Axially symmetric solutions of the Allen-Cahn equation with finite Morse index

Abstract We consider nonlinear 2nd-order elliptic problems of the type $$\begin{align*} & -\Delta u=f(u)\ \textrm{in}\ \Omega, \qquad u=0\ \textrm{on}\ \partial \Omega, \end{align*}$$where $\Omega $ is an open $C^{1,1}$–domain in ${{\mathbb{R}}}^N$, $N\geq 2$, under some general assumptions on the nonlinearity that include the case of a sublinear pure power $f(s)=|s|^{p-1}s$ with $0<p<1$ and of Allen–Cahn type $f(s)=\lambda (s-|s|^{p-1}s)$ with $p>1$ and $\lambda>\lambda _2(\Omega )$ (the second Dirichlet eigenvalue of the Laplacian). We prove the existence of a least energy nodal (i.e., sign changing) solution and of a nodal solution of mountain-pass type. We then give explicit examples of domains where the associated levels do not coincide. For the case where $\Omega $ is a ball or annulus and $f$ is of class $C^1$, we prove instead that the levels coincide and that least energy nodal solutions are nonradial but axially symmetric functions. Finally, we provide stronger results for the Allen–Cahn type nonlinearities in case $\Omega $ is either a ball or a square. In particular, we give a complete description of the solution set for $\lambda \sim \lambda _2(\Omega )$, computing the Morse index of the solutions.

Download Full-text

Some remarks on the structure of finite Morse index solutions to the Allen–Cahn equation in $${\mathbb R}^2$$ R 2

Nonlinear Differential Equations and Applications NoDEA ◽

10.1007/s00030-017-0481-7 ◽

2017 ◽

Vol 24 (5) ◽

Cited By ~ 1

Author(s):

Kelei Wang

Keyword(s):

Morse Index ◽

Cahn Equation

Download Full-text

Spiraling solutions of nonlinear Schrödinger equations

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/prm.2021.18 ◽

2021 ◽

pp. 1-34

Author(s):

Oscar Agudelo ◽

Joel Kübler ◽

Tobias Weth

Keyword(s):

Manuscripta Math ◽

Axially Symmetric ◽

Screw Motion ◽

Nonlinear Schrödinger ◽

Cahn Equation ◽

New Family ◽

Symmetry Properties ◽

Variational Structure ◽

Image Position ◽

Sign Changing Solutions

We study a new family of sign-changing solutions to the stationary nonlinear Schrödinger equation \[ -\Delta v +q v =|v|^{p-2} v, \qquad \text{in}\,{ {\mathbb{R}^{3}},} \] with $2 < p < \infty$ and $q \ge 0$ . These solutions are spiraling in the sense that they are not axially symmetric but invariant under screw motion, i.e., they share the symmetry properties of a helicoid. In addition to existence results, we provide information on the shape of spiraling solutions, which depends on the parameter value representing the rotational slope of the underlying screw motion. Our results complement a related analysis of Del Pino, Musso and Pacard in their study (2012, Manuscripta Math., 138, 273–286) for the Allen–Cahn equation, whereas the nature of results and the underlying variational structure are completely different.

Download Full-text

Morse index of layered solutions to the heterogeneous Allen–Cahn equation

Journal of Differential Equations ◽

10.1016/j.jde.2007.03.024 ◽

2007 ◽

Vol 238 (1) ◽

pp. 87-117 ◽

Cited By ~ 5

Author(s):

Yihong Du ◽

Kimie Nakashima

Keyword(s):

Morse Index ◽

Cahn Equation

Download Full-text

Solar Internal Rotation and Dynamo Waves: A Two-Dimensional Asymptotic Solution in the Convection Zone

International Astronomical Union Colloquium ◽

10.1017/s0252921100064861 ◽

2000 ◽

Vol 179 ◽

pp. 379-380

Author(s):

Gaetano Belvedere ◽

Kirill Kuzanyan ◽

Dmitry Sokoloff

Keyword(s):

Magnetic Field ◽

Asymptotic Solution ◽

Internal Rotation ◽

Convection Zone ◽

General Idea ◽

Dynamo Model ◽

Axially Symmetric ◽

Dynamo Action ◽

Regeneration Rate ◽

Dynamo Waves

Extended abstractHere we outline how asymptotic models may contribute to the investigation of mean field dynamos applied to the solar convective zone. We calculate here a spatial 2-D structure of the mean magnetic field, adopting real profiles of the solar internal rotation (the Ω-effect) and an extended prescription of the turbulent α-effect. In our model assumptions we do not prescribe any meridional flow that might seriously affect the resulting generated magnetic fields. We do not assume apriori any region or layer as a preferred site for the dynamo action (such as the overshoot zone), but the location of the α- and Ω-effects results in the propagation of dynamo waves deep in the convection zone. We consider an axially symmetric magnetic field dynamo model in a differentially rotating spherical shell. The main assumption, when using asymptotic WKB methods, is that the absolute value of the dynamo number (regeneration rate) |D| is large, i.e., the spatial scale of the solution is small. Following the general idea of an asymptotic solution for dynamo waves (e.g., Kuzanyan & Sokoloff 1995), we search for a solution in the form of a power series with respect to the small parameter |D|–1/3(short wavelength scale). This solution is of the order of magnitude of exp(i|D|1/3S), where S is a scalar function of position.

Download Full-text