## Bulletin of the Belgian Mathematical Society - Simon Stevin

- Bull. Belg. Math. Soc. Simon Stevin
- Volume 20, Number 3 (2013), 519-534.

### Multiplicity of solutions for a biharmonic equation with subcritical or critical growth

Giovany M. Figueiredo and Marcos T. O. Pimenta

#### Abstract

We consider the fourth-order problem $$ \left\{ \begin{array}{l} \epsilon^4\Delta^2u + V(x)u = f(u) + \gamma|u|^{2_{**}-2}u \mbox{in $\mathbb{R}^N$}\\ u\in H^2(\mathbb{R}^N), \end{array} \right. $$ where $\epsilon > 0$, $N\geq 5$, $V$ is a positive continuous potential, $f$ is a function with subcritical growth and $\gamma \in \{0,1\}$. We relate the number of solutions with the topology of the set where $V$ attain its minimum values. We consider the subcritical case $\gamma=0$ and the critical case $\gamma=1$. In the proofs we apply Ljusternik-Schnirelmann theory.

#### Article information

**Source**

Bull. Belg. Math. Soc. Simon Stevin, Volume 20, Number 3 (2013), 519-534.

**Dates**

First available in Project Euclid: 4 September 2013

**Permanent link to this document**

https://projecteuclid.org/euclid.bbms/1378314513

**Digital Object Identifier**

doi:10.36045/bbms/1378314513

**Mathematical Reviews number (MathSciNet)**

MR3129056

**Zentralblatt MATH identifier**

1282.35152

**Subjects**

Primary: 35J30: Higher-order elliptic equations [See also 31A30, 31B30]

Secondary: 35J35: Variational methods for higher-order elliptic equations

**Keywords**

variational methods biharmonic equations nontrivial solutions

#### Citation

Figueiredo, Giovany M.; Pimenta, Marcos T. O. Multiplicity of solutions for a biharmonic equation with subcritical or critical growth. Bull. Belg. Math. Soc. Simon Stevin 20 (2013), no. 3, 519--534. doi:10.36045/bbms/1378314513. https://projecteuclid.org/euclid.bbms/1378314513