Brezis-Kamin Type Results Involving Locally Integrable Weights

We study existence and asymptotic behavior of entire positive bounded solutions for the following class of semilinear elliptic problem (Formula Presented) where 0 ≤ ϱ ∈ Lp<inf>loc</inf>(RN), for some N < p ≤ ∞. Here, L is a local uniform elliptic operator and f(x, s) is a nonlinearity with sublinear behavior at zero and at +∞. This type of result has already been studied in the celebrated work by H. Brezis and S. Kamin for the case when L = −∆ and ϱ ∈ L∞<inf>loc</inf>(RN). Our approach allows us to include for instance −div ((1 + |x|µ)ν∇u) = uq(|x|α + |x|β)−1 with suitable α, β > 0, µ, ν ∈ R and 0 < q < 1. Here, we include two local uniform elliptic situations: µ > 0 with ν = 1 or ν = −1. © 2024 The Author(s).