The Neumann problem for the Hénon equation,trace inequalities and Steklov eigenvalues

We consider the Neumann problem for the Hénon equation. We obtain existence results and we analyze the symmetry properties of the ground state solutions. We prove that some symmetry and variational properties can be expressed in terms of eigenvalues of a Steklov problem. Applications are also given to extremals of certain trace inequalities.     

Multiple positive and sign-changing solutions for a singular Schrödinger equation with critical growth

Variational methods are used to prove the existence of multiple positive and sign-changing solutions for a Schrödinger equation with singular potential having prescribed finitely many singular points. Some exact local behavior for positive solutions obtained here are also given. The interesting aspects are two. One is that one singular point of the potential V(x)$V(x)$ and one positive solution can produce one sign-changing solution of the problem. The other is that each sign-changing solution changes its sign exactly once.     

Multiple positive and 2-nodal symmetric solutions of elliptic problems with critical nonlinearity

We consider the problem −Δu+a(x)u=f(x)|u|^2*−2u in Ω, u=0 on ∂Ω, where Ω is a bounded smooth domain in RN, N?4, is the critical Sobolev exponent, and a,f are continuous functions. We assume that Ω, a and f are invariant under the action of a group of orthogonal transformations. We obtain multiplicity results which contain information about the symmetry and symmetry-breaking properties of the solutions, and about their nodal domains. Our results include new multiplicity results for the Brezis-Nirenberg problem −Δu+λu=|u|^2*−2u in Ω, u=0 on ∂Ω.     

Multiple solutions for a semilinear equation involving singular potential and critical exponent

Variational methods are used to prove the existence of positive and sign-changing solutions for a semilinear equation involving singular potential and critical exponent in any bounded domain.*supported in part by Tian Yuan Foundation of NNSF (A0324612)**Supported by 973 Chinese NSF and Foundation of Chinese Academy of Sciences.***Supported in part by NNSF of China.Received: September 23, 2002; revised: November 30, 2003     

Positive solutions to semilinear polyharmonic Dirichlet problems involving critical Sobolev exponents

We are concerned with the semilinear polyharmonic model problem (–)K v = v +v|v| ^s–1 inB,D v|B = 0 for ¦|<-K – 1. HereK ,B is the unit ball in ⁿ,n >2K, is the critical Sobolev exponent. Let ₁ denote the first Dirichlet eigenvalue of (-)^K inB. The existence of a positive radial solutionv is shown for

Existence of type II blowup solutions for a semilinear heat equation with critical nonlinearity

We consider the blowup rate of solutions for a semilinear heat equation

     

Nontrivial solutions for resonant cooperative elliptic systems via computations of the critical groups

In this paper, we consider a class of resonant cooperative elliptic systems. Based on some new results concerning the computations of the critical groups and the Morse theory, we establish some new results about the existence and multiplicity of solutions under new classes of conditions. It turns out that our main results sharply improve some known results in the literature.     

Existence and uniqueness of positive solutions for singular nonlinear elliptic boundary value problems

In this paper we establish the existence and the uniqueness of positive solutions for Dirichlet boundary value problems of nonlinear elliptic equations with singularity. We obtain the existence and the uniqueness by using the mixed monotone method in the cone theory. Moreover, we give an iterative method of constructing the solution. The rate of convergence of the iterative sequence is analyzed.     

Multiple solutions for a Hénon-like equation on the annulus

For the equation −Δu=||xα|−2|up−1, 1<|x|<3, we prove the existence of two solutions for α large, and of two additional solutions when p is close to the critical Sobolev exponent ^2∗=2N/(N−2). A symmetry-breaking phenomenon appears, showing that the least-energy solutions cannot be radial functions.     

Nontrivial solutions for critical potential elliptic systems

We consider potential elliptic systems involving p-Laplace operators, critical nonlinearities and lower-order perturbations. Suitable necessary and sufficient conditions for existence of nontrivial solutions are presented. In particular, a number of results on Brezis-Nirenberg type problems are extended in a unified framework.     

Existence and nonexistence results for critical growth polyharmonic elliptic systems

In this work, we consider semilinear elliptic systems for the polyharmonic operator having a critical growth nonlinearity. We establish conditions for existence and nonexistence of nontrivial solutions to these systems.     

On a biharmonic equation involving nearly critical exponent

This paper is concerned with a biharmonic equation under the Navier boundary condition , u > 0 in Ω and u = Δu = 0 on ∂Ω, where Ω is a smooth bounded domain in , n ≥ 5, and ε > 0. We study the asymptotic behavior of solutions of (P _−ε) which are minimizing for the Sobolev quotient as ε goes to zero. We show that such solutions concentrate around a point x ₀ ∈Ω as ε → 0, moreover x ₀ is a critical point of the Robin’s function. Conversely, we show that for any nondegenerate critical point x ₀ of the Robin’s function, there exist solutions of (P _−ε) concentrating around x ₀ as ε → 0. Finally we prove that, in contrast with what happened in the subcritical equation (P _−ε), the supercritical problem (P _+ε) has no solutions which concentrate around a point of Ω as ε → 0. Work finished when the authors were visiting Mathematics Department of the University of Roma “La Sapienza”. They would like to thank the Mathematics Department for its warm hospitality. The authors also thank Professors Massimo Grossi and Filomena Pacella for their constant support.