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

• Published in: Proceedings of the Royal Society of Edinburgh. Section A. Mathematics 2000-02, Vol.130 (1), p.141-166
• Main Authors: Molle, R., Musso, M., Passaseo, D.
• Format: Article
• Language: English
• ISSN: 0308-2105; 1473-7124
• DOI: 10.1017/S0308210500000081

We are concerned with positive solutions decaying at infinity for a class of semilinear elliptic equations in all of R N having superlinear subcritical nonlinearity. The corresponding variational problem lacks compactness because of the unboundedness of the domain and, in particular, it cannot be solved by minimization methods. However, we prove the existence of a positive solution, corresponding to a higher critical value of the related functional, under a suitable fast decay condition on the coefficient of the linear term. Moreover, we analyse the behaviour of the solution as this coefficient 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 2 k − 1 distinct positive solutions, when this coefficient splits as the sum of k bumps sufficiently far apart.