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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 C^2 -domain in R n \mathbb {R}^n of the following form ∫ Ω d β ( x ) | ∇ u ( x ) | 2 d x ≥ C ( α , β ) ∫ Ω | u ( x ) | 2 d α ( x ) d x with ∫ Ω u ( x ) d α ( x ) d x = 0 , \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 ) d(x) is the distance from x ∈ Ω x \in \Omega to the boundary ∂ Ω \partial \Omega and α , β ∈ R \alpha ,\beta \in \mathbb {R} . We classify all ( α , β ) ∈ R 2 (\alpha ,\beta ) \in \mathbb {R}^2 for which C ( α , β ) > 0 C(\alpha ,\beta ) > 0 . Then, we study whether an optimal constant C ( α , β ) C(\alpha ,\beta ) is attained or not. Our study on C ( α , β ) C(\alpha ,\beta ) for general ( α , β ) ∈ R 2 (\alpha ,\beta ) \in \mathbb {R}^2 shows that the (classical) Hardy inequality can be regarded as a special case of the Neumann version.

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.

We are concerned with a nonnegative solution to the scalar field equation $$\Delta u + f(u) = 0{\rm in }{\open R}^N,\quad \mathop {\lim }\limits_{|x|\to \infty } u(x) = 0.$$ A classical existence result by Berestycki and Lions [3] considers only the case when f is continuous. In this paper, we are interested in the existence of a solution when f is singular. For a singular nonlinearity f, Gazzola, Serrin and Tang [8] proved an existence result when $f \in L^1_{loc}(\mathbb {R}) \cap \mathrm {Lip}_{loc}(0,\infty )$ with $\int _0^u f(s)\,{\rm d}s < 0$ for small $u>0.$ Since they use a shooting argument for their proof, they require the property that $f \in \mathrm {Lip}_{loc}(0,\infty ).$ In this paper, using a purely variational method, we extend the previous existence results for $f \in L^1_{loc}(\mathbb {R}) \cap C(0,\infty )$. We show that a solution obtained through minimization has the least energy among all radially symmetric weak solutions. Moreover, we describe a general condition under which a least energy solution has compact support.

We consider the following singularly perturbed problem \begin{document}$ \begin{equation*} \varepsilon ^2 \Delta u - u + f(u) = 0,\ \ \ \, u>0 \text{ in } \Omega, \ \ \ \ \frac{\partial u}{\partial \nu} = 0 \text{ on } \partial\Omega. \end{equation*} $\end{document} Existence of a solution with a spike layer near a min-max critical point of the mean curvature on the boundary \begin{document}$ \partial \Omega $\end{document} is well known when a nondegeneracy for a limiting problem holds. In this paper, we use a variational method for the construction of such a solution which does not depend on the nondengeneracy for the limiting problem. By a purely variational approach, we construct the solution for an optimal class of nonlinearities \begin{document}$ f $\end{document} satisfying the Berestycki-Lions conditions.

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.

For $n≥ 3$ and $p = (n+2)/(n-2), $ we consider the Henon equation with the homogeneous Neumann boundary condition \begin{document}$ -Δ u + u = |x|^{α}u^{p}, \; u > 0 \;\text{in} \; Ω,\ \ \frac{\partial u}{\partial ν} = 0 \; \text{ on }\;\partial Ω,$ \end{document} where \begin{document}$Ω \subset B(0,1) \subset \mathbb{R}^n, n ≥ 3$, $α≥ 0$ and $\partial^*Ω \equiv \partialΩ \cap \partial B(0,1) \ne \emptyset.$\end{document} It is well known that for \begin{document}$α = 0,$\end{document} there exists a least energy solution of the problem. We are concerned on the existence of a least energy solution for \begin{document}$α > 0$\end{document} and its asymptotic behavior as the parameter \begin{document}$α$\end{document} approaches from below to a threshold \begin{document}$α_0 ∈ (0,∞]$\end{document} for existence of a least energy solution.

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 this paper, we are interested in standing waves with a vortex for the nonlinear Chern–Simons–Schrodinger 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.

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 $M_1\cup \dots \cup M_K$.

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.