Nodal solutions of nonlinear boundary value problems with p-Laplacian.

摘要

We study the second order nonlinear boundary value problem with p-Laplacian consisting of the equation -fy' '+qt fy=wt fy with &phis(y) = |y| p-1y for p > 0 on [a, b] and a general separated boundary condition. By comparing it with a half-linear Sturm-Liouville problem we obtain conditions for the existence and nonexistence of nodal solutions of this problem. More specifically, let lambdan, n = 0, 1, 2,..., be the n-th eigenvalue of the corresponding half-linear Sturm-Liouville problem. Then the boundary value problem has a pair of solutions with exactly n zeros in (a, b) if lambdan is in the interior of the range of f(y)/&phis(y) and does not have any solution with exactly n zeros in ( a, b) if lambdan is outside the range. These conditions become necessary and sufficient when f(y)/&phis(y) is monotone on (-infinity, 0) and (0, infinity). We also study the changes of the number of different types of nodal solutions as the equation or the boundary condition changes. Moreover, we extend the above results to a boundary value problem with p-Laplacian consisting of the equation -fy' '+qt fy=w1 tf1y +w2tf2 y on [a, b] and the general separated boundary condition. Our main results are obtained based on the global existence and uniqueness of solutions of the corresponding initial value problems, which are derived by the establishment of nonlinear integral inequalities and the application of a generalized energy function and a generalized Prufer transformation.