Nonlinear singular problems with indefinite potential term

We consider a nonlinear Dirichlet problem driven by a nonhomogeneous differential operator plus an indefinite potential. In the reaction we have the competing effects of a singular term and of concave and convex nonlinearities. In this paper the concave term will be parametric. We prove a bifurcation-type theorem describing the changes in the set of positive solutions as the positive parameter λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document} varies.


Introduction 2 Mathematical background and hypotheses
In this section we present the main mathematical tools which we will use in the analysis of problem (P λ ). We also fix our notation and state the hypotheses on the data of the problem.
So, let X be a Banach space, X * its topological dual, and let ϕ ∈ C 1 (X ). We say that ϕ(·) satisfies the "C-condition", if the following property holds: "Every sequence {u n } n 1 ⊆ X such that {ϕ(u n )} n 1 ⊆ R is bounded and (1 + ||u n || X )ϕ (u n ) → 0 in X * as n → ∞, admits a strongly convergent subsequence . This is a compactness-type condition on the functional ϕ(·), which leads to the minimax theory of the critical values of ϕ(·) (see, for example, Papageorgiou et al. [15]). We denote by K ϕ the critical set of ϕ, that is, The main spaces in the analysis of problem (P λ ) are the Sobolev space W This cone has a nonempty interior given by int C + = u ∈ C + : u(z) > 0 for all z ∈ , ∂u ∂n | ∂ < 0 , with n(·) being the outward unit normal on ∂ .
We will also use two additional ordered Banach spaces. The first one is This cone is ordered with positive (order) cone K + = {u ∈ C 0 ( ) : u(z) 0 for all z ∈ }.
This cone has a nonempty interior given by int K + = {u ∈ K + : c ud u for some c u > 0}, whered(z) = d(z, ∂ ) for all z ∈ . On account of Lemma 14.16 of Gilbarg and Trudinger [6, p. 355], we have "c ud u for some c u > 0 if and only ifĉ uû1 u for someĉ u > 0 , (1) withû 1 being the positive, L p -normalized (that is, ||û 1 || p = 1) eigenfunction corresponding to the principal eigenvalueλ 1 > 0 of the Dirichlet p-Laplacian. The nonlinear regularity theory and the nonlinear maximum principle (see, for example, Gasinski and Papageorgiou [4, pp. 737-738]), imply thatû 1 ∈ int C + . The second ordered space is C 1 ( ) with positive (order) conê Clearly, this cone has a nonempty interior. Concerning ordered Banach spaces with an order cone which has a nonempty interior (solid order cone), we have the following result which will be useful in our analysis (see Papageorgiou et al. [15,Proposition 4.1.22]).

Proposition 1
If X is an ordered Banach space with positive (order) cone K, int K = ∅, and e ∈ int K , then for every u ∈ X we can find λ u > 0 such that λ u e − u ∈ K .
for all t > 0, and some c 1 , c 2 > 0, 1 s < p. ( Then the conditions on the map a(·) are the following: H (a) : a(y) = a 0 (|y|)y for all y ∈ R N , with a 0 (t) > 0 for all t > 0 and (ii) there exists c 3 > 0 such that and 0 pG 0 (t) − a 0 (t)t 2 for all t > 0.

Remark 1 Hypotheses H (a)(i), (ii), (iii)
are dictated by the nonlinear regularity theory of Lieberman [10] and the nonlinear maximum principle of Pucci and Serrin [21]. Hypothesis H (a)(iv) serves the needs of our problem, but in fact, it is a mild condition and it is satisfied in all cases of interest (see the examples below). These conditions were used by Papageorgiou and Rȃdulescu [13] and by Papageorgiou et al. [16].

Lemma 2 If hypotheses H (a)(i), (ii), (iii) hold, then
(a) the map y → a(y) is continuous, strictly monotone (hence maximal monotone, too); Using this lemma and (3), we obtain the following growth estimates for the primitive G(·). The examples that follow confirm that the framework provided by hypotheses H (a) is broad and includes many differential operators of interest (see [13]).
This map corresponds to the p-Laplace differential operator defined by This map corresponds to the ( p, q)-Laplace differential operator defined by Such operators arise in models of physical processes. We mention the works of Cherfils and Ilyasov [1] (reaction-diffusion systems) and Zhikov [22] (homogenization of composites consisting of two materials with distinct hardening exponent in elasticity theory).
The hypotheses on the potential term ξ(·) and on the singular part ϑ(·) of the reaction are the following: (ii) ϑ(·) is nonincreasing.

Remark 2
In the literature we almost always encounter the following particular singular term Of course, hypotheses H (ϑ) provide a much more general framework and can accomodate also singularities like the ones that follow: The following strong comparison principle can be found in Papageorgiou et al. [16,Proposition 6] (see also Papageorgiou and Smyrlis [17,Proposition 4]).

Proposition 4 If hypotheses H (a),
In what follows, p * is the critical Sobolev exponent corresponding to p, that is, Now we introduce our hypotheses on the nonlinearity f (z, x).
Carathéodory function such that f (z, 0) = 0 for almost all z ∈ and (i) f (z, x) a(z)(1 + x r −1 ) for almost all z ∈ , and all x 0, with a ∈ L ∞ ( ), x σ uniformly for almost all z ∈ ; (iv) lim sup x→0 + f (z,x) x r −1 η 0 uniformly for almost all z ∈ ; (v) for every ρ > 0, there existsξ ρ > 0 such that for almost all z ∈ the function

Remark 3
Since our aim is to find positive solutions and the above hypotheses concern the positive semiaxis R + = [0, +∞), we may assume that f (z, x) = 0, for almost all z ∈ , and all x 0. Hypotheses So, the nonlinearity f (z, ·) is ( p − 1)-superlinear near +∞. However, this superlinearity of f (z, ·) is not formulated using the AR-condition. We recall that the AR-condition (unilateral version due to (4)), says that there exist γ > p and M > 0 such that If we integrate (6a) and use (6b), we obtain the weaker condition Therefore the AR-condition implies that f (z, ·) exhibits at least (γ −1)-polynomial growth. Evidently, (7) implies the much weaker condition (5). In this work instead of the standard AR-condition, we employ the less restrictive hypothesis H ( f )(iii).
In this way we incorporate in our framework also ( p − 1)-superlinear terms with "slower" growth near +∞, which fail to satisfy the AR-condition. The following function satisfies hypotheses H ( f ) but fails to satisfy the AR-condition (for the sake of simplicity we drop the z-dependence) Finally, let us fix the notation which we will use throughout this work. For x ∈ R we set x ± = max{±x, 0}. Then for u ∈ W We know (see Gasinski and Papageorgiou [4]), that A(·) is continuous, strictly monotone (hence maximal monotone, too) and of type (S) + , that is, We introduce the following two sets related to problem (P λ ): We let λ * = sup L.

Positive solutions
We start by considering the following purely singular problem: From Papageorgiou et al. [16,Proposition 10], we have the following property.

Proposition 5 If hypotheses H (a), H (ξ ), H (ϑ) hold, then problem (8) admits a unique positive solution
Let β > ||ξ || ∞ . Then hypotheses H ( f )(i), (iv) and since 1 < q < p < r , imply that we can find c 10 , c 11 > 0 such that With v ∈ int C + from Proposition 5, we consider the following auxiliary Dirichlet problem: For this problem we prove the following result.
We will useū λ ∈ int C + from Proposition 6 to show the nonemptiness of L.
Proof From (9) we have
Note that ϑ λ (u λ ) ϑ(v) (see (28) and hypothesis H (ϑ)(ii)) and ϑ(v) ∈ L s ( ). So, as before (see the proof of Proposition 6), we infer that Therefore we have seen that The proof is now complete.
For η > 0, letũ η ∈ int C + be the unique solution of the following Dirichlet problem By Proposition 9 of Papageorgiou et al. [16], we see that given u ∈ S λ ⊆ int C + (that is, λ ∈ L), we can find small η > 0 such that u η u and η ϑ(ũ η ). (29) We will use this to obtain a lower bound for the elements of S λ .
Proof Let u ∈ S λ ⊆ int C + . Then on account of (29) we can define the following Carathéodory function We set E(z, x) = x 0 e(z, s)ds and consider the functional μ : W As before, Proposition 3 of Papageorgiou and Smyrlis [17] implies that μ ∈ C 1 (W 1, p 0 ( )). The coercivity of μ(·) (see (30)) and the sequential weak lower semicontinuity guarantee the existence ofṽ ∈ W 1, p So, we have proved thatṽ It follows from (30), (31), (32) thatṽ is a positive solution of (18). Then on account of Proposition 5, we haveṽ The proof is now complete.
Next, we show a structural property of the set L, namely that L is an interval. Moreover, we establish a kind of strong monotonicity property for the solution set S λ . H (a), H (ξ ), H (ϑ), H ( f ) hold, λ ∈ L, 0 < μ < λ and u λ ∈ S λ ⊆ int C + , then μ ∈ L and there exists u μ ∈ S μ ⊆ int C + such that u λ − u μ ∈ int C + .
Then from (40) and Proposition 4, we see that for small enough > 0 we have which contradicts the definition of m 0 . Hence λ / ∈ L and so λ * λ 0 < +∞.
By Propositions 9 and 10 it follows that
This proves Claim 1.
Passing to the limit as n → ∞ in (71) and using (72), we have u * ∈ S λ * ⊆ int C + and so λ * ∈ L.
The proof is now complete.