Multiplicity Of Radial Solutions Research Articles

Quasilinear Schrödinger equations with bounded coefficients

We study quasilinear elliptic equations of the form , where is a bounded function, V(x) is allowed to be singular at the origin, and is a general nonlinearity. Such type of equations has been derived as models of several physical phenomena corresponding to various types of . Despite the lack of a priori compactness condition for the energy functional, we develop a new variational approach to deal with the issues of multiple radial solutions, nonradial solutions and sign-changing solutions.

Multiple radial solutions for Dirichlet problem involving two mean curvature equations in Euclidean and Minkowski spaces

In this paper, we establish the existence of multiple positive radial solutions for the Dirichlet problem involving two mean curvature equations of spacelike graphs in Euclidean and Minkowski spaces.

Normalized solutions to the mixed dispersion nonlinear Schrödinger equation in the mass critical and supercritical regime

In this paper, we study the existence of solutions to the mixed dispersion nonlinear Schrödinger equation \[ γ Δ 2 u − Δ u + α u = | u | 2 σ u , u ∈ H 2 ( R N ) , \gamma \Delta ^2 u -\Delta u + \alpha u=|u|^{2 \sigma } u, \qquad u \in H^2({\mathbb {R}}^N), \] under the constraint \[ ∫ R N | u | 2 d x = c > 0. \int _{{\mathbb {R}}^N}|u|^2 \, dx =c>0. \] We assume that γ > 0 , N ≥ 1 , 4 ≤ σ N > 4 N ( N − 4 ) + \gamma >0, N \geq 1, 4 \leq \sigma N > \frac {4N}{(N-4)^+} , whereas the parameter α ∈ R \alpha \in {\mathbb {R}} will appear as a Lagrange multiplier. Given c ∈ R + c \in {\mathbb {R}}^+ , we consider several questions including the existence of ground states and of positive solutions and the multiplicity of radial solutions. We also discuss the stability of the standing waves of the associated dispersive equation.

Dynamical system modeling fermionic limit

The existence of multiple radial solutions to the elliptic equation modeling fermionic cloud of interacting particles is proved for the limiting Planck constant and intermediate value of mass parameters. It is achieved by considering the related nonautonomous dynamical system for which the passage to the limit can be established due to the continuity of the solutions with respect to the parameter going to zero.

An autonomous Kirchhoff-type equation with general nonlinearity in [formula omitted

We consider the following autonomous Kirchhoff-type equation −(a+b∫RN|∇u|2)Δu=f(u),u∈H1(RN), where a≥0,b>0 are constants and N≥1. Under general Berestycki–Lions type assumptions on the nonlinearity f, we establish the existence results of a ground state and multiple radial solutions for N≥2, and obtain a nontrivial solution and its uniqueness, up to a translation and up to a sign, for N=1. The proofs are mainly based on a rescaling argument, which is specific for the autonomous case, and a new description of the critical values in association with the level sets argument.

Multiple solutions for Neumann and periodic problems with singular ϕ-Laplacian

We use the critical point theory for convex, lower semicontinuous perturbations of C 1 -functionals to establish existence of multiple radial solutions for some one parameter Neumann problems involving the operator v ↦ div ( ∇ v 1 − | ∇ v | 2 ) . Similar results for periodic problems are also provided.

Multiple radial solutions for a class of quasilinear elliptic problems

In this note, we revisit a class of p -Laplacian boundary value problems. By means of the Leggett–Williams fixed-point theorem, we establish the existence of at least three positive radial solutions without the sublinear and superlinear growths imposed on the nonlinearity, which improves the main results in Ma (1996) [5], and Marcos do Ó and Ubilla (2003) [6].

Existence of radial solutions for an asymptotically linear p-Laplacian problem

We prove that an asymptotically linear Dirichlet problem which involves the p-Laplacian operator has multiple radial solutions when the nonlinearity has a positive zero and the range of the ‘ p-derivative’ of the nonlinearity includes at least the first j radial eigenvalues of the p-Laplacian operator. The main tools that we use are a uniqueness result for the p-Laplacian operator and bifurcation theory.

On Multiple Radial Solutions of a Singularly Perturbed Nonlinear Elliptic System

We study radial solutions of a singularly perturbed nonlinear elliptic system of the FitzHugh–Nagumo type. In a particular parameter range, we find a large number of layered solutions. First we show the existence of solutions whose layers are well separated from each other and also separated from the origin and the boundary of the domain. Some of these solutions are local minimizers of a related functional while the others are critical points of saddle type. Although the local minimizers may be studied by the Γ‐convergence method, the reduction procedure presented in this paper gives a more unified approach that shows the existence of both local minimizers and saddle points. Critical points of both types are all found in the reduced finite dimensional problem. The reduced finite dimensional problem is solved by a topological degree argument. Next we construct solutions with odd numbers of layers that cluster near the boundary, again using the reduction method. In this case the reduced finite dimensional p...

The p-Laplacian equation with superlinear and supercritical growth, multiplicity of radial solutions

This paper deals with the existence and multiplicity results for radial symmetric solutions of the p-Laplacian equation with nonlinearities having combined superlinear and supercritical growth.

A Robin problem for a class of quasilinear operators and a related minimizing problem

In this paper we establish the existence of multiple radial solutions for a class of quasilinear operators with nonlinear boundary Robin conditions. Besides other conditions, we consider the nonlinearities having a behavior at -∞ at least like a linearity of slope less than first eigenvalue λ1(R). The technical approach is by variational methods, which is mainly based on a version of Mountain Pass Theorem due to Ghoussoub and Preiss.

A Robin problem for a class of quasilinear operators and a related minimizing problem

In this paper we establish the existence of multiple radial solutions for a class of quasilinear operators with nonlinear boundary Robin conditions. Besides other conditions, we consider the nonlinearities having a behavior at - ∞ at least like a linearity of slope less than first eigenvalue λ 1 ( R ) . The technical approach is by variational methods, which is mainly based on a version of Mountain Pass Theorem due to Ghoussoub and Preiss.

Existence and Multiplicity of Radial Solutions Describing the Equilibrium of Nematic Liquid Crystals on Annular Domains

Existence and Multiplicity of Radial Solutions Describing the Equilibrium of Nematic Liquid Crystals on Annular Domains

Multiple radial solutions for a class of elliptic systems with singular nonlinearities

We study radial solutions u=(u1, u2) in an exterior domain ofR N (N⩾3)of the elliptic system−Δu+V⊂u)=0, where V is a positive and singular potential. We look for solutions which satisfy Dirichlet boundary conditions and vanish at infinity. We prove existence of infinitely many radial solutions, which can be topologically classified by their winding numbers around the singularity of V. Furthermore, we study some qualitative properties of such solutions.

Bifurcation theory and radial solutions to elliptic boundary value problems

Using basic tools from general bifurcation theory we explain, through three well-known problems, how to establish the existence of multiple radial solutions to elliptic boundary value problems.