Existence of Solutions for Nonlocal Supercritical Elliptic Problems

Utilizing a new variational principle, we prove the existence of a weak solution for the following nonlocal semilinear elliptic problem {(-Delta)(s)u = vertical bar u vertical bar(p-2)u + f (x), in Omega, u = 0, on R-n\Omega, where (-Delta)(s) represents the fractional Laplace operator with s is an element of(0,1], n > 2s, Omega is an open bounded domain in R-n and f is an element of L-d(Omega) where d >= 2. We are particularly interested in problems where the nonlinear term is supercritical by means of fractional Sobolev spaces. As opposed to the usual standard variational methods, this new variational principle allows one to effectively work with problems beyond the standard weakly compact structure. Rather than working with the problem on the entire appropriate Sobolev space, this new principle enables one to deal with this problem on appropriate convex weakly compact subsets.

Automated summary

By utilizing a new variational principle, the study proves the existence of weak solution for a nonlocal semilinear elliptic problem, particularly focusing on supercritical cases, and utilizing fractional Sobolev spaces for analysis. This new variational principle allows effective handling of problems beyond standard weakly compact structure. Instead of working on the entire appropriate Sobolev space, this principle enables dealing with the problem on appropriate convex weakly compact subsets.