Comparison principle for elliptic equations with mixed singular nonlinearities

We deal with existence and uniqueness of positive solutions of an elliptic boundary value problem modeled by \begin{equation*} \begin{cases} \displaystyle -\Delta_p u= \frac{f}{u^\gamma} + g u^q&\mbox{in $\Omega$,} \\ u = 0&\mbox{on $\partial\Omega$,} \end{cases} \end{equation*} where $\Omega$ is an open bounded subset of $\mathbb{R}^N$, $\Delta_p u:=\text{div}(|\nabla u|^{p-2}\nabla u)$ is the usual $p$-Laplacian operator, $\gamma\geq 0$ and $0\leq q\leq p-1$; $f$ and $g$ are nonnegative functions belonging to suitable Lebesgue spaces.

where Ω is an open bounded subset of R N , ∆ := div(|∇ | −2 ∇ ) is the -Laplacian operator (1 < < N), γ ≥ 0 are such that < − 1 or = − 1, which correspond to the sublinear and to the linear behaviour in case = 2; here are nonnegative functions belonging to suitable Lebesgue spaces. Clearly the Dirichlet problem (1.1) is singular since the request that the solution is zero on the boundary of the set implies that the right hand side blows up. For (1.1) we are mainly interested to existence and uniqueness of possibly unbounded solutions with finite energy (i.e. ∈ W 1 0 (Ω)). Let us briefly recall the mathematical framework concerning problem (1.1); we start with the non-singular case, namely ≡ 0. The main idea of this paper comes from the seminal paper [9] where the authors show existence and uniqueness of a solution ∈ H 1 0 (Ω) ∩ L ∞ (Ω) to (1.1) in case = 2, ≡ 0, < 1 and as a bounded nonnegative function. Let us also mention that classical arguments apply once that is bounded in order to get a C 1 -solution, at least when the set Ω is smooth enough. Later, in [6], in presence of a possibly unbounded and if < − 1, the existence of a solution is proven through an approximation process; here, even in the nonvariational case, it is proven existence of a solution with infinite energy (i.e. ∈ W 1 0 (Ω)) for rough data .
For what concerns the weak theory of the purely singular case, existence of a distributional solution to (1.1) when the is only a nonnegative function in L (Ω) ( ≥ 1) is established in [7]. This solution, if γ ≤ 1 (i.e. the mild singular case), attains the boundary datum in the classical sense of Sobolev traces; otherwise, when γ > 1 (i.e. the strong singular case), only a power of the solution has zero Sobolev trace and the solution is shown to be locally in the same space. Later, in [18,33,19], existence of solutions to (1.1) is given when the right hand side is of the general form ( ) , with as a nonnegative and not necessarily monotone function such that ( ) ≤ −γ near zero and just bounded at infinity. For the nonhomogeneous case in which = 0 and ≡ 0 we mention [32].
Dealing with uniqueness is more tricky; in [5] the authors show that the solution is unique in the class of H 1 0 (Ω) and this kind of result has been extended to general nonincreasing nonlinearities and nonlinear operators in [31] for solutions in W 1 0 (Ω). In [8], when = 2, the authors show that there is at most one solution to (1.1) belonging to W 1 1 0 (Ω). In [33], uniqueness of a distributional solution belonging to W 1 1 loc (Ω) (with suitable boundary conditions) is shown for a general measure datum and a nonincreasing nonlinearity. Finally in presence of a very general nonlinear operator and a nonincreasing it is shown in [19] the existence and uniqueness of a renormalized solution for a diffuse measure datum . For further reading on singular problems we refer to [10,11,12,21,23,24,32] As one should expect the literature concerning (1.1) in presence of both and not identically zero is less investigated. Already in [38] the author proves existence of a classical solution to (1.1) when both and are regular enough, = 2 and < 1. In the same direction we refer to [15] where it is also investigated the superlinear case, which is a completely different framework. The uniqueness of classical solutions to (1.1) is shown in [36] in presence of the Laplacian operator and < − 1; we also refer to [13] where, in case of regular and constant, it is proved existence and uniqueness of solutions to (1.1) if ≤ 1; here in the linear case it is proved existence under a smallness assumption on and nonnexistence otherwise. Then in [30], for > 1, through a sub and supersolution argument it is shown existence of solutions to (1.1) when the right hand side is of the form ( ) + ( ) and no monotonicity is assumed on . In [14] it is investigated the existence of a solution to (1.1) in case = 2 when and are functions in suitable Lebesgue spaces. Let us mention that in [35], for > 1, the authors show existence and uniqueness of finite energy solutions to (1.1) under suitable assumptions on . We finally refer to [22,26] for more interesting results.
The aim of this work is twofold. Firstly, we deal with uniqueness of finite energy solutions by employing the idea contained in [9]. More precisely we want to prove it for positive solutions to the Dirichlet problem associated to where > 1 and F is a Carathéodory function which is possibly unbounded both at the origin and at the infinity and such that F ( ) 1− decreases with respect to for a.e. ∈ Ω (1.3) Here the major difficult is dealing with a nonlinear operator when looking for comparison principles. Another issue which needs to be underlined is that the solutions are not required to be bounded; this implying the need of a suitable truncation arguments. It is also worth mentioning that (1.3) allows to deal with the case ≤ − 1, at least for positive if one considers the model case given by (1.1). This result is presented as the comparison principle given by Theorem 2.2 which, as a simple corollary, takes to uniqueness of finite energy solutions. Other than uniqueness, we are interested to instances of finite energy solutions to (1.2); this is done both in the mild and in the strongly singular case by means of approximation arguments firstly if < − 1; then we also give an existence result in case = − 1. Summarizing, if < − 1, we provide existence of finite energy solutions to equations as in (1.1) (Ω) where we mean L 1 (Ω) once that γ = 1. Otherwise, we show that if ∈ L (Ω) with 1 < γ < 2 − 1 then the existence is guaranteed under the same assumptions on . Let us also highlight that, as remarked in Section 3.2, there are instances in which one could expect finite energy solutions up to γ < 1 + ( −1) ( −1) . Finally, once again if ∈ L * 1−γ

′
(Ω), we also show the existence of a solution in case = − 1 under a smallness assumption on .
Let us mention that formally the change of variable = γ+1 γ + 1 for = 2 takes (1.1) to the following equation which, for = 0, was extensively studied in the past, see for instance [1,2,3,20,25]. The previous discussion could be formalized and the existence and uniqueness results given in the current paper could provide information regarding problem (1.4).
The plan of the paper is the following: in Section 2 we state and prove the comparison principle and the associated uniqueness result for problems as in (1.1) (Theorem 2.2 and Corollary 2.3). In Section 3 we give some existence results; precisely we investigate both the mild and the strongly singular case when < − 1 (Theorem 3.2 and Theorem 3.4); moreover we also treat a case in which = − 1 (Theorem 3.5).
1.1. Notation. In the entire paper Ω is an open and bounded subset of R N , with N ≥ 1. We denote by ∂A the boundary and by |A| the Lebesgue measure of a subset A of R N . By C (Ω), with ≥ 1, we mean the space of C functions with compact support in Ω.
For any > 1, ′ := −1 is the Hölder conjugate exponent of , while for any 1 ≤ < N, * = N N− is the Sobolev conjugate exponent of . We denote by χ E the characteristic function of E ⊂ Ω, namely ∈ Ω \ E and by + := max( 0) − := − min( 0) the positive and the negative part of a function . We will widely use the following function defined for a fixed > 0 and ∈ R T ( ) = max(− min( )) (1.5) and If no otherwise specified, we will denote by C several constants whose value may change from line to line. These values will only depend on the data (for instance C may depend on Ω, N and ) but they will never depend on the indexes of the sequences we will often introduce.

C
Let 1 < < N and let us consider the following problem We start specifying the notion of weak solution to (2.1).
In order to deal with uniqueness of solutions, we present a comparison principle for solutions to (2.1) provided the right hand side enjoys some monotonicity condition. In particular let us consider 1 2 We state the main result of this section.
Theorem 2.2 (Comparison Principle). Let us assume G 1 G 2 are nonnegative functions such that either G 1 ( ) 1− or G 2 ( ) 1− is decreasing with respect to and for almost every ∈ Ω and for almost every ∈ Ω and for all ∈ (0 ∞). Let 1 and 2 be weak solutions to problem (2.3) with data, respectively, As a simple corollary of the previous result, one has that uniqueness holds for weak solutions to (2.1).

2.1.
Proof of the comparison principle. In this section we prove the comparison principle for weak solutions to problem (2.1) and, as a consequence, we deduce the uniqueness result, namely Corollary 2.3.

Proof of Theorem 2.2.
First of all we need to show that for any weak solution to (2.1), the formulation (2.2) can be extended for W 1 -test functions. We consider a nonnegative ∈ W 1 0 (Ω) and a sequence of nonnegative functions where ρ η is a smooth mollifier and φ is a sequence of nonnegative functions in C 1 (Ω) which converges to in W 1 0 (Ω). Hence let us take η as a test function in (2.2), yielding to We want to pass first η to zero and then to infinity in the previous. Since ∈ W 1 0 (Ω) one can pass to the limit the first term recalling that η converges to in W 1 0 (Ω). For the right hand side one has that F ( ) ∈ L 1 loc (Ω) that gives that we can pass η → 0 since η converges * -weakly in L ∞ (Ω) to which has compact support in Ω. Hence we deduce Now let observe that by the Young inequality and by the Fatou Lemma with respect to , one gets Now we take → ∞ in (2.5). For the term on the left hand side we can reason as already done when η → 0. For the right hand side of (2.5) one can easily apply the Lebesgue Theorem since for every ∈ W 1 0 (Ω).
Since 1 and 2 are weak solutions to problem (2.3) with data G 1 G 2 then, recalling (2.7), one can test both equations with W 1 0 -functions. From here we suppose that G 1 ( ) 1− is decreasing with respect to for almost every ∈ Ω; if one is in the other case, then slight modifications will be needed. Let us fix ε > 0 and ∈ N and let us define We consider the following two functions: where T is defined by (1.5). Let us also underline that ψ 1 ψ 2 ∈ W 1 0 (Ω) (see Remark 2.5 below). One has We choose ψ 1 and ψ 2 as test functions in equations solved by, respectively, 1 and 2 and we subtract them yielding to Now using the following classical estimate due to the convexity of the power function (recall that > 1) Noting that the first term of (2.9) is nonnegative, we have Since 1 2 are positive then one has that + ε ( − ε ) converges to + ( − resp.) and˜ ε converges to˜ almost everywhere in Ω, where Moreover, using that T ( ) ≤ for ≥ 0, we deduce that and where the last inequality holds by means of the Langrange Theorem. It follows that˜ ε (2.12) (2.13) Since 1 ∈ W 1 0 (Ω) and from (2.6) one has that the right hand side of (2.14) belongs to L 1 (Ω). This implies, applying the Lebesgue Theorem, that˜ ε strongly converges to˜ in L 1 (Ω). Now starting from (2.11) and applying the Vitali Theorem, we obtain that + ε → + strongly in L 1 (Ω) ( Now we pass to the limit in (2.17) as tends to infinity. We note that χ A tends to 0 as tends to infinity. Moreover, using (2.12) with ε = 0, we have . This implies, applying the Lebesgue Theorem, that As regards the second term in the right hand side of (2.17), from G 1 ( ) 1− decreasing with respect to , one has that where the right hand side of (2.19) is increasing in . Applying Beppo Levi's Theorem, we obtain that By passing to the limit as tends to infinity in (2.17), using (2.18) and (2.20), we have Furthermore from the fact that G 1 ( ) 1− is decreasing with respect to , one yields to in Ω which, gathered with (2.21), gives that ( 1 − 2 ) + ≡ 0, that is 1 ≤ 2 almost everywhere in Ω.

Remark 2.5.
Here we show that ψ 1 ψ 2 defined by (2.8) belong to W 1 0 (Ω). We focus on ψ 2 . As a consequence of Lemma 1 1 contained in [37] and the fact the 1 2 have finite energy, we have that the function ψ defined as belongs to W 1 0 (Ω) for every ≥ 0. Moreover, by computing its gradient, we get It follows from the definition of A ε that with C ( ε) a positive constant dependent only on ε. Hence, using 1 2 ∈ W 1 0 (Ω), we deduce that {ψ } is bounded in W 1 0 (Ω) uniformly in . Moreover ψ converges to ψ 2 almost everywhere in Ω. So that ψ converges to ψ 2 weakly in W 1 0 (Ω) and ψ 2 belongs to W 1 0 (Ω). As regards , in a similar way it is possible to prove that ψ 1 belongs to W 1 0 (Ω).

E
In this section we give existence results to (2.1) for some explicit nonlinearities F of the following form where are nonnegative functions belonging to suitable Lebesgue space, with ≡ 0, and and Remark 3.1. Let us observe that (3.3) implies that can be extended by continuity at 0 defining (0) = 0.
We underline that we are not assuming any kind of monotonicity on the functions but just some control from the above. Moreover, the case of continuous and bounded are well contained in our existence result. For the sake of clarity we reformulate the problem under the assumption (3.1): At first we state an existence result in case γ ≤ 1 and < − 1, which we recall that corresponds to the sublinear case when = 2 ; let us explicitly note that in the sequel we define * 1−γ ′ := 1 if γ = 1.
In particular one has the following result.  is contained in [6].
Next we deal with the more difficult case of a strong singularity; here, in order to deduce an existence result, we need some regularity on the Ω. Finally we also dealt with = − 1. In the next result we denote by C the best constant for the Poincaré inequality in Ω; we also recall that C is the one defined by (3.3).  (Ω) is guaranteed by [28]. Moreover, by Theorem 4.2 of [37], we get that is bounded and, since the right hand side of (3.5) is nonnegative, that is nonnegative. This allows to apply Theorem 2.2, yielding to We divide the proof in two steps. In the first one, we show a priori estimates on , solutions to (3.5). In the second one we pass to the limit our approximation in order to deduce the existence of a weak solution to (3.4).
Step 2. In this second step we prove that obtained in the first step is a weak solution to (3.4).
Choosing it as test function in the weak formulation of (3.5) we have We want to pass to the limit in (3.10) as tends to infinity. We fix δ > 0 and we decompose the right hand side in the following way: Therefore we have, thanks to Lemma 1 1 contained in [37], that V δ ( ) belongs to W 1 0 (Ω), where V δ ( ) is defined by (1.6). So we take it as test function in the weak formulation of (3.5) and we obtain Using that V δ is bounded we deduce that |∇ | −2 ∇ V δ ( ) converges to |∇ | −2 ∇ V δ ( ) weakly in L ′ (Ω) N as tends to infinity. This implies that in Ω as δ tends to 0 and since ∈ W 1 0 (Ω), then |∇ | −2 ∇ · ∇ V δ ( ) converges to 0 a.e. in Ω as δ tends to 0. Applying the Lebesgue Theorem on the right hand side of (3.12) we obtain that lim δ→0 + lim →∞ { ≤δ} ( ( ) + ( )) = 0 (3.13) As regards the second term in the right hand side of (3.11) we have Thanks to the a priori estimates on and using the Rellich-Kondrakov Theorem, we deduce, up to subsequence, that converges to strongly in L * 1+ (Ω). Since belongs to L * 1+
Proof of Theorem 3.4. We take as a test function in (3.5) yielding to Hence, we just need an estimate on the first term of the right hand side of (3.17). First of all let us observe that there exists a nonincreasing and continuous function : For the construction of such we refer to [18]. Hence let us consider ∈ W Once again, reasoning as in [18,19], one has that is nondecreasing with respect to and also that ≥ ≥ 1 . Moreover, it follows from the Hopf Lemma (see Lemma A 3 of [34]) that where δ( ) is the distance function from the boundary ∂Ω. Thanks to the previous we can finally estimate the term on the right hand side of (3.17) as follows: which is finite since γ < 2 − 1 This allows to have an estimate on in W 1 0 (Ω) which is independent on . Hence one can reason as in Step 2 of Theorem 3.2 in order to deduce the existence of a weak solution.
Finally we prove Theorem 3.5.
Proof. In order to show the existence of a solution we employ (3.20) with 0 = δ( ) for some > 1 2 and where δ( ) is the distance function from the boundary ∂Ω. Indeed, one can show that an application of the Hölder inequality We also remark that, in [16], Theorem 3.8 is extended for the case of the -Laplacian operator with > 2. In this case one can show that a similar result to Theorem 3.9 with 1 < γ < 1 + ( −1) ( −1) . @ . .