On a fourth-order quasilinear elliptic equation of concave-convex type (Q839521)

Let \(\Omega\subset\mathbb R^N\), \(N\geq1\), be a bounded domain with a smooth boundary. This paper deals with the following system: \[ \begin{cases} -\Delta\big((-\Delta u)^{\frac{1}{p}}\big)=u^q+\lambda g(x)u^r &\text{in }\Omega,\\ u,-\Delta u=0 &\text{ on }\partial \Omega, \end{cases}\tag{1} \] where \(u^q=|u|^{q-1}u,u^r=|u|^{r-1}u\), and \((-\Delta u)^{\frac{1}{p}}=|-\Delta u| ^{\frac{1}{p}-1}(-\Delta u)\), \(\lambda\geq0\) is a parameter and \(g\in C^1(\overline{\Omega})\) is positive inside \(\Omega\), under the hypotheses \(H_1: 0<r<\frac{1}{p}<q\) and \(H_2:\frac{1}{p+1}+\frac{1}{q+1}\geq 1-\frac{2}{N}\). When \(\lambda\) is small enough, the author studies, by variational methods, the existence of at least two positive and infinitely many solutions to (1). The critical case, when the identity occurs in \(H_2\), and the subcritical case, which corresponds to the strict inequality in \(H_2\), are investigated separately. Some open problems are mentioned at the end of the paper.