Existence of ground state solutions to the nonlinear kirchho-type equations with potentials

Norihisa Ikoma

doi:10.3934/dcds.2015.35.943

Existence of ground state solutions to the nonlinear kirchho-type equations with potentials

Norihisa Ikoma

Research output: Contribution to journal › Article › peer-review

24 Citations (Scopus)

Abstract

In this paper, we study the existence of ground state solutions to the nonlinear Kirchho- type equations -m(||∇_u||² L²(R^N)) Δu + V (x)u = |u|^p-1u in R^N; u ∈ H¹(R^N); N ≥ 1 where 1 < p < ∞ when N = 1; 2, 1 < p < (N + 2)=(N - 2) when N ≥ 3, m : [0,∞) → (0,∞) is a continuous function and V : R^N → R a smooth function. Under suitable conditions on m(s) and V , it is shown that a ground state solution to the above equation exists.

Original language	English
Pages (from-to)	943-966
Number of pages	24
Journal	Discrete and Continuous Dynamical Systems- Series A
Volume	35
Issue number	3
DOIs	https://doi.org/10.3934/dcds.2015.35.943
Publication status	Published - 2015 Mar 1
Externally published	Yes

Keywords

Ground state solutions
Kirchho- type equations
Monotonicity trick
The Pohozaev identity
Variational methods

ASJC Scopus subject areas

Analysis
Discrete Mathematics and Combinatorics
Applied Mathematics

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10.3934/dcds.2015.35.943

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abstract = "In this paper, we study the existence of ground state solutions to the nonlinear Kirchho- type equations -m(||∇u||2 L2(RN)) Δu + V (x)u = |u|p-1u in RN; u ∈ H1(RN); N ≥ 1 where 1 < p < ∞ when N = 1; 2, 1 < p < (N + 2)=(N - 2) when N ≥ 3, m : [0,∞) → (0,∞) is a continuous function and V : RN → R a smooth function. Under suitable conditions on m(s) and V , it is shown that a ground state solution to the above equation exists.",

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AB - In this paper, we study the existence of ground state solutions to the nonlinear Kirchho- type equations -m(||∇u||2 L2(RN)) Δu + V (x)u = |u|p-1u in RN; u ∈ H1(RN); N ≥ 1 where 1 < p < ∞ when N = 1; 2, 1 < p < (N + 2)=(N - 2) when N ≥ 3, m : [0,∞) → (0,∞) is a continuous function and V : RN → R a smooth function. Under suitable conditions on m(s) and V , it is shown that a ground state solution to the above equation exists.

KW - Ground state solutions

KW - Kirchho- type equations

KW - Monotonicity trick

KW - The Pohozaev identity

KW - Variational methods

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Existence of ground state solutions to the nonlinear kirchho-type equations with potentials

Abstract

Keywords

ASJC Scopus subject areas

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