Multiplicity of solutions for $p$-biharmonic problems with critical growth

Source abstract

We prove the existence of infinitely many solutions for $p$-biharmonic problems in a bounded, smooth domain $\Omega$ with concave-convex nonlinearities dependent upon a parameter $\lambda$ and a positive continuous function $f\colon \overline {\Omega }\to \mathbb {R}$. We simultaneously handle critical case problems with both Navier and Dirichlet boundary conditions by applying the Ljusternik-Schnirelmann method. The multiplicity of solutions is obtained when $\lambda$ is small enough. In the case of Navier boundary conditions, all solutions are positive, and a regularity result is proved.