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

This paper is concerned with the existence and multiplicity of the solutions for the fourth-order boundary value problem u ( 4 ) ( t ) + η u ″ ( t ) − ζ u ( t ) = λ f ( t , u ( t ) ) , 0 < t < 1 , u ( 0 ) = u ( 1 ) = u ″ ( 0 ) = u ″ ( 1 ) = 0 , where f : [ 0 , 1 ] × R → R is continuous, ζ , 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.