Bulletin of the Belgian Mathematical Society - Simon Stevin

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

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

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

Digital Object Identifier
doi:10.36045/bbms/1378314513

Mathematical Reviews number (MathSciNet)
MR3129056

Zentralblatt MATH identifier
1282.35152

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