Positive solutions for nonlinear singular superlinear elliptic equations

We consider a nonlinear nonparametric elliptic Dirichlet problem driven by the p-Laplacian and reaction containing a singular term and a $$(p-1)$$(p-1)-superlinear perturbation. Using variational tools together with suitable truncation and comparison techniques we produce two positive, smooth, ordered solutions.


Introduction
Let ⊂ R N be a bounded domain with a C 2 -boundary ∂ and let 1 < p < +∞. In this paper we study the following nonlinear Dirichlet problem with a singular reaction term: The Leszek Gasiński was supported by the National Science Center of Poland under Project No. 2015/19/B/ST1/01169. − p u(z) = u(z) −μ + f (z, u(z)) in , u| ∂ = 0, u > 0. (1.1) In this problem p stands for the p-Laplace differential operator defined by p u = div (|Du| p−2 Du) ∀u ∈ W 1, p 0 ( ), for 1 < p < +∞. Also μ ∈ (0, 1) and f : × R −→ R is a Carathéodory perturbation of the singular term (that is, for all x ∈ R, z −→ f (z, x) is measurable and for almost all z ∈ , x −→ f (z, x) is continuous). We assume that f (z, ·) is ( p − 1)-superlinear near +∞ but need not satisfy the usual in such cases Ambrosetti-Rabinowitz condition. We are looking for positive solutions and we prove the existence of at least two positive smooth solutions. Our approach is variational based on the critical point theory, together with truncation and comparison techniques.
In the past multiplicity theorems for positive solutions of singular problems were proved by Hirano et al. [20], Sun et al. [31] (semilinear problems driven by the Dirichlet Laplacian) and Giacomoni et al. [18], Kyritsi-Papageorgiou [21], Papageorgiou et al. [27], Papageorgiou-Smyrlis [28,29], Perera-Zhang [30], Zhao et al. [32]. In all aforementioned works, there is a parameter λ > 0 in the reaction term. The presence of the parameter λ > 0 permits a better control of the right-hand side nonlinearity as the parameter becomes small. In particular in [29] the authors also deal with superlinear singular problems. However, the assumptions lead to a different geometry. More precisely, in [29] the perturbation function f (z, x) has a fixed sign, that is, f (z, x) > 0. We do not assume this here. In fact our conditions here force f (z, ·) to be sign-changing by requiring an oscillatory behaviour near zero (see hypothesis H ( f )(i)). Our work here complements that of [27], where the authors deal with the resonant case, that is, in [27] the perturbation f (z, ·) is ( p −1)-linear. The present work and [27] cover a broad class of parametric nonlinear singular Dirichlet problems. We mention also the parametric work of Aizicovici et al. [2] on singular Neumann problems. For other parametric problems see also . Nonparametric singular Dirichlet problems were examined by Canino-Degiovanni [4], Gasiński-Papageorgiou [6] and Mohammed [25]. In [4,25] we have existence but not multiplicity while in [6] we have also multiplicity results (the methods of proofs in all these papers are different).

Preliminaries and Hypotheses
Let X be a Banach space and X * its topological dual. By ·, · we denote the duality brackets for the pair (X * , X ). Given ϕ ∈ C 1 (X ) we say that ϕ satisfies the Cerami condition, if the following property holds: "Every sequence {u n } n 1 ⊆ X such that {ϕ(u n )} n 1 is bounded and admits a strongly convergent subsequence." Evidently this is a kind of compactness-type condition on the functional ϕ. Using the Cerami condition one can prove a deformation theorem from which follows the minimax theory of the critical values of ϕ. A basic result in that theory is the mountain pass theorem which we will use in the sequel.
then c m r and c is a critical value of ϕ (that is, there exists u ∈ X such that ϕ(u) = c and ϕ (u) = 0).
The Sobolev space W 1, p 0 ( ) and the Banach space C 1 0 ( ) = {u ∈ C 1 ( ) : u| ∂ = 0} will be the two main spaces of this work. By · we will denote the norm of W 1, p 0 ( ). On account of Poincaré's inequality, we have The Banach space C 1 0 ( ) is an ordered Banach space with positive (order) cone This cone has a nonempty interior given by Here ∂u ∂n denotes the normal derivative of u defined by with n being the outward unit normal on ∂ .
In the next proposition, we recall the main properties of this map (see Motreanu By p * we denote the critical Sobolev exponent corresponding to p, i.e., The hypotheses on the perturbation term f are the following: and for every compact set K ⊆ , there exists c K > 0 such that and there exists β λ ∈ L 1 ( ), β λ (z) 0 for a.a. z ∈ such that (iv) for every > 0, there exists ξ > 0 such that for a.a. z ∈ the function

Remark 2.3
Since we look for positive solutions and the above hypotheses concern the positive semiaxes R + = [0, +∞), without any loss of generality, we assume that Hypothesis We stress that for the superlinearity of f (z, ·) we do not use the Ambrosetti-Rabinowitz condition which says that there exist r > p and M > 0 such that This condition implies that f (z, ·) has at least x r −1 -growth near +∞, that is for some c 0 > 0. This excludes from consideration ( p − 1)-superlinear nonlinearities with "slower" growth near +∞ (see Example 2.4). Here we replace the Ambrosetti-Rabinowitz condition with a quasimonotonicity condition on ξ(z, ·) (see hypothesis H ( f )(ii)), which incorporates in our framework more superlinear nonlinearities.
Hypothesis H ( f )(ii) is a slight generalization of a condition used by Li-Yang [23].
is nondecreasing on [M, +∞) and this in turn is equivalent to saying that for a.a. z ∈ , ξ(z, ·) is nondecreasing on [M, +∞). For details see Li-Yang [23]. Hypotheses H ( f )(i) and (iii) imply that for a.a. z ∈ , f (z, ·) exhibits a kind of oscillatory behaviour near zero. In hypothesis Evidently the condition with w(·) in hypothesis oscillatory behaviour for f (z, ·) near zero.

Example 2.4
The following function satisfies hypotheses H ( f ). For the sake of simplicity we drop the z-dependence: . Note that f although ( p − 1)superlinear, it fails to satisfy the Ambrosetti-Rabinowitz condition.
Finally let us fix our notation. If x ∈ R, we set x ± = max{± x, 0}. Then given We also mention that when we want to emphasize the domain D on which the cones C + and int C + are considered, we write C + (D) and int C + (D).
Moreover, by | · | N we denote the Lebesgue measure on R N and if ϕ ∈ C 1 (X ), then (the "critical set" of ϕ).

Positive Solutions
In this section we prove the existence of two positive smooth solution for problem (1.1).

Proposition 3.1 If hypotheses H ( f )(i) and (iii) hold, then there exists u
Proof We consider the following auxiliary singular Dirichlet problem From Proposition 5 of Papageorgiou-Smyrlis [29], we know that this problem has a unique positive solution u ∈ int C + .
With c > 0 and δ > 0 as postulated by hypotheses H ( f )(i) and (iii) respectively, we choose We set u = t u ∈ int C + . We have u(z)) for a.a. z ∈ (recall that t 1 and see hypothesis H ( f )(iii) and Papageorgiou-Smyrlis [29]). Moreover, we have u w.
Using u ∈ int C + , from Proposition 3.1 and w ∈ C 1 ( ) from hypothesis H ( f )(i), we introduce the following truncation of f (z, ·): Proposition 3 of Papageorgiou-Smyrlis [29] implies that ϕ ∈ C 1 (W 1, p 0 ( )) and we have From (3.1) it is clear that ϕ is coercive. Also, the Sobolev embedding theorem implies that ϕ is sequentially weakly lower semicontinuous. So, by the Weierstrass-Tonelli theorem, we can find u 0 ∈ W 1, p

1)] and hypothesis H ( f )(i)), so
hence u 0 w. So, we have proved that Recall that u ∈ int C + (see Proposition 3.1). So, on account of Proposition 2.1 of Marano-Papageorgiou [24], we can find 0 < c 1 < c 2 such that

From (3.1), (3.2) and (3.3), we have
Invoking Theorem B.1 of Giacomoni-Schindler-Takáč [18], we have that u 0 ∈ int C + . Therefore finally we can say that u 0 ∈ [u, w] ∩ C 1 0 ( ). If we strengthen the conditions on the perturbation term f (z, x) we can improve the condition of Proposition 3.2.

Proposition 3.3 If hypotheses H ( f )(i), (iii) and (iv) hold, then
Proof From Proposition 3.2 we already know that Let = w ∞ and let ξ > 0 be as postulated by hypothesis H ( f )(iv). We have [see (3.3)], hypotheses H ( f )(iv), (iii) and Proposition 3.1). Then (3.6) and Proposition 4 of Papageorgiou-Smyrlis [29], imply that Let D 0 = {z ∈ : u 0 (z) = w(z)}. The hypothesis on the function w (see hypothesis H ( f )(i)), implies that D 0 ⊆ is compact. So, we can find an open set U ⊆ with C 2 -boundary ∂U such that We have (3.3) and hypotheses H ( f )(i) and (iv)]. Then Proposition 5 of Papageorgiou-Smyrlis [29] (the "singular" strong comparison principle) implies that Since D 0 ⊆ U , it follows that D 0 = ∅ and so we have Therefore, we conclude that u 0 ∈ int C 1 0 ( ) [u, w]. Next we produce a second positive solution for problem (1.1).

Proposition 3.4 If hypotheses H ( f ) hold, then problem (1.1) admits a second positive solution u ∈ int C + .
Proof We introduce the following truncation of the reaction term in problem (1.1): Clearly this is a Carathéodory function. We set E(z, x) = x 0 e(z, s) ds and consider the functional ϕ * : W We know that ϕ * ∈ C 1 (W From (3.9) we have for some c 5 > 0, thus We use (3.11) in (3.8) and we have We add (3.12) and (3.13) and obtain (e(z, u + n )u + n − pE(z, u + n )) dz M 3 ∀n ∈ N, (3.14) Suppose that the sequence {u + n } n 1 ⊆ W 1, p 0 ( ) is not bounded. By passing to a subsequence if necessary, we may assume that u + n −→ +∞.
Hypothesis H ( f )(ii) implies that We can find n 0 ∈ N such that 0 < (kp) Let t n ∈ [0, 1] be such that  (3.24)]. But k > 0 is arbitrary. So, we infer that ϕ * (t n u + n ) −→ +∞ as n → +∞. Then we have 0 = d dt ϕ * (tu + n )| t=t n = ϕ * (t n u + n ), u + n (by the chain rule), so We have e(z, t n u + n )(t n u + n ) dz [see (3.7) and hypothesis H ( f )(ii)]. We use (3.28) in (3.27) and recall that u(z) −μ + f (z, u(z)) 0 for a.a. z ∈ (see hypothesis H ( f )(iii)). We have for some c 9 , c 10 > β 1  (here ϕ is as in the proof of Proposition 3.2). From the proof of Proposition 3.2, we know that u 0 ∈ int C + is a minimizer of ϕ, (3.37) while from Proposition 3.3, we know that Therefore we may assume that K ϕ * is finite or otherwise we already have an infinity of positive smooth solutions of (1.1) [see (3.7)] all bigger than u 0 and so we are done. The finiteness of K ϕ * and (3.39) imply that we can find ∈ (0, 1) small such that we conclude that u ∈ int C + , u = u 0 , u is a positive solution of (1.1) and u u 0 .
We can state the following multiplicity theorem for problem (1.1).