Uniqueness of positive solutions for a class of elliptic systems

In this article, we consider uniqueness of positive radial solutions to the elliptic system Δu+a(|x|)f(u,v)=0, Δv+b(|x|)g(u,v)=0, subject to the Dirichlet boundary condition on the open unit ball in RN (N?2). Our uniqueness results applies to, for instance, f(u,v)=uqvp, g(u,v)=upvq, p,q>0, p+q<1 or more general cases.     

Multiple and sign changing solutions of an elliptic eigenvalue problem with constraint

We prove the existence of sign changing solutions of a semilinear elliptic eigenvalue problem with constraint by using variational methods. Among those three solutions we obtained, one is positive, one negative and one sign changing. We also prove the existence of multiple sign changing solutions under some additional condition.     

Strongly indefinite systems with critical Sobolev exponents

We consider an elliptic system of Hamiltonian type on a bounded domain. In the superlinear case with critical growth rates we obtain existence and positivity results for solutions under suitable conditions on the linear terms. Our proof is based on an adaptation of the dual variational method as applied before to the scalar case.

     

Existence of three nontrivial solutions for an elliptic system

In this paper we consider the existence of nontrivial solutions for an elliptic system, where the nonlinear term is superlinear in one equation and sublinear in the other equation. By constructing two cones and computing the fixed point index in K₁, K₂ and K₁×K₂, we obtain that the elliptic system has three nontrivial solutions (u,0), (0,v) and (u^∗,v^∗). It is remarkable that the third nontrivial solution (u^∗,v^∗) is established on the Cartesian product of two cones, in which the feature of two equations can be exploited better.     

On the number of positive solutions of elliptic systems

The paper deals with the existence, multiplicity and nonexistence of positive radial solutions for the elliptic system div(|?|^{p –2}?) + λk_i (|x |) fⁱ (u₁, …,u_n) = 0, p > 1, R₁ < |x | < R₂, u_i (x) = 0, on |x | = R₁ and R₂, i = 1, …, n, x ∈ ?^N , where k_i and fⁱ, i = 1, …, n, are continuous and nonnegative functions. Let u = (u₁, …, u_n), φ (t) = |t |^{p –2}t, fⁱ₀ = lim_{‖ u ‖→0}((fⁱ ( u ))/(φ (‖ u ‖))), fⁱ_∞= lim_{‖ u ‖→∞}((fⁱ ( u ))/(φ (‖ u ‖))), i = 1, …, n, f = (f¹, …, fⁿ), f ₀ = ∑ⁿ _{i =1} fⁱ ₀ and f _∞ = ∑ⁿ _{i =1} fⁱ _∞. We prove that either f ₀ = 0 and f _∞ = ∞ (superlinear), or f ₀ = ∞and f _∞ = 0 (sublinear), guarantee existence for all λ > 0. In addition, if fⁱ ( u ) > 0 for ‖ u ‖ > 0, i = 1, …, n, then either f ₀ = f _∞ = 0, or f ₀ = f _∞ = ∞, guarantee multiplicity for sufficiently large, or small λ, respectively. On the other hand, either f₀ and f_∞ > 0, or f₀ and f_∞ < ∞ imply nonexistence for sufficiently large, or small λ, respectively. Furthermore, all the results are valid for Dirichlet/Neumann boundary conditions. We shall use fixed point theorems in a cone. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the existence and shape of least energy solutions for some elliptic systems

We establish an existence result for strongly indefinite semilinear elliptic systems with Neumann boundary condition, and we study the limiting behavior of the positive solutions of the singularly perturbed problem.     

Sign-changing solutions for some fourth-order nonlinear elliptic problems

In this paper, we consider the existence and multiplicity of sign-changing solutions for some fourth-order nonlinear elliptic problems and some existence and multiple are obtained. The weak solutions are sought by means of sign-changing critical theorems.     

非线性Schodinger方程多重正解的一个注记

Some results about the multiplicity of positive solutions for a nonlinear SchrOdinger equation are given by means of variational methods.     

Nonnegative solutions for nonlinear elliptic systems

In this paper, we show that the semilinear elliptic systems of the form

(0.1)     

Entire large solutions to semilinear elliptic systems

We consider the problem of existence of positive solutions to the elliptic system Δu=p(|x|)vα, Δv=q(|x|)uβ on Rn (n?3) which satisfies . The parameters α and β are positive, and the nonnegative functions p and q are continuous and min{p(r),q(r)} does not have compact support. We show that if αβ?1, then such a solution exists if and only if the functions p and q satisfy

     

Infinitely many solutions to elliptic systems involving critical exponents and Hardy potential

In this paper, we consider the following elliptic systems involving critical Sobolev growth and Hardy potential: where N ≥ 3,λ₁,λ₂ ∈ [0,Λ_N), is the best Hardy constant. is the critical Sobolev exponent. a₁,a₂, b are positive parameters, α,β > 0 and 1 < α + β : = q < 2 < 2*. with . By means of the concentration‐compactness principle and Kajikiya's new version of symmetric mountain pass lemma, we obtain infinitely many solutions that tend to zero for suitable positive parameters a₁,a₂,b and λ₁,λ₂. Copyright © 2012 John Wiley & Sons, Ltd.     

Positive solutions for a class of multiparameter ordinary elliptic systems

Using a fixed point theorem due to M.A. Krasnosel'skii, the upper-lower solutions method and fixed point index theory, we study existence, non-existence and multiplicity of positive solutions for a class of systems of second-order ordinary differential equations. We give a new notion of superlinearity for nonlinearities. Observe that the nonlinearities may even be, in some sense, singular. As an application, we show multiplicity results for a class of semilinear elliptic systems in both bounded annular domains and exterior domains. In addition, we apply our results to fourth-order boundary value problems.     

Nontrivial solutions for impulsive fractional differential systems through variational methods

Fractional differential equations (FDEs) as a generalization of ordinary differential equations and integration to arbitrary noninteger orders have gained importance due to their numerous applications in many fields of science and engineering. Indeed, there are a large number of phenomena, including fluid flow, diffusive transport akin to diffusion, rheology, probability, and electrical networks, that are modeled by different equations involving fractional order derivatives. This paper deals with multiplicity results of solutions for a class of impulsive fractional differential systems. The approach is based on variational methods and critical point theory. Indeed, we establish existence results for our system under some algebraic conditions on the nonlinear part with the classical Ambrosetti–Rabinowitz (AR) condition on the nonlinear and the impulsive terms. Moreover by combining two algebraic conditions on the nonlinear term, which guarantee the existence of two weak solutions, applying the mountain pass theorem, we establish the existence of third weak solution for our system.     

Critical points theorems concerning strongly indefinite functionals and infinite many periodic solutions for a class of Hamiltonian systems

Based on new deformation theorems concerning strongly indefinite functionals, we give some new min-max theorems which are useful in looking for critical points of functionals which are strongly indefinite and satisfy Cerami condition instead of Palais-Smale condition. As one application of abstract results, we study existence of multiple periodic solutions for a class of non-autonomous first order Hamiltonian system

     

Infinitely many solutions for some nonlinear scalar system of two elliptic equations

In this paper, we consider the elliptic system of two equations in H¹(RN)×H¹(RN):

     

Existence of positive bounded solutions for some nonlinear elliptic systems

We prove some existence results of positive bounded continuous solutions to the semilinear elliptic system Δu=λp(x)g(v), Δv=μq(x)f(u) in domains D with compact boundary subject to some Dirichlet conditions, where λ and μ are nonnegative parameters. The functions f,g are nonnegative continuous monotone on (0,∞) and the potentials p, q are nonnegative and satisfy some hypotheses related to the Kato class K(D).     

Infinitely many solutions to elliptic systems with critical exponents and Hardy potentials

In this paper, we consider the following elliptic systems with critical Sobolev growth and Hardy potentials: where N ≥ 3, η > 0, λ₁,λ₂ ∈ [0,Λ_N), and is the best Hardy constant. is the critical Sobolev exponent. a₁, a₂, b₁, and b₂ are positive parameters, and α,β > 1 satisfy 2 < α + β < 2*. h(x) ? 0, h(x) ≥ 0, , , and with . By means of the concentration–compactness principle and R. Kajikiya's new version of symmetric mountain pass lemma, we obtain infinitely many solutions that tend to zero. Copyright © 2013 John Wiley & Sons, Ltd.