Multiple solutions to fourth-order boundary value problems
Computers & Mathematics with Applications
Positive solutions of operator equations and nonlinear beam quations with a perturbed loading force
WSEAS Transactions on Mathematics
| Hi-index | 0.98 |
This paper is concerned with the existence and multiplicity of the solutions for the fourth-order boundary value problem u^(^4^)(t)+@hu^''(t)-@zu(t)=@lf(t,u(t)), 0R is continuous, @z,@h and @l@?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 (@h,@z) and @l@?R lead to different existence results for the above boundary value problem. More precisely, if the pair (@h,@z) is on the left side of the first eigenvalue line, then the above boundary value problem has only the trivial solution for @l@?(-~,0) and has infinitely many solutions for @l@?(0,~); if (@h,@z) is on the right side of the first eigenvalue line and @l@?(-~,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 (@h,@z) is located in the second or fourth (first) quadrant.