Existence and concentration of a nonlinear biharmonic equation with sign-changing potentials and indefinite nonlinearity

We consider the following nonlinear biharmonic equations: Δ 2 u − Δ u + V λ ( x ) u = f ( x , u ) , in  R N , where V λ ( x ) is allowed to be sign-changing and f is an indefinite function. Under some suitable assumptions, the existence of nontrivial solutions and the high energy solutions are obtai...

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Bibliographic Details
Published in:Advances in difference equations 2018-10, Vol.2018 (1), p.1-18, Article 384
Main Authors: Xiao, Qizhen, Liu, Hongliang, Ouyang, Zigen
Format: Article
Language:English
Subjects:
Analysis
Biharmonic equation
Biharmonic equations
Concentration of solutions
Difference and Functional Equations
Functional Analysis
High energy solutions
Mathematics
Mathematics and Statistics
Nonlinear equations
Nonlinearity
Ordinary Differential Equations
Partial Differential Equations
Variational methods
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Summary:We consider the following nonlinear biharmonic equations: Δ 2 u − Δ u + V λ ( x ) u = f ( x , u ) , in  R N , where V λ ( x ) is allowed to be sign-changing and f is an indefinite function. Under some suitable assumptions, the existence of nontrivial solutions and the high energy solutions are obtained by using variational methods. Moreover, the phenomenon of concentration of solutions is explored. The results extend the main conclusions in recent literature.
ISSN:1687-1847
1687-1839
1687-1847
DOI:10.1186/s13662-018-1782-9