Martin kernel of Schrödinger operators with singular potentials and applications to B.V.P. for linear elliptic equations

Gkikas, Konstantinos T.; Nguyen, Phuoc-Tai

doi:10.1007/s00526-021-02102-6

Cited by 15 publications

(40 citation statements)

References 23 publications

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“…The space X µ (Ω \ Σ) was introduced in [14] to study linear problem (1.3). From (1.7), it is easy to see that the first term on the left-hand side of (1.6) is finite.…”

Section: Main Results Let Us Assume Throughout the Paper Thatmentioning

confidence: 99%

“…From (1.7), it is easy to see that the first term on the left-hand side of (1.6) is finite. By [14,Lemma 7.3], for any ζ ∈ X µ (Ω \ Σ), we have |ζ| φ µ , hence the second term on the left-hand side and the first term on the right-hand side of (1.6) are finite. Finally, since K µ [ν] ∈ L 1 (Ω; φ µ ), the second term on the right-hand side of (1.6) is also finite.…”

Section: Main Results Let Us Assume Throughout the Paper Thatmentioning

confidence: 99%

“…In view of the proof of (3.14) in [14,Lemma 3.3] and by A.22 , we can show that there exists a constant c > 0 depending only on N, µ, β 0 such that…”

mentioning

confidence: 91%

“…However, in general, this does not hold true. Under the assumption λ µ > 0, the authors of the present paper obtained the existence and sharp two-sided estimates of the Green function G µ and Martin kernel K µ associated to −L µ (see [14]) which are crucial tools in the study of the boundary value problem with measures data for linear equations involving L µ…”

mentioning

confidence: 91%

“…In (1.3), tr(u) denotes the boundary trace of u on ∂Ω ∪ Σ which was defined in [14] in terms of harmonic measures of −L µ (see Subsection 2.4). A highlighting property of this where Σ is of dimension 0 ≤ k < N − 2, g : R → R is a nondecreasing continuous function such that g(0) = 0, τ ∈ M(Ω \Σ; φ µ ) and ν ∈ M(∂Ω ∪ Σ).…”

mentioning

confidence: 99%

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Semilinear elliptic Schrödinger equations with singular potentials and absorption terms

Gkikas¹,

Nguyen²

2022

Preprint

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Let Ω ⊂ R N (N ≥ 3) be a C 2 bounded domain and Σ ⊂ Ω be a compact, C 2 submanifold without boundary, of dimensionin Ω \ Σ, where dΣ(x) = dist(x, Σ) and µ is a parameter. We investigate the boundary value problem (P) −Lµu + g(u) = τ in Ω \ Σ with condition u = ν on ∂Ω ∪ Σ, where g : R → R is a nondecreasing, continuous function, and τ and ν are positive measures. The complex interplay between the competing effects of the inverse-square potential d −2 Σ , the absorption term g(u) and the measure data τ, ν discloses different scenarios in which problem (P) is solvable. We provide sharp conditions on the growth of g for the existence of solutions. When g is a power function, namely g(u) = |u| p−1 u with p > 1, we show that problem (P) admits several critical exponents in the sense that singular solutions exist in the subcritical cases (i.e. p is smaller than a critical exponent) and singularities are removable in the supercritical cases (i.e. p is greater than a critical exponent). Finally, we establish various necessary and sufficient conditions expressed in terms of appropriate capacities for the solvability of (P).

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“…The space X µ (Ω \ Σ) was introduced in [14] to study linear problem (1.3). From (1.7), it is easy to see that the first term on the left-hand side of (1.6) is finite.…”

Section: Main Results Let Us Assume Throughout the Paper Thatmentioning

confidence: 99%

Section: Main Results Let Us Assume Throughout the Paper Thatmentioning

confidence: 99%

“…In view of the proof of (3.14) in [14,Lemma 3.3] and by A.22 , we can show that there exists a constant c > 0 depending only on N, µ, β 0 such that…”

mentioning

confidence: 91%

mentioning

confidence: 91%

mentioning

confidence: 99%

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Semilinear elliptic Schrödinger equations with singular potentials and absorption terms

Gkikas¹,

Nguyen²

2022

Preprint

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Singular solutions for Lane–Emden equations with CKN operator

Chen,

Sun,

Wang

2024

Mathematische Nachrichten

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In this paper, we concern the isolated singular solutions for Lane–Emden equations involving the CKN operator We classify the isolated singularities by building the connection with the equations involving the Dirac mass in a weighted distributional sense. Then, we obtain the minimal singular solution with given singularities at the origin, analyze the stability of the minimal singular solutions. Finally, we can derive a second solution by the variational method with the given singularities.

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Heat and Martin Kernel estimates for Schrödinger operators with critical Hardy potentials

Barbatis,

Gkikas,

Tertikas

2023

Math. Ann.

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Let $$\Omega $$ Ω be a bounded domain in $${{\mathbb {R}}}^N$$ R N with $$C^2$$ C 2 boundary and let $$K\subset \partial \Omega $$ K ⊂ ∂ Ω be either a $$C^2$$ C 2 submanifold of the boundary of codimension $$k<N$$ k < N or a point. In this article we study various problems related to the Schrödinger operator $$L_{\mu } =-\Delta - \mu d_K^{-2}$$ L μ = - Δ - μ d K - 2 where $$d_K$$ d K denotes the distance to K and $$\mu \le k^2/4$$ μ ≤ k 2 / 4 . We establish parabolic boundary Harnack inequalities as well as related two-sided heat kernel and Green function estimates. We construct the associated Martin kernel and prove existence and uniqueness for the corresponding boundary value problem with data given by measures. To prove our results we introduce among other things a suitable notion of boundary trace. This trace is different from the one used by Marcus and Nguyen (Math Ann 374(1–2):361–394, 2019) thus allowing us to cover the whole range $$\mu \le k^2/4$$ μ ≤ k 2 / 4 .

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Martin kernel of Schrödinger operators with singular potentials and applications to B.V.P. for linear elliptic equations

Cited by 15 publications

References 23 publications

Semilinear elliptic Schrödinger equations with singular potentials and absorption terms

Semilinear elliptic Schrödinger equations with singular potentials and absorption terms

Singular solutions for Lane–Emden equations with CKN operator

Heat and Martin Kernel estimates for Schrödinger operators with critical Hardy potentials

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