Symmetric Positive Solutions Research Articles

Multiplicity of radially symmetric solutions for a p-harmonic equation in R N

This paper is concerned with the multiplicity of radially symmetric positive solutions of the Dirichlet boundary value problem for the following N-dimensional p-harmonic equation of the form Δ ( | Δ u | p − 2 Δ u ) =λg(x)f(u),x∈ B 1 , where B 1 is a unit ball in R N (N≥3). We apply the fixed point index theory and the upper and lower solutions method to investigate the multiplicity of radially symmetric positive solutions. We find that there exists a threshold λ ∗ <+∞ such that if λ> λ ∗ , the problem has no radially symmetric positive solution; while if 0<λ≤ λ ∗ , the problem admits at least one radially symmetric positive solution. Especially, there exist at least two radially symmetric positive solutions for 0<λ< λ ∗ .

The Symmetric Solutions for Three-Point Singular Boundary Value Problems of Differential Equation

In this paper, by constructing a special operator and using fixed point index theorem of cone, we get the sufficient conditions for symmetric positive solution of a class of nonlinear singular boundary value problems with p-Laplace operator, which improved and generalized the result of related paper. Keywords—Banach space, cone, fixed point index, singular differential equation, p-Laplace operator, symmetric solutions.

The Symmetric Solutions for Boundary Value Problems of Second-Order Singular Differential Equation

In this paper, by constructing a special operator and using fixed point index theorem of cone, we get the sufficient conditions for symmetric positive solution of a class of nonlinear singular boundary value problems with p-Laplace operator, which improved and generalized the result of related paper. Keywords—Banach space, cone, fixed point index, singular differential equation, p-Laplace operator, symmetric solutions.

Bifurcation method for computing the multiple positive solutions to p-Henon equation

In this paper, we proposed the algorithms to solving the D4 symmetric positive solutions to the boundary value problem of p-Henon equation. Taking r in p-Henon equation as a bifurcation parameter, the symmetry-breaking bifurcation point on the branch of the D4 symmetric positive solutions is found via the extended systems. Based on the Liapunov–Schmidt reduction, other symmetric positive solutions are computed by the branch switching method.

Symmetric positive solutions to singular system with multi-point coupled boundary conditions

In this paper, we study the existence and multiplicity of symmetric positive solutions for a nonlinear system with multi-point coupled boundary conditions. The arguments are based upon a specially constructed cone and the fixed point index theorem in cones. An example is then given to demonstrate the applicability of our results.

Symmetric positive solutions to a second-order boundary value problem with integral boundary conditions

This paper investigates the existence of concave symmetric positive solutions and establishes corresponding iterative schemes for a second-order boundary value problem with integral boundary conditions. The main tool is a monotone iterative technique. Meanwhile, an example is worked out to demonstrate the main results.

Erratum to: The retreat of the less fit allele in a population-controlled model for population genetics

The proof of Lemma 2.4 is correct when the habitat is one-dimensional. However, the fact that the function ρ1 defined by formula (6.14) approaches zero on the part of the cone |x| = ct where x1 = 0 invalidates the argument that ρ1 ≤ ρ1 for |x| ≤ ct when the habitat is multidimensional. This gap is easily repaired by replacing the function coshμx1 in (6.14) by a radially symmetric positive solution (|x|) of the equation ∇2 = μ2 . When the habitat has dimension 2, one can let := I0(μ|x|) where I0 is the usual Bessel function with imaginary argument. In three dimensions, one can let := [sinhμ|x|]/[μ|x|].

Existence of symmetric positive solutions for a multipoint boundary value problem with sign-changing nonlinearity on time scales

In this paper, we make use of the four functionals fixed point theorem to verify the existence of at least one symmetric positive solution of a second-order m-point boundary value problem on time scales such that the considered equation admits a nonlinear term f whose sign is allowed to change. The discussed problem involves both an increasing homeomorphism and homomorphism, which generalizes the p-Laplacian operator. An example which supports our theoretical results is also indicated.MSC:34B10, 39A10.

Existence and uniqueness of symmetric positive solutions of2n-order nonlinear singular boundary value problems

By applying an iterative technique, a necessary and sufficient condition is obtained for the existence of symmetric positive solutions of 2 n -order nonlinear singular boundary value problems. At the same time, we also show the uniqueness of the symmetric positive solution.

Positive symmetric solutions of a second-order difference equation

Using an extension of the Leggett‐Williams fixed-point theorem due to Avery, Henderson, and Anderson, we prove the existence of solutions for a class of second-order difference equations with Dirichlet boundary conditions, and discuss a specific example.

Three positive solutions for a generalized Laplacian boundary value problem with a parameter

Concerned is a generalized Laplacian boundary value problem with a positive parameter. First we apply the Leggett–Williams fixed point theorem to establish sufficient conditions on the existence of at least three positive solutions for the parameter belonging to an explicit interval. Then, under a little bit stronger assumptions, we show that there are at least three positive symmetric solutions for the parameter in an open interval. The obtained results are new even for Laplacian boundary value problems and they are illustrated with an example.

Symmetric positive solutions of boundary value problems with integral boundary conditions

Under conditions weaker than those used by Ma, we establish various results on the existence and nonexistence of symmetric positive solutions to fourth-order boundary value problems with integral boundary conditions. The arguments are based upon a specially constructed cone and the fixed point theory in a cone. Our results improve and extend that obtained in [H. Ma, Symmetric positive solutions for nonlocal boundary value problems of fourth order, Nonlinear Anal. 68 (2008) 645–651]. We illustrate our results by one example, which cannot be handled using the existing results.

On symmetric positive homoclinic solutions of semilinear p-Laplacian differential equations

In this paper we study the existence of even positive homoclinic solutions for p-Laplacian ordinary differential equations (ODEs) of the type ( u ′ | u ′ | p − 2 ) ′ −a(x)u | u | p − 2 +λb(x)u | u | q − 2 =0, where 2≤p<q, λ>0 and the functions a and b are strictly positive and even. First, we prove a result on symmetry of positive solutions of p-Laplacian ODEs. Then, using the mountain-pass theorem, we prove the existence of symmetric positive homoclinic solutions of the considered equations. Some examples and additional comments are given.MSC:34B18, 34B40, 49J40.

Existence and multiplicity of solutions for critical elliptic equations with multi-polar potentials in symmetric domains

In this paper, we consider the elliptic equations with critical Sobolev exponents and multi-polar potentials in bounded symmetric domains and prove the existence and multiplicity of symmetric positive solutions by using the Ekeland variational principle and the Lusternik–Schnirelmann category theory.

Positive solutions for a system of nonlinear Hammerstein integral equations and applications

We are concerned with the existence and multiplicity of positive solutions for the system of nonlinear Hammerstein integral equationsu(t)=∫abk1(t,s)g1(s)f1(s,u(s),v(s))ds,v(t)=∫abk2(t,s)g2(s)f2(s,u(s),v(s))ds,where ki∈C[a,b]×[a,b],R+,fi∈C[a,b]×R+2,R+, and gi∈C[a,b] is almost everywhere positive on [a,b](i=1,2). We overcome the difficulty arising from the difference between the two kernels k1(t,s)g1(s) and k2(t,s)g2(s) by defining certain integral constants, and use fixed point index theory to establish our main results, based on a priori estimates achieved by utilizing Jensen’s inequality for concave functions and nonnegative matrices. Our nonlinearities f and g cover the following three cases: the first with both superlinear, the second with both sublinear, and the last with one sublinear and the other superlinear. Our main results are applied to establish the existence and multiplicity of positive symmetric solutions for an elliptic system in an annulus.

Positive solutions of a fourth-order boundary value problem involving derivatives of all orders

This paper is mainly concerned with the existence, multiplicity and uniqueness of positive solutions for the fourth-order boundary value problem \begin{eqnarray*} u^{(4)}=f(t,u,u^\prime,-u^{\prime\prime},-u^{\prime\prime\prime}),\\ u(0)=u^\prime(1)=u^{\prime\prime}(0)=u^{\prime\prime\prime}(1)=0, \end{eqnarray*} where $f\in C([0,1]\times\mathbb R_+^4,\mathbb R_+)(\mathbb R_+:=[0,\infty))$. Based on a priori estimates achieved by utilizing some integral identities and inequalities, we use fixed point index theory to prove the existence, multiplicity and uniqueness of positive solutions for the above problem. Finally, as a byproduct, our main results are applied to establish the existence, multiplicity and uniqueness of symmetric positive solutions for the fourth order Lidstone problem.

Output regulation problem of multi-agents in networked systems

In this study, the output regulation problem of multi-agent systems in fixed topologies is analysed. The agents studied are identical general linear systems that aim at tracking a desired trajectory in presence of certain determinate disturbances. Under a mild assumption that the augmented communication topology has a spanning tree, a distributed control law based on relative output measurements between the agents and state measurements within the agent itself is proposed. It is shown that this output regulation problem can be solved if a Riccati equation with parameters has a positive definite symmetric solution. Moreover, by parameter determination of this Riccati equation, we find that the lower bound of the second smallest eigenvalue associated with the augmented graph Laplacian of the communication topology is of great importance on both the stability property and the convergence rate. A lower bound for this eigenvalue is given for N-node graphs.

Existence and bounds for nonlinear Schrödinger systems with strong competition

We consider systems of coupled Schrodinger equations which appear in nonlinear optics and binary Bose-Einstein condensation. Namely, we prove that for most μ, ν ∈ R, a, b> 0, the system −Δu = μu + au 3 − βuv 2 , −Δv = νv + bv 3 − βvu 2 , u, v ∈ H 1 (B1(0)), where B1(0) is the unit ball of R 3 , admits a family of radially symmetric positive solutions (uβ, vβ) provided the interaction parameter β> 0 is sufficiently large. By using a Morse index technique we deduce that these solutions are bounded uniformly in β, hence their limit functions as β →∞ undergo the phenomenon of phase segregation.

Symmetric positive solutions of fourth order boundary value problems for an increasing homeomorphism and homomorphism on time-scales

Let T ⊂ R be a symmetric bounded time-scale, with a = min T , b = max T . We consider the following fourth order boundary value problem ϕ ( − p x Δ ∇ ) Δ ∇ ( t ) + f ( t , x ( t ) ) = 0 , t ∈ T κ 2 κ 2 , x ( a ) = x ( b ) = 0 , x Δ ∇ ( σ ( a ) ) = x Δ ∇ ( ρ ( b ) ) = 0 for a suitable function p and an increasing homeomorphism and homomorphism ϕ . By using the Krasnosel’skii fixed point theorem, we present sufficient conditions for the existence of at least one or two symmetric positive solutions of the above problem on time-scales. As applications, two examples are given to illustrate the main results.

Positive solutions of a focal problem for one-dimensional [formula omitted]-Laplacian equations

This paper mainly deals with the existence and multiplicity of positive solutions for the focal problem involving both the p-Laplacian and the first order derivative: {((u′)p−1)′+f(t,u,u′)=0,t∈(0,1),u(0)=u′(1)=0. The main tool in the proofs is the fixed point index theory, based on a priori estimates achieved by using Jensen’s inequality and a new inequality. Finally the main results are applied to establish the existence of positive symmetric solutions to the Dirichlet problem: {(|u′|p−2u′)′+f(u,u′)=0,t∈(−1,0)∪(0,1),u(−1)=u(1)=0.