Existence, uniqueness and qualitative properties of positive solutions of quasilinear elliptic equations

We study the following quasilinear elliptic equationequation(PβPβ)−Δpu+(βΦ(x)−a(x))up−1+b(x)g(u)=0in RN, where p>1p>1, a,b∈L∞(RN)a,b∈L∞(RN), β,b,g≥0,b≢0β,b,g≥0,b≢0 and Φ∈Lloc∞(RN), infRN⁡Φ>−∞infRN⁡Φ>−∞. We provide a sharp criterion in term of generalized principal eigenvalues   for existence/nonexistence of positive solution of (PβPβ) in suitable classes of functions. Uniqueness result for (PβPβ) in those classes is also derived. Under additional conditions on Φ, we further show that:i) either for every β≥0β≥0 nonexistence phenomenon occurs,ii) or there exists a threshold value β⁎>0β⁎>0 in the sense that for every β∈[0,β⁎)β∈[0,β⁎) existence and uniqueness phenomenon occurs and for every β≥β⁎β≥β⁎ nonexistence phenomenon occurs.In the latter case, we study the limits, as β→0β→0 and β→β⁎β→β⁎, of the sequence of positive solutions of (Pβ)(Pβ).Our results are new even in the case p=2p=2.