Nonnegative Entire Bounded Solutions to some Semilinear Equations Involving the Fractional Laplacian

Abstract

The main goal is to establish necessary and sufficient conditions under which the fractional semilinear elliptic equation ∆ α 2 u = ρ(x) ϕ(u) admits nonnegative nontrivial bounded solutions in the whole space R N .

Characterization for the existence of bounded solutions to elliptic equations

“…Our methods have the advantage of being intrinsic, they will essentially still work to give similar results for some differential operators such as second order elliptic operators, parabolic differential operators, fractional Laplace operators, etc. In this context, we note that an analogous characterization result for nonlocal operators was given in [3] where ∆ is substituted by its fractional powers ∆ α 2 , 0 < α < 2.…”

Blow up boundary solutions of some semilinear fractional equations in the unit ball

“…In the following proposition, we would like to prove the existence of a solution u ∈ C + (D) to the same problem but dropping the boundedness of the boundary datum f, thus extending the result in [5].…”

“…holds for every nonnegative function ϕ belonging to the space C ∞ c (U ). We first quote from [5] the following lemma which states a straightforward and useful fact.…”

“…It should be noticed that the comparison principle as stated in the above lemma become more or less a classical result and it is widely used in the literature ( [5,14]...). Also, as we alluded to before, the minimum principle and comparison results require information on the solutions in the whole complement of the domain and not only at the boundary, consistently with the nonlocal character of the operator ∆ As a matter of fact, the continuity of f in the whole complement of D is not vital for the existence of a solution u ∈ C(D) but it guarantees rather the continuity of the solution in the whole space R N and this is not important for our purposes.…”

Semilinear equations for non-local operators: beyond the fractional Laplacian

“…For the nonlinearity f throughout the paper we assume the condition Semilinear problems for the Laplacian have been studied for at least 40 years and we refer the reader to the monograph [36] for a detailed account. The study of semilinear problems for non-local operators is more recent and is mostly focused on the fractional Laplacian, see [21,14,1,2,5,4,6,12,20]. One of the important differences between the local and nonlocal equations is that in the non-local case the boundary blow-up solutions are possible even for linear equations.…”