require just continuous nonlinearities and no cooperative conditions for systems. Thus, in particular, our results cannot be obtained by using the moving planes method. In the case of scalar equations, we also prove that any least energy solution has a constant sign and is monotone with respect to the radial variable. Our proofs rely on results in Brothers and Ziemer (J Reine Angew Math 384:153–179, 1988) and Mariş (Arch Ration Mech Anal, 192:311–330, 2009) and answer questions from Brézis and Lieb (Comm Math Phys 96:97–113, 1984) and Lions (Ann Inst H Poincaré Anal Non Linéaire 1:223–283, 1984). [ABSTRACT FROM AUTHOR]