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

Advances in Difference Equations

Qizhen Xiao¹  Zigen Ouyang¹  Hongliang Liu¹ 

[1] School of Mathematics and Physics, University of South China;

关键词: Biharmonic equation;    Variational methods;    High energy solutions;    Concentration of solutions;   

DOI  :  10.1186/s13662-018-1782-9

来源: DOAJ

【 摘 要 】

Abstract We consider the following nonlinear biharmonic equations: Δ2u−Δu+Vλ(x)u=f(x,u),in RN, $$ \Delta^{2} u-\Delta u+ V_{\lambda }(x)u=f(x,u),\quad \text{in } \mathbb{R}^{N}, $$ where Vλ(x) $V_{\lambda }(x)$ 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.

【 授权许可】

Unknown