## Rocky Mountain Journal of Mathematics

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

#### 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.

#### Article information

Source
Rocky Mountain J. Math., Volume 48, Number 2 (2018), 425-442.

Dates
First available in Project Euclid: 4 June 2018

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1528077624

Digital Object Identifier
doi:10.1216/RMJ-2018-48-2-425

Mathematical Reviews number (MathSciNet)
MR3810206

Zentralblatt MATH identifier
06883474

#### Citation

Bueno, H.; Paes-Leme, L.; Rodrigues, H. Multiplicity of solutions for $p$-biharmonic problems with critical growth. Rocky Mountain J. Math. 48 (2018), no. 2, 425--442. doi:10.1216/RMJ-2018-48-2-425. https://projecteuclid.org/euclid.rmjm/1528077624

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