Some recent results on singular p-Laplacian equations

3.1 The case p = 2

Let Ω be a bounded domain in R N , N ≥ 3 , with smooth boundary ∂ Ω ; let a : Ω → R 0 + be nontrivial measurable; and let γ > 0 . The simplest singular elliptic Dirichlet problem is written as:

Since the pioneering papers [14,15, 16,17,18], a wealth of existence, uniqueness or multiplicity, and regularity results concerning (3.1) have been published. We refer the reader to the monograph [4] as well as the surveys [1,2] for an exhaustive account. Roughly speaking, four basic questions can be identified:

• Find the right conditions on the datum a . Usually, a ∈ L q ( Ω ) with q ≥ 1 is enough for existence. However, starting from the works [19,20], the case when a is a bounded Radon measure took interest.

• Consider nonmonotone singular terms. This is a difficult task, mainly when we want to guarantee the uniqueness of solutions.

• Insert convective terms on the right-hand side. For equations driven by the Laplacian, good references are [4, section 9] and [21]. Otherwise, cf. [22,23, 24,25].

• Substitute the Laplacian with more general elliptic operators. Obviously, the first attempt might be to consider equations driven by the p-Laplacian, and this section aims to provide a short account of the recent literature. However, further possibly nonhomogeneous operators have been considered; see, e.g., [14,15,20,26,27,28, 29,30,31].

3.2 Existence and multiplicity

Consider the model problem

where a : Ω → R 0 + denotes a nonzero measurable function, γ , λ > 0 , while f : Ω × R 0 + → R satisfies Carathéodory’s conditions. Let us stress that, here, the parameter λ multiplies the nonsingular term.

In 2006, Perera and Silva investigated (3.2) under the following assumptions, where f is allowed to change sign.

1. There exist φ 0 ∈ C 0 1 ( Ω ¯ ) + and q ˆ > N such that a φ 0 − γ ∈ L q ˆ ( Ω ) .

2. With appropriate δ , c 1 > 0 , one has

   f ( x , t ) ≥ − c 1 a ( x ) in Ω × [ 0 , δ ] .

3. To every M > 0 , there are corresponding h ∈ L 1 ( Ω ) and c 2 > 0 such that

   − h ( x ) ≤ f ( x , t ) ≤ c 2 ∀ ( x , t ) ∈ Ω × [ 0 , M ] .

4. With appropriate q ∈ ] 1 , p ∗ [ and c 3 > 0 , one has

   f ( x , t ) ≤ c 3 ( t q − 1 + 1 ) in Ω × R 0 + .

5. There are t 0 > 0 and μ > p such that

   0 < μ ∫ 0 t f ( x , τ ) d τ ≤ t f ( x , t ) ∀ ( x , t ) ∈ Ω × [ t 0 , + ∞ [ .

Proofs employ perturbation arguments and variational methods, which were previously introduced in [36]. An immediate but hopefully useful consequence of Theorem 3.1 is as follows:

Further results concerning (3.3) can be found in the work of Aranda and Godoy [37], where a continuous nonincreasing function g ( u ) takes the place of u − γ and, from a technical point of view, fixed point theorems for nonlinear eigenvalue problems are exploited.

The case λ = 0 in (3.3) was well investigated by Canino, Sciunzi, and Trombetta [38], with a special attention to uniqueness (see Section 3.3). Here, given u ∈ W loc 1 , p ( Ω ) ,

The proof of this result relies on a technique that was previously introduced in [29] for the semi-linear case. It employs truncation and regularization arguments. The work [39] contains a version of Theorem 3.3 for the so-called Φ -Laplacian. A more general problem patterned after

where μ denotes a nonnegative bounded Radon measure on Ω while γ ≥ 0 is thoroughly studied in [40]; see also [41] and references therein.

Finally, as regards Problem (3.2) again, papers [42, 43,44,45] do not require Ambrosetti-Rabinowitz’s condition ( a 5 ) , whereas [46] establishes the existence of at least three weak solutions. Moreover, a possibly nonhomogeneous elliptic operator is considered in [44], but λ = 1 .

The nice paper [47] investigates the problem

where 0 < γ < 1 and 1 < p < q < p ∗ . It should be noted that, here, contrary to above, the parameter λ multiplies the singular term. Combining known variational methods with a C 1 , α ( Ω ¯ ) -regularity result [47, Theorem 2.2] for solutions to (3.5) and a strong comparison principle [47, Theorem 2.3], the authors obtain the following.

The case q = p ∗ is also studied, and it is shown that γ < 1 is a reasonable sufficient (and likely optimal) condition to obtain C 1 ( Ω ¯ ) -solutions of (3.5).

If p = 2 and, roughly speaking, a ≡ − 1 while f does not depend on u , then Problem (3.2) was fruitfully studied in [48].

We end this section by pointing out two very recent works, namely [49], which deals with possibly nonmonotone singular reactions (see also [50,51], essentially based on sub-super-solution methods), and [31], which is devoted to singular equations driven by the ( p , q ) -Laplace operator u ↦ Δ p u + Δ q u .

3.3 Uniqueness

Surprisingly enough, if p ≠ 2 , then the uniqueness of solutions looks a difficult matter, even for the model problem

As observed in [38], this is mainly caused by the fact that, in general, solutions do not belong to W 0 1 , p ( Ω ) once γ ≥ 1 . The paper [38] provides two different results. The first one (Theorem 1.4) holds in star-shaped domains, while the other is as follows.

We next point out that, for γ ≤ 1 , Theorem 3.4 of [40] establishes the uniqueness of renormalized solutions to (3.4).

The situation becomes quite clear when p = 2 and one seeks sufficiently regular solutions. Denote by φ 1 a positive eigenfunction corresponding to the first eigenvalue λ 1 of the problem − Δ u = λ u in Ω , u = 0 on ∂ Ω .

See also the nice paper [52]. As regards weak solutions, one has the following:

Another uniqueness case occurs when γ > 1 .

3.4 Equations with convective terms

Consider the problem

where p < N while f : Ω × R 0 + × R N → R 0 + and g : Ω × R + → R 0 + satisfy Carathéodory’s conditions. In 2019, Liu, Motreanu, and Zheng established the existence of solutions u ∈ W 0 1 , p ( Ω ) to (3.7) under the following hypotheses, where λ 1 stands for the first eigenvalue of − Δ p in W 0 1 , p ( Ω ) .

1. There exist c 0 , c 1 , c 2 > 0 such that c 1 + c 2 λ 1 1 − 1 / p < λ 1 and

   f ( x , t , ξ ) ≤ c 0 + c 1 t p − 1 + c 2 ∣ ξ ∣ p − 1 ∀ ( x , t , ξ ) ∈ Ω × R 0 + × R N .

2. g ( x , ⋅ ) is nonincreasing on ( 0 , 1 ] for all x ∈ Ω and g ( ⋅ , 1 ) ≢ 0 .

3. With appropriate θ ∈ int ( C 0 1 ( Ω ¯ ) + ) , q ˆ > max { N , p ′ } , and ε 0 > 0 , the map x ↦ g ( x , ε θ ( x ) ) belongs to L q ˆ ( Ω ) for any ε ∈ ( 0 , ε 0 ) .

We think it is worthwhile to sketch the main ideas of the proof. For every fixed w ∈ C 0 1 ( Ω ¯ ) , an intermediate problem, where ∇ w replaces ∇ u in f ( x , u , ∇ u ) and the singular term remains unchanged, is considered. The authors construct a positive sub-solution u ̲ ∈ int ( C 0 1 ( Ω ¯ ) + ) independently of w and show the existence of a solution greater than u ̲ . If S ( w ) denotes the set of such solutions, then, via suitable properties of the multi-function w ↦ S ( w ) , it is proved that the map Γ , which assigns to every w the minimal element of S ( w ) , is completely continuous. Now, applying Leray-Schauder’s alternative principle to Γ yields a solution u ∈ int ( C 0 1 ( Ω ¯ ) + ) to (3.7).

The recent paper [24], partially patterned after [55], treats the Robin problem

where A : R N → R N denotes a continuous strictly monotone map having suitable properties, which basically stem from Lieberman’s nonlinear regularity theory [56] and Pucci-Serrin’s maximum principle [57]. By the way, the conditions on A include classical nonhomogeneous operators such as the ( p , q ) -Laplacian. Moreover, β is a positive constant while ∂ ∂ ν A indicates the co-normal derivative associated with A . If p = 2 , then a uniqueness result is also presented; cf. [24, Theorem 4.2].

The special case A ( ξ ) ≔ ∣ ξ ∣ p − 2 ξ , g ( x , t ) ≔ t − γ for some 0 < γ < 1 , and β = 0 (which reduces (3.8) to a Neumann problem) has been investigated in [8] without imposing any global growth condition on t ↦ f ( x , t , ξ ) . Instead, a kind of oscillatory behavior near zero is taken on. For such an f , the work [25] establishes the existence of a solution u ∈ C 0 1 ( Ω ¯ ) to the parametric problem

provided λ > 0 is small enough.

Finally, the very recent paper [58] treats Φ -Laplacian equations with strongly singular reactions perturbed by gradient terms.

4.1 The case p = 2

Let N ≥ 3 , let a : R N → R 0 + be nontrivial measurable, and let γ > 0 . The simplest singular elliptic problem in the whole space is written as:

Sometimes it is also required that u ( x ) → 0 as ∣ x ∣ → ∞ . Since the pioneering papers [59,60, 61,62], some existence and uniqueness results concerning (4.1) have been published. We refer the reader to the monograph [4] for a deep account. Roughly speaking, four basic questions can be identified:

• Find the right hypotheses on a . Usually, a ∈ C loc 0 , α ( R N ) + and

  ∫ 1 ∞ r max ∣ x ∣ = r a ( x ) d r < ∞

  (cf. condition ( a 8 ) below) guarantee both existence and uniqueness of solutions u ∈ C loc 2 , α ( R N ) .

• Replace u − γ with a function f ( u ) such that lim t → 0 + f ( t ) = ∞ . This was done in [63,64] for decreasing f . Later on, nonmonotone singular reactions were also fruitfully treated [65,66,67].

• Put convective terms on the right-hand side. For equations driven by the Laplacian, a good reference is [4, section 9.8]; cf. in addition [68,69].

• Generalize the left-hand side of the equation. The case of a second-order uniformly elliptic operator is treated in [27,70], while [71] deals with u ↦ − Δ u + c ( x ) u , where c ∈ L loc ∞ ( R N ) + .

4.2 Existence and multiplicity

To the best of our knowledge, the first paper treating singular p-Laplacian equations on the whole space is that of Goncalves and Santos [76], published in 2004. The authors consider the problem

where a ∈ C 0 ( R N ) + is radially symmetric while f ∈ C 1 ( R + , R + ) , and assume that:

1. The function t ↦ f ( t ) t p − 1 is nonincreasing on R + .

2. lim inf t → 0 + f ( t ) > 0 as well as lim t → ∞ f ( t ) t p − 1 = 0 .

3. If Φ ( r ) ≔ max ∣ x ∣ = r a ( x ) , r > 0 , then

   0 < ∫ 1 ∞ [ r Φ ( r ) ] 1 p − 1 d r < ∞ for 1 < p ≤ 2 , 0 < ∫ 1 ∞ r ( p − 2 ) N + 1 p − 1 Φ ( r ) d r < ∞ for p > 2 .

The proof exploits fixed point arguments, the shooting method, and sub-super-solution techniques.

One year later, Covei [77] did not assume a to be radially symmetric but locally Hölder continuous and positive, replaced conditions ( a 6 ) and ( a 7 ) with those below, and obtained similar results. See also [78], where the asymptotic behavior of solutions is described.

1. The function t ↦ f ( t ) ( t + β ) p − 1 turns out decreasing on R + for some β > 0 .

2. lim t → 0 + f ( t ) t p − 1 = ∞ and f ( t ) ≤ c for any t that is large enough.

where 1 < p < N , 0 < γ < 1 , λ > 0 , max { p , 2 } < q < p ∗ , and the coefficients fulfill

It may be of interest to point out that this result is proved by combining sub-super-solution methods with the mountain pass theorem for continuous functionals.

A meaningful case occurs when a , b : R N → R 0 + turn out nonzero locally Hölder continuous functions. In fact, define

From [81, Remarks 1–2] it follows:

Via sub-super-solution techniques, Lemma 4.4 gives rise to the following:

This result was next generalized under various aspects by the same author and Rezende [82]; cf. also [80].

Finally, infinite semi-positone problems, i.e., lim t → 0 + f ( t ) = − ∞ , were fruitfully investigated in [83]. Precisely, given a ∈ L ∞ ( R N ) and f ∈ C 0 ( R + ) , consider the problem

where λ > 0 and 1 < p < N . The following conditions will be posited.

1. There exists γ ∈ ] 0 , 1 [ such that lim t → 0 + t γ f ( t ) = c 0 ∈ R − .

2. lim t → ∞ f ( t ) = ∞ but lim t → ∞ f ( t ) t p − 1 = 0 .

3. inf ∣ x ∣ = r a ( x ) > 0 for all r > 0 and 0 < a ( x ) < C 0 ∣ x ∣ σ in R N ⧹ { 0 } with suitable C 0 > 0 , σ > N + γ N − p p − 1 .

4.3 Uniqueness

As far as we know, uniqueness has been addressed only in [76, Remark 1.2] and [77, Section 2] under the key assumption ( a 6 ′ ) above. The arguments of both papers rely on a famous result by Diaz and Saa [84]. Theorem 1.3 of [85] contains a nice idea to achieve uniqueness for singular problems in exterior domains.

4.4 Equations with convective terms

To the best of our knowledge, there is only one paper concerning singular quasi-linear elliptic equations in the whole space and with convective terms, namely [86]. It treats the problem

where N ≥ 2 and 1 < p < N . The differential operator u ↦ div A ( ∇ u ) is as in (3.8), while f : R N × R + → R 0 + and g : R N × R N → R 0 + fulfill Carathéodory’s conditions. Moreover,

and

Here, x 0 ∈ R N , σ ∈ ] 0 , 1 [ , γ ≥ 1 , r ∈ [ 0 , p − 1 [ , as well as

To prove this result, the authors first use sub-super-solution techniques to solve some auxiliary problems obtained by shifting the singular term and working in balls. A compactness result, jointly with a fine local energy estimate on super-level sets of solutions, then yields the conclusion.