Existence of Positive Solution for a Singular Elliptic Problem with an Asymptotically Linear Nonlinearity
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Existence of Positive Solution for a Singular Elliptic Problem with an Asymptotically Linear Nonlinearity Ricardo Lima Alves Abstract. In this paper, we consider the existence of positive solutions for a singular elliptic problem involving an asymptotically linear nonlinearity and depending on one positive parameter. Using variational methods, together with comparison techniques, we show the existence, uniqueness, non-existence, and regularity of the solutions. We also obtain a bifurcation-type result. Mathematics Subject Classification. 35B09, 35B32, 35B33, 35B38, 35B65. Keywords. Strong singularity, nondifferentiable functional, global minima, regularity, bifurcation from infinity.
1. Introduction In this paper, we deal with the following semilinear elliptic problem involving a singular term: −Δu = a(x)u−γ + λf (u) in Ω, (Pλ ) u > 0 in Ω, u(x) = 0 on ∂Ω, where 0 < γ, Ω ⊂ RN (N ≥ 3) is a bounded domain with smooth boundary ∂Ω and a is a positive function that belongs to L1 (Ω). The continuous function f : R+ → R+ (R+ = [0, ∞[) satisfies: (f )1 lims→∞ f (s) s = θ for some θ ∈ (0, ∞). (f )2 the function s → f (s) s is non-increasing in (0, ∞).
We say that u ∈ H01 (Ω) is a solution of (Pλ ) if u > 0 almost everywhere (a.e.) in Ω, and for every φ ∈ H01 (Ω): au−γ φ ∈ L1 (Ω) The author was supported by CNPq/Brazil Proc. No. 141110/2017-1.
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R. L. Alves
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The study of singular elliptic problems started with the pioneering work of Fulks–Maybee [8] and received a considerable attention after the paper of Crandall–Rabinowitz–Tartar [7] (see, e.g., [1,6,11–16,19,20] and the references therein). Note that, due to the presence of the singular term, some difficulties arise to solve problem (Pλ ). For example, problem (Pλ ) does not have a variational structure to apply classical results of critical point theory which are useful in the study of nonlinear boundary value problems (see, e.g., [2,4,5]). Problem (Pλ ) was studied by Anello–Faraci [3] when a(x) ≡ 1 and 0 < γ < 1. By combining truncation techniques with variational methods, together with comparison techniques, they proved the existence, non-existence, and uniqueness of solution to (Pλ ), and obtained a bifurcation-type result. In this work, we complete the study done by [3] considering the case γ ≥ 1, and proving new results even when 0 < γ < 1. We would like to point out that the approach used in [3] can not be applied when γ ≥ 1. Here, we intend to use variational methods as well, but in a different way from previous works. Indeed, we give a direct method to obtain solutions of (Pλ ). In our approach, we do not use truncation as in previous works, see, for example, [3,14,15]. We do not invoke sets constraint to use the variational principle of Ekeland as in Sun [17] and Sun–Zhang [18]. In fact, the technique used in [17,18] is more efficient when the nonlinearity f (t) is homogeneous and sublinear, such as f (t) = tp , 0 < p < 1, and it may not be applied to a more general n
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