A Semilinear Problem with a Gradient Term in the Nonlinearity

We consider the following semilinear problem with a gradient term in the nonlinearity (equation presented) where λ, p, q > 0 and be a bounded, smooth domain in RN. We prove that when is a unit ball and p = 1 for q ∈ (0, q∗(N)) with q∗(N) ∈ (1, 2), we have infinitely many radial solutions for 2 ≤N < 2 6-q+2p8-2q (2-q)2 + 1 and λ= λ. On the other hand, for N > 2 6-q+2p8-2q (2-q)2 +1 there exists a unique radial solution for 0 < λ< λ. © 2022 American Institute of Mathematical Sciences. All rights reserved.