Abstract
We study existence and stability of solutions of (E 1) −∆u + µ |x| 2 u + g(u) = ν in Ω, u = 0 on ∂Ω, where Ω is a bounded, smooth domain of R N , N ≥ 2, containing the origin, µ ≥ − (N −2) 2 4 is a constant, g is a nondecreasing function satisfying some integral growth assumption and ν is a Radon measure on Ω. We show that the situation differs according ν is diffuse or concentrated at the origin. When g is a power we introduce a capacity framework to find necessary and sufficient condition for solvability.
Highlights
Schrodinger operators with singular potentials under the form u → H(u) := −∆u + V (x)u x ∈ R3 (1.1)are at the core of the description of many aspects of nuclear physics
When g is a power we introduce a capacity framework to find necessary and sufficient conditions for solvability
In quantum physics there are reasons arising from its mathematical formulation which leads, at least in the case of the hydrogen atom, to V (x) =
Summary
Schrodinger operators with singular potentials under the form u → H(u) := −∆u + V (x)u x ∈ R3. It is easy to see that u satisfies uL∗μξ + g(u)ξ dγμ(x) = cμγξ(0), ∀ξ ∈ C01,1(RN ) In view of these results and identity (1.13), we introduce a definition of weak solutions adapted to the operator Lμ in a measure framework. We have avoided to use the estimates on the Green kernel for Hardy operators which are not tractable when 0 > μ ≥ μ0, and our main idea is to separate the measure ν∗ in M(Ω; Γμ) and the Dirac mass at the origin, and to glue the solutions with above measures respectively This technique requires this new weak ∆2-condition.
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