Combined effects of asymptotically linear and singular nonlinearities in bifurcation problems of Lane–Emden–Fowler type

Multi-parameter bifurcation and asymptotics for the singular Lane–Emden–Fowler equation with a convection term

“…(f 4) lim s→∞ f (x, s) s = 0, uniformly for x ∈ Ω, then problem (1.2) has at least one solutions for all µ > 0 (see [10,11,18,35] and the references therein). The same assumptions will be used in the study of (1.1).…”

Nonnegative solutions to some singular semilinear elliptic problems

“…Theorems 1.3 and 1.4, can be viewed as partial generalizations of the already mentioned existence results contained in [8]. Let us briefly compare those results with ours: On the one hand, our assumptions on a and f are weaker than those imposed in [8]: we allow that |{x ∈ Ω : a (x) = 0}| > 0; and we do not require f > 0. Notice also that we allow f to depend on (x, u) ; and that we do not require monotonicity, either on f , or on s → f (s) /s.…”

“…was considered by Cîrstea, Ghergu and Rȃdulescu [8] under the following assumptions: Ω is a regular enough bounded domain in R n , 0 ≤ a ∈ C β Ω , 0 < f ∈ C 0,β [0, ∞) for some β ∈ (0, 1) , f is nondecreasing on [0, ∞) , f (s) /s is nonincreasing for s > 0, g is nonincreasing on (0, ∞) , lim s→0 + g (s) = +∞; and there exist α ∈ (0, 1) , σ 0 > 0, and c > 0, such that g (s) ≤ cs −α for s ∈ (0, σ 0 ). Under these hypothesis, and defining µ := lim s→∞ f (s) /s, λ * := λ 1 /µ (where λ 1 stands for the first Dirichlet eigenvalue of −∆ in Ω ), and E := u ∈ C 2 (Ω) ∩C 1,1−α Ω : ∆u ∈ L 1 (Ω) , the following results were proved:…”

“…( [8], Theorem 1): If µ = 0 and min Ω a > 0 (respectively min Ω a = 0), then, for all λ ∈ R (resp. λ ≥ 0), problem (1.3) has a unique solution u λ ∈ E , the map λ → u λ is strictly increasing, and each u λ satisfies c 1 d Ω ≤ u λ ≤ c 2 d Ω for some positive constants c 1 and c 2 , where d Ω := dist (., ∂ Ω) ( [8], Theorem 2): If µ > 0 and λ ≥ λ * , then (1.3) has no solutions in E .…”

“…λ ≥ 0), problem (1.3) has a unique solution u λ ∈ E , the map λ → u λ is strictly increasing, and each u λ satisfies c 1 d Ω ≤ u λ ≤ c 2 d Ω for some positive constants c 1 and c 2 , where d Ω := dist (., ∂ Ω) ( [8], Theorem 2): If µ > 0 and λ ≥ λ * , then (1.3) has no solutions in E . Furthermore, if µ > 0 and min Ω a > 0 (respectively min Ω a = 0), then (1.3) has a unique solution u λ ∈ E for any λ < λ * (resp.…”