Nonradial Solutions Research Articles

Nonradial Solutions to Critical Elliptic Equations of Caffarelli–Kohn–Nirenberg Type

Journal Article Nonradial Solutions to Critical Elliptic Equations of Caffarelli–Kohn–Nirenberg Type Get access Monica Musso, Monica Musso 1Departamento de Matemática, Pontificia Universidad Catolica de Chile, Avda. Vicuña Mackenna 4860, Macul, Chile Search for other works by this author on: Oxford Academic Google Scholar Juncheng Wei Juncheng Wei 2Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong Correspondence to be sent to: wei@math.cuhk.edu.hk Search for other works by this author on: Oxford Academic Google Scholar International Mathematics Research Notices, Volume 2012, Issue 18, 2012, Pages 4120–4162, https://doi.org/10.1093/imrn/rnr179 Published: 14 September 2011 Article history Received: 13 April 2011 Revision received: 18 August 2011 Accepted: 22 August 2011 Published: 14 September 2011

Non-radial least energy solutions of the generalized Hénon equation

In this paper, we study the generalized Hénon equation with a radial coefficient function in the unit ball and show the existence of a positive non-radial solution. Our result is applicable to a wide class of coefficient functions. Our theorem ensures that if the ratio of the density of the coefficient function in | x | < a to that in a < | x | < 1 is small enough and a is sufficiently close to 1, then a least energy solution is not radially symmetric.

Infinitely many solutions for nonlinear Schrödinger equations with electromagnetic fields

In this paper, we study the nonlinear Schrödinger equation with electromagnetic fields ( ∇ i − A ( | y | ) ) 2 u + V ( | y | ) u = | u | p − 1 u , u : R N ↦ C , where the vector A ( r ) = ( A 1 ( r ) , A 2 ( r ) , … , A N ( r ) ) is such that A j ( r ) ( j = 1 , 2 , … , N ) is a real function on R + and V ( r ) is a positive function on R + , 1 < p < N + 2 N − 2 if N ⩾ 3 and 1 < p < + ∞ if N = 2 . We prove that the equation has infinitely many non-radial complex-valued solutions under conditions ( H 1 ) and ( H 2 ) which are given in Section 1.

Nonradial Solutions of a Mixed Concave-convex Elliptic Problem

Nonradial Solutions of a Mixed Concave-convex Elliptic Problem

Anisotropic entire large solutions

Given q ∈ ( 0 , 1 ] , we construct nonradial entire large solutions to the equation Δ u = u q in R N .

Bifurcation and asymptotic analysis for a class of supercritical elliptic problems in an exterior domain

We consider the problem either in an annulus AR or in the domain , N ⩾ 3, and .We prove that the unique radial solution up,R in AR converges, as R → +∞, to the unique fast-decay radial solution in Ω and show several related asymptotic estimates, in particular spectral convergence. Analogous asymptotic estimates are also proved for nonradial uniformly bounded solutions in AR.From this we deduce that bifurcation of nonradial solutions occurs at the fast-decay degenerate radial solutions of the problem in Ω and that the bifurcation branches are limits, in a suitable sense, of the bifurcation branches already found in (Gladiali et al 2011 Calc. Var. Partial Diff. Eqns 40 295–317).

Divergent solutions to the 5D Hartree equations

We consider the Cauchy problem for the focusing Hartree equation $iu_{t}+\varDelta u+(|\cdot|^{-3}\ast|u|^{2})u=0$ in $\mathbb{R}^{5}$ with initial data in $H^1$, and study the divergence property of infinite-variance and nonradial solutions. For the

Radial and Nonradial Solutions for a Semilinear Elliptic System of Schrödinger Type

Radial and Nonradial Solutions for a Semilinear Elliptic System of Schrödinger Type

Stability Properties of Non-Radial Steady Ferrofluid Patterns

We consider the dynamic of a fixed volume of ferrofluid in a Hele-Shaw cell under the influence of centrifugal and magnetic forces. The steady-state solutions of the associated moving boundary problem are the periodic solutions of a generalized Laplace-Young equation. We use bifurcation theory to find analytic curves consisting of non-radial steady-state solutions of the problem. The stability of these solutions is discussed by using the exchange of stability theorem.

Infinitely many positive solutions for a Schrödinger–Poisson system

We are interested in the existence of infinitely many positive non-radial solutions of a Schrödinger–Poisson system with a positive radial bounded external potential decaying at infinity.

INFINITELY MANY POSITIVE SOLUTIONS FOR THE NONLINEAR SCHRÖDINGER–POISSON SYSTEM

We consider the following nonlinear Schrödinger–Poisson system in ℝ3[Formula: see text] where K(r) and Q(r) are bounded and positive functions, 1 &lt; p &lt; 5. Assume that K(r) and Q(r) have the following expansions (as r → +∞): [Formula: see text] where a &gt; 0, b ∈ ℝ, m &gt; 1/2, n &gt; 1, θ &gt; 0, κ &gt; 0, and Q0&gt; 0 are some constants. We prove that (0.1) has infinitely many non-radial positive solutions if b &lt; 0, or if b ≥ 0 and 2m &lt; n.

Concentration and compactness in nonlinear Schrödinger–Poisson system with a general nonlinearity

In this paper we use a concentration and compactness argument to prove the existence of a nontrivial non-radial solution to the nonlinear Schrödinger–Poisson equations in R 3 , assuming on the nonlinearity the general hypotheses introduced by Berestycki and Lions.

Bifurcation and symmetry breaking for a class of semilinear elliptic equations in an annulus

In this paper we consider the problem $$\left\{ \begin{array}{ll} -\Delta u=u^p+\lambda u & \quad\hbox{ in }A,\\ u > 0&\quad \hbox{ in }A,\\ u=0 &\quad \hbox{ on }\partial A, \end{array}\right. $$ where A is an annulus of \({\mathbb{R}^N,N\ge2}\) and p > 1. We prove bifurcation of nonradial solutions from the radial solution in correspondence of a sequence of exponents {pk} and for expanding annuli.

Nonradial Entire Large Solutions of Semilinear Elliptic Equations

Nonradial Entire Large Solutions of Semilinear Elliptic Equations

Divergence of Infinite-Variance Nonradial Solutions to the 3D NLS Equation

We consider solutions u(t) to the 3d NLS equation i∂ t u + Δu + |u|2 u = 0 such that ‖xu(t)‖ L 2 = ∞ and u(t) is nonradial. Denoting by M[u] and E[u], the mass and energy, respectively, of a solution u, and by Q(x) the ground state solution to −Q + ΔQ + |Q|2 Q = 0, we prove the following: if M[u]E[u] < M[Q]E[Q] and ‖u 0‖ L 2 ‖∇u 0‖ L 2 > ‖Q‖ L 2 ‖∇Q‖ L 2 , then either u(t) blows-up in finite positive time or u(t) exists globally for all positive time and there exists a sequence of times t n → + ∞ such that ‖∇u(t n )‖ L 2 → ∞. Similar statements hold for negative time.

A global bifurcation result for a semilinear elliptic equation

We consider the problem { − Δ u = u p + λ u in A , u > 0 in A , u = 0 on ∂ A , where A is an annulus of R N , N ⩾ 2 , p ∈ ( 1 , + ∞ ) and λ ∈ ( − ∞ , 0 ] . Recent results (Gladiali et al., 2009 [5]) ensure that there exists a sequence of values of the exponent { p k } at which nonradial bifurcation from the radial solution occurs. We prove the existence of global branches of nonradial solutions bifurcating from the curve of radial ones.

Infinitely many solutions for the prescribed scalar curvature problem on [formula omitted

We consider the following prescribed scalar curvature problem on S N (∗) { − Δ S N u + N ( N − 2 ) 2 u = K ˜ u N + 2 N − 2 on S N , u > 0 where K ˜ is positive and rotationally symmetric. We show that if K ˜ has a local maximum point between the poles then Eq. (∗) has infinitely many non-radial positive solutions, whose energy can be made arbitrarily large.

Liouville-type theorems and universal bounds for nonnegative solutions of the porous medium equation with source

We prove universal bounds for nonnegative weak solutions of the porous medium equation with source $u_t-\Delta u^m=u^p$ where $1 < m < p$. These bounds imply initial and final blow-up rate estimates, as well as a~priori estimates or decay rates for global solutions. We consider both radial and nonradial solutions, and in the radial case we cover all Sobolev-subcritical values of $p/m$, which is the best possible range. Our bounds are proved as a consequence of Liouville-type theorems for entire solutions and doubling and rescaling arguments. In this connection, we use known Liouville-type theorems for radial solutions, along with some new Liouville-type theorems that are here established for nonradial solutions in RN and for solutions on a half-line.

QUASILINEAR ELLIPTIC EQUATIONS OF THE HENON-TYPE: EXISTENCE OF NON-RADIAL SOLUTIONS

In this work, we prove results on existence and multiplicity of non-radial solutions for a class of singular quasilinear elliptic problems of the form [Formula: see text] where B = {x ∈ ℝN: |x| &lt; 1} (N ≥ 3) is a unit open ball centered at the origin, -∞ &lt; a &lt; (N - p)/p, β &gt; 0 and [Formula: see text].

Infinitely many positive solutions for the nonlinear Schrödinger equations in $${\mathbb{R}^N}$$

We consider the following nonlinear problem in \({\mathbb {R}^N}\) $$- \Delta u +V(|y|)u = u^{p},\quad u > 0 \quad {\rm in}\, \mathbb {R}^N, \quad u \in H^1(\mathbb {R}^N), \quad \quad \quad (0.1)$$ where V(r) is a positive function, \({1< p < {\frac{N+2}{N-2}}}\). We show that if V(r) has the following expansion: $$V(r) = V_0+\frac a {r^m} +O \left(\frac1{r^{m+\theta}}\right),\quad {\rm as} \, r\to +\infty,$$ where a > 0, m > 1, θ > 0, and V 0 > 0 are some constants, then (0.1) has infinitely many non-radial positive solutions, whose energy can be made arbitrarily large.