Positive solutions for a class of nonlinear elliptic problems in R-N

We are concerned with positive solutions decaying at infinity for a class o f semilinear elliptic equations in all of R-N having superlinear subcritica l nonlinearity. The corresponding variational problem lacks compactness bec ause of the unboundedness of the domain and, in particular, it cannot be so lved by minimization methods. However, we prove the existence of a positive solution, corresponding to a higher critical value of the related function al, under a suitable fast decay condition on the coefficient of the linear term. Moreover, we analyse the behaviour of the solution as this coefficien t goes to infinity and show that the solution tends to split as the sum of two positive functions sliding to infinity in opposite directions. Finally, we use this property to prove the existence of at least 2k - 1 distinct po sitive solutions, when this coefficient splits as the sum of k bumps suffic iently far apart.