Abstract not found

The existence of positive weak solutions is related to spectral information on the corresponding partial differential operator.

Abstract. We present an introduction to the framework of strongly local Dirichlet forms and discuss connections between the existence of certain generalized eigenfunctions and spectral properties within this framework. The range of applications is illustrated by a list of examples.

In this paper we prove a sufficient condition, in terms of the behavior of a ground state of a symmetric critical operator, such that a nonzero solution of another symmetric nonnegative operator is a ground state. In particular, if Pj: = − ∆ + Vj, for j = 0,1, are two nonnegative Schrödinger operators defined on Ω ⊆ R d such that P1 is critical in Ω with a ground state ϕ, the function ψ ≰ 0 solves the equation P0u = 0 in Ω and satisfies |ψ | ≤ Cϕ in Ω, then P0 is critical in Ω and ψ is its ground state. In particular, ψ is (up to a multiplicative constant) the unique positive solution of the equation P0u = 0 in Ω. Similar results hold for general symmetric operators, and also on Riemannian manifolds.

Abstract. We study the existence of positive solutions for a class of nonlinear Schrödinger equations of the type −ε 2 ∆u + V u = u p in R N, where N ≥ 3, p> 1 is subcritical and V is a nonnegative continuous potential. Amongst other results, we prove that if V has a positive local minimum, and N N+2 < p <, then N−2 N−2 for small ε the problem admits positive solutions which concentrate as ε → 0 around the local minimum point of V. The novelty is that no restriction is imposed on the rate of decay of V. In particular, we cover the case where V is compactly supported. 1.

: The purpose of this paper is to extend existence results for HamiltonJacobi -Bellman equations, i.e. sup k2IN (L k u(x) + f k (x)) = 0 in !, u = 0 on @!, where L k is a family of uniformly elliptic operators and ! is a bounded regular domain in IR p . The classical results give existence, uniqueness and Holder regularity when all the operators L k have nonpositive zero-order term (see Lions [15], Krylov-Safonov [14], [18]). We want to handle here the more general case where L k have principal eigenvalues bounded below by a positive constant. As a motivation for this work, we give an application to the study of the Maximum Principle in infinite cylinders, following a work by Berestycki, Caffarelli and Nirenberg [2]. This is used to extend the cylindrical symmetry result in [2] to a more general class of operators. 1 Introduction and main results We consider the following Hamilton-Jacobi-Bellman equation: 8 ! : sup k2N (L k u(x) + f k (x)) = 0 in ! u = 0 on @!; (1.1...

Abstract not found

Let a be a quadratic form associated with a Schrödinger operator L = − ∇ · (A∇) + V on a domain Ω ⊂ Rd. If a is nonnegative on C ∞ 0 (Ω), then either there is W> 0 such that ∫ W |u | 2 dx ≤ a[u] for all C ∞ 0 (Ω; R), or there is a sequence ϕk ∈ C ∞ 0 (Ω) and a function ϕ> 0 satisfying Lϕ = 0 such that a[ϕk] → 0, ϕk → ϕ locally uniformly in Ω \ {x0}. This dichotomy is equivalent to the dichotomy between L being subcritical resp. critical in Ω. In the latter case, one has an inequality of Poincaré type: there exists W> 0 such that for every ψ ∈ C ∞ 0 (Ω; R) satisfying ∫ ψϕdx ̸ = 0 there exists a constant C> 0 such that C−1 ∫ W |u | 2 dx ≤ a[u] + C ∣ ∫ uψ dx ∣ 2 for all u ∈ C ∞ 0 (Ω; R).

Positive solutions to singular semilinear elliptic equations with critical potential on cone–like domains

Let Ω be a bounded C 2 domain in R n, where n is any positive integer, and let Ω ∗ be the Euclidean ball centered at 0 and having the same Lebesgue measure as Ω. Consider the operator L = −div(A∇) + v · ∇ + V on Ω with Dirichlet boundary condition, where the symmetric matrix field A is in W 1, ∞ (Ω), the vector field v is in L ∞ (Ω, R n) and V is a continuous function in Ω. We prove that minimizing the principal eigenvalue of L when the Lebesgue measure of Ω is fixed and when A, v and V vary under some constraints is the same as minimizing the principal eigenvalue of some operators L ∗ in the ball Ω ∗ with smooth and radially symmetric coefficients. The constraints which are satisfied by the original coefficients in Ω and the new ones in Ω ∗ are expressed in terms of some distribution functions or some integral, pointwise or geometric quantities. Some strict comparisons are also established when Ω is not a ball. To these purposes, we associate to the principal eigenfunction ϕ of L a new symmetric rearrangement defined on Ω ∗ , which is different from the classical Schwarz symmetrization, and which preserves the integral of div(A∇ϕ) on suitable equi-measurable sets. A substantial part of the paper is devoted to the proofs of pointwise and integral inequalities of independent interest which are satisfied by this rearrangement. The comparisons for the eigenvalues hold for general operators of the type L and they are new even for symmetric operators. Furthermore they generalize, in particular, and provide an alternative proof of the well-known Rayleigh-Faber-Krahn isoperimetric inequality about the principal eigenvalue of the Laplacian under Dirichlet boundary condition on a domain with fixed Lebesgue measure.