Asymptotically Linear Problems and Antimaximum Principle for the Square Root of the Laplacian
David Arcoya, Eduardo Colorado, Tommaso Leonori · Advanced Nonlinear Studies · 2012
Abstract This work deals with bifurcation of positive solutions for some asymptotically linear problems, involving the square root of the Laplacian (−Δ) 1/2 . A simplified model problem is the following: with Ω ⊂ ℝ N a smooth bounded domain, N ≥ 2, λ > 0, m ∈ L ∞ (Ω), m + ≢ 0 and g is a continuous function which is super-linear at 0 and sub-linear at infinity. As a consequence of our bifurcation theory approach we prove some existence and multiplicity results. Finally, we also show an anti-maximum principle in the corresponding functional setting.