We consider the fractional Laplacian with positive Dirichlet data {
(-?)?/2 u = ?up in ?, u > 0 in ?, u = ? in Rn\?, where p > 1,0 < ? < min{2,n}, ? ? Rn is a smooth bounded domain, ? is a nonnegative function, positive
somewhere and satisfying some other conditions. We prove that there exists
?* > 0 such that for any 0 < ? < ?*, the problem admits at least one
positive classical solution; for ? > ?*, the problem admits no classical
solution. Moreover, for 1 < p ? n+?/n-?, there exists 0 < ?? ? ?* such that
for any 0 < ? < ??, the problem admits a second positive classical solution.
From the results obtained, we can see that the existence results of the
fractional Laplacian with positive Dirichlet data are quite different from
the fractional Laplacian with zero Dirichlet data.
Spectral Fractional Laplacian with Inhomogeneous Dirichlet Data: Questions, Problems, Solutions
