Non-radial Positive Solutions Research Articles

Multiple non-radial solutions for coupled Schrödinger equations

The paper deals with the existence of non-radial solutions for an N-coupled nonlinear elliptic system. In the repulsive regime with some structure conditions on the coupling and for each symmetric subspace of rotation symmetry, we prove the existence of an infinite sequence of non-radial positive solutions and an infinite sequence of non-radial nodal solutions.

Infinitely many solutions for Hamiltonian system with critical growth

Abstract In this article, we consider the following elliptic system of Hamiltonian-type on a bounded domain: − Δ u = K 1 ( ∣ y ∣ ) ∣ v ∣ p − 1 v , in B 1 ( 0 ) , − Δ v = K 2 ( ∣ y ∣ ) ∣ u ∣ q − 1 u , in B 1 ( 0 ) , u = v = 0 on ∂ B 1 ( 0 ) , \left\{\begin{array}{ll}-\Delta u={K}_{1}\left(| y| ){| v| }^{p-1}v,\hspace{1.0em}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{B}_{1}\left(0),\\ -\Delta v={K}_{2}\left(| y| ){| u| }^{q-1}u,\hspace{1.0em}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{B}_{1}\left(0),\\ u=v=0\hspace{1.0em}& \hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}\partial {B}_{1}\left(0),\end{array}\right. where K 1 ( r ) {K}_{1}\left(r) and K 2 ( r ) {K}_{2}\left(r) are positive bounded functions defined in [ 0 , 1 ] \left[0,1] , B 1 ( 0 ) {B}_{1}\left(0) is the unit ball in R N {{\mathbb{R}}}^{N} , and ( p , q ) \left(p,q) is a pair of positive numbers lying on the critical hyperbola 1 p + 1 + 1 q + 1 = N − 2 N . \frac{1}{p+1}+\frac{1}{q+1}=\frac{N-2}{N}. Under some suitable further assumptions on the functions K 1 ( r ) {K}_{1}\left(r) and K 2 ( r ) {K}_{2}\left(r) , we prove the existence of infinitely many nonradial positive solutions whose energy can be made arbitrarily large. Our proof is based on the reduction method. The most ingredients of the article are using the Green representation and estimating the Green function and its regular part very carefully. For this purpose, some more extra ideas and techniques are needed. We believe that our method and techniques can be applied to other related problems.

Infinitely many nonradial positive solutions for multi-species nonlinear Schrödinger systems in [formula omitted

In this paper, we consider the multi-species nonlinear Schrödinger systems in RN:{−Δuj+Vj(x)uj=μjuj3+∑i=1;i≠jdβi,jui2ujin RN,uj(x)>0in RN,uj(x)→0as |x|→+∞,j=1,2,⋯,d, where N=2,3, μj>0 are constants, βi,j=βj,i≠0 are coupling parameters, d≥2 and Vj(x) are potentials. By Ljapunov-Schmidt reduction arguments, we construct infinitely many nonradial positive solutions of the above system under some mild assumptions on potentials Vj(x) and coupling parameters {βi,j}, without any symmetric assumptions on the limit case of the above system. Our result, giving a positive answer to the conjecture in Pistoia and Viara [50] and extending the results in [50,52], reveals new phenomenon in the case of N=2 and d=2 and is almost optimal for the coupling parameters {βi,j}.

Infinitely many solutions for nonlinear fourth-order Schrödinger equations with mixed dispersion

ABSTRACT In this paper, we first show the nondegeneracy and asymptotic behavior of ground states for the nonlinear fourth-order Schrödinger equation with mixed dispersion: δ Δ 2 u − Δu + u = | u | 2 σ u , u ∈ H 2 ( R N ) , where δ > 0 is sufficiently small, 0 < σ < 2 ( N − 2 ) + , 2 ( N − 2 ) + = 2 N − 2 for N ≥ 3 and 2 ( N − 2 ) + = + ∞ for N=2,3. This work extends some results in Bonheure, Casteras, Dos Santos, and Nascimento [Orbitally stable standing waves of a mixed dispersion nonlinear Schrödinger equation. SIAM J Math Anal. 2018;50:5027–5071]. Next, suppose P ( x ) and Q ( x ) are two positive, radial and continuous functions satisfying that as r = | x | → + ∞ , P ( r ) = 1 + a 1 r m 1 + O ( 1 r m 1 + θ 1 ) , Q ( r ) = 1 + a 2 r m 2 + O ( 1 r m 2 + θ 2 ) , where a 1 , a 2 ∈ R , m 1 , m 2 > 1 , θ 1 , θ 2 > 0 . We use the Lyapunov–Schmidt reduction method developed by Wei and Yan [Infinitely many positive solutions for the nonlinear Schrödinger equations in RN. Calc Var. 2010;37:423–439] to construct infinitely many nonradial positive and sign-changing solutions with arbitrary large energy for the following equation: δ Δ 2 u − Δu + P ( x ) u = Q ( x ) | u | 2 σ u , u ∈ H 2 ( R N ) .

Doubling the equatorial for the prescribed scalar curvature problem on {{mathbb {S}}}^N

We consider the prescribed scalar curvature problem on {{mathbb {S}}}^N ΔSNv-N(N-2)2v+K~(y)vN+2N-2=0onSN,v>0inSN,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\Delta _{{{\\mathbb {S}}}^N} v-\\frac{N(N-2)}{2} v+{\ ilde{K}}(y) v^{\\frac{N+2}{N-2}}=0 \\quad \ ext{ on } \\ {{\\mathbb {S}}}^N, \\qquad v >0 \\quad {\\quad \\hbox {in } }{{\\mathbb {S}}}^N, \\end{aligned}$$\\end{document}under the assumptions that the scalar curvature {tilde{K}} is rotationally symmetric, and has a positive local maximum point between the poles. We prove the existence of infinitely many non-radial positive solutions, whose energy can be made arbitrarily large. These solutions are invariant under some non-trivial sub-group of O(3) obtained doubling the equatorial. We use the finite dimensional Lyapunov–Schmidt reduction method.

New concentrated solutions for the nonlinear Schrödinger–Newton system

ABSTRACT In this paper, we study the following nonlinear Schrödinger–Newton system { Δ u − V ( x ) u + Ψ u = 0 , x ∈ R 3 , Δ Ψ + 1 2 u 2 = 0 , x ∈ R 3 , which is a nonlinear system obtained by coupling the linear Schrödinger equation of quantum mechanics with the gravitation law of Newtonian mechanics. Assuming that V ( y ) is radial and satisfies some algebraic decay at the infinity, we construct infinitely many non-radial positive solutions which have polygonal symmetry with respect to y 1 and y 2 and are even in y 2 for the system above by the Lyapunov-Schmidt reduction method. Moreover, we have overcame some new difficulties caused by the non-local term.

Infinitely many solutions for Schrödinger–Newton equations

We prove the existence of infinitely many non-radial positive solutions for the Schrödinger–Newton system [Formula: see text] provided that [Formula: see text] has the following behavior at infinity: [Formula: see text] where [Formula: see text] and [Formula: see text] are some positive constants. In particular, for any [Formula: see text] large we use a reduction method to construct [Formula: see text]-bump solutions lying on a circle of radius [Formula: see text].

Non-radial normalized solutions for a nonlinear Schrodinger equation

This article concerns the existence of multiple non-radial positive solutions of the L&lt;sup&gt;2&lt;/sup&gt;-constrained problem $$\displaylines{-\Delta{u}-Q(\varepsilon x)|u|^{p-2}u=\lambda{u},\quad \text{in }\mathbb{R}^N, \\ \int_{\mathbb{R}^N}|u|^2dx=1,}$$ where \(Q(x)\) is a radially symmetric function, &amp;epsilon;&amp;gt;0 is a small parameter, \(N\geq 2\), and \(p \in (2, 2+4/N)\) is assumed to be mass sub-critical. We are interested in the symmetry breaking of the normalized solutions and we prove the existence of multiple non-radial positive solutions as local minimizers of the energy functional.

Infinitely many positive multi-bump solutions for fractional Kirchhoff equations

The purpose of this paper is to study the existence of infinitely many non-radial positive solutions to the following Kirchhoff equation involving the fractional Laplacian in RN(K)(a+b∫RN×RN|u(x)−u(y)|2|x−y|N+2sdxdy)(−Δ)su+V(|x|)u=up,u>0,inRN where a,b>0, 0<s<1, N>2s, 1<p<2s⁎−1, 2s⁎=2NN−2s is the critical Sobolev exponent and V(|x|) is a positive function. Here we are interested in the case N≥3. For the case N=3, by reducing (K) to a finite dimensional problem by the Lyapunov-Schmidt reduction method, we show that (K) has infinitely solutions whose energy tends to infinity. In contrast to most existing multiplicity results in the literature, we do not need the restriction p>3. For the case N≥4, we use a scaling technique to prove that the energy on the set of solutions to (K) is bounded. It turns out that N≤3 is necessary for the existence of solutions with large energy.

Multiple positive and sign-changing solutions for a class of Kirchhoff equations

This paper is concerned with the existence of multiple non-radial positive and sign-changing solutions for the following Kirchhoff equation: [Formula: see text] where [Formula: see text] are constants, [Formula: see text], [Formula: see text] is a parameter, and [Formula: see text] is a potential function. Under the assumption on [Formula: see text] with exponential decay at infinity, we construct multi-peak positive and sign-changing solutions for problem (0.1) as [Formula: see text] (or [Formula: see text]), where the peaks concentrate at infinity.

Infinitely many non-radial positive solutions for Choquard equations

This paper is concerned with the following Choquard equation:(0.1)−Δu+V(|y|)u=(∫R3Q(|x|)up(x)|x−y|dx)Q(|y|)up−2u. Here p∈(2,2+δ), δ∈(0,13), V(r) and Q(r) carry the following behavior:V(r)=V0+arn+O(1rn+ρ),Q(r)=Q0+brm+O(1rm+θ)asr→∞, where 12≤n,m<1, a,b∈R and V0, Q0, ρ, θ are positive constants. We prove that the equation (0.1) has infinitely many non-radial positive solutions when a and b satisfy some suitable conditions.

Non-degeneracy of synchronized vector solutions for weakly coupled nonlinear schrödiner systems

AbstractIn this paper, we consider the following two-component Schrödiner system \[ \begin{cases} -\Delta u_1 +V_1(x) u_1= \mu_1 u_1^{3} +\beta u_1u_2^{2} &amp; \text{in}\;\mathbb{R}^{3},\\ -\Delta u_2+V_2(x) u_2= \mu_2 u_2^{3} +\beta u_1^{2}u_2 &amp; \text{in}\;\mathbb{R}^{3}. \end{cases} \]First, we revisit the proof of the existence of an unbounded sequence of non-radial positive vector solutions of synchronized type obtained in S. Peng and Z. Wang [Segregated and synchronized vector solutions for nonlinear Schrödinger systems, Arch. Rational Mech. Anal. 208 (2013), 305–339] to give a point-wise estimate of the solutions. Taking advantage of these estimates, we then show a non-degeneracy result of the synchronized solutions in some suitable symmetric space by use of the locally Pohozaev identities. The main difficulties of BEC systems come from the interspecies interaction between the components, which never appear in the study of single equations. The idea used to estimate the coupling terms is inspired by the characterization of the Fermat points in the famous Fermat problem, which is the main novelty of this paper.

Existence of multi-bump solutions for the Schrödinger-Poisson system

This paper deals with the existence of solutions for the following Schrödinger-Poisson system{−Δu+u+βV(x)Φu=|u|p−1u,x∈RN,−ΔΦ=(N−2)ωN−1V(x)u2,x∈RN, where β is a parameter, 3≤N≤6, 1<p<N+2N−2, ωN−1 is the surface area of a unit ball in RN, and V(x) is a potential function. We construct non-radial positive and sign-changing solutions for the Schrödinger-Poisson system through Lyapunov-Schmidt reduction process when V(x) decays exponentially at infinity.

Infinitely many non-radial positive solutions to the double-power nonlinear Schrödinger equations

We consider the following problem with combined nonlinearities: where when ; when N = 2. By use of the Lyapunov-Schmidt reduction argument, under some expansion conditions of the potentials and , we establish infinitely many non-radial solutions. Especially, the nonlinearity is allowed to be (super-)critical, that is, .

Existence of nonradial positive and nodal solutions to a critical Neumann problem in a cone

We study the critical Neumann problem $ \begin{equation*} \begin{cases} -\Delta u = |u|^{2^*-2}u &amp;\text{in }\Sigma_\omega,\\ \quad\frac{\partial u}{\partial\nu} = 0 &amp;\text{on }\partial\Sigma_\omega, \end{cases} \end{equation*} $ in the unbounded cone $\Sigma_\omega: = \{tx:x\in\omega\text{ and }t&gt;0\}$, where $\omega$ is an open connected subset of the unit sphere $\mathbb{S}^{N-1}$ in $\mathbb{R}^N$ with smooth boundary, $N\geq 3$ and $2^*: = \frac{2N}{N-2}$. We assume that some local convexity condition at the boundary of the cone is satisfied. If $\omega$ is symmetric with respect to the north pole of $\mathbb{S}^{N-1}$, we establish the existence of a nonradial sign-changing solution. On the other hand, if the volume of the unitary bounded cone $\Sigma_\omega\cap B_1(0)$ is large enough (but possibly smaller than half the volume of the unit ball $B_1(0)$ in $\mathbb{R}^N$), we establish the existence of a positive nonradial solution.

Infinitely many Solutions with Peaks for a Fractional System in ℝN

In this article, we consider the following coupled fractional nonlinear Schrodinger system in ℝN$$\begin{cases}(-\Delta)^s u+P(x)u=\mu _1 |u|^{2p-2}u+\beta|v|^p|u|^{p-2}u, & x \in \mathbb{R}^N, \\ (-\Delta)^s v+Q(x)v=\mu_2|v|^{2p-2}v+\beta|u|^p|v|^{p-2}v, & x \in \mathbb{R}^N, \\ u, v \in H^s(\mathbb{R}^N), \end{cases}$$ where $$N \ge 2,0 0,\mu _{2} > 0$$ and β ϵ ℝ is a coupling constant. We prove that it has infinitely many non-radial positive solutions under some additional conditions on P(x), Q(x), p and β. More precisely, we will show that for the attractive case, it has infinitely many non-radial positive synchronized vector solutions, and for the repulsive case, infinitely many non-radial positive segregated vector solutions can be found, where we assume that P(x) and Q(x) satisfy some algebraic decay at infinity.

The Neumann Problem for the Generalized Hénon Equation

We study the behavior of radial solutions to the boundary value problem $$ -{\varDelta}_pu+{u}^{p-1}={\left|x\right|}^a{u}^{q-1}\; in\;B,\kern1em \frac{\partial u}{\mathrm{\partial n}}=0\; on\;\partial B,\kern1em q>p, $$ in the unit ball B and prove the existence of nonradial positive solutions for some values of parameters. We obtain multiplicity results which are new even in the case p = 2.

Critical exponent of a simple model of spot replication

This paper is concerned with a semilinear elliptic inhomogeneous equationΔu−u+(1+a|x|q)up=0 introduced in Chen and Kolokolnikov (2012) [2] as a simple prototype of self-replication in more complex reaction–diffusion systems. Under certain conditions on p, q, it was previously shown by Chen–Kolokolnikov that the equation has no radial ground state solution when the control parameter a is increased above some threshold. This property is important for the existence of a saddle-node bifurcation proposed in the Nishiura–Ueyema conditions, which is believed to be necessary for an initiation of a self-replication event. In this paper, we generalize Chen–Kolokolnikov's result to non-radial positive solutions by proving a Liouville-type nonexistence theorem. Furthermore we derive a local version of this nonexistence theorem for solutions defined on a bounded ball. Our result indicates that critical values of q derived in Ding and Ni (1986) [3] are also crucial for the existence and nonexistence problem of positive solutions when the space dimension N≥3.

Non-radial multipeak positive solutions for the Schrödinger–Poisson problem

In this paper, we consider the following Schrödinger–Poisson problem{−ε2Δu+V(x)u+Φu=up,x∈RN,−ΔΦ=u2,x∈RN, where ε>0 is a small parameter, N≥3, and V(x) is a potential function. This system describes some physical phenomena such as quantum mechanics models. We construct non-radial positive solutions, whose components may have spikes clustering at the same point as ε→0+, but the distance between them divided by ε will go to infinity.

Infinitely many non-radial solutions for the polyharmonic Hénon equation with a critical exponent

We study the following polyharmonic Hénon equation:where (m)* = 2N/(N – 2m) is the critical exponent, B1(0) is the unit ball in ℝN, N ⩾ 2m + 2 and K(|y|) is a bounded function. We prove the existence of infinitely many non-radial positive solutions, whose energy can be made arbitrarily large.