Existence Of Radial Solutions Research Articles

Symmetry and symmetry breaking for Hénon-type problems involving the 1-Laplacian operator

In this work, we study a class of Hénon-type equations which involve the 1-Laplacian operator in the unit ball. Under mild assumptions on the nonlinearity, the existence of radial solutions is proved and, for a parameter in a certain range, the existence of symmetry breaking is proved, through the presence of non-radial solutions. The approach is based on an approximation scheme, where a thorough analysis of the solutions of the associated p-Laplacian problems is necessary.

Radial Solutions for p-Laplacian Neumann Problems Involving Gradient Term Without Growth Restrictions

We study the existence of radial solutions for the p-Laplacian Neumann problem with gradient term of the type $$\begin{aligned} \left\{ \begin{array}{l} -\Delta _{p}u=f(|x|,u,x\cdot \nabla u)\quad \text {in} ~\varOmega ,\\ \displaystyle \frac{\partial u}{\partial \mathbf{n} }=0\quad \text {on}~ \partial \varOmega , \end{array} \right. \end{aligned}$$ where $$\Delta _pu=\text {div}(|\nabla u|^{p-2}\nabla u)$$ is the p-Laplace operator with $$p>1$$ , $$\varOmega \subset \mathbb {R}^N(N\ge 2)$$ is a ball. We do not impose any growth restrictions on the nonlinearity. By using the topological transversality method together with the barrier strip technique, the existence of radial solutions to the above problem is obtained.

The Keller\u2013Osserman-type conditions for the study of a semilinear elliptic system

We study the following system of equations: \t\t\t0.1{Δu1=p1(|x|)f1(u1,u2)in RN,Δu2=p2(|x|)f2(u1,u2)in RN.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\textstyle\\begin{cases} \\Delta u_{1}=p_{1} ( \\vert x \\vert ) f_{1} ( u_{1},u_{2} ) &\\text{in }\\mathbb{R}^{N}, \\\\ \\Delta u_{2}=p_{2} ( \\vert x \\vert ) f_{2} ( u_{1},u_{2} )& \\text{in }\\mathbb{R}^{N}. \\end{cases} $$\\end{document} Here f_{1}, f_{2} are continuous nonlinear functions that satisfy Keller–Osserman-type conditions, and p_{1} and p_{2} are continuous weight functions. We establish the existence of radial solutions for this system under various boundary conditions.

Shooting from singularity to singularity and a semilinear Laplace–Beltrami equation

For surfaces of revolution we prove the existence of infinitely many rotationally symmetric solutions to a wide class of semilinear Laplace–Beltrami equations. Our results extend those in Castro and Fischer (Can Math Bull 58(4):723–729, 2015) where for the same equations the existence of infinitely many even (symmetric about the equator) rotationally symmetric solutions on spheres was established. Unlike the results in that paper, where shooting from a singularity to an ordinary point was used, here we obtain regular solutions shooting from a singular point to another singular point. Shooting from a singularity to an ordinary point has been extensively used in establishing the existence of radial solutions to semilinear equations in balls, annulii, or $$\mathbb {R}^N$$ .

A fractional Kirchhoff-type problem in ℝN without the (AR) condition

The aim of this paper was to investigate the existence of radial solutions for a Kirchhoff-type problem driven by the fractional Laplacian, that iswhere is the fractional Laplacian operator with and , and are constants, is a parameter and without the Ambrosetti–Rabinowitz condition. The existence of nontrivial nonnegative radial solutions is obtained using variational methods combined with a cut-off function technique.

Elliptic systems in unbounded domains

We study a nonlinear elliptic system of Lane–Emden type which is equivalent to a fourth-order elliptic equation. By using variational methods the existence of radial solutions of the given problem is proved. To this aim, new compact embeddings are stated.

Existence of radial solutions with prescribed number of zeros for elliptic equations and their Morse index

In this paper we consider radially symmetric solutions of the nonlinear Dirichlet problem Δ u + f ( | x | , u ) = 0 in Ω, where Ω is a ball in R N , N ⩾ 3 and f satisfies some appropriate assumptions. We prove existence of radially symmetric solutions with k prescribed number of zeros. Moreover, when f ( | x | , u ) = K ( | x | ) | u | p − 1 u , using the uniqueness result due to Tanaka (2008) [21], we verify that these solutions are non-degenerate and we prove that their radial Morse index is exactly k.

Existence of multiple solutions for semilinear elliptic equations in the annulus

The existence of radial solutions of Δu + λg(|x|)f(u) = 0 in annuli with Dirichlet (Dirichlet/Neumann) boundary conditions is investigated. It is proved that the problems have at least two positive radial solutions on any annulus if f is superlinear at 0 and sublinear at ∞.

INFINITELY MANY NODAL SOLUTIONS FOR A WEAKLY COUPLED NONLINEAR SCHRÖDINGER SYSTEM

Existence of radial solutions with a prescribed number of nodes is established, via variational methods, for a system of weakly coupled nonlinear Schrödinger equations. The main goal is to obtain a nodal solution with all vector components not identically zero and an estimate on their energies.

A Neumann problem in exterior domain

We are concerned with the existence of radial solutions for the following Neumann problem $$$$ where Ω is an exterior domain in ℝ N , \(\) denotes the normal interior derivative on ∂Ω and g satisfies certain assumptions.

Existence of multiple positive solutions of discrete two-point

We consider the following boundary value problem: , where is fixed. With the aid of fixed point theorems for operators on a cone, existence criteria are developed for multiple (at least three) positive solutions of the boundary value problem. As as application, we establish the existence of radial solutions of certain partial difference equation. Several examples are also included to dwell upon the importance of the results obtained.

Sturm type theorems for general quasi-variational nonlinear problems

Consider the general equation \( [G_p(u^{'})] + f(t,u) = Q (t,u, u^{'}), t \in (a,b) \) associated to the initial-value problem \( u(a) \neq {0} , u^{'}(a) = u(b) = 0 \) where \( f \) is a restoring force and Q represents a nonlinear damping term. Through an analysis of the equation, we give precise estimates of b in terms of the initial data that extend some results derived from Sturm comparison theorems for linear differential equations of second order. In particular, we show some important theorems of non existence of radial solutions of Dirichlet problems in \( {\Bbb R}^n \) that either significantly improve the former ones, with the m-Laplacian operator, or cover cases never appeared before, with the mean curvature operator.

On the existence of radial solutions of a nonlinear elliptic equation on the unit ball

On the existence of radial solutions of a nonlinear elliptic equation on the unit ball

Multiple Solutions of Semi-linear Elliptic Equations on [formula omitted

We consider the existence of radial solutions with finite energy of the equation −Δu = ƒ(|x|)g(u) on RN.