Bifurcation and decay of solutions for a class of elliptic equations on R-N

We consider a class of nonlinear eigenvalue problems including equations such as −Δu(x) + q(x)u(x) + γ u(x)2 ξ(x)2 + u(x)2 u = λu(x) for x ∈ R , where γ > 0, q ∈ L∞(RN ) and ξ ∈ L2(RN ) are given and we are interested in eigenvalues λ ∈ R for which this equation admits a bound state, that is, a non-trivial solution in L2(RN ). The formal linearisation of this problem is −Δu+ qu = λu but we find that bound states can bifurcate at values of λ which are not in the L2−spectrum of this linear problem. It turns out that all bound states belong to C1(RN ) and decay to zero as |x| → ∞. However, for given q, ξ and λ, some bound states may decay exponentially fast whereas others do not.