Existence and multiplicity of solutions for fourth-order boundary value problems with three parameters

✍ Scribed by Xi-Lan Liu; Wan-Tong Li

• Publisher: Elsevier Science
• Year: 2007
• Tongue: English
• Weight: 267 KB
• Volume: 46
• Category: Article
• ISSN: 0895-7177
• DOI: 10.1016/j.mcm.2006.11.018

✦ Synopsis

This paper is concerned with the existence and multiplicity of the solutions for the fourth-order boundary value problem

η and λ ∈ R are parameters. Using the variational structure of the above boundary value problem and critical point theory, it is shown that the different locations of the pair (η, ζ ) and λ ∈ R lead to different existence results for the above boundary value problem. More precisely, if the pair (η, ζ ) is on the left side of the first eigenvalue line, then the above boundary value problem has only the trivial solution for λ ∈ (-∞, 0) and has infinitely many solutions for λ ∈ (0, ∞); if (η, ζ ) is on the right side of the first eigenvalue line and λ ∈ (-∞, 0), then the above boundary value problem has two nontrivial solutions or has at least n * (n * ∈ N) distinct pairs of solutions, which depends on the fact that the pair (η, ζ ) is located in the second or fourth (first) quadrant.