Principal eigenvalues and the Dirichlet problem for fully nonlinear elliptic operators

We study uniformly elliptic fully nonlinear equations of the type F(D²u,Du,u,x)=f(x). We show that convex positively 1-homogeneous operators possess two principal eigenvalues and eigenfunctions, and study these objects; we obtain existence and uniqueness results for nonproper operators whose principal eigenvalues (in some cases, only one of them) are positive; finally, we obtain an existence result for nonproper Isaac's equations.     

Conformally invariant fully nonlinear elliptic equations and isolated singularities

We study properties of solutions with isolated singularities to general conformally invariant fully nonlinear elliptic equations of second order. The properties being studied include radial symmetry and monotonicity of solutions in the punctured Euclidean space and the asymptotic behavior of solutions in a punctured ball. Some results apply to more general situations including more general fully nonlinear elliptic equations of second order, and some have been used in a companion paper to establish comparison principles and Liouville type theorems for degenerate elliptic equations.     

Comparison principle for some classes of nonlinear elliptic equations

We prove a comparison principle for second order quasilinear elliptic operators in divergence form when a first order term appears. We deduce uniqueness results for weak solutions to Dirichlet problems when data belong to the natural dual space.     

Critical eigenvalues for a nonlinear problem

We study the application, , where is the supremum of positive s such that the problem admits a solution. Where B ₁ is the unit ball in We show that is a decreasing function, with where is the unique solution of the problem . We also give the explicit solutions of the problem , when and show that . We show that the problem doesnt admit a solution. In the end, we give a numerical approximation of , when .     

Multiplicity of solutions to uniformly elliptic fully nonlinear equations with concave-convex right-hand side

We deal with existence, non-existence and multiplicity of solutions to the model problem

(P)     

Monotonicity of solutions of Fully nonlinear uniformly elliptic equations in the half-plane

In this paper we study the monotonicity of positive (or non-negative) viscosity solutions to uniformly elliptic equations F(∇u,D²u)=f(u) in the half plane, where f is locally Lipschitz continuous (with f(0)?0) and zero Dirichlet boundary conditions are imposed. The result is obtained without assuming the u or |∇u| are bounded.     

Multiple solutions of nonlinear elliptic systems

We proved a multiplicity result for a nonlinear elliptic system in R^N. The functional related to the system is strongly indefinite. We investigated the relation between the number of solutions and the topology of the set of the global maxima of the coefficients.     

On radial solutions of certain semi-linear elliptic equations

We consider the semi-linear elliptic equation Δu+f(x,u)+g(|x|)x·∇u=0$\mathrm{\Delta }u+f(x,u)+g(|x|)x·\nabla u=0$, in some exterior region of Rⁿ,n?3${\mathbb{R}}^n,n?3$. It is shown that if f$f$ depends radially on its first argument and is nonincreasing in its second, boundary conditions force the unique solution to be radial. Under different conditions, we prove the existence of a positive radial asymptotic solution to the same equation.     

Structure of least-energy solutions for quasilinear elliptic equations on convex domains

In this paper we study the existence and structure of the least-energy solutions for a class of singularly perturbed quasilinear elliptic equations. Using the moving plane method and a geometric lemma we show that any least-energy solution develops to a single spike-layer solution on convex domains.     

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

     

Optimal boundary regularity for nonlinear singular elliptic equations

In this paper we prove the optimal boundary regularity under natural structural conditions for a large class of nonlinear elliptic equations with singular terms near the boundary. By a careful construction of super- and sub-solutions, we obtain precise growth estimates for solutions at the boundary and reduce the boundary regularity to the interior one by a rescaling argument.     

Nonlinear eigenvalues for anisotropic quasilinear degenerate elliptic equations

We consider the Dirichlet problem for a class of anisotropic degenerate elliptic equations.     

Exponential decay of solutions to nonlinear elliptic equations with potentials

Exponential decay estimates are obtained for complex-valued solutions to nonlinear elliptic equations in where the linear term is given by Schr?dinger operators H = − Δ + V with nonnegative potentials V and the nonlinear term is given by a single power with subcritical Sobolev exponent in the attractive case. We describe specific rates of decay in terms of V, some of which are shown to be optimal. Moreover, our estimates provide a unified understanding of two distinct cases in the available literature, namely, the vanishing potential case V = 0 and the harmonic potential case V(x) = |x|². Dedicated to Professor Jun Uchiyama on the occasion of his sixtieth birthday Received: May 4, 2004     

Eigenvalue problems for nonlinear elliptic equations with unilateral constraints

In this paper we study eigenvalue problems for hemivariational and variational inequalities driven by the p$p$-Laplacian differential operator. Using topological methods (based on multivalued versions of the Leray–Schauder alternative principle) and variational methods (based on the nonsmooth critical point theory), we prove existence and multiplicity results for the eigenvalue problems that we examine.     

Maximum and anti-maximum principlesand eigenfunctions estimates via perturbation theory of positive solutions of elliptic equations

In this paper we discuss some new results concerning perturbation theory for second order elliptic partial differential equations related to positivity properties of such equations. We continue the study of some different notions of “small” perturbations and discuss their relations to comparisons of Green's functions, refined maximum and anti-maximum principles, ground state, and the decay of eigenfunctions. In particular, we show that if V is a positive function which is a semismall perturbation of a subcritical Schr?dinger operator H defined on a domain , and are the (Dirichlet) eigenfunctions of the equation , then for any , the function is bounded and has a continuous extension up to the Martin boundary of the pair , where is the ground state of H with a principal eigenvalue . Received: 29 November 1998     

Eigenvalues for radially symmetric fully nonlinear singular or degenerate operators

In this paper we present a one dimensional and radial theory for the existence of eigenvalues and eigenfunctions for fully nonlinear elliptic (α+1)$(\alpha +1)$-homogeneous operators, α>−1$\alpha >-1$. A general theory for the first eigenvalue and eigenfunction exists in the frame of viscosity solutions, but in this particular case a simpler theory can be established, that extends, via degree theory, to obtain the complete set of eigenvalues and eigenfunctions characterized by the number of zeros.     

Bounded positive entire solutions of singular quasilinear elliptic equations

Suppose that β?0 is a constant and that is a continuous function with R₊:=(0,∞). This paper investigates N-dimensional singular, quasilinear elliptic equations of the form