Concentrating Bound States for Kirchhoff Type Problems in ℝ3 Involving Critical Sobolev Exponents

By Yi He, Gongbao Li, Shuangjie Peng. 2013

196 citations recorded by the source corpus.

Abstract We study the concentration and multiplicity of weak solutions to the Kirchhoff type equation with critical Sobolev growth, where ε is a small positive parameter and a, b > 0 are constants, f ∈ C1(ℝ+,ℝ) is subcritical, V : ℝ3 → ℝ is a locally Hölder continuous function. We first prove that for ε0 > 0 sufficiently small, the above problem has a weak solution uε with exponential decay at infinity. Moreover, uε concentrates around a local minimum point of V in Λ as ε → 0. With minimax theorems and Ljusternik-Schnirelmann theory, we also obtain multiple solutions by employing the topological construction of the set where the potential V(z) attains its minimum.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Control and Systems Engineering

View Paper: https://www.semanticscholar.org/paper/893f9eea3cfc3c57f2a07da211ff19cd9ed16f82

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In Applied Mathematics

In Computational Theory and Mathematics

In Control and Systems Engineering

Citations in the corpus

A Nonhomogeneous Fractional p-Kirchhoff Type Problem Involving Critical Exponent in ℝN (2016, background): “For instance, in bounded domains we refer to [17, 22, 30]; in the whole space, see [21, 24, 26].”
Existence and Concentration Result for the Kirchhoff Type Equations with General Nonlinearities (2014, background): “In [35], He, Li and Peng studied (Pε) in a similar situation to [62], namely, N = 3, M(t) = a + bt and f (s) = g(s)+ |s|4s where g(s) = o(s3) as s → 0 and g(s)/s3 is strictly increasing.”