EXISTENCE AND L ∞ ESTIMATES OF SOME MOUNTAIN-PASS TYPE SOLUTIONS

Abstract. We prove the existence of a positive solution to the BVP (Φ(t)u′(t)) ′ = f(t, u(t)), u′(0) = u(1) = 0, imposing some conditions on Φ and f. In particular, we assume Φ(t)f(t, u) to be decreasing in t. Our method combines variational and topological arguments and can be applied to some elliptic problems in annular domains. An L ∞ bound for the solution is provided by the L ∞ norm of any test function with negative energy