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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.

For a compact smooth manifold $$(M,g_0)$$ with a boundary, we study the conformal rigidity and non-rigidity of the scalar curvature in the conformal class. It is known that the sign of the first eigenvalue for a linearized operator of the scalar curvature by a conformal change determines the rigidity/non-rigidity of the scalar curvature by conformal changes when the scalar curvature $$R_{g_0}$$ is positive. In this paper, we show the sign condition of $$R_{g_0}$$ is not necessary, and a reversed rigidity of the scalar curvature in the conformal class does not hold if there exists a point $$x_0 \in M$$ with $$R_{g_0}(x_0) > 0.$$

In this paper we study the structure of positive vector solutions for nonlinear Schrodinger 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.

This paper is concerned with the asymptotic behavior of least energy vector solutions for nonlinear Schrödinger systems with mixed couplings of attractive and repulsive forces. We focus here on the radially symmetric case while the general studies were already conducted in our earlier work [J. Byeon, Y. Sato, and Z.-Q. Wang, J. Math. Pures Appl. (9), 106 (2016), pp. 477--511], [J. Byeon, Y. Sato, and Z.-Q. Wang, J. Fixed Point Theory Appl., 19 (2017), pp. 559--583]. Though there is still the general phenomenon of component-wise pattern formation with co-existence of partial synchronization and segregation for positive least energy vector solutions as in [J. Byeon, Y. Sato, and Z.-Q. Wang, J. Math. Pures Appl. (9), 106 (2016), pp. 477--511], [J. Byeon, Y. Sato, and Z.-Q. Wang, J. Fixed Point Theory Appl., 19 (2017), pp. 559--583], in our case of radially symmetric domains, it turns out that the energy of synchronization part may be concentrated either on the center of the domain or on the boundary of the domain depending on the spatial dimension of the domain. This is a distinct new feature from [J. Byeon, Y. Sato, and Z.-Q. Wang, J. Math. Pures Appl. (9), 106 (2016), pp. 477--511], [J. Byeon, Y. Sato, and Z.-Q. Wang, J. Fixed Point Theory Appl., 19 (2017), pp. 559--583] due to the radially symmetric property. Our approach develops techniques of multiscale asymptotic estimates.

Consider the Henon 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.

In this paper, we prove a partly clustering phenomenon for nonlinear Schrodinger 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] , [11] , [12] , [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.

In this paper we study the existence, regularity, radial symmetry and decay property of a mountain pass solution for nonlinear scalar field equations involving the fractional Laplacian under an almost optimal class of continuous nonlinearities.

This paper is a sequel to Byeon et al. (J Math Pures Appl 106(9):477–511, 2016) concerning the asymptotic behavior of positive least energy vector solutions to nonlinear Schrodinger systems with mixed couplings that arise from models in Bose–Einstein condensates and nonlinear optics. In [8] we treated homogeneous Dirichlet boundary condition. In the current paper we investigate the case of homogeneous Neumann boundary condition. We show that due to mixed attractive and repulsive interactions the least energy solutions exhibit component-wise pattern formations, in particular, co-existence of partial synchronization and segregation. We employ multiple scalings to carry out a refined asymptotic analysis of convergence to a multiple scaled limiting system.

The paper is concerned with the asymptotic behavior of positive least energy vector solutions to nonlinear Schrodinger 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.

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.

In this paper, we study Hardy’s inequality in a limiting case: [Formula: see text] for functions [Formula: see text], where [Formula: see text] is a bounded domain in [Formula: see text] with [Formula: see text]. We study the attainability of the best constant [Formula: see text] in several cases. We provide sufficient conditions that assure [Formula: see text] and [Formula: see text] is attained, here [Formula: see text] is the [Formula: see text]-dimensional ball with center the origin and radius [Formula: see text]. Also, we provide an example of [Formula: see text] such that [Formula: see text] and [Formula: see text] is not attained.

For \(N \le 3\) and \(\beta > 0,\) we consider the following singularly perturbed elliptic system $$\begin{aligned} \left\{ \begin{array}{rl} \varepsilon ^2\Delta u_1 - W_1(x)u_1 + \mu _1 (u_1)^3 +\beta u_1 (u_2)^2= 0,\ u_1 > 0 &{}\quad \text {in }\mathbf{R}^N,\\ \varepsilon ^2 \Delta u_2 - W_2(x)u_2 +\mu _2 (u_2)^3 +\beta u_2(u_1)^2 = 0,\ u_2 > 0 &{}\quad \text {in }\mathbf{R}^N.\\ \end{array} \right. \end{aligned}$$ 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 \(\frac{\partial W_1}{\partial n} > \max _{x \in \partial O}W_i(x)\) 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.

For $k =1,\cdots,K,$ let $M_k$ be a $q_k$-dimensional smooth compact framed manifold in $R^N$ with $q_k \in \{1,\cdots,N-1\} $. We consider the equation $-\varepsilon^2\Delta u + V(x)u - u^p = 0$ in $R^N$ where for each $k \in \{1,\cdots,K\}$ and some $m_k > 0,$ $V(x)=|\textrm{dist}(x,M_k)|^{m_k}+O(|\textrm{dist}(x,M_k)|^{m_k+1})$ as $\textrm{dist}(x,M_k) \to 0 $. For a sequence of $\varepsilon$ converging to zero, we will find a positive solution $u_{\varepsilon}$ of the equation which concentrates on $M_1\cup \dots \cup M_K$.

In this paper, the Allen–Cahn model equation has 0 and 1 as solutions and as minima of the associated functional. The existence of solutions that are local minima and are close to 0 and 1 on a prescribed pattern of sets has been established in [Calc. Var. Partial Differential Equations 47 (2013), 1–23]. Here related solutions of higher topological type are obtained using minimax and heat flow arguments on a family of invariant sets.

For any N ⩾ 1 and sufficiently small ε > 0, we find a positive solution of a nonlinear elliptic equationwhen lim|x| → ∞V(x) = m > 0 and some optimal conditions on f are satisfied. Furthermore, we investigate the asymptotic behaviour of the solution as ε → 0.

We study the existence of a positive solution of a nonlinear elliptic equation where k ≥ 2 and D is a bounded domain domain in Rℓ, ℓ ≥ 1. We give almost necessary and sufficient condition of f(ξ) for the existence of a positive solution, which is inspired by the works of Berestycki-Lions [1] and Berestycki-Gallouët-Kavian [3].

We study standing waves for nonlinear Schrodinger 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 A μ 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.

A semilinear elliptic system of PDEs with a nonlinear term of double well potential type is studied in a cylindrical domain. The existence of solutions heteroclinic to the bottom of the wells as minima of the associated functional is established. Further applications are given, including the existence of multitransition solutions as local minima of the functional.

An Allen–Cahn phase transition model with a periodic nonautonomous term is presented for which an infinite number of transition states is shown to exist. A constrained minimization argument and the analysis of a limit problem are employed to get states having a finite number of transitions. A priori bounds and an approximation procedure give the general case. Decay properties are also studied and a sharp transition result with an arbitrary interface is proved.

Let $\Omega $ be a smooth bounded domain. We are concerned about the following nonlinear elliptic problem: $\Delta u + |x|^{\alpha}u^{p} = 0, \ u > 0 \quad$ in $\Omega$, $\ u = 0 \quad$ on $\partial \Omega $, where $\alpha > 0, p \in (1,\frac{n+2}{n-2}).$ In this paper, we show that for $n \ge 8$, a maximum point $x_{\alpha }$ of a least energy solution of above problem converges to a point $x_0 \in \partial^$*$ \Omega $ satisfying $H(x_0) = \min_$$\w \in \partial ^$*$ \Omega$$H( w )$ as $\alpha \to \infty,$ where $H$ is the mean curvature on $\partial \Omega $ and $\partial ^$*$\Omega \equiv \{ x \in \partial \Omega : |x| \ge |y|$ for any $y \in \Omega \}.$

We glue together standing wave solutions concentrating around critical points of the potential V with different energy scales. We devise a hybrid method using simultaneously a Lyapunov–Schmidt reduction method and a variational method to glue together standing waves concentrating on local minimum points which possibly have no corresponding limiting equations and those concentrating on general critical points which converge to solutions of corresponding limiting problems satisfying a non-degeneracy condition.

LetΩ\Omegabe a bounded domain inRn,\mathbf {R}^n,n≥3,n \ge 3,with a boundary∂Ω∈C2.\partial \Omega \in C^2.We consider the following singularly perturbed nonlinear elliptic problem onΩ\Omega:\[ε2Δu−u+f(u)=0, u>0 on Ω,u=0 on ∂Ω,\varepsilon ^2 \Delta u - u + f(u) = 0, \ \ u > 0 \textrm { on }\Omega , \quad u = 0 \textrm { on } \partial \Omega ,\]where the nonlinearityffis of subcritical growth. Under rather strong conditions onf,f,it has been known that for smallε>0,\varepsilon > 0,there exists a mountain pass solutionuεu_\varepsilonof above problem which exhibits a spike layer near a maximum point of the distance functionddfrom∂Ω\partial \Omegaasε→0.\varepsilon \to 0.In this paper, we construct a solutionuεu_\varepsilonof above problem which exhibits a spike layer near a maximum point of the distance function under certain conditions onff, which we believe to be almost optimal.

We study semiclassical states of nonlinear Schrodinger equations with anisotropic type potentials which may exhibit a combination of vanishing and singularity while allowing decays and unboundedness at infinity. We give existence of spike type standing waves concentrating at the singularities of the potentials.

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., 12 pages

For N = 1,2, we consider singularly perturbed elliptic equations ϵ2Δ u − V(x) u + f(u)= 0, u(x)> 0 on R N , lim|x|→∞ u(x)= 0. For small ϵ > 0, we show the existence of a localized bound state solut...

Let Ω be a bounded domain in Rn, n⩾3, with the boundary ∂Ω∈C3. We consider the following singularly perturbed nonlinear elliptic problem on Ωε2Δu−u+f(u)=0,u>0on Ω,∂u∂ν=0on ∂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 ε>0, there exists a solution uε of the above problem which exhibits a spike layer near local maximum points of the mean curvature H on ∂Ω as ε→0. In this paper, we obtain the same result under some conditions on f (Berestycki–Lions conditions), which we believe to be almost optimal.

This paper is concerned with the qualitative property of the ground state solutions for the Henon equation. By studying a limiting equation on the upper half space R + N , we investigate the asymptotic energy and the asymptotic profile of the ground states for the Henon equation. The limiting problem is related to a weighted Sobolev type inequality which we establish in this paper.

For singularly perturbed Schrodinger equations with decaying potentials at infinity we construct semiclassical states of a critical frequency concentrating on spheres near zeroes of the potentials. The results generalize some recent work of Ambrosetti-Malchiodi-Ni[3] which gives solutions concentrating on spheres where the potential is positive. The solutions we obtain exhibit different behaviors from the ones given in [3].

Let \({\cal M}\) be a connected compact smooth Riemannian manifold of dimension \(n \ge 3\) with or without smooth boundary \(\partial {\cal M}.\) We consider the following singularly perturbed nonlinear elliptic problem on \({\cal M}\) $$ \varepsilon^2 \Delta_{{\cal M}} u - u + f(u)=0, \ \ u > 0 \quad {\rm on} \quad {\cal M}, \quad \frac{\partial u}{\partial \nu}=0 {\rm on } \partial {\cal M} $$ where \(\Delta_{{\cal M}}\) is the Laplace-Beltrami operator on \({\cal M} \), \(\nu\) is an exterior normal to \(\partial {\cal M}\) and a nonlinearity \(f\) of subcritical growth. For certain \(f,\) there exists a mountain pass solution \(u_\varepsilon\) of above problem which exhibits a spike layer. We are interested in the asymptotic behaviour of the spike layer. Without any non-degeneracy condition and monotonicity of \(f(t)/t,\) we show that if \(\partial {\cal M} =\emptyset(\partial {\cal M} \ne \emptyset),\) the peak point \(x_\varepsilon\) of the solution \(u_\varepsilon\) converges to a maximum point of the scalar curvature \(S\) on \({\cal M}\)(the mean curvature \(H\) on \(\partial {\cal M})\) as \(\varepsilon \to 0,\)respectively.

We consider the following singularly perturbed nonlinear elliptic problem ɛ 2 Δ u - u + f ( u ) = 0 , u > 0 in Ω , u = 0 on ∂ Ω where the nonlinearity f is of subcritical growth and Ω is a bounded domain in R n . Under certain conditions of f , there exists a mountain pass solution u ɛ for above problem and the solution u ɛ exhibits a spike layer as ɛ → 0 . To see the location of the spike layer for small ɛ > 0 , some additional assumptions, certain nondegeneracy for a limiting problem or monotonicity of f ( t ) / t , have been assumed. In this paper, we characterize the location of the spike layer without such additional assumptions for n ⩾ 3 .

For elliptic equations of the form $\Delta u -V(\varepsilon x) u + f(u)=0, x\in {\bf R}^N$ , where the potential V satisfies $\liminf_{\vert x\vert\to \infty} V(x) > \inf_{{\bf R}^N} V(x) =0$ , we develop a new variational approach to construct localized bound state solutions concentrating at an isolated component of the local minimum of V where the minimum value of V can be positive or zero. These solutions give rise to standing wave solutions having a critical frequency for the corresponding nonlinear Schrodinger equations. Our method allows a fairly general class of nonlinearity f(u) including ones without any growth restrictions at large.

This paper is concerned with the existence and qualitative property of standing wave solutions \(\) for the nonlinear Schrodinger equation \(\) with E being a critical frequency in the sense that \(\). We show that there exists a standing wave which is trapped in a neighbourhood of isolated minimum points of V and whose amplitude goes to 0 as \(\). Moreover, depending upon the local behaviour of the potential function V(x) near the minimum points, the limiting profile of the standing-wave solutions will be shown to exhibit quite different characteristic features. This is in striking contrast with the non-critical frequency case \(\) which has been extensively studied in recent years.

For the Lp(p > 1) version of the Caffarelli–Kohn–Nirenberg inequalities we prove a symmetry breaking result for the extremal functions.

Consider the following semilinear Neumann problem: where Ω is a bounded smooth domain in R N , and ν is the unit outward normal to ∂Ω. We assume that 1 < p

We consider the problem: Δu+hu+f(u)=0 in Ω R u=0 on ∂Ω R u>0 in Ω R , where ΩR≡{x∈ R N ∣ R−1

Erratum: Arch. Rational Mech. Anal. DOI 10.1007/s00205-006-0019-3Through answering to some questions from Joao Marcos Bezzerra do Oon the paper above, we found some mistakes in the paper. We thank him forhis interest and careful reading. Here we correct these mistakes.First at the beginning of the proof of Proposition 4, 14-17 line on page193, the correct version is:By compactness of S

We consider the nonlinear elliptic problem − Δ u = u p in Ω R , u > 0 in Ω R , u = 0 in Ω R where p > 1 and Ω R = { x ∈ R N : R | x | R + 1 } with N ⩾ 3 . It is known that as R → ∞ , the number of nonequivalent solutions of the above problem goes to ∞ when p ∈ ( 1 , ( N + 2 ) / ( N − 2 ) ) , N ⩾ 3 . Here we prove the same phenomenon for any p > 1 by finding O ( N − 1 ) -symmetric clustering bump solutions which concentrate near the set { ( x 1 , … , x N ) ∈ Ω R : x N = 0 } for large R > 0 .

For elliptic equations e2Δu − V(x) u + f(u) = 0, x ∈ RN, N ≧ 3, we develop a new variational approach to construct localized positive solutions which concentrate at an isolated component of positive local minimum points of V, as e → 0, under conditions on f which we believe to be almost optimal.

Let Ω be a bounded domain in R N with the boundary ∂ Ω ∈ C 3 . We consider the following singularly perturbed nonlinear elliptic problem on Ω, e 2 Δ v − v + f ( v ) = 0 , v > 0 on Ω , ∂ v ∂ ν = 0 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 f ∈ C 1 ( R ) and N ⩾ 3 , there exists a solution v e of the problem which develops a spike layer near a local maximum point of the mean curvature H on ∂Ω for small e > 0 . In this paper, we extend the previous result for f ∈ C 0 ( R ) and N ⩾ 2 .

We consider the problem: Δu+hu+f(u)=0 in Ω R u=0 on ∂Ω R u>0 in Ω R , where Ω R ≡{ x ∈ R N ∣ R −1 x | R +1}, and the function f and the constant h satisfy suitable assumptions. This problem is invariant under the orthogonal coordinate transformations, in other words, O ( N )-symmetric. We investigate how the symmetry affects to the structure of positive solutions. For a closed subgroup G of O ( N ), we consider a natural group action G × S N −1 → S N −1 . Then, we give a partial order on the space of G -orbits. Then, with respect to the partial order, a critical ( locally minimal ) orbital set will be defined. As a main result of this paper, we show that, when R →∞, a critical orbital set produces a solution of our problem whose energy is concentrated around a scaled critical orbital set .

We consider a semilinear elliptic equation,Δu+up=0 onΩR≡{x∈Rn∣R−12. We prove that, when the space dimension n is three, the number of nonequivalent nonradial positive solutions of the equation goes to ∞ asR→∞. The same result has been known forn=2 andn⩾4; in those cases, the result was obtained by showing that the minimal energy solutions in various symmetry classes have different energy levels. As we will show in this paper, this is not true ifn=3. This makes the casen=3 highly exceptional, and explains why past attempts failed in this case. In this paper we will prove the above result by considering local—rather than global—minimizers in some symmetry classes.