Positive solutions for nonlinear singular elliptic equations of p-Laplacian type with dependence on the gradient

In this paper, we study a nonlinear Dirichlet problem of p-Laplacian type with combined effects of nonlinear singular and convection terms. An existence theorem for positive solutions is established as well as the compactness of solution set. Our approach is based on Leray–Schauder alternative principle, method of sub-supersolution, nonlinear regularity, truncation techniques, and set-valued analysis.


Introduction
Let ⊂ R N (N ≥ 3) be a bounded domain with C 2 boundary. In this paper, we investigate the following singular elliptic equation with Dirichlet boundary condition, p-Laplace differential operator, and a nonlinear convection term (i.e., the reaction function depends on the solution u and its gradient ∇u): (x, u(x), ∇u(x)) + g(x, u(x)) in u > 0 i n u = 0 o n ∂ . Here p stands for the p-Laplace differential operator defined by p u = div(|∇u| p−2 ∇u) for all u ∈ W 1, p 0 ( ) with 1 < p < ∞ and gradient operator ∇. For the convection term f : × R × R N → R, a suitable growth condition H ( f ) in Sect. 3 is required. The semilinear function g : × (0, ∞) → R is singular at s = 0, that is, lim s→0 + g(x, s) = +∞.
In order to emphasize the main ideas, we suppose that p < N . The case N ≤ p can be handled along the same lines. As usual, we denote p * := N p N − p , which is the Sobolev critical exponent. The solution of problem (1) is understood in the weak sense as described below.
Definition 1 We say that u ∈ W 1, p 0 ( ) is a (weak) solution of problem (1) if If p = 2, problem (1) reduces to the semilinear Dirichlet elliptic equation with a singular term and gradient dependence considered by Faraci and Puglisi [14]: A typical case in (1) and (2) Elliptic equations with singular terms represent a class of hot-point problems because they appear in applications to chemical catalysts processes, non-Newtonian fluids, and in models for the temperature of electrical conductors, see, e.g., [4,11]. An extensive literature is devoted to such problems, especially from the point of view of theoretical analysis. For instance, Ghergu and Rȃdulescu [21] established several existence and nonexistence results for boundary value problems with singular term and parameters; Gasínski and Papageorgiou [20] studied a nonlinear Dirichlet problem with a singular term, a ( p − 1)-sublinear term, and a Carathéodory perturbation; Hirano et al. [23] proved Brezis-Nirenberg type theorems for a singular elliptic problem. More details on the topics related to singular problems can be found in Crandall et al. [8], Cîrstea et al. [7], Dupaigne et al. [12], Kaufmann and Medri [25], D'Ambrosio and Mitidieri [9], Carl et al. [6], Giacomoni et al. [22], Gasiński and Papageorgiou [19], Bai et al. [2], Carl [5] and the references therein.
On the other hand, as another challenging topic, elliptic problems with convection terms have been considered in various frameworks. Amongst the results we mention: Faraci et al. [13] proved the existence of a positive solution and of a negative solution for a quasilinear elliptic problem with dependence on the gradient; Faria et al. [15] proved the existence of a positive solution for a quasi-linear elliptic problem involving the ( p, q)-Laplacian and a convection term; Zeng et al. [39] proved the existence of positive solutions for a generalized elliptic inclusion problem driven by a nonhomogeneous partial differential operator with Dirichlet boundary condition and a convection multivalued term; Papageorgiou et al. [35] proved that a nonlinear boundary value problem driven by a nonhomogeneous differential operator has at least five nontrivial smooth solutions, four of constant sign, and one nodal. For other results in this area the reader may consult: Motreanu et al. [32], Motreanu and Tanaka [33], Averna et al. [1], Faria et al. [16], Gasiński and Papageorgiou [18], and the references therein.
In this paper, under verifiable conditions, we provide the existence of positive solutions for problem (1). It is for the first time when such a result is obtained for problem (1), in particular (3), exhibiting singular and convection terms in the nonlinear case p = 2. The approach uses the method of subsolution-supersolution, truncation techniques, nonlinear regularity theory, Leray-Schauder alternative principle, and set-valued analysis. It is worth mentioning that in our analysis of problem (1) we strongly rely on multi-valued mappings arguments. Specifically, the multi-valued setting offers an efficient way to handle the smallest solution of the constructed auxiliary problem. This is another trait of novelty in our paper. The compactness of the solution set of problem (1) is proved, too.
We briefly describe the main ideas in our approach. Corresponding to a fixed smooth function w, we associate to the original statement (1) an intermediate problem replacing the gradient ∇u in f (x, u, ∇u) with ∇w and keeping unchanged the singular term. For the intermediate problem, a positive subsolution u is constructed independently of w and is shown the existence of a solution greater than u. We are thus enabled to consider the set-valued mapping S assigning to w the set S (w) of all such solutions of the intermediate problem. On the basis of the properties of the set-valued mapping S we can prove that the mapping defined by (w) equal to the minimal element of S (w) is compact. The positive solution of the original problem is obtained by applying Leray-Schauder alternative principle to the mapping . At this point we need the following smallness condition where the constants c 1 > 0 and c 2 > 0 are the coefficients of |u| and |∇u|, respectively, in the subcritical growth condition of f (x, u, ∇u), while λ 1 is the first eigenvalue of − u on W 1, p 0 ( ). This condition requires a certain compatibility between the growth of f (x, u, ∇u) and the geometry of the bounded domain imposing some restrictions on as can be seen from known estimates from above and from below for λ 1 . For instance, if is the ball B(0, R) in R N of radius R > 0 and centered at the origin, we have the estimates We refer to Benedikt and Drábek [3] and Kajikiya [24] for estimates of λ 1 on different bounded domains ⊂ R N related to geometric quantities. The rest of the paper is organized as follows. In Sect. 2 we present the needed preliminary material. Section 3 is devoted to establishing our results.

Mathematical background
Let 1 < p < ∞ and p defined by 1 p + 1 p = 1. The Lebesgue space L p ( ) is endowed with the standard norm We denote by C k ( ) for k ∈ N the space of real-valued k-times continuously differentiable functions u in such that the partial derivatives D α u continuously extend to for all |α| ≤ k.
The space C k ( ) is endowed with the norm We shall also use the Banach space where the notation ∂u/∂ν stands for the normal derivative of u with the unit outer normal ν to ∂ . For clarity regarding arguments that involve order we recall the following notions.
Definition 2 Let (P, ≤) be a partially ordered set.
For any s ∈ R, we set s ± = max{±s, 0}. If u ∈ W 1, p 0 ( ), one has The gradients of these functions are equal to Given the functions u 1 , u 2 : → R, we utilize the notation {u 1 > u 2 } = {x ∈ : u 1 (x) > u 2 (x)}, and accordingly {u 1 ≥ u 2 }. For a subset K ⊂ , its characteristic function is denoted by χ K , which means We recall the eigenvalue problem for the p-Laplacian with Dirichlet boundary condition The first eigenvalue denoted λ 1 is positive, isolated, simple, and has the following variational characterization Finally, we review some background material of set-valued analysis. More details can be found in [10,17,31,34,38].
when this holds for every x ∈ X , F is called upper semicontinuous; (ii) lower semicontinuous (l.s.c., for short) at when this holds for every x ∈ X , F is called lower semicontinuous; (iii) continuous at x ∈ X , if F is both upper semicontinuous and lower semicontinuous at x ∈ X ; when this holds for every x ∈ X , F is called continuous.

Proposition 4
The following properties are equivalent:

Proposition 5
The following properties are equivalent:

Proposition 6 Let X , Y be topological spaces and let F
An essential tool in the sequel is the Leray-Schauder alternative principle (or Schaefer's fixed point theorem), see, e.g., Gasiński and Papageorgiou [17, p. 827].
Theorem 7 Let X be a Banach space and let C ⊂ X be nonempty and convex. Assume that : C → C is a compact mapping, i.e., is continuous and maps bounded sets into relatively compact sets. Then it holds exactly one of the following statements: (a) has a fixed point;

Existence of positive solutions
Our assumptions on the data in problem (1) are as follows.

Remark 8
Hypotheses H ( f ) and H (g) permit to construct a sub-supersolution for intermediate problem (4), see below. Condition H ( f ) was employed in Faraci et al. [13], whereas condition H (g) was dealt with in Faraci and Puglisi [14] and goes back to Perera and Silva [36] and Perera and Zhang [37].
Examples of singular functions fulfilling all the requirements in H (g) can be constructed with any γ > 0 and h ∈ L q ( ) + . For instance, one can take to be an open ball in R N and choose any function as with s ∈ (0, 1) and appropriately extending for s > 1, and suitable corresponding functions h on (see [36,37]).
For w ∈ C 1 0 ( ) fixed, we first focus on an intermediate singular Dirichlet (4)

Definition 9
We say that The next lemma is essential for our development.
Proof Let u 1 , u 2 ∈ W 1, p ( ) be supersolutions for problem (4). Corresponding to any ε > 0, consider the truncation η ε : R → R given by which is Lipschitz continuous. From Marcus and Mizel [30], we know about the composition The definition of supersolution for problem (4) yields , and then summing up the resulting inequalities, we get We note that Altogether, we obtain Now we pass to the limit as ε → 0 + . Using Lebesgue's Dominated Convergence Theorem (see, e.g., [31,Theorem 2.38]) and Combining with (5) leads to (6) holds true for all v ∈ W 1, p 0 ( ) + , so u is also a supersolution of problem (4).
Similarly, we can prove the corresponding statement for subsolutions.
Denote by U w ⊂ W 1, p ( ) and U w ⊂ W 1, p ( ) the supersolution set and subsolution set of problem (4), respectively. The following result is a direct consequence gathering Lemmata 10 and 11.

Corollary 12
The sets U w and U w are upward directed and downward directed, respectively.
Next we establish the existence of subsolutions of problem (4).
Condition H (g)(iii) entitles that for each ε ∈ (0, ε 0 ) the function x → g(x, εϑ(x)) belongs to L q ( ) for some q > N , which results in According to H (g)(iii), the function x → g(x, 1) is not identically zero. Then there exists a unique u * ∈ int(C 1 0 ( ) + ) that resolves the Dirichlet problem Using the monotonicity of g on (0, 1) again, it turns out Because of ϑ, u ∈ int(C 1 0 ( ) + ), we can choose ε > 0 small enough to fulfill u − εϑ ∈ int(C 1 0 ( ) + ). Taking into account hypothesis H (g), we derive Since q > N > (p * ) , we have q < p * , where for r > 1, we denote r = r /(r − 1). Therefore we can use the Sobolev embedding theorem (see, e.g., [17, Theorem 2.5.3]), to infer that the embedding of W 1, p 0 ( ) into L q ( ) is continuous. On account of (8) we deduce and due to α ≤ 1 and (7), one has In view of f (x, s, ξ) ≥ 0 for a.e. x ∈ , all s ∈ R and ξ ∈ R N , by (9) it holds 0 and for a.e. x ∈ . Consequently, u is a subsolution of problem (4), which completes the proof.

Remark 14
From the proof Lemma 13 it is clear that the obtained subsolution u is independent of function w and belongs to int(C 1 0 ( ) + ).
We are able to show the existence of positive solutions to auxiliary problem (4).

Lemma 15 Assume that conditions H (g) and H ( f ) hold. Then problem (4) admits a positive solution u with regularity u ∈ int(C 1 0 ( ) + ), which is greater than the subsolution u.
Proof Consider the nonlinear singular truncated Dirichlet problem where f : × R → R and g : × R → R are truncated functions corresponding to f and g defined by for a.e. x ∈ and s ∈ R. Consider also the primitives G : × R → R and F : × R → R given by for a.e. x ∈ and s ∈ R. The energy functional E w : W 1, p 0 → R associated to problem (10) has the expression

Claim 1 The energy functional E w is of class C 1 .
Let u, v ∈ W 1, p 0 ( ) and t > 0. By Mean Value Theorem we may write with some τ ∈ (0, 1). Using Lebesgue's Dominated Convergence Theorem entails The expressions of G and F imply with some τ 1 , τ 2 ∈ (0, t). Invoking Lebesgue's Dominated Convergence Theorem again, we obtain Thus it holds true We can conclude that E w is of class C 1 because g(x, ·) and f (x, ·) are continuous.

Claim 2 The energy functional E w is coercive.
Through the definition of g, hypothesis H (g)(ii) and u ≤ 1, the following estimate is valid The definition of f and growth condition with a constant C 1 > 0. Therefore we get ) determines that the energy functional E w is coercive.

Claim 3 The energy functional E w is weakly sequentially lower semicontinuous.
Let u n u in W Claim 3 ensues. On the basis of Claims 1-3 we are able to apply Weierstrass-Tonelli Theorem finding (4).

Claim 4 If u is a critical point of E w , then u ≥ u and u is a solution of problem
This reads as (10). Inserting v = (u − u) + in the above equality and in (9) produces x ∈ , all s ∈ R and ξ ∈ R N . We are led to On the basis of Claim 4, by virtue of the definitions of g and f , the solution u of (10) becomes a solution of problem (4). This completes the proof.
Via Lemma 15 and Remark 16 we see that S is well-defined meaning that its values are nonempty.

Lemma 17
Assume that H (g) and H ( f ) hold. Then the set-valued mapping S is compact, that is, S maps the bounded sets in C 1 0 ( ) into relatively compact subsets of C 1 0 ( ).
Proof. Let B be a bounded subset of C 1 0 ( ), so there is a constant M > 0 such that For w ∈ B and u ∈ S (w), we have (8), and Hölder's inequality that Thanks to u p p ≤ u p /λ 1 , we obtain The smallness condition d M < λ 1 allows us to derive that S (B) is bounded in W 1, p 0 ( ). Through the nonlinear regularity theory in [26,28,29], there exists α ∈ (0, 1) such that S (B) ⊂ C 1,α ( ) is bounded as well. Since C 1,α ( ) is compactly embedded in C 1 ( ), we infer that S (B) is relatively compact in C 1 0 ( ). The next results establish the continuity of S .

Lemma 18 Assume that H (g) and H ( f ) hold. Then the set-valued mapping S is upper semicontinuous.
Proof According to Proposition 4, we must prove that for any closed subset C of C 1 0 ( ), the set is closed in C 1 0 ( ). To this end, let {w n } ⊂ S − (C) satisfy w n → w in C 1 0 ( ). For each n ∈ N there exists u n ∈ S (w n ) ∩ C, so for all v ∈ W 1, p 0 ( ). It follows from Lemma 17 that the sequence {u n } is relatively compact in C 1 0 ( ). Passing to a relabeled subsequence, we may assume that u n → u in C 1 0 ( ). Recall that u n ≥ u and C is closed in C 1 0 ( ). Hence we have The continuity of f (x, ·, ·) and g(x, ·) implies for a.e. x ∈ because u n → u and w n → w in C 1 0 ( ). From (8), for a.e. x ∈ . Letting n → ∞ in (11), by means of Lebesgue's Dominated Convergence Theorem, we see that for all v ∈ W 1, p 0 ( ), thus u is a solution of problem (4). The latter and (12) reveal that u ∈ S (w) ∩ C, or in other terms, w ∈ S − (C), achieving the proof that S is upper semicontinuous.

Corollary 19
Assume that H (g) and H ( f ) hold. If {w n } and {u n } are sequences in C 1 0 ( ) satisfying w n → w as n → ∞ and u n ∈ S (w n ) for all n ∈ N, then there exist u ∈ S (w) and a subsequence {u n k } of {u n } such that u n k → u in C 1 0 ( ) as k → ∞.
Proof It is straightforward to check that S has closed values. Then Lemma 17 guarantees that S has compact values. The desired conclusion is readily obtained from Lemma 18 and Proposition 6.

Lemma 20 Assume that H (g) and H ( f ) hold. Then the set-valued mapping S is lower semicontinuous.
Proof In order to invoke Proposition 5, let {w n } ⊂ C 1 0 ( ) satisfy w n → w in C 1 0 ( ) and let v ∈ S (w). For each n ∈ N, we formulate the Dirichlet problem In view of v ≥ u, (8) and it is clear that problem (13) has a unique solution u 0 n ∈ int(C 1 0 ( ) + ). As in the proof of Lemma 17, we can verify that, since w n → w in C 1 0 ( ), then the sequence {u 0 n } is relatively compact in C 1 0 ( ). So, there exists a subsequence {u 0 n k } of {u n } such that u 0 n k → u in C 1 0 ( ) as k → ∞ and u is the unique solution of the problem A simple comparison gives u = v. Since every subsequence {u 0 n k } of {u n } converges to the same limit v, it is true that Next, for each n ∈ N, we consider the Dirichlet problem Proceeding as before, we show that this problem has a unique solution u 1 n , which belongs to int(C 1 0 ( ) + ), and lim n→∞ u 1 n = v.
Continuing the process, we generate a sequence {u k n } k,n≥1 such that Fix n ≥ 1. As in the proof of Lemma 17, we notice that the sequence {u k n } k≥1 is relatively compact in C 1 0 ( ), so we may suppose u k n → u n in C 1 0 ( ) as k → ∞. Then it appears that ⎧ ⎨ ⎩ − p u n (x) = g(x, u n (x)) + f (x, u n (x), ∇w n (x)) in u n > 0 i n u n = 0 o n ∂ , and u n ≥ u (see Lemma 15), which amounts to saying that u n ∈ S (w n ).
We carry on the proof by the nonlinear regularity [26,28,29], the convergence in (14), and the double limit lemma (see, e.g., [17,p. Proposition A.2.35]) to obtain u n → v in C 1 0 ( ) as n → ∞. We conclude that for every sequence {w n } in C 1 0 ( ) such that w n → w in C 1 0 ( ) and for every v ∈ S (w) we can find a sequence {u n } ⊂ C 1 0 ( ) satisfying u n ∈ S (w n ) for each n ∈ N and u n → u in C 1 0 ( ). Consequently, by Proposition 5, S is lower semicontinuous.

Corollary 21 Assume that H (g)
and H ( f ) hold. Then the set-valued mapping S : C 1 0 ( ) → 2 C 1 0 ( ) is continuous in the sense of Definition 3(iii) and has compact values.
For each w ∈ C 1 0 ( ), the set S (w) has a rich order structure.

Lemma 22
Assume that H (g) and H ( f ) hold. Then for each w ∈ C 1 0 ( ), the set S (w) is downward directed in the sense of Definition 2.
Proof For any w ∈ C 1 0 ( ), let u 1 , u 2 ∈ S (w) and u := min{u 1 , where f : × R → R is defined by Arguing as in the proof of Lemma 15, we see that problem (15) admits a positive solution u with u ≥ u.
We now show that u ≤ u. Since The last inequality holds because, by Lemma 10, u is a supersolution of problem (4). Observe that the obtained inequality ensures that u ≤ u. Then from (15) and the definition of f we deduce that u ∈ S (w), which completes the proof.

Theorem 23 Assume that H (g)
and H ( f ) hold. Then, for each w ∈ C 1 0 ( ) problem (4) admits a smallest solution u w greater than the subsolution u.
Proof Lemma 22 asserts that for each w ∈ C 1 0 ( ) the ordered set S (w) is downward directed. Let B be a chain in S (w). We can find a sequence {u n } ⊂ B such that lim n→∞ u n = inf B.
Since every u n is a solution of (4) with u n ≥ u, Lemma 17 claims that the sequence {u n } is relatively compact in C 1 0 ( ). So, passing to a subsequence if necessary, there exists v ∈ C 1 0 ( ) such that u n → v in C 1 0 ( ) and v ≥ u. Therefore v = inf B, which allows us to apply Zorn's Lemma (see, e.g., [38]) to provide a minimal element u w for S (w).
We check that u w is the smallest solution of (4) greater than the subsolution u. Let u ∈ S (w). Since, as known from Lemma 22, the ordered set S (w) is downward directed, we can find u ∈ S (w) verifying u ≤ min{u w , u}. However, the minimality of u w ∈ S (w) entails which yields that u w is the smallest solution greater than the subsolution u.
Proof The fact that maps the bounded subsets of C 1 0 ( ) into relatively compact subsets in C 1 0 ( ) is the direct consequence of Lemma 17. Indeed, if B is a bounded subset of C 1 0 ( ), then S (B) is relatively compact in C 1 0 ( ), so does (B) ⊂ S (B). It remains to verify that is continuous. Let {w n } ⊂ C 1 0 ( ) satisfy w n → w and denote u n = (w n ), which reads as ⎧ ⎨ ⎩ − p u n (x) = f (x, u n (x), ∇w n (x)) + g(x, u n (x)) in u n > 0 i n u n = 0 o n ∂ .
Invoking Lemma 17 again, the sequence {u n } is relatively compact in C 1 0 ( ). Up to a subsequence, we may assume that u n → u in C 1 0 ( ). It is obvious that u ≥ u owing to u n ≥ u. On the other hand, in the limit (16) yields ⎧ ⎨ ⎩ − p u(x) = f (x, u(x), ∇w(x)) + g(x, u(x)) in u > 0 i n u = 0 o n ∂ , thus u ∈ S (w). The lower semicontinuity of S proved in Lemma 20 and the characterization of semicontinuity in Proposition 5 ensure that there exists a sequence {v n } ⊂ C 1 0 ( ) with the properties v n ∈ S (w n ) for each n ∈ N, and v n → (w) ∈ S (w).
Notice that u n = (w n ) ≤ v n and u ∈ S (w). Letting n → ∞ implies (w) ≤ u = lim n→∞ u n ≤ lim n→∞ v n = (w), that is u = (w), so the map is continuous.
We are now in a position to prove our main result. Proof First, let us emphasize that every solution of problem (1) must be positive. We claim that each solution of problem (1) is greater than the subsolution u of problem (4) constructed in Lemma 13. Let u be a solution of (1). This is expressed by From the above inequality we deduce that u ≥ u.
In order to justify that problem (1) possesses a (positive) solution we make use of Theorem 7. From Lemma 24, we know that is a compact map. It remains to prove that the set ( ) := u ∈ C 1 0 ( ) : u = t (u) for some t ∈ (0, 1) is bounded in C 1 0 ( ). For any u ∈ ( ), we have u = t (u) for some t ∈ (0, 1), or equivalently