EXISTENCE OF NONNEGATIVE SOLUTIONS TO SINGULAR ELLIPTIC PROBLEMS, A VARIATIONAL APPROACH

. We consider the problem − ∆ u = χ { u> 0 } g ( .,u )+ f ( .,u ) in Ω , u = 0 on ∂ Ω , u ≥ 0 in Ω , where Ω is a bounded domain in R n , f : Ω × [0 , ∞ ) → R and g : Ω × (0 , ∞ ) → [0 , ∞ ) are Carath´eodory functions, with g ( x,. ) nonnegative, nonincreasing, and singular at the origin. We establish suﬃcient conditions for the existence of a nonnegative weak solution 0 (cid:54)≡ u ∈ H 10 (Ω) to the stated problem. We also provide conditions that guarantee that the found solution is positive a.e. in Ω. The problem with a parameter ∆ u = χ { u> 0 } g ( .,u )+ λf ( .,u ) in Ω , u = 0 on ∂ Ω , u ≥ 0 in Ω is also studied. For both problems, the special case when g ( x,s ) := a ( x ) s − α ( x ) , i.e., a singularity with variable exponent, is also considered.

Multiplicity results for positive solutions of singular elliptic Dirichlet problems were obtained in [18] and [25]; in both articles the singular term of the considered nonlinearity has the form a (x) s −α , with 0 ≤ a ∈ L ∞ (Ω) , a ≡ 0 in Ω, and α positive.
For additional references, and a systematic study of singular problems, we refer the reader to [21], [26], see also [14].
The aim of this work is to prove, under suitable rather general hypothesis on g, and for a wide class of sublinear (at ∞) functions f, existence results for nonnegative weak solutions (which may be zero on a subset of Ω with positive measure) to the following analogous of problems (1) and (2): and where Ω is a bounded domain in R n with C 1,1 boundary; λ ∈ R, f, g are functions defined on Ω × [0, ∞) and Ω × (0, ∞) respectively; with s → g (x, s) singular at the origin for x in a subset of Ω with positive measure. By a weak solution of (3) we mean a solution in the sense of the following definition: The following conditions (g 1 )-(g 5 ) and (f 1 )-(f 3 ) will be assumed throughout the paper, except in Theorem 1.3 where a weaker condition (g 2 ) will be imposed; and also in Theorem 1.6. Additional conditions on f will be required in Theorem 1.2, and in Theorems 1.3-1.6. ( is a Carathéodory function, and g (., s) ∈ L ∞ (Ω) for any s > 0. ( There exists a positive constant c such that the following inequality holds a.e. x ∈ Ω: g (x, s) ≤ c (g (x, 2s) + 1) for all s > 0 (g 4 ) D M = ∅ for any M > 0, where As we will see in Remark 2.4, conditions (g 2 ) and (g 3 ) imply that, for any s 0 > 0, there exist positive constants A and β such that, a.e. in Ω, we have g (., s) ≤ As −β for any s ∈ (0, s 0 ] . Recall that λ ∈ R is called a principal eigenvalue for −∆ in Ω, with homogeneous Dirichlet boundary condition and weight function b ∈ L ∞ (Ω) , if the problem −∆φ = λbφ in Ω, φ = 0 on ∂Ω has a solution φ (called a principal eigenfunction) such that φ > 0 in Ω. It is a well known fact that if b + ≡ 0 then this problem has a unique positive principal eigenvalue, denoted by λ 1 (b) (see e.g., Remark 2.8 below and the references therein).
Remark 2. Let us stress that the strength of the singularity has to be limited if one expects weak solutions in H 1 0 (Ω) . Indeed, Lazer and McKenna [24] considered the problem −∆u = au −α in Ω, u = 0 on ∂Ω, u > 0 in Ω, under the assumptions a ∈ C γ Ω , min Ω a > 0, α > 0, and Ω a bounded regular domain. They proved that there exists a unique solution u ∈ C 2 (Ω) ∩ C Ω ; and that u ∈ H 1 0 (Ω) if, and only if, α < 3. A clear-cut simple condition like that is elusive when dealing with a more general singular function g; condition (g 4 ) is our core hypothesis in this regard. In examples 1 and 2 of the Section 3, we apply Theorem 1.2 to show that it is not necessary that (5) holds for some β ∈ (0, 3) for a weak solution in H 1 0 (Ω) to exist.
Remark 3. Let us recall that, under the assumptions of Theorem 1.2, it is possible that no strictly positive solution exist, whereas a nonnegative solution does exist (see ([22], Example 3.7)).
The existence of nonnegative solutions for restricted versions of problem (3) was investigated in [22] and in [23]. In [22], existence results were obtained for the case g (., u) = au −α , with 0 ≤ a ∈ L ∞ (Ω) , a ≡ 0, 0 < α < 1, and f (., u) = −bu p , with 0 < p < n+2 n−2 , and 0 ≤ b ∈ L r (Ω) for suitable values of r. Those results were extended in [23] (where problem (4) was also considered) to include the case 0 < α < 3, with f satisfying conditions similar to those assumed here. Our objective in this manuscript is to obtain similar results for more general singular nonlinearities (including, for example, singular nonlinearities with variable exponent g (x, u) = a (x) u −α(x) ). In order to achieve our objective, in this work we improve the variational approach introduced in [22] and [23]. As in those works, a major obstacle is posed by the fact that, because of the singularity, the energy functional may not be Gateaux differentiable, which prevents the variational method from being applied in its standard form. A further challenge is posed by the fact that the domain of the energy functional is not an open subset of H 1 0 (Ω) . Additionally, expressing G M in terms of elementary functions was possible in [22] and in [23], but not here. To overcome these difficulties we consider, for any positive number M , the energy functional J M : D M → R. We prove that J M has a nonnegative minimizer u M ≡ 0; and that u M ∞ ≤ M, for some constant M independent of M. Using these results, and some auxiliary lemmas, we prove Theorem 1.2 in Section 3 by showing that, for M large enough, u M is a weak solution of (3) (in spite of the fact that J M may not be Gateaux differentiable at u M ). Finally, at the end of Section 3, we use Theorem 1.2 to obtain Theorems 1.3 -1.6.

Preliminaries. For any real number
where A straightforward application of Lebesgue's dominated convergence theorem and the mean value theorem give the following lemma (for a proof see e.g., [23], Lemma 2.3) Lemma 2.1. i) Let M > 0, and let {u j } j∈N be a sequence of measurable functions on Ω such that 0 ≤ u j ≤ M for all j ∈ N , and lim j→∞ u j = u a.e. in Ω for some u : Ω → R. Then lim j→∞ Ω F (., u j ) = Ω F (., u) .
In order to see that f satisfies (f 4 ), consider the number σ 0 and the function b given, for the function f, in (f 4 ). Let s 1 be as in (g 2 ). Note that g L ∞ (Ω×(σ,∞)) < ∞ for any σ > 0.
Since . Then, since u λ is nonnegative, from (47) we get u λ