Nonlinear nonhomogeneous Dirichlet problems with singular and convection terms

We consider a nonlinear Dirichlet problem driven by a general nonhomogeneous differential operator and with a reaction exhibiting the combined effects of a parametric singular term plus a Carathéodory perturbation f(z,x,y)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$f(z,x,y)$\end{document} which is only locally defined in x∈R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$x \in {\mathbb {R}} $\end{document}. Using the frozen variable method, we prove the existence of a positive smooth solution, when the parameter is small.

In this problem, the map a : R N → R N in the differential operator is continuous and strictly monotone (thus, maximal monotone too) and satisfies certain other growth and regularity conditions listed in hypotheses H 0 below (see Sect. 2). These conditions are general and provide a broad framework in which we can fit many differential operators of interest, such as the p-Laplacian and the (p, q)-Laplacian (that is, the sum of a p-Laplacian and of q-Laplacian). In the reaction (right-hand side) of (P λ ), we have the competing effects of a parametric singular term u → λu -η with λ > 0 being the parameter and of a perturbation u → f (z, u, Du) which is a Carathéodory function (that is, for all x ∈ R, y ∈ R N the function z → f (z, x, y) is measurable and for a.a. z ∈ , (x, y) → f (z, x, y) is continuous). So, this perturbation is gradient dependent and on f (z, ·, y) we do not impose any growth condition. Instead we assume that near zero the function x → f (z, x, y) exhibits a kind of oscillatory behavior.
The gradient dependence of the reaction means that problem (P λ ) is not variational and so eventually our method of proof is going to be topological. We use the so-called "frozen variable method". According to this approach, in the perturbation f (z, x, y), we fix ("freeze") the y-variable. This leads to a variational problem, which a priori can be solved using tools from the critical point theory. However, the presence of the singular term leads to an energy functional which is not C 1 and so we have difficulty in applying the minimax theorems of critical point theory. So, we need to find a way to bypass the singularity and deal with C 1 -functionals. This is done by considering the purely singular problem (that is, f ≡ 0), which we show that for every λ > 0 it has a unique positive smooth solution which converges to zero in C 1 0 ( ) as λ → 0 + . We use this solution and its properties and truncation techniques, to show that, for all small values of the parameter λ > 0, the "frozen problem" has at least one positive smooth solution. In order to use topological tools (fixed point theory), we need to find a canonical way to choose such a positive solution. This is done by showing that the frozen problem has a smallest positive solution (minimal positive solution). Then we show that the minimal solution map satisfies all the requirements of Leray-Schauder alternative principle (see Sect. 2) and so we can produce a positive solution for (P λ ) when λ > 0 is small.

Mathematical background-hypotheses
The main spaces in the analysis of our problem (P λ ), are the Sobolev space W 1,p 0 ( ) and the Banach space C 1 0 ( ) = {u ∈ C 1 ( ) : u| ∂ = 0}. By · we denote the norm of the Sobolev space W 1,p 0 ( ). On account of the Poincaré inequality, we have The Banach space C 1 0 ( ) is an ordered Banach space with positive (order) cone C + = {u ∈ C 1 0 ( ) : u(z) ≥ 0 for all z ∈ }. This cone has a nonempty interior given by with n(·) being the outward unit normal on ∂ . Let X be a Banach space and ξ : X → X. We say that ξ (·) is "compact" if it is continuous and for every B ⊆ X bounded, the set ξ (B) ⊆ X is compact. The "Leray-Schauder alternative principle" asserts the following. Theorem 1 If X is a Banach space, ξ : X → X is compact and D(ξ ) = u ∈ X : u = tξ (u) for some 0 < t < 1 , then the following alternative holds: (a) D(ξ ) is unbounded; or (b) ξ (·) has a fixed point.
Let β ∈ C 1 (0, ∞) with β(t) > 0 for all t > 0 and assume that Then our hypotheses on the map a(·) are the following. H 0 : a(y) = a 0 (|y|)y for all y ∈ R N , with a 0 (t) > 0 for all t > 0 and Remark 1 Such conditions on the differential operator were used in the context of singular or convection problems, also by Papageorgiou-Rădulescu-Repovš [23] and Candito-Gasinski-Papageorgiou [4]. Hypotheses H 0 (i), (ii), (iii) are motivated by the nonlinear regularity theory of Lieberman [17] and the nonlinear maximum principle of Pucci-Serrin [28] (p. 111). Hypothesis H 0 (iv) serves the needs of our problem and it is mild. As we will see in the examples listed below, it is satisfied in all cases of interest.
Clearly the above hypotheses imply that the primitive G 0 (·) is strictly convex and strictly increasing. Let G(y) = G 0 (|y|) for all y ∈ R N . Evidently G ∈ C 1 (R N , R), it is convex and we have So, G(·) is the primitive of a(·) and on account of the convexity of G(·), we have Hypotheses H 0 lead easily to the following properties of the map a(·).

Lemma 2
The mapping a(·) is continuous, strictly monotone (hence maximal monotone too) and (a) |a(y)| ≤ c 4 (|y| s-1 + |y| p-1 ) for some c 4 > 0, all y ∈ R N ; Using this lemma and (1) we are led to the following growth restrictions on G(·).
(ii) a(y) = |y| p-2 y + |y| q-2 y with 1 < q < p. This map corresponds to the (p, q)-Laplace differential operator defined by Such operators arise often in the mathematical models of physical processes and recently there have been published several works dealing with equations driven by such operators. We mention the works of Bobkov-Tanaka [3], Papageorgiou-Zhang [26,27], Rădulescu [29], Ragusa-Tachikawa [30].

be the nonlinear operator defined by
From Gasinski-Papageorgiou [9] (Problem 2.192, p. 279), we have the following properties for this operator.

Proposition 4
The operator A(·) is bounded (that is, maps bounded sets to bounded sets), continuous, strictly monotone (hence maximal monotone too) and of type (S) + , that is, In the sequel byλ 1 (r) we denote the first (principal) eigenvalue of (r , W 1,r 0 ( )) (1 < r < ∞). We know that it is simple and isolated.
Now we introduce the hypotheses on perturbation f (z, x, y).
Carathéodory function, f (z, 0, y) = 0 for a.a. z ∈ , all y ∈ R N and (i) there exist 0 < δ 0 < θ and c 6 > 0 such that Remark 2 The above hypotheses concern only the behavior of f (z, ·, y) near zero. No global growth condition is imposed on f (z, ·, y). Hypothesis H 1 (i) dictates an oscillatory behavior for f (z, ·, y). It starts positive and by the time we have reached x = θ , the perturbation f (z, ·, y) has become negative. In the case of the equation driven by the q-Laplacian, hypothesis H 1 (ii) is a nonuniform nonresonance condition at zero. Hypothesis H 1 (iii) is satisfied if for a.a. z ∈ and all y ∈ R N the quotient function x → f (z,x,y) x p-1 is nonincreasing on (0, θ ].
Example The following function satisfies hypotheses H 1 above. For the sake of simplicity, we drop the z-dependence. We have As we already mentioned in the Introduction, due to the presence of the singular term, we have an energy functional which is not C 1 and this prevents us from using the tools of critical point theory. So, we need to find a way to bypass the singularity and deal with C 1functionals. For this reason in the next section we deal with the purely singular problem (f ≡ 0). A solution of this problem will help us isolate the singularity.

Purely singular problem
In this section we examine the following purely singular problem:
Next, let ε n = 1 n and u n = u ε n ∈ int C + for all n ∈ N. We have u 1 ≤ u n for all n ∈ N (by the claim).
We return to (9) and use h = u n ∈ W Note that (14)).
On account of (15), we can also say (at least for a subsequence) that From (16) and (17) it follows that (see , Problem 1.19, p. 38).
Again we return to (9) and choose h = u nu λ ∈ W 1,p 0 ( ), pass to the limit as n → ∞ and use (15) and (18). We obtain Therefore if in (9) we pass to the limit as n → ∞ and use (19), then (10)).
From (20) we infer that u λ is a positive solution of (Q λ ).

The frozen variable method
In this section we develop the method described in the Introduction (the frozen variable method). So, fix v ∈ C 1 0 ( ) and consider the following Dirichlet problem (the "frozen problem"): Since we have fixed the gradient variable in f , the resulting problem (P v λ ) has a variational structure. However, as we already mentioned in the Introduction, the presence of the singular term u → λu -η leads to an energy functional which is not C 1 and so we cannot use the minimax theorems of the critical point theory. To remedy this situation, we use the solution u λ of (Q λ ) to bypass the singularity and deal with a C 1 -functional.
Let S λ v be the set of positive solutions of the "frozen problem" (P v λ ). We have just proved that for v ∈ C 1 0 ( ) and 0 < λ ≤λ * we have We will show that each of these solution sets has a smallest element. In this way we have a canonical procedure to choose an element from S λ v as v ∈ C 1 0 ( ) varies. So, we define the minimal solution map on which we will use Theorem 1 (the Leray-Schauder alternative principle).
To produce a minimal element for the solution set S λ v , we need the following result providing a lower bound for the set S λ v .
Proof Let u ∈ S λ v and consider the Carathéodory functionμ λ (z, x) defined on ×R + = × (0, ∞) bŷ We consider the following singular Dirichlet problem Reasoning as in the proof of Proposition 5, we show that problem (30) has a solution u λ ∈ int C + and using (29), we show that But then (29) and Proposition 5 imply that This proof is now complete.
Using this lower bound, we show the existence of a minimal element for the set S λ v .

Proposition 8
If hypotheses H 0 , H 1 hold and 0 < λ ≤λ * , then problem (P v λ ) has a smallest positive solution u * v ∈ int C + such that Proof From Proposition 18 of Papageorgiou-Rădulescu-Repovš [23], we know that S λ v is downward directed. Then using Lemma 3.10, p. 178, of Hu-Papageorgiou [14], we can find {u n } n∈N ⊆ S λ v decreasing such that From the proof of Proposition 6 we know that S λ v ∩ [0, θ ] = ∅. So, without any loss of generality, we may assume that {u n } n∈N ⊆ [0, θ ]. We have u λ ≤ u n ≤ u 1 for all n ∈ N (see Proposition 7).
In (31) we choose h = u n ∈ W 1,p 0 ( ). Using Lemma 2(b), (32) and hypothesis H 1 (i), we infer that So, we may assume that In (31) we choose h = u nu * v ∈ W 1,p 0 ( ), pass to limit as n → ∞ and use (33). We obtain Then, if in (31) we pass to the limit as n → ∞ and use (34), we obtain It follows that This proof is now complete.
Using Proposition 8, we define the minimal solution map by setting Evidently a fixed point of ξ λ (·) is a solution of problem (P λ ). To produce a fixed point of (P λ ), we will use Theorem 1 (the Leray-Schauder alternative principle). This is done in the next section.

Positive solution
To apply Theorem 1, we need to know that ξ λ (·) is compact. The next proposition will be helpful in this respect.

Proposition 9 If hypotheses
, then we can find u n ∈ S λ v n , n ∈ N such that u n → u in C 1 0 ( ).
Proof Choosing M ≥ sup n∈N Dv n ∞ , we see that all the previous results are valid. We consider the following Dirichlet problem: div a Dy(z) = τ λ v n z, u(z) in , y| ∂ = 0 (see (24)).
We set y n = y 0 n and consider the following Dirichlet problem: div a Dy(z) = τ λ v n z, y 0 n (z) in , y| ∂ = 0.
As above this problem has a unique solution y 1 n ∈ C 1 0 ( ) and we have y 1 n → u in C 1 0 ( ) as n → ∞.
We continue this way and generate a sequence {y k n } k,n∈N ⊆ C 1 0 ( ) such that div a Dy k n (z) = τ λ v n z, y k-1 n (z) in , y k n | ∂ = 0, We will show that for every n ∈ N y k n k∈N ⊆ W 1,p 0 ( ) is bounded.
Arguing by contradiction, suppose that (40) is not true. By passing to a subsequence if necessary, we may assume that We set w k = y k n y k n , k ∈ N. Then w k = 1 for all k ∈ N and so we may assume that From (38) we have On account of (24), (41) and (42), we have τ λ v n (z, y k-1 n ) y k n p-1 w k dz → 0 as k → ∞ ⇒ w k → 0 in W 1,p 0 ( ) as k → ∞ (see (43)), which contradicts the fact that w k = 1 for all k ∈ N.
Therefore (40) is true. Then on account of (40) and the nonlinear regularity theory (see [17] and [21]), we infer that {y k n } k∈N ⊆ C 1 0 ( ) is relatively compact. Hence we may assume that From (38) and hypothesis H 1 (i), we have div a(Du n ) = τ λ v n (z, u n ) in , u n | ∂ = 0, u λ ≤ u n ≤ θ for all n ∈ N (see (24)) (46) ⇒ u n ∈ S λ v n for all n ∈ N.
This proof is now complete.
Using this proposition, we can establish the compactness of the minimal solution map ξ λ (·). From this as before using the nonlinear regularity theory of Lieberman [17], we infer that ξ λ (B) ⊆ C 1 0 ( ) is relatively compact.
Next, we show that ξ λ (·) is continuous. So, let v n → v in C 1 0 ( ) and let u n = ξ λ (v n ), n ∈ N and u = ξ λ (v). From (47) we see that {u n } n∈N ⊆ C 1 0 ( ) is relatively compact. Hence we may assume that u n → u in C 1 0 ( ) as n → ∞.
Evidently u ∈ S λ v and so we have On the other hand, from Proposition 9, we know that we can find u n ∈ S λ v n , n ∈ N such that u n → u in C 1 0 ( ) as n → ∞.
We have u n ≤ u n for all n ∈ N and so from (48) and (50), we obtain u ≤ u ⇒ u = u (see (49)).
This proof is now complete.

It follows that
for all h ∈ W 1,p 0 ( ). From Proposition 7, we know that In (51) we choose h = u ∈ W 1,p 0 ( ). Using Lemma 2(b), (52) (recall that u λ -η ∈ L s ( )) and hypothesis H 1 (i), we obtain Then as before via the nonlinear regularity theory of Lieberman [17], we see that D λ ⊆ C 1 0 ( ) is bounded (in fact relatively compact). This proof is now complete.
So we can use Theorem 1 on the map ξ λ (·) and produce a fixed point which is a positive solution of problem (P λ ), for λ ∈ (0,λ * ]. Concluding we can state the following existence theorem for problem (P λ ).