Bifurcation and one-sign solutions for semilinear elliptic problems in RN

In this work, we study the existence of one-sign solutions without signum condition for the following problem:(-Delta u = lambda a(x)f(u), x is an element of RN, u(x) -> 0, as |x| -> +infinity,where N >= 3, lambda is a real parameter and a is an element of C alpha loc(RN, R) for some alpha is an element of (0, 1) is a weighted function, f is an element of C alpha(R, R), and there exist two constants s2 < 0 < s1, such that f(s1) = f (s2) = f (0) = 0 and sf(s) > 0 for s is an element of R\{s1, 0, s2}. Furthermore, we consider the exact multiplicity of one-sign solutions for above problem under more strict hypotheses. We use bifurcation techniques and the approximation of connected components to prove our main results.