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Estimate of the Hausdorff measure of the singular set of a solution for a semi-linear elliptic equation associated with superconductivity

Junichi Aramaki (2010)

Archivum Mathematicum

We study the boundedness of the Hausdorff measure of the singular set of any solution for a semi-linear elliptic equation in general dimensional Euclidean space ℝ n . In our previous paper, we have clarified the structures of the nodal set and singular set of a solution for the semi-linear elliptic equation. In particular, we showed that the singular set is ( n - 2 ) -rectifiable. In this paper, we shall show that under some additive smoothness assumptions, the ( n - 2 ) -dimensional Hausdorff measure of singular set...

Existence and multiplicity results for a nonlinear stationary Schrödinger equation

Danila Sandra Moschetto (2010)

Annales Polonici Mathematici

We revisit Kristály’s result on the existence of weak solutions of the Schrödinger equation of the form -Δu + a(x)u = λb(x)f(u), x ∈ ℝ N , u ∈ H ¹ ( ℝ N ) , where λ is a positive parameter, a and b are positive functions, while f : ℝ → ℝ is sublinear at infinity and superlinear at the origin. In particular, by using Ricceri’s recent three critical points theorem, we show that, under the same hypotheses, a much more precise conclusion can be obtained.

Existence of positive radial solutions for the elliptic equations on an exterior domain

Yongxiang Li, Huanhuan Zhang (2016)

Annales Polonici Mathematici

We discuss the existence of positive radial solutions of the semilinear elliptic equation ⎧-Δu = K(|x|)f(u), x ∈ Ω ⎨αu + β ∂u/∂n = 0, x ∈ ∂Ω, ⎩ l i m | x | → ∞ u ( x ) = 0 , where Ω = x ∈ ℝ N : | x | > r ₀ , N ≥ 3, K: [r₀,∞) → ℝ⁺ is continuous and 0 < ∫ r ₀ ∞ r K ( r ) d r < ∞ , f ∈ C(ℝ⁺,ℝ⁺), f(0) = 0. Under the conditions related to the asymptotic behaviour of f(u)/u at 0 and infinity, the existence of positive radial solutions is obtained. Our conditions are more precise and weaker than the superlinear or sublinear growth conditions. Our discussion is based on the fixed point...

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