Concentration Compactness at the Mountain Pass Level in Semilinear Elliptic Problems

Tintarev, Kyril

doi:10.1007/s00030-008-7046-8

Cited by 3 publications

(2 citation statements)

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“…Note that even without assuming the penalty conditions (F 3 ), (V 1 ) and (V 2 ) one always has the non-strict inequality c c ∞ (we recall that generally these are multiple inequalities), since one can consider the mountain path statement (5.4) for c restricted to nearly-optimal paths for c ∞ , under suitable dislocations, that is, paths of the form γ N−2 2 j v t (γ j · +y) with large | j| or |y|. The proof of this statement is analogous to that given in [18] in the case V = 0, and is omitted.…”

Section: Now Let Us Consider a Limiting Quadratic Formmentioning

confidence: 85%

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Schrödinger equations with critical nonlinearity, singular potential and a ground state

Costa¹,

Ó²,

Tintarev³

2010

Journal of Differential Equations

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We study semilinear elliptic equations in a generally unbounded domain Ω ⊂ R N 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 f ∞ (λ j s) = λ N+2 N−2 j f ∞ (s), j ∈ Z, s ∈ R. 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 R N with the radial Hardy potential −μ|x| −2 with μ = μ * in the first case, μ < μ * in the second case, where μ * = (N−2) 2 4 is the largest constant for which the energy form remains nonnegative.

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Section: Now Let Us Consider a Limiting Quadratic Formmentioning

confidence: 85%

“…Here, we follow the version of concentration-compactness presented in [19] and which was applied in [18] to the case V = 0 of the present problem. By contrast with [18] this paper is focused on the role of the potential.…”

Section: Introductionmentioning

confidence: 99%

Schrödinger equations with critical nonlinearity, singular potential and a ground state

Costa¹,

Ó²,

Tintarev³

2010

Journal of Differential Equations

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Concentration-compactness principle for nonlocal scalar field equations with critical growth

Ferraz

2017

Journal of Mathematical Analysis and Applications

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Concentration-compactness for singular nonlocal Schrödinger equations with oscillatory nonlinearities

Ó¹,

Ferraz²

2018

TMNA

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Concentration Compactness at the Mountain Pass Level in Semilinear Elliptic Problems

Abstract: Abstract. Let c be the usual mountain pass level for the semilinear elliptic functional

Cited by 3 publications

References 29 publications

Schrödinger equations with critical nonlinearity, singular potential and a ground state

Schrödinger equations with critical nonlinearity, singular potential and a ground state

Concentration-compactness principle for nonlocal scalar field equations with critical growth

Concentration-compactness for singular nonlocal Schrödinger equations with oscillatory nonlinearities

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