We investigate the existence and nonexistence of solutions to the Dirichlet problem (Figure presented.) where Ω⊂RN is a smooth bounded domain, p∈(1,∞), λ>0 and g∈C(R). Our main assumption is that f:R→R is a continuous function such that f(s)>0 for all s∈(α,β), where 0<α<β are two zeros of f. If f(0)≥0, we show that an area condition involving f and g is both sufficient and necessary in order to have a pair (λ,u)∈R+×C01(Ω¯), with u≥0 and ‖u‖C(Ω¯)∈(α,β], solving (P). We also study how the presence of the gradient term affects the existence of solution. Roughly speaking, the more negative g is, the stronger its regularizing effect on (P). We prove that, regardless of the shape of f, for any fixed λ, there always exists a function g such that (P) admits a nonnegative solution with maximum in (α,β].
Regularizing Effect of the Natural Growth Term in Quasilinear Problems with Sign-Changing Nonlinearities
Malanchini, Paolo;
2026-01-01
Abstract
We investigate the existence and nonexistence of solutions to the Dirichlet problem (Figure presented.) where Ω⊂RN is a smooth bounded domain, p∈(1,∞), λ>0 and g∈C(R). Our main assumption is that f:R→R is a continuous function such that f(s)>0 for all s∈(α,β), where 0<α<β are two zeros of f. If f(0)≥0, we show that an area condition involving f and g is both sufficient and necessary in order to have a pair (λ,u)∈R+×C01(Ω¯), with u≥0 and ‖u‖C(Ω¯)∈(α,β], solving (P). We also study how the presence of the gradient term affects the existence of solution. Roughly speaking, the more negative g is, the stronger its regularizing effect on (P). We prove that, regardless of the shape of f, for any fixed λ, there always exists a function g such that (P) admits a nonnegative solution with maximum in (α,β].I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
