Results 1  10
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17
Marcus M.: Large solutions of semilinear elliptic equations: existence, uniqueness and asymptotic behaviour
 J. Anal. Math
, 1992
"... Abstract. On a bounded smooth domain Ω ⊂ R N we study solutions of a semilinear elliptic equation with an exponential nonlinearity and a Hardy potential depending on the distance to ∂Ω. We derive global a priori bounds of the Keller–Osserman type. Using a Phragmen–Lindelöf alternative for generalize ..."
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Cited by 21 (2 self)
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Abstract. On a bounded smooth domain Ω ⊂ R N we study solutions of a semilinear elliptic equation with an exponential nonlinearity and a Hardy potential depending on the distance to ∂Ω. We derive global a priori bounds of the Keller–Osserman type. Using a Phragmen–Lindelöf alternative for generalized sub and superharmonic functions we discuss existence, nonexistence and uniqueness of socalled large solutions, i.e., solution which tend to infinity at ∂Ω. The approach develops the one used by the same authors [2] for a problem with a power nonlinearity instead of the exponential nonlinearity. 1.
The AllegrettoPiepenbrink Theorem for Strongly Local Dirichlet Forms
 DOCUMENTA MATH.
, 2009
"... The existence of positive weak solutions is related to spectral information on the corresponding partial differential operator. ..."
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Cited by 7 (4 self)
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The existence of positive weak solutions is related to spectral information on the corresponding partial differential operator.
Ground state alternative for pLaplacian with potential term, http://arxiv.org/PS cache/math/pdf/0511/0511039.pdf
"... term ..."
I.: Generalized eigenfunctions and spectral theory for strongly local Dirichlet forms
"... 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. ..."
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Cited by 4 (3 self)
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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.
A Liouvilletype theorem for Schrödinger operators
, 2005
"... 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 operato ..."
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Cited by 2 (1 self)
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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.
Existence results for Bellman equations and maximum principles in unbounded domains
, 1997
"... : 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, ..."
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Cited by 2 (0 self)
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: 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 zeroorder term (see Lions [15], KrylovSafonov [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 HamiltonJacobiBellman equation: 8 ! : sup k2N (L k u(x) + f k (x)) = 0 in ! u = 0 on @!; (1.1...
SEMICLASSICAL STATIONARY STATES FOR NONLINEAR SCHRÖDINGER EQUATIONS WITH FAST DECAYING POTENTIALS
, 902
"... 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+ ..."
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Cited by 1 (0 self)
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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.
”BOUNDARY BLOWUP ” TYPE SUBSOLUTIONS TO SEMILINEAR ELLIPTIC EQUATIONS WITH HARDY POTENTIAL
, 708
"... Abstract. Semilinear elliptic equations which give rise to solutions blowing up at the boundary are perturbed by a Hardy potential µ/δ(x, ∂Ω) 2. The size of this potential effects the existence of a certain type of solutions (large solutions): if µ is too small, then no large solution exists. The pr ..."
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Cited by 1 (0 self)
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Abstract. Semilinear elliptic equations which give rise to solutions blowing up at the boundary are perturbed by a Hardy potential µ/δ(x, ∂Ω) 2. The size of this potential effects the existence of a certain type of solutions (large solutions): if µ is too small, then no large solution exists. The presence of the Hardy potential requires a new definition of large solutions, following the pattern of the associated linear problem. Nonexistence and existence results for different types of solutions will be given. Our considerations are based on a PhragmenLindelöf type theorem which enables us to classify the solutions and subsolutions according to their behavior near the boundary. Nonexistence follows from this principle together with the KellerOsserman upper bound. The existence proofs rely on sub and supersolution techniques and on estimates for the Hardy constant derived in Marcus, Mizel and Pinchover [9]. 1.
Abstract
, 2008
"... 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 t ..."
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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).