High energy solutions of modified quasilinear fourth-order elliptic equation

Abstract

Abstract This paper focuses on the following modified quasilinear fourth-order elliptic equation: {△2u−(a+b∫R3|∇u|2dx)△u+λV(x)u−12△(u2)u=f(x,u),in R3,u(x)∈H2(R3), $$\textstyle\begin{cases} \triangle^{2}u-(a+b\int_{\mathbb{R}^{3}} \vert \nabla u \vert ^{2}\,dx)\triangle u+\lambda V(x)u-\frac{1}{2}\triangle(u^{2})u=f(x,u),& \mbox{in }\mathbb{R}^{3}, \\ u(x)\in H^{2}(\mathbb{R}^{3}), \end{cases} $$ where △2=△(△) $\triangle^{2}=\triangle(\triangle)$ is the biharmonic operator, a>0 $a>0$, b≥0 $b\geq 0$, λ≥1 $\lambda\geq 1$ is a parameter, V∈C(R3,R) $V\in C(\mathbb{R}^{3},\mathbb{R})$, f(x,u)∈C(R3×R,R) $f(x,u)\in C(\mathbb{R}^{3}\times\mathbb{R}, \mathbb{R})$. V(x) $V(x)$ and f(x,u)u $f(x,u)u$ are both allowed to be sign-changing. Under the weaker assumption lim|t|→∞∫0tf(x,s)ds|t|3=∞ $\lim_{ \vert t \vert \rightarrow\infty}\frac{\int^{t}_{0}f(x,s)\,ds}{ \vert t \vert ^{3}}=\infty$ uniformly in x∈R3 $x\in\mathbb{R}^{3}$, a sequence of high energy weak solutions for the above problem are obtained.

Keywords

• Super-quadratic
• High energy solutions
• Sign-changing potential
• Fountain theorem