On the logistic equation for the fractional p‐Laplacian

Abstract

We consider a Dirichlet problem for a nonlinear, nonlocal equation driven by the degenerate fractional p‐Laplacian, with a logistic‐type reaction depending on a positive parameter. In the subdiffusive and equidiffusive cases, we prove existence and uniqueness of the positive solution when the parameter lies in convenient intervals. In the superdiffusive case, we establish a bifurcation result. A new strong comparison result, of independent interest, plays a crucial role in the proof of such bifurcation result.

On an Anisotropic Logistic Equation

“…Since u µ is a critical point of ψ µ (see (20)), from ( 19) and ( 20), we see that u µ ∈ S µ and so u * µ ⩽ u µ ⩽ u * µ . This proves (17).…”

“…The inequality u µ ⩽ u * µ follows from Proposition 7. Next, we show the inequality u * µ ⩽ u * λ in (17). Note that…”

“…We mention the works of Afrouzi-Brown [4], Ambrosetti-Lupo [5], Ambrosetti-Mancini [6], Papageorgiou-Rǎdulescu-Repovš [7], Rȃdulescu-Repovš [8], (semilinear equations) and by Aizicovici-Papageorgiou-Staicu [9], Dong [10], Gasi ński-O'Regan-Papageorgiou [11], Iannizzotto-Papageorgiou [12], Papageorgiou-Rǎdulescu [13], and Takeuchi [14,15] (nonlinear equations). We also mention the works of Gasi ński-Papageorgiou [16] (double phase equations) and Iannizzotto-Mosconi-Papageorgiou [17] (fractional equations). All the aforementioned works deal with superdiffusive problems.…”