We study the boundary value problem with Radon measures for nonnegative solutions of L V u := −∆u + V u = 0 in a bounded smooth domain Ω, when V is a locally bounded nonnegative function. Introducing some specific capacity, we give sufficient conditions on a Radon measure µ on ∂Ω so that the problem can be solved. We study the reduced measure associated to this equation as well as the boundary trace of positive solutions. In the appendix A. Ancona solves a question raised by M. Marcus and L. Véron concerning the vanishing set of the Poisson kernel of L V for an important class of potentials V .

In this paper we establish the existence of bound state solutions having a prescribed number of sign change forwhere ∆ m u = ∇·(|∇u| m−2 ∇u). Our result is new even for the case of the Laplacian (m = 2).

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We establish the uniqueness of the higher radial bound state solutions ofWe assume that the nonlinearity f ∈ C(−∞, ∞) is an odd function satisfying some convexity and growth conditions, and either has one zero at b > 0, is non positive and not identically 0 in (0, b), and is differentiable and positive [b, ∞), or is positive and differentiable in [0, ∞).

We establish the uniqueness of the second radial bound state solution ofWe assume that the nonlinearity f ∈ C(−∞, ∞) is an odd function satisfying some convexity and growth conditions of superlinear type, and either has one zero at b > 0, is nonpositive and not identically 0 in (0, b), and is differentiable and positive [b, ∞), or is positive and differentiable in [0, ∞).