Lie point symmetries and some group invariant solutions of the quasilinear equation involving the infinity Laplacian

Using the classical Lie method we obtain the full Lie point symmetry group of the Aronsson equation in two independent variables. Some group invariant solutions of this equation are found and a conjecture on the Lie point symmetry group of the Aronsson equation in Rⁿ is presented.     

On maps with given Jacobians involving the heat equation

In this paper we present and analyze two new algorithms to construct a smooth diffeomorphism of a domain with prescribed jacobian function. The first one is free from any restriction on the boundary, while the second one produces a diffeomorphism that coincides with the identity map on the boundary of the domain. Both are based on the solution of an initial value problem for the linear heat equation, and the second also uses solutions of the Stokes system of Fluid Mechanics.     

Asymptotically linear Schrödinger equation with potential vanishing at infinity

We are concerned with the existence of bound states and ground states of the following nonlinear Schrödinger equation

(0.1)     

Positive radial solutions to a ‘semilinear’ equation involving the Pucci's operator

In this article we prove existence of positive radially symmetric solutions for the nonlinear elliptic equation

     

A boundary blow-up elliptic problem with an inhomogeneous term

By a perturbation method and constructing comparison functions, we reveal how the inhomogeneous term h$h$ affects the exact asymptotic behaviour of solutions near the boundary to the problem △u=b(x)g(u)+λh(x)$△u=b\left(x\right)g\left(u\right)+\lambda h\left(x\right)$, u>0$u>0$ in Ω$\Omega$, u_|∂Ω=∞$u{|}_{\partial \Omega }=\infty$, where Ω$\Omega$ is a bounded domain with smooth boundary in R^N${\mathbb{R}}^N$, λ>0$\lambda >0$, g∈C¹[0,∞)$g\in C^1[0,\infty )$ is increasing on [0,∞)$[0,\infty )$, g(0)=0$g\left(0\right)=0$, g^′$g^{\prime }$ is regularly varying at infinity with positive index ρ$\rho$, the weight b$b$, which is non-trivial and non-negative in Ω$\Omega$, may be vanishing on the boundary, and the inhomogeneous term h$h$ is non-negative in Ω$\Omega$ and may be singular on the boundary.     

Nontrivial solutions to semilinear elliptic problems involving two critical Hardy-Sobolev exponents

In this paper, we investigate a semilinear elliptic equation, which involves doubly critical Hardy-Sobolev exponents and a Hardy-type term. By means of the Linking Theorem and delicate energy estimates, the existence of nontrivial solutions to the problem is established.     

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     

On uniqueness for nonlinear elliptic equation involving the Pucci's extremal operator

In this article we study uniqueness of positive solutions for the nonlinear uniformly elliptic equation in RN, limr→∞u(r)=0, where denotes the Pucci's extremal operator with parameters 0<λ?Λ and p>1. It is known that all positive solutions of this equation are radially symmetric with respect to a point in RN, so the problem reduces to the study of a radial version of this equation. However, this is still a nontrivial question even in the case of the Laplacian (λ=Λ). The Pucci's operator is a prototype of a nonlinear operator in no-divergence form. This feature makes the uniqueness question specially challenging, since two standard tools like Pohozaev identity and global integration by parts are no longer available. The corresponding equation involving is also considered.     

Bound states of nonlinear Schrödinger equations with potentials tending to zero at infinity

In this paper, we are concerned with the existence of solutions to the N-dimensional nonlinear Schrödinger equation −ε²Δu+V(x)u=K(x)up with u(x)>0, u∈H¹(RN), N?3 and . When the potential V(x) decays at infinity faster than ⁻²(1+|x|) and K(x)?0 is permitted to be unbounded, we will show that the positive H¹(RN)-solutions exist if it is assumed that G(x) has local minimum points for small ε>0, here with denotes the ground energy function which is introduced in [X. Wang, B. Zeng, On concentration of positive bound states of nonlinear Schrödinger equations with competing potential functions, SIAM J. Math. Anal. 28 (1997) 633-655]. In addition, when the potential V(x) decays to zero at most like (1+|x|)^−α with 0<α?2, we also discuss the existence of positive H¹(RN)-solutions for unbounded K(x). Compared with some previous papers [A. Ambrosetti, A. Malchiodi, D. Ruiz, Bound states of nonlinear Schrödinger equations with potentials vanishing at infinity, J. Anal. Math. 98 (2006) 317-348; A. Ambrosetti, D. Ruiz, Radial solutions concentrating on spheres of NLS with vanishing potentials, Proc. Roy. Soc. Edinburgh Sect. A 136 (2006) 889-907; A. Ambrosetti, Z.Q. Wang, Nonlinear Schrödinger equations with vanishing and decaying potentials, Differential Integral Equations 18 (2005) 1321-1332] and so on, we remove the restrictions on the potential function V(x) which decays at infinity like (1+|x|)^−α with 0<α?2 as well as the restrictions on the boundedness of K(x)>0. Therefore, we partly answer a question posed in the reference [A. Ambrosetti, A. Malchiodi, Concentration phenomena for NLS: Recent results and new perspectives, preprint, 2006].     

Bernoulli jets and the zero mean curvature equation

We consider an elliptic PDE problem related with fluid mechanics. We show that level sets of rescaled solutions satisfy the zero mean curvature equation in a suitable weak viscosity sense. In particular, such level sets cannot be touched from below (above) by a convex (concave) paraboloid in a suitably small neighborhood.     

The heterogeneous Allen-Cahn equation in a ball: Solutions with layers and spikes

Let u? be a single layered radially symmetric unstable solution of the Allen-Cahn equation −?²Δu=u(u−a(|x|))(1−u) over the unit ball with Neumann boundary conditions. Based on our estimate of the small eigenvalues of the linearized eigenvalue problem at u? when ? is small, we construct solutions of the form u?+v?, with v? non-radially symmetric and close to zero in the unit ball except near one point x₀ such that |x₀| is close to a nondegenerate critical point of a(r). Such a solution has a sharp layer as well as a spike.     

Multiple solutions of an inhomogeneous Neumann problem for an elliptic system with critical Sobolev exponent

In this paper we prove the existence of two solutions for the inhomogeneous Neumann problem with critical Sobolev exponent.     

Subcritical perturbations of a singular quasilinear elliptic equation involving the critical Hardy–Sobolev exponent

In this work we improve some known results for a singular operator and also for a wide class of lower-order terms by proving a multiplicity result. The proof is made by applying the generalized mountain-pass theorem due to Ambrosetti and Rabinowitz. To do this, we show that the minimax levels are in a convenient range by combining a special class of approximating functions, due to Gazzola and Ruf, with the concentrating functions of the best Sobolev constant.     

Elliptic systems involving the critical exponents and potentials

In this paper, we investigate a singular elliptic system, which involves the critical Sobolev exponent and multiple Hardy-type terms. By employing variational methods, the existence of its positive solutions is established. By the Moser iteration method, some asymptotic properties of its nontrivial solutions at the singular points are verified.     

Large solution to nonlinear elliptic equation with nonlinear gradient terms

The aim of this paper is to study the qualitative behavior of large solutions to the following problem