Multiple solutions for a Kirchhoff equation with critical growth

• Published in: Zeitschrift für angewandte Mathematik und Physik Vol. 70; no. 1; pp. 1 - 15
• Main Authors: Furtado, Marcelo F., de Oliveira, Luan D., da Silva, João Pablo P.
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.02.2019; Springer Nature B.V
• Subjects: Engineering; Mathematical Methods in Physics; Theoretical and Applied Mechanics; Variational methods; Multiple solutions; Critical nonlinearities; Kirchhoff-type problems; Primary 35J60; Secondary 35J20
• ISSN: 0044-2275; 1420-9039
• DOI: 10.1007/s00033-018-1045-3

We consider the problem - m ∫ Ω | ∇ u | 2 d x Δ u = λ f ( x , u ) + μ | u | 2 ∗ - 2 u , x ∈ Ω , u ∈ H 0 1 ( Ω ) , where Ω ⊂ R N , N ≥ 3 , is a bounded smooth domain, 2 ∗ = 2 N / ( N - 2 ) , λ , μ > 0 and m is an increasing positive function. The function f is odd in the second variable and has superlinear growth. In our first result we obtain, for each k ∈ N , the existence of k pairs of nonzero solutions for all μ > 0 fixed and λ large. Under weaker assumptions of f , we also obtain a similar result if N = 3 , λ > 0 is fixed and μ is close to 0. In the proofs, we apply variational methods.