2022/10/29 by Badiale, Marino, Guida, Michela, Rolando, Sergio
#35J20 #46E30 #Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2210.16696
In this note we complete a previous study, where we got existence results for the quasilinear elliptic equation -Δw+ V( | x| ) w - w ( Δw2 )= K(|x|) g(w) in ℝN, with singular or vanishing continuous radial potentials V(r), K(r). In our previuos study we assumed, for technical reasons, that K(r) was vanishing as r → 0, while in the present paper we remove this obstruction. To face the problem we apply a suitable change of variables w=f(u) and we find existence of non negative solutions by the application of variational methods. Our solutions satisfy a weak formulations of the above equation, but they are in fact classical solutions in ℝN ∖ \0\. The nonlinearity g has a double-power behavior, whose standard example is g(t) = min \ tq1 -1, tq2 -1 \ (t>0), recovering the usual case of a single-power behavior when q1 = q2.