Multiplicity Of Positive Radial Solutions Research Articles

Multiple positive radial solutions for Dirichlet problem of the prescribed mean curvature spacelike equation in a Friedmann–Lemaître–Robertson–Walker spacetime

In this paper, we consider the radially symmetric spacelike solutions of a nonlinear Dirichlet problem for the prescribed mean curvature spacelike equation in a Friedmann–Lemaître–Robertson–Walker spacetime. By using a conformal change of variable, this problem can be translated an equivalent problem in the Minkowski spacetime. By using the lower and upper solution method, fixed point, a priori bounds and topological degree method, we obtain the existence, nonexistence and multiplicity of radially symmetric spacelike solutions.

Positive radial solutions for Dirichlet problems in the ball

We are concerned with existence, localization and multiplicity of positive radial solutions to Dirichlet problems with ϕ-Laplacians in a ball, in both scalar and system cases. Our approach essentially relies on fixed point index computations and a main feature is that it avoids any Harnack type inequality. Applications to some problems involving operators with Uhlenbeck structure are discussed.

Multiplicity of positive radial solutions for semilinear elliptic equation with locally concave-convex variable exponent

Multiplicity of positive radial solutions for semilinear elliptic equation with locally concave-convex variable exponent

Positive radial solutions to singular nonlinear elliptic problems involving nonhomogeneous operators

We consider singular nonlinear elliptic problems involving nonhomogeneous operators on annular domains −div(A(|x|)B(|∇u|)∇u)=λK(|x|)f(u),x∈Ω,u=0,|x|=r1,a∂u∂n+c(λ,u)=0,|x|=r2,where λ>0, a≥0, N>1, Ω≔{x∈RN∣0<r1<|x|<r2<∞} and ∂u∂n is the outward normal derivative of u on ∂Br2. Here A∈C([r1,r2],(0,∞)), K∈C((r1,r2),(0,∞)), c∈C((0,∞)×R,R), B∈C([0,∞),[0,∞)) is such that B(s)s is a homeomorphism from [0,∞) onto [0,∞), and f∈C((0,∞),(0,∞)) has a singularity at 0. The aim of this paper is to analyze the existence and multiplicity of positive radial solutions according to the behavior of f near ∞. In particular, we discuss sufficient conditions for at least three positive radial solutions to exist. The results are obtained via a Krasnoselskii type fixed point theorem. Finally, we provide examples including Gelfand-type problems to illustrate each result.

Multiplicity of positive radial solutions for systems with mean curvature operator in Minkowski space

In this paper, we are considered with the Dirichlet problem of quasilinear differential system with mean curvature operator in Minkowski space $ \mathcal{M}(w): = \text{div}\Big(\frac{\nabla w}{\sqrt{1-|\nabla w|^2}}\Big), $ in a ball in $ \mathbb{R}^N $. In particular, we deal with this system with Lane-Emden type nonlinearities in a superlinear case, by using the Leggett-Williams' fixed point theorem, we obtain the existence of three positive radial solutions.

Semipositone nonlocal Neumann elliptic system depending on the gradient in exterior domains

We study existence and multiplicity of positive radial solutions for a semipositone coupled elliptic system in exterior domains where the nonlinearities depend on the gradients and the Neumann boundary conditions are nonlocal. We use a nonstandard cone to establish existence of multiple solutions by means of fixed point index theory.

Coupled elliptic systems depending on the gradient with nonlocal BCs in exterior domains

We study the existence and multiplicity of positive radial solutions for a coupled elliptic system in exterior domains where the nonlinearities depend on the gradients and the boundary conditions are nonlocal. We use a new cone to establish the existence of solutions by means of fixed point index theory.

Existence and multiplicity of positive radial solutions to the Dirichlet problem for nonlinear elliptic equations on annular domains

In this paper, we study the existence and nonexistence of mono- tone positive radial solutions of elliptic boundary value problems on bounded annular domains subject to local boundary condition. By using Krasnoselskii's xed point theorem of cone expansion-compression type we show that there exists > 0 such that the elliptic equation has at least two, one and no radial positive solutions for 0 respectively. We include an example to illustrate our results.

Multiplicity of positive radial solutions of singular Minkowski-curvature equations

This paper is about a singular Dirichlet problem involving the mean curvature operator in Minkowski space. We get the existence of triple and arbitrarily many positive radial solutions by using the Leggett-Williams fixed point theorem. Recent results in the literature are significantly improved.

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.

Existence results for Dirichlet problems about two mean curvature equations in Euclidean and Minkowski spaces

Let $$H_L$$ and $$H_R$$ be the extrinsic mean curvatures of a spacelike graph in Minkowski space $$\mathbb {L}^{N+1}$$ and Euclidean space $$\mathbb {R}^{N+1}$$ , respectively. In this paper, we establish the existence of multiple positive radial solutions for the Dirichlet problem of two nonlinear partial differential equations involving $$H_L$$ and $$H_R$$ .

Existence and multiplicity of positive radial solutions for singular superlinear elliptic systems in the exterior of a ball

We prove the existence and multiplicity of positive radial solutions to the nonlinear system{−Δui=λKi(|x|)fi(uj) in Ω,di∂ui∂n+c˜i(ui)ui=0 on |x|=r0,ui(x)→0 as |x|→∞, for a certain range of λ>0, where i,j∈{1,2},i≠j, Ω={x∈RN:|x|>r0>0}, N>2,di≥0, Ki:[r0,∞)→(0,∞), c˜:[0,∞)→[0,∞),fi:(0,∞)→R are continuous with possible singularity ±∞ at 0 and satisfy a combined superlinear condition at ∞.

Multiple positive radial solutions for Neumann elliptic systems with gradient dependence

We provide new results on the existence, nonexistence, and multiplicity of nonnegative radial solutions for semilinear elliptic systems with Neumann boundary conditions on an annulus. Our approach is topological and relies on the classical fixed‐point index. We present an example to illustrate our theory.

Multiplicity of positive radial solutions of p-Laplacian problems with nonlinear gradient term

In the present paper, we prove the existence of at least three radial solutions of the p-Laplacian problem with nonlinear gradient term {Δpv+f(|x|,v,|∇v|)=0in Ω,v=0on ∂Ω,\\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}$$\\textstyle\\begin{cases} \\Delta_{p}v+f(\\vert x\\vert ,v,\\vert \\nabla v\\vert )=0\\quad \\mbox{in }\\Omega, \\\\ v=0\\quad \\mbox{on }\\partial\\Omega, \\end{cases} $$\\end{document} and the corresponding one-parameter problem. Here Ω is a unit ball in mathbb{R}^{N}. Our approach relies on the Avery-Peterson fixed point theorem. In contrast with the usual hypotheses, no asymptotic behavior is assumed on the nonlinearity f with respect to phi_{p}(cdot).

Multiple positive solutions to elliptic boundary blow-up problems

We prove the existence of multiple positive radial solutions to the sign-indefinite elliptic boundary blow-up problem{Δu+(a+(|x|)−μa−(|x|))g(u)=0,|x|<1,u(x)→∞,|x|→1, where g is a function superlinear at zero and at infinity, a+ and a− are the positive/negative part, respectively, of a sign-changing function a and μ>0 is a large parameter. In particular, we show how the number of solutions is affected by the nodal behavior of the weight function a. The proof is based on a careful shooting-type argument for the equivalent singular ODE problem. As a further application of this technique, the existence of multiple positive radial homoclinic solutions toΔu+(a+(|x|)−μa−(|x|))g(u)=0,x∈RN, is also considered.

Multiplicity of positive radial solutions of a singular mean curvature equations in Minkowski space

In this article, by using the Leggett–Williams’ fixed point theorem, we prove the existence of at least three positive radial solutions of the singular Dirichlet problem for the prescribed mean curvature equation in Minkowski space {div(∇v1−|∇v|2)+f(|x|,v)=0inΩ;v=0on∂Ω, and the corresponding one-parameter problem. Here Ω is a unit ball in RN.

Multiple positive radial solutions to some Kirchhoff equations

In this paper, we discuss the existence and multiplicity of positive radial solutions to some Kirchhoff equations and elliptic equations. By using the Leggett–Williams three solutions theorem skillfully, we obtain two positive solutions to Kirchhoff equations under some appropriate conditions. As a direct corollary, we also obtain the same result to elliptic equations. In order to facilitate the proof of main results, we first establish a new simple three solutions theorems for the equation [a+bφ(x)]x=Ax, where A is a completely continuous operator, a>0, b⩾0 and φ(x)⩽k(‖x‖) for all x∈P, where k is a nondecreasing nonnegative continuous function, then apply it to prove the main results. In addition, we discover and prove a new inequality in the article. In the end of the paper, we present an example which makes the elliptic equation has infinite many positive radial solutions and give the approximate images of the nonlinear term.

Positive solutions for some nonlocal and nonvariational elliptic systems

Using fixed point techniques, we study the existence and multiplicity of positive radial solutions for two classes of nonlocal elliptic systems defined on bounded annular domains or exterior domains. To this end, we reduce our problem to second-order functional ordinary elliptic systems. Our approach also allows us to study systems involving various orders, which serve as models for the suspension bridge equations.

On positive radial solutions for a class of elliptic equations.

A class of elliptic boundary value problem in an exterior domain is considered under some conditions concerning the first eigenvalue of the relevant linear operator, where the variables of nonlinear term f(s, u) need not to be separated. Several new theorems on the existence and multiplicity of positive radial solutions are obtained by means of fixed point index theory. Our conclusions are essential improvements of the results in Lan and Webb (1998), Lee (1997), Mao and Xue (2002), Stańczy (2000), and Han and Wang (2006).

Multiplicity of positive radial solutions for the weighted [formula omitted]-Laplacian in [formula omitted

This paper is concerned with the multiplicity of positive radial solutions for the weighted p-Laplacian in Rn∖{0}. By constructing suitable Banach spaces, special cones and combining with the Avery–Peterson fixed point theorem, we obtain the existence of at least three positive radial solutions.