WE ANALYZE THE EXISTENCE OF T ?PERIODIC SOLUTIONS TO THE SECOND-ORDER INDEFINITE SINGULAR EQUATION U?? = ? H(T) COS ² U WHICH DEPENDS ON A POSITIVE PARAMETER ? > 0. HERE, H IS A SIGN-CHANGING FUNCTION WITH H = 0 AND WHERE THE NONLINEAR TERM OF THE EQUATION HAS TWO SINGULARITIES. FOR THE FIRST TIME, THE DEGENERATE CASE IS STUDIED, DISPLAYING AN UNEXPECTED FEATURE WHICH CONTRASTS WITH THE RESULTS KNOWN IN THE LITERATURE FOR INDEFINITE SINGULAR EQUATIONS. © 2019 AMERICAN INSTITUTE OF MATHEMATICAL SCIENCES. ALL RIGHTS RESERVED.

IN THIS PAPER, WE STUDY THE EXISTENCE, MULTIPLICITY AND DYNAMICS OF POSITIVE PERIODIC SOLUTIONS TO A GENERALIZED LIÉNARD EQUATION WITH REPULSIVE SINGULARITIES. THE AMBROSETTI-PRODI TYPE RESULT IS PROVED IN THE ABSENCE OF THE SO-CALLED ANTICOERCIVITY CONDITION. FURTHERMORE, WITH S AS A PARAMETER, UNDER SOME CONDITIONS ON THE FUNCTION H, IT HAS BEEN SHOWN THAT FOR ANY M > 1 THERE EXISTS SM ? R SUCH THAT THE EQUATION X?? + F (X)X? + H(T, X) = S HAS TWO POSITIVE T -PERIODIC SOLUTIONS U1(·; S) AND U2(·; S) SATISFYING MIN{U1(T; S) : T ? [0, T ]} > M AND MIN{U2(T; S) : T ? [0, T ]} < 1/M FOR EVERY S < S M . AS A BY-PRODUCT OF THE PROPERTY, WE OBTAIN SUFFICIENT CONDITIONS TO GUARANTEE THE EXISTENCE OF POSITIVE T -PERIODIC SOLUTIONS OF INDEFINITE DIFFERENTIAL EQUATIONS.

WE STUDY THE DIRICHLET BOUNDARY VALUE PROBLEM U '' = H(T)/SIN(2)U, U(0+) = C(1), U(T-) = C(2), WHERE C(1), C(2) IS AN ELEMENT OF [0,PI] AND H : [0,T] -> R IS A LEBESGUE INTEGRABLE FUNCTION. THE FORCING TERM UNDER CONSIDERATION IS THE PRODUCT OF A NONLINEARITY WHICH IS SINGULAR AT TWO POINTS WITH A WEIGHT FUNCTION H. WE PROVE THAT THE CORRESPONDING SINGULAR BOUNDARY VALUE PROBLEM IS SOLVABLE ONLY IF THE WEIGHT FUNCTION DOES NOT CHANGE ITS SIGN. THEREFORE, OUR MAIN RESULT IS STATED UNDER THIS SETTING: SUPPOSING THAT H : [0, T] [0, +INFINITY), THE EXISTENCE AND MULTIPLICITY OF SOLUTIONS TO THE AFOREMENTIONED PROBLEM IS GUARANTEED IF AND ONLY IF (H) OVER BAR IS SMALL ENOUGH.