Multi-peak Solutions Research Articles

Multi-Peak Positive Solutions for Nonlinear Fractional Schr\"{o}dinger Systems in ℝ N

Abstract In this paper, we study the nonlinear fractional Schrödinger system in ℝ N ${\mathbb{R}^{N}}$ . Applying the finite reduction method, we prove that the nonlinear fractional Schrödinger system has multi-peak positive solutions under some suitable conditions.

Existence of multi-peak solutions for p-Laplace problems in ℝ N

In this paper we consider the p-Laplace problem \( - {\varepsilon ^p}{\Delta _p}u + V\left( x \right){u^{p - 1}} = {u^{q - 1}},u{\text{ > }}0\) in R N where 2 ≤ p 0 and \(p{\text{ 0 is small.

Multi-peak solutions for nonlinear Schrödinger systems with magnetic potentials in [formula omitted

We study the following nonlinear Schrödinger system with magnetic potentials in R3{(ϵ∇i−A(x))2u+P(x)u=μ1|u|2u+β|v|2u,x∈R3,(ϵ∇i−A(x))2v+Q(x)v=μ2|v|2v+β|u|2v,x∈R3, where ϵ>0 is a small parameter, μ1,μ2>0 and β>0 is a coupling constant. A(x), P(x) and Q(x) are potential functions. Applying the finite reduction method, we prove that the nonlinear Schrödinger system has multi-peak solutions under some suitable conditions which are given in Section 1.

Multi-peak nodal solutions for a two-dimensional elliptic problem with large exponent in weighted nonlinearity

We consider the boundary value problem \(\Delta u + \left| x \right|^{2\alpha } \left| u \right|^{p - 1} u = 0, - 1 < \alpha \ne 0\), in the unit ball B with the homogeneous Dirichlet boundary condition, when p is a large exponent. By a constructive way, we prove that for any positive integer m, there exists a multi-peak nodal solution up whose maxima and minima are located alternately near the origin and the other m points \(\widetilde{q_l } = (\lambda \cos \frac{{2\pi (l - 1)}} {m},\lambda \sin \frac{{2\pi (l - 1)}} {m}),l = 2, \cdots ,m + 1 \), such that as p goes to +∞, $$p\left| x \right|^{2\alpha } \left| {u_p } \right|^{p - 1} u_p \rightharpoonup 8\pi e(1 + \alpha )\delta _0 + \sum\limits_{l = 2}^{m + 1} {8\pi e( - 1)^{l - 1} \delta _{\widetilde{q_l }} } $$ , where λ ∈ (0, 1), m is an odd number with (1+α)(m+2)−1 > 0, or m is an even number. The same techniques lead also to a more general result on general domains.

Multi-peak positive solutions for a fractional nonlinear elliptic equation

In this paper we study the existence of positive multi-peaksolutions to the semilinear equation\begin{eqnarray*} \varepsilon^{2s}(-\Delta)^{s}u + u= Q(x)u^{p-1}, \hskip0.5cm u >0, \hskip 0.2cm u\in H^{s}(\mathbb{R}^{N})\end{eqnarray*}where $(-\Delta)^{s} $ stands for the fractional Laplacian, $s\in(0,1)$, $\varepsilon$ is a positive small parameter, $2 < p <\frac{2N}{N-2s}$, $Q(x)$ is a bounded positive continuous function.For any positive integer $k$, we prove the existence of a positivesolution with $k$-peaks and concentrating near a given local minimumpoint of $Q$. For $s=1$ this corresponds to the result of [22].

Multi-peak solutions of non-linear elliptic singularly perturbed reaction-diffusion equations using finite element simulation

The present paper deals with a review of nonlinear reaction-diffusion models appearing in different branches of science and engineering. The multi-peak solutions of non-linear elliptic singularly perturbed reaction-diffusion equations do not appear to have been previously studied in detail, computationally. The finite element method for the solution of non-linear elliptic singularly perturbed reaction-diffusion equations (subject to appropriate Dirichlet's boundary condition) is discussed, and a variant of Newton's method having fifth order of convergence is used to linearize the nonlinear system of equations. Examples of nonlinear elliptic singularly perturbed reaction-diffusion equation having non-linearity in homogeneous/non-homogeneous form are considered to show the existence of multi-peak solutions.

Non-radial sign-changing solutions for the Schrödinger–Poisson problem in the semiclassical limit

We study a system of equations known as Schrodinger–Poisson problem proving that the set of sign-changing solutions has a rich structure in the semiclassical limit. Indeed we construct non-radial multi-peak solutions with an arbitrary large number of positive and negative peaks which are displaced in suitable symmetric configurations and which collapse to the same point as ϵ ⟶ 0. The proof is based on the Lyapunov–Schmidt reduction.

Multi-peak Solutions of a Nonlinear Schrödinger Equation with Magnetic Fields

Abstract In this paper, we study a nonlinear Schrödinger equation with magnetic fields involving subcritical growth. Applying the finite reduction method, we prove that the equation has multi-peak solutions under some suitable conditions which are given in section 1.

Multi-peak solutions to two types of free boundary problems

We consider the existence of multi-peak solutions to two types of free boundary problems arising in confined plasma and steady vortex pair under conditions on the nonlinearity we believe to be almost optimal. Our results show that the “core” of the solution has multiple connected components, whose boundary called free boundary of the problems consists approximately of spheres which shrink to distinct single points as the parameter tends to zero.

Multi-peak solution for nonlinear magnetic Choquard type equation

In this paper, we study a class of nonlinear magnetic Choquard type equation involving a magnetic potential and nonlocal nonlinearities. By using the method of penalization argument, we show that there exists a family of solutions having multiple concentration regions which are concentrate at the minimum points of potential V.

Multi-peak solutions to coupled Schrödinger systems with Neumann boundary conditions

In this paper, we are concerned with the following two coupled Schrödinger systems in a bounded domain Ω⊂RN(N=2,3) with Neumann boundary conditions {−ε2Δu+u=μ1u3+βuv2,−ε2Δv+v=μ2v3+βu2v,u>0,v>0,∂u/∂n=0,∂v/∂n=0,on ∂Ω. Suppose the mean curvature H(P) of the boundary ∂Ω has several local minimums or local maximums, we obtain the existence of solutions with multi-peaks to the system with all peaks being on the boundary and all peaks locate either near the local maxima or near the local minima of the mean curvature at the boundary of the domain.

Concentration of solutions for the Yamabe problem on half-spaces

By variational methods, for a type of Yamabe problem with boundary, we construct multi-peak solutions whose maxima tend to the boundary as the parameter tends to 0+ under certain assumptions. We also obtain the asymptotic behaviours of the solutions.

Sign-Changing Multi-Peak Solutions for Nonlinear Schrödinger Equations with Compactly Supported Potential

In this paper, we aim to prove the existence and concentration of sign-changing H 1(Â? N ) solutions to the following nonlinear Schrodinger equations $$-\varepsilon^2\Delta u_{\varepsilon}+V(x)u_{\varepsilon}=K(x)|u_{\varepsilon}|^{p-1}u_{\varepsilon} $$ with NÂ?3, $1 0. When the potential function V(x) has compact support, $V(x)\not\equiv 0$ and V(x)Â?0, K(x) is permitted to be unbounded under some necessary restrictions, we will show that one sign-changing H 1(Â? N ) solution exists and exhibits concentration profile around local minimum points of the ground energy function $G(\xi)\equiv V^{\theta}(\xi) K^{-\frac{2}{p-1}}(\xi)$ with $\theta=\frac{p+1}{p-1}-\frac{N}{2}$ in the semiclassical limit Â?Â?0.

Integrated Isolation Niche and Immune Genetic Algorithm for solving Bid-Based Dynamic Economic Dispatch

This paper presents a novel algorithm-Isolation Niche Immune Genetic Algorithm for solving power system Bid-Based Dynamic Economic Dispatch (INIGA–BDED). Economic Dispatch determines the electrical power to be generated by the committed generating units in a power system so that the generation cost can minimized, while simultaneously satisfying various load demands. The Bid-Based Dynamic Economic Dispatch model is proposed in order to maximize the social profit under a competitive electricity market environment. This model synthetically considers various constraints on ramp rates, transmission line capacity and emission constraints. The Isolation Niche Immune Genetic Algorithm was induced as a new solution for this model. With the introduction of niche technology, the immune genetic algorithm capability in dealing with multi-peak model function optimization was enhanced. This paper proposes the Niche based on the Isolation mechanism which is based on biological possesses. The proposed method effectively ensures diverse group solutions and also has a strong ability to guide evolution. Using the immune genetic algorithm itself is a very good and innovative method for multi-peak model function solutions. A new improved method for this algorithm is also presented in this paper. This research integrated these two methods to enhance the evolutionary capability in seeking a more optimal solution for solving nonlinear programming. The test results from this integrated method were very good.

On the Webster Scalar Curvature Problem on the CR Sphere with a Cylindrical-Type Symmetry

By variational methods, for a kind of Webster scalar curvature problems on the CR sphere with cylindrically symmetric curvature, we construct some multi-peak solutions as the parameter is sufficiently small under certain assumptions. We also obtain the asymptotic behaviors of the solutions.

Single and multi-peak solutions for a nonlinear Maxwell–Schrödinger system with a general nonlinearity

For the following elliptic system in R 3 − ε 2 Δ v + V ( x ) v + ϕ ( x ) v = f ( v ) , − Δ ϕ = v 2 , lim | x | → ∞ ϕ ( x ) = 0 , we construct positive solutions u and ϕ which concentrate at several given isolated local minimum components of V as ε → 0 . The potential V is a strictly positive continuous function and the nonlinearity f is subcritical near infinity and superlinear near zero and satisfies only the Berestycki–Lions condition.

Multipeak solutions for asymptotically critical elliptic equations on Riemannian manifolds

Let ( M , g ) be a smooth compact Riemannian manifold of dimension n ≥ 3 . We are concerned with the following asymptotically critical elliptic problem (0.1) Δ g u + a ( x ) u = u 2 ∗ − 1 − ε , u > 0 in M , where Δ g = − div g ( ∇ ) is the Laplace–Beltrami operator on M , a ( x ) is a C 1 function on M , 2 ∗ = 2 n n − 2 denotes the Sobolev critical exponent, ε is a small real parameter such that ε goes to 0. We use the Lyapunov–Schmidt reduction procedure to obtain that the problem (0.1) has a k -peaks solution for positive integer k ≥ 2 , which blow up and concentrate at some points in M .

Positive and sign-changing clusters around saddle points of the potential for nonlinear elliptic problems

We study the existence and asymptotic behavior of positive and sign-changing multipeak solutions for the equation $$ -\varepsilon^2\Delta v+V(x)v=f(v)\quad{\rm in}\,\,\,\mathbb{R}^N, $$ where e is a small positive parameter, f a superlinear, subcritical and odd nonlinearity, V a uniformly positive potential. No symmetry on V is assumed. It is known (Kang and Wei in Adv Differ Equ 5:899–928, 2000) that this equation has positive multipeak solutions with all peaks approaching a local maximum of V. It is also proved that solutions alternating positive and negative spikes exist in the case of a minimum (see D’Aprile and Pistoia in Ann Inst H. Poincare Anal Non Lineaire 26:1423–1451, 2009). The aim of this paper is to show the existence of both positive and sign-changing multipeak solutions around a nondegenerate saddle point of V.

Construction of multi-peak solutions to the Gierer–Meinhardt system with saturation and source term

In this paper, we are concerned with stationary solutions to the following Gierer–Meinhardt system with saturation and source term under the homogeneous Neumann boundary condition: { A t = ε 2 Δ A − A + A 2 H ( 1 + k A 2 ) + σ 0 in Ω × ( 0 , ∞ ) , τ H t = D Δ H − H + A 2 in Ω × ( 0 , ∞ ) . Here, ε > 0 , τ ≥ 0 , k ≥ 0 , and Ω is a bounded smooth domain in R N . In this paper, we suppose Ω is an x N -axially symmetric domain and σ 0 is an x N -axially symmetric nonnegative function of class C α ( Ω ¯ ) , α ∈ ( 0 , 1 ) . For sufficiently small ε and sufficiently large D , we construct a multi-peak stationary solution peaked at arbitrarily chosen intersections of the x N -axis and ∂ Ω under the condition that 4 k ε − 2 N | Ω | 2 converges to some k 0 ∈ [ 0 , ∞ ) as ε → 0 . This extends the results of Kurata and Morimoto [K. Kurata, K. Morimoto, Construction and asymptotic behavior of the multi-peak solutions to the Gierer–Meinhardt system with saturation, Commun. Pure Appl. Anal. 7 (2008) 1443–1482] to the case σ 0 ( x ) ≥ 0 . Moreover, we study an effect of the source term σ 0 on a precise asymptotic behavior of the solution as ε → 0 .

Asymptotic behaviour of ground state solutions for the Henon equation

Let B1(0) ⊂ RN be the unit ball centred at the origin, N ≥ 3. In this paper, we analyse the profile of the ground state solution of the Hénon equation – Δu = |x|αup−1 in B1(0), u = 0 on∂B1(0). We prove that for fixed p ∈ (2, 2*), (2* = 2N/(N − 2)), the maximum point xα of the ground state solution uα satisfies α(1 − |xα|) → l ∈ (0, +∞) as α → ∞. We also obtain the asymptotic behaviour of uα, which shows that the ground state solution is non-radial. Moreover, we prove the existence of multi-peaked solutions and give their asymptotic behaviour.