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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.

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.

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 $k =1,\cdots,K,$ 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 $M_1\cup \dots \cup M_K$.

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 consider the singularly perturbed nonlinear elliptic problem \begin{eqnarray*} \varepsilon^2 \Delta v - V(x)v + f(v) =0, v > 0, \lim_{|x|\to \infty} v(x) = 0. \end{eqnarray*} Under almost optimal conditions for the potential $V$ and the nonlinearity $f$, we establish the existence of single-peak solutions whose peak points converge to local minimum points of $V$ as $\varepsilon \to 0$. Moreover, we exhibit a threshold on the condition of $V$ at infinity between existence and nonexistence of solutions.

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.

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

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].

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 .

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.

We consider singularly perturbed elliptic equations $\varepsilon^2\Delta u - V(x) u + f(u)=0, x\in R^N, N \ge 3.$ For small $\varepsilon > 0,$ we glue together localized bound state solutions concentrating at isolated components of positive local minimum of $V$ under conditions on $f$ we believe to be almost optimal.