NDLI: Standing waves for a class of Kirchhoff type problems in $${\mathbb {R}^3}$$ involving critical Sobolev exponents

Content Provider	SpringerLink
Author	He, Yi Li, Gongbao
Copyright Year	2015
Abstract	We are concerned with the following Kirchhoff type equation with critical nonlinearity: $$\begin{aligned} \left\{ \begin{array}{ll} - \Bigl ( {{\varepsilon ^2}a + \varepsilon b\int _{{\mathbb {R}^3}} {{{\| {\nabla u} \|}^2}} } \Bigr )\Delta u + V(x)u = \lambda {\| u \|^{p - 2}}u + {\| u \|^4}u{\text { in }}{\mathbb {R}^3}, \\ u > 0,u \in {H^1}({\mathbb {R}^3}), \\ \end{array} \right. \end{aligned}$$ where $$\varepsilon $$ is a small positive parameter, $$a,b>0$$ , $$\lambda > 0$$ , $$2 < p \le 4$$ . Under certain assumptions on the potential V, we construct a family of positive solutions $${u_\varepsilon } \in {H^1}({\mathbb {R}^3})$$ which concentrates around a local minimum of V as $$\varepsilon \rightarrow 0$$ . Although, critical growth Kirchhoff type problem $$\begin{aligned} \left\{ \begin{array}{ll} - \Bigl ( {{\varepsilon ^2}a + \varepsilon b\int _{{\mathbb {R}^3}} {{{\| {\nabla u} \|}^2}} } \Bigr )\Delta u + V(x)u = f(u)+{u^5}{\text { in }}{\mathbb {R}^3}, \\ u > 0,u \in {H^1}({\mathbb {R}^3}) \\ \end{array} \right. \end{aligned}$$ has been studied in e.g. He et al. [18], where the assumption for f(u) is that $$f(u) \sim \|u{\|^{p - 2}}u$$ with $$4 < p < 6$$ and satisfies the Ambrosetti-Rabinowitz condition which forces the boundedness of any Palais-Smale sequence of the corresponding energy functional of the equation. As $$g(u): = \lambda {\| u \|^{p - 2}}u + {\| u \|^4u}$$ with $$2 4, 0 < \mu \int _0^u {g(s)ds \le g(u)u}$$ ), the boundedness of Palais–Smale sequence becomes a major difficulty in proving the existence of a positive solution. Also, the fact that the function $$g(s)/{s^3}$$ is not increasing for $$s > 0$$ prevents us from using the Nehari manifold directly as usual. Our result extends the main result in He et al. [18] concerning the existence and concentration of positive solutions to the case where $$f(u) \sim \|u{\|^{p - 2}}u$$ with $$4 < p < 6$$ .
Ending Page	3106
Page Count	40
Starting Page	3067
File Format	PDF
ISSN	09442669
e-ISSN	14320835
Journal	Calculus of Variations and Partial Differential Equations
Issue Number	3
Volume Number	54
Language	English
Publisher	Springer Berlin Heidelberg
Publisher Date	2015-07-07
Publisher Place	Berlin, Heidelberg
Access Restriction	Subscribed
Subject Keyword	Systems Theory, Control Calculus of Variations and Optimal Control; Optimization Analysis Theoretical, Mathematical and Computational Physics Variational methods for second-order elliptic equations Nonlinear elliptic equations Quasilinear elliptic equations with $p$ -Laplacian
Content Type	Text
Resource Type	Article
Subject	Analysis Applied Mathematics

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