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Schrödinger equations with critical nonlinearity, singular potential and a ground state
Authors:David G Costa  João Marcos do Ó
Institution:a Department of Mathematical Sciences, University of Nevada Las Vegas, P.O. Box No. 454020, NV, USA
b Department of Mathematics, Universidade Federal de Paraiba, 58051-900 João Pessoa, PB, Brazil
c Department of Mathematics, Uppsala University, P.O. Box 480, 751 06 Uppsala, Sweden
Abstract:We study semilinear elliptic equations in a generally unbounded domain ΩRN when the pertinent quadratic form is nonnegative and the potential is generally singular, typically a homogeneous function of degree −2. We prove solvability results based on the asymptotic behavior of the potential with respect to unbounded translations and dilations, while the nonlinearity is a perturbation of a self-similar, possibly oscillating, term f of critical growth satisfying View the MathML source, jZ, sR. This paper focuses on two qualitatively different cases of this problem, one when the quadratic form has a generalized ground state and another where the presence of potential does not change the energy space. In the latter case we allow nonlinearities with oscillatory critical growth. An important example of such quadratic form is the one on RN with the radial Hardy potential −μ|x|−2 with μ=μ in the first case, μ<μ in the second case, where View the MathML source is the largest constant for which the energy form remains nonnegative.
Keywords:Nonlinear Schrö  dinger equations  Generalized ground state  Hardy potential  Criticality theory  Sign-changing solutions  Linking geometry  Minimax  Critical points
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