The effect of the domain topology on the number of positive solutions of nonlinear elliptic problems

The effect of the graph topology on the existence ofmultipeak solutions for nonlinear Schrödinger equations

“…See, for example, [2,3,4,10,11,12,14,15]. In [3], the category of the domain was used to estimate the number of the single peak solutions, while in [12,14,15], the effect of the domain topology on the existence of multipeak solutions was studied. The domain in problem (1.1) has trivial topology, so our results here emphasize the effect of the topology of the level set of the potential V (y) on the existence and multiplicity of multipeak solutions for (1.1).…”

Positive solutions for the fractional Laplacian in the almost critical case in a bounded domain

“…Beside the case when p → 2 * , the papers [1,2] treat also when a certain parameter λ appearing in the equation tends to +∞. We do not enter in details here, but the same type of result is obtained: for large λ the domain topology gives a lower bound on the number of positive solutions of the problem.…”

On the number of positive solutions of a quasilinear elliptic problem

“…In the second step, by using a technique due to Benci and Cerami [6], we relate the category of the set M with the number of positive solutions of the modified problem. It is worthwhile to mention that, since we deal with a nonhomogeneous term f (u), we cannot apply the concentration compactness principle [21] directly as in [6,13,19].…”

“…It is worthwhile to mention that, since we deal with a nonhomogeneous term f (u), we cannot apply the concentration compactness principle [21] directly as in [6,13,19]. This difficulty is overcame by a detailed study of the energy functional restricted to its Nehari manifold.…”