Multiplicity of solutions for quasilinear elliptic systems with critical growth

We study the existence of multiple solutions for a quasilinear elliptic system of gradient type with critical growth and the possibility of coupling on the subcritical term. The solutions are obtained from a version of the Symmetric Mountain Pass Theorem. The Concentration-Compactness Principle allows to verify that the Palais-Smale condition is satisfied below a certain level. The authors were partially supported by CNPq/Brazil     

Infinitely many solutions of dirichlet problem for <Emphasis Type="Italic">p</Emphasis>-mean curvature operator

The existence of infinitely many solutions of the following Dirichlet problem for p-mean curvature operator: is considered, where Θ is a bounded domain in R ⁿ (n>p>1) with smooth boundary ∂Θ. Under some natural conditions together with some conditions weaker than (AR) condition, we prove that the above problem has infinitely many solutions by a symmetric version of the Mountain Pass Theorem if . Supported by the National Natural Science Foundation of China (10171032) and the Guangdong Provincial Natural Science Foundation (011606).     

Existence of translating solutions to the flow by powers of mean curvature on unbounded domains

In this paper, we prove the existence of classical solutions to the Dirichlet problem of a class of quasi-linear elliptic equations on an unbounded cone and a U-type domain in Rn(n?2). This problem comes from the study of mean curvature flow or its generalization, the flow by powers of mean curvature. Our approach is a modified version of the classical Perron method, where the solutions to the minimal surface equation are used as sub-solutions and a family auxiliary functions are constructed as super-solutions.     

COMPLEX MONGE-AMPEREEQUATIONS ON GENERAL DOMAINS

Abstract. The smooth solutions of the Dirichlet problems for the complex Monge-Ampere equa-tions on general smooth domains are found ,provided that there exists a C3 strictly plurisub har-monic subsolution with prescribed boundary value. It is the smooth version of an existence theo-rem given by Bedford and Taylor.     

The Fujita exponent for the Cauchy problem in the hyperbolic space

It is well known that the heat kernel in the hyperbolic space has a different behavior for large times than the one in the Euclidean space. The main purpose of this paper is to study its effect on the positive solutions of Cauchy problems with power nonlinearities. Existence and non-existence results for local solutions are derived. Emphasis is put on their long time behavior and on Fujita?s phenomenon. To have the same situation as for the Cauchy problem in RN, namely finite time blow up for all solutions if the exponent is smaller than a critical value and existence of global solutions only for powers above the critical exponent, we must introduce a weight depending exponentially on the time. In this respect the situation is similar to problems in bounded domains with Dirichlet boundary conditions. Important tools are estimates for the heat kernel in the hyperbolic space and comparison principles.     

Parabolic equations of Von Karman type on Kähler manifolds

We study a parabolic version of a system of Von Karman type on a compact Kähler manifold of arbitrary dimension. We provide local in time regular solutions, which can be extended to global bounded ones if the data of the problem are small.     

Conditions for two-peaked solutions of singularly perturbed elliptic equations

We obtain necessary conditions for the existence of two-peaked solutions of singularly perturbed elliptic equations. These conditions are related to the geometry of the domain. In particular, we prove there are no two-peaked solutions in a strictly convex domain. Received: 20 January 1997 / Revised version: 2 December 1997     

Entire solutions for a semilinear fourth order elliptic problem with exponential nonlinearity

We investigate entire radial solutions of the semilinear biharmonic equation Δ²u=λexp(u) in Rn, n?5, λ>0 being a parameter. We show that singular radial solutions of the corresponding Dirichlet problem in the unit ball cannot be extended as solutions of the equation to the whole of Rn. In particular, they cannot be expanded as power series in the natural variable s=log|x|. Next, we prove the existence of infinitely many entire regular radial solutions. They all diverge to −∞ as |x|→∞ and we specify their asymptotic behaviour. As in the case with power-type nonlinearities [F. Gazzola, H.-Ch. Grunau, Radial entire solutions for supercritical biharmonic equations, Math. Ann. 334 (2006) 905-936], the entire singular solution x?−4log|x| plays the role of a separatrix in the bifurcation picture. Finally, a technique for the computer assisted study of a broad class of equations is developed. It is applied to obtain a computer assisted proof of the underlying dynamical behaviour for the bifurcation diagram of a corresponding autonomous system of ODEs, in the case n=5.     

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.     

Construction of pseudo-gradient vector field and sign-changing multiple solutions involving p-Laplacian

Existence of multiple and sign-changing solutions for a problem involving p-Laplacian and jumping nonlinearities are considered via the construction of descent flow in . Sign-changing and multiple solutions are obtained under additional assumption on the nonlinearity. The uniqueness of positive (negative) solution theorem is included too.     

Interior estimates for some semilinear elliptic problem with critical nonlinearity

We study compactness properties for solutions of a semilinear elliptic equation with critical nonlinearity. For high dimensions, we are able to show that any solutions sequence with uniformly bounded energy is uniformly bounded in the interior of the domain. In particular, singularly perturbed Neumann equations admit pointwise concentration phenomena only at the boundary.     

Blow-up and symmetry of sign-changing solutions to some critical elliptic equations

In this paper we continue the analysis of the blow-up of low energy sign-changing solutions of semi-linear elliptic equations with critical Sobolev exponent, started in [M. Ben Ayed, K. El Mehdi, F. Pacella, Blow-up and nonexistence of sign-changing solutions to the Brezis-Nirenberg problem in dimension three, Ann. Inst. H. Poincaré Anal. Non Linéaire, in press]. In addition we prove axial symmetry results for the same kind of solutions in a ball.     

Symmetry breaking in a bounded symmetry domain

In this article, we prove that there are three positive solutions of a semilinear elliptic equation in a bounded symmetric domain _t for large t>0 in which one is axially symmetric and the other two are nonaxially symmetric. Main tools are the Palais-Smale theory.     

Estimates and existence results for a fully nonlinear Yamabe problem on manifolds with boundary

In this paper we consider a fully nonlinear version of the Yamabe problem on compact Riemannian manifold with boundary. Under various conditions we derive local estimates for solutions and establish some existence results. Partially supported by NSF grant DMS-0401118.     

A priori estimate for non-uniform elliptic equations

A priori estimate for non-uniform elliptic equations with periodic boundary conditions is concerned. The domain considered consists of two sub-regions, a connected high permeability region and a disconnected matrix block region with low permeability. Let ? denote the size ratio of one matrix block to the whole domain. It is shown that in the connected high permeability sub-region, the Hölder and the Lipschitz estimates of the non-uniform elliptic solutions are bounded uniformly in ?. But Hölder gradient estimate and Lp estimate of the second order derivatives of the solutions in general are not bounded uniformly in ?.     

Solution set splitting at low energy levels in Schrödinger equations with periodic and symmetric potential

The time-independent superlinear Schrödinger equation with spatially periodic and positive potential admits sign-changing two-bump solutions if the set of positive solutions at the minimal nontrivial energy level is the disjoint union of period translates of a compact set. Assuming a reflection symmetric potential we give a condition on the equation that ensures this splitting property for the solution set. Moreover, we provide a recipe to explicitly verify the condition, and we carry out the calculation in dimension one for a specific class of potentials.     

<Emphasis Type="Bold">C</Emphasis><Superscript><Emphasis Type="Bold">1,</Emphasis><Emphasis Type="Italic">α</Emphasis></Superscript> regularity for infinity harmonic functions in two dimensions

We propose a new method for showing C ^{1, α} regularity for solutions of the infinity Laplacian equation and provide full details of the proof in two dimensions. The proof for dimensions n ≥ 3 depends upon some conjectured local gradient estimates for solutions of certain transformed PDE. LCE is supported in part by NSF Grant DMS-0500452. OS was supported in part by the Miller Institute for Basic Research in Science, Berkeley.     

Oblique boundary value problems for equations of Monge-Ampère type

We prove the existence of classical solutions of elliptic equations of Monge-Ampère type subject to a semilinear oblique boundary condition which is a perturbation of the Neumann boundary condition. Our techniques also allow us to treat fully nonlinear strictly oblique boundary conditions satisfying a concavity condition. Examples show that the above restrictions on the boundary condition are generally necessary for the existence of classical solutions. Received May 22, 1996 / Accepted April 10, 1997     

On ground state solutions for singular and semi-linear problems including super-linear terms at infinity

We establish a result concerning the existence of entire, positive, classical and bounded solutions which converge to zero at infinity for the semi-linear equation −Δu=λf(x,u),x∈R^N, where f:R^N×(0,∞)→[0,∞) is a suitable function and λ>0 is a real parameter. This result completes the principal theorem of A. Mohammed [A. Mohammed, Ground state solutions for singular semi-linear elliptic equations, Nonlinear Analysis (2008) doi:10.1016/j.na.2008.11.080] mainly because his result does not address the super-linear terms at infinity. Penalty arguments, lower-upper solutions and an approximation procedure will be used.     

Concentration of solutions for some singularly perturbed mixed problems: Asymptotics of minimal energy solutions

In this paper we carry on the study of asymptotic behavior of some solutions to a singularly perturbed problem with mixed Dirichlet and Neumann boundary conditions, started in the first paper [J. Garcia Azorero, A. Malchiodi, L. Montoro, I. Peral, Concentration of solutions for some singularly perturbed mixed problems: Existence results, Arch. Ration. Mech. Anal., in press]. Here we are mainly interested in the analysis of the location and shape of least energy solutions when the singular perturbation parameter tends to zero. We show that in many cases they coincide with the new solutions produced in [J. Garcia Azorero, A. Malchiodi, L. Montoro, I. Peral, Concentration of solutions for some singularly perturbed mixed problems: Existence results, Arch. Ration. Mech. Anal., in press].