About existence and regularity of positive solutions for a Quasilinear Schr\"odinger equation with singular nonlinearity

This paper deals with the existence of positive solution for the singular quasilinear Schr\"odinger equation $-\Delta u -\Delta (u^{2})u=h(x) u^{-\gamma} + f(x,u)~\mbox{in} ~ \Omega,$ where $\gamma>1$, $\Omega \subset \mathbb{R}^{N}, (N\geq 3)$ is a bounded smooth domain, $0<h\in L^{1}(\Omega)$, $f$ is a measurable function that can change signal and can be sublinear or has critical growth. Inspired by Sun \cite{Y} we derive a compatible condition on the couple $(h(x),\gamma)$, which is optimal for the existence of $H_{0}^{1}$-solution for this problem.


Introduction
In this article we are concerned with the existence of solution for the following quasilinear Schrödinger equation where Ω ⊂ R N is a bounded smooth domain and f is non-singular. See for example [1,13,14,16] and its references, where the authors used variational methods to prove the existence of solution. In these works the nonlinearity is non-singular and therefore the functional energy associated to the problem has a good regularity to use the usual techniques for functional of class C 1 . When f is singular, problems of type (1.1) was studied by Do Ó-Moameni [6], Liu-Liu-Zhao [12] and Wang [18]. In [6] the authors studied the problem where Ω is a ball in R N centered at the origin, 0 < γ < 1 and N ≥ 2. They show, using the Nehari manifold method, that problem (1.2) has a radially symmetric solution u ∈ H 1 0 (Ω) for all λ ∈ I, where I is a open and bounded interval. 1 Ricardo Lima Alves acknowledges the support of CNPq/Brazil 1 In 2016 Liu-Liu-Zhao in [12] considered the problem where ∆ s is the s-Laplacian operator, s > 1,γ > 0, Ω ⊂ R N (N ≥ 3) is a bounded smooth domain, 2s < p + 1 < ∞ and h(x) ≥ 0 is a nontrivial measurable function satisfying the following condition: there exists a function φ 0 ≥ 0 in C 1 0 (Ω) and q > N such that h(x)φ −γ 0 ∈ L q (Ω). Combining the sub and supersolution method, truncation argument and variational methods, they proved the existence of λ * > 0 such that the problem (1.3) has at least two solutions for λ ∈ (0, λ * ).
Recently Wang in [18] proved the existence and uniqueness of solution to the following quasilinear Schrödinger equation where Ω ⊂ R N (N ≥ 3) is a bounded smooth domain, γ ∈ (0, 1), p ∈ [2, 22 * ] and 0 < h ∈ L 22 * 22 * −1+γ (Ω). The author used global minimization arguments to prove the existence of solution. Here after the use of variable change developed in Colin-Jeanjean [4] the functional associated to the dual problem is well defined in H 1 0 (Ω) and continuous.
Before stating our results we would like to cite here the work of Sun [17]. In this work the following problem was considered − ∆u = h(x)u −γ + b(x)u p in Ω, u = 0 on ∂Ω, (1.5) where Ω ⊂ R N is a bounded smooth domain, b ∈ L ∞ (Ω) is a non-negative function, 0 < p < 1, γ > 1 and 0 < h ∈ L 1 (Ω). The author has proved, using variational methods, that the existence of positive solution in H 1 0 (Ω) of the problem (1.5) is related to a compatibility hypothesis between on the couple (h(x), γ), more precisely it has been proved that the problem (1.5) has a solution in The main difficulty there was because of the strong silgularity that causes a serious loss of regularity of the functional energy associated with which it is not continuous. In order to deal with these difficulty she worked with appropriate constrainsts sets to restore the integrability of singular term. Moreover, it was essential in its approach that nonlinearity was homogeneous.
Motivated by [17] a natural question arises: the hypothesis of compatibility between on the couple (h(x), γ) given by (1.6) remains necessary and sufficient for the existence of solution for our class of problems (P )? In this paper we will give a positive answer to this question. Also requesting more regularity in the function h we prove that the solution have C 1,α regularity and as a consequence of this the solution is unique.
Our main results are then the solution u given in a) belongs to C 1,α (Ω) for some α ∈ (0, 1). In particular the problem (P ) has a unique solution in H 1 0 (Ω).
To prove our main result let us use the method of changing variables developed by Colin-Jeanjean [4]. After this the functional associated with the dual problem is not homogeneous and this causes several difficulties. For example, the techniques used by previous work do not apply directly. To cover this difficulty we will make a careful analysis of the fiber maps associated to the functional of the dual problem and will approach the problem in a new way to prove the existence of a solution to the problem (P ). Now let us mention some contributions from our work. In this work we consider the most general potentials, for instance the potential b can change signal. The regularity of solution (and hence uniqueness) for problem with strong singularity has not been treated yet. Theorem 1.2 completes the study made by Wang in [18] in the sense that we now consider the case γ > 1, while [18] consider the case 0 < γ < 1. Moreover in our work we consider the more general potentials also. This paper is organized as follows: In the next section we reformulate the problem (P ) into a new one which finds its natural setting in the Sobolev space H 1 0 (Ω) and we will present some preliminary lemmas. In section 3, we give the proof of Theorem 1.1 and in section 4 the proof of Theorem 1.2. The last section consists of an appendix to which we will study the problem (P ) with b(x) ≡ λb(x), λ ≥ 0 and prove that the solutions found in the Theorem 1.1 vary continuously with respect to λ and the norm from H 1 0 (Ω). N otation. In the rest of the paper we make use of the following notation: • c, C denote positive constants, which may vary from line to line,

Variational framework and Preliminary Lemmas
In this section we provide preliminary results wich will be used to prove the existence of a solution of the problem (P ). By solutions we mean here weak solutions in H 1 0 (Ω), that is u ∈ H 1 0 (Ω) satisfying u(x) > 0, in Ω and for every ϕ ∈ H 1 0 (Ω), which is formally the variational formulation of the following functional J : However, this functional is not well-defined, because Ω u 2 |∇u| 2 dx is not finite for all u ∈ H 1 0 (Ω), hence it is difficult to apply variational methods directly. Firstly we use the method developed in [4] introducing the unknown variable v := g −1 (u), where g is defined by It is easy to see that if v is solution of if and only if u = g(v) is solution of (P ). We will call the problem (P A ) of dual problem to (P ).
The weak form of the equation for every φ ∈ H 1 0 (Ω) and therefore v is a critical point of functional f (x, t)dt. Now, we list some properties of g, whose proofs can be found in Liu [11].
The next lemma gives us a relation of duality between the compatibility hypothesis for the problems (P ) and (P A ).
The following conditions are equivalent: , ∀x ∈ A 2 , and this implies that To prove that (b) ⇒ (c) note that by Lemma 2.1 (4), Finally to prove that (c) ⇒ (a) we use the Lemma 2.1 (5) to obtain that and the proof is completed.
Note that if v 0 satisfies the condintion (1.6) then, since |v 0 | ∈ H 1 0 (Ω) we have that |v 0 | satisfies the condition (1.6). Hence we may assume that v 0 ≥ 0. Also as we are interested in positive solution let us work on the following subset of H 1 0 (Ω) and consider the fiber map The understanding of the fibering maps will be extremely important in the next section. Let us start by proving that for each v satisfying (1.9) the fiber map associated to v has a good regularity.
Proof. We have that prove just that Γ : (0, ∞) −→ R defined by is of classe C 1 . To do this we fix t > 0 and note that for each s > 0 by Mean Value Theorem there exits a mensurable function θ = θ(s, x) ∈ (0, 1) such that, . Furthermore, as consequence of the Lemma 2.2, h(g(tv)) −γ g ′ (tv)v ∈ L 1 (Ω). Hence we are able to apply the Lebesgue's dominated convergence theorem to infer that that is, the derivative Γ ′ (t) there exists for all t > 0 and is given by the last expression above. Now, using the Lemma 2.2 and the Lebesgue's dominated convergence theorem we have that Γ ′ : (0, ∞) −→ R is a continuous function.
The next lemma guarantees that for each v satisfying (1.9) the fiber map φ v assumes its minimum value and therefore φ v has a critical point.
and therefore there exists t(v) > 0 such that Proof. Firstly we will consider the sublinear case, that is (f ) 1 with p ∈ (0, 1). By the Lemma 2.1 (5) we have that The continuity of the function φ v and the limits lim .
The following picture give the possible graph of the fiber map.
Remark 2.1. When (f ) 1 is satisfied the graph of φ v can be as in Figures 1 and 2.
On the other hand if (f ) 2 is satisfied then the graph of φ v can only be as in Figure  2.
Motivated by [17] we define the following constraint sets for the problem (P A ) It should be noted that for γ > 1, N 2 is not closed as usual (certainly not weakly closed).
Next lemma ensures that any function v ∈ V + satisfying the following condition of dual compatibility can be projected over N 2 .
Proof. As v satisfied (1.10) it follows from the Lemma 2.2 that v satisfied (1.9) also and therefore by Lemma 2.3 we have that φ v ∈ C 1 ((0, ∞), R). Follows from the Lemma 2.4 that there exists t(v) > 0 such that The following lemmas will be used to prove the regularity of the solution.
Lemma 2.6. Let Ω be a bounded domain in R N with smooth boundary ∂Ω. Let u ∈ L 1 loc (Ω) and assume that, for some k ≥ 0, u satisfies, in the sense of distributions, Then either u ≡ 0, or there exists ǫ > 0 such that u(x) ≥ ǫd(x, ∂Ω), x ∈ Ω.
Proof In this section we will show the Theorem 1.1. First we wil given some preliminary lemmas.
Lemma 3.1. The set N 1 = ∅ and the functional Φ is coercive in N 1 .
Proof. Since (1.6) is satisfied it follows from Lemmas 2.2, 2.5 that N 1 = ∅. Now, let us prove that Φ is coercive. Indeed for every v ∈ N 1 , and by Lemma 2.1 (5) and Sobolev embedding we have for some constant C > 0. Since p ∈ (0, 1) follows that Φ is coercive.
By the Lemma 3.1 we have that are well defined with J 1 , J 2 ∈ R and J 2 ≥ J 1 .
Proof. By Lemma 3.1 there exists a sequence {v n } ⊂ N 1 such that Φ(v n ) −→ J 1 , and may assume that {v n } is bounded and exist Since that v n > 0 follows that v ≥ 0 in Ω. Moreover, there exists a constant C > 0 such that ||v n || ≤ C for every n ∈ N. By definition of N 1 and Lemma 2.1 (3), (4), (5) we have that where we used Sobolev embedding. Therefore using the Fatou's lemma in the last inequality we have that Since that g(0) = 0 (by Lemma 2.1 (2)) and Ω θ(x) < ∞ follows that v > 0 in Ω.
Once again by Fatou's lemma we obtain Consequently by the Lemma 2.4 and Lemma 2.5 there exists t(v) > 0 such that φ v (t(v)) = inf t>0 φ v (t) and t(v)v ∈ N 2 . Taking advantage of this information it follows that Now let us prove the Theorem 1.1.
Proof. Firstly we will to prove a). Suppose that (P ) has a solution v 0 . Taking v 0 as test function is easy to see that (1.6) is satisfied. Now assume that (1.6) is satisfied. Let v as in the Lemma 3.2. Let us prove that v is a solution of problem (P A ). Consider ϕ ∈ H 1 0 (Ω) such that ϕ ≥ 0 in Ω and let ǫ > 0. By the Lemma 2.1 (10) and consequently by Lemmas 2.4, 2.5 there exist t(ǫ) > 0 such that φ v+ǫϕ (t(ǫ)) = inf t>0 φ v+ǫϕ (t) and t(ǫ)(v + ǫϕ) ∈ N 2 . Therefore Thus, dividing the last inequality by ǫ > 0 and passing to the lim inf as ǫ −→ 0, by Fatou's Lemma we have To end the proof of item a) we will use an argument inspired by Graham-Eagle [8]. Since that v ∈ N 2 we have Since the measures of the domains of integration [v + ǫϕ < 0] and Ω ǫ 1 tends to zero as ǫ → 0, we then divide the last expression above by ǫ > 0 to obtain as ǫ → 0. Replacing ϕ by −ϕ we conclude: and therefore v is a solution of (P A ).
Defining u = g(v) we have that u is a solution of problem (P ). Now, let us prove b). Suppose that v is a solution of the problem (P A ). We will show that v ∈ C 1,α (Ω). Hence as g ∈ C ∞ we obtain that u = g(v) ∈ C 1,α (Ω). Note that the function v satisfies in the sense of distributions, Since that v ∈ H 1 0 (Ω) and v ≡ 0 by Lemma 2.6 there exists ǫ > 0 such that v(x) ≥ ǫd(x, ∂Ω), x ∈ Ω.
Let us end this section by making some remarks.  [10] proved that the solution of the following semilinear problem does not belong to C 1 (Ω), if 0 < h ∈ C α (Ω), α ∈ (0, 1) and 1 < γ. Note that in this case inf In this section we assume that f (x, is not compact, the proof of the Theorem 1.2 can not be applied directly. To cover this difficulty we use the Brezis-Lieb Theorem (see [2]). Now the functional associated with the problem (P A ) is If (1.6) is satisfied, using the Lemmas 2.2, 2.5 we have N 1 = ∅. So it follows that. Proof. Indeed, for every v ∈ N 1 , we have Φ(v) ≥ ||v|| 2 2 , and hence Φ is coercive.
Let us prove the Theorem 1.2.
Proof. (Theorem 1.2) Suppose that (P ) has a solution v 0 . Taking v 0 as test function is easy to see that (1.6) is satisfied. Now assume that (1.6) is satisfied.
Defining u = g(v) we have that u is a solution of problem (P ).
To prove that the solution is unique, let us denote by for x ∈ Ω, t > 0. Note that j(., t) is decreasing by Lemma 2.1 (9), (10). Suppose that v 1 and v 2 are solutions from (P A ). Then, Therefore the solution is unique.

Appendix
In this section we will study the stabilty of the solutions of the following problem with parameter where λ ≥ 0, 0 < p < 1 and 0 b ∈ L ∞ (Ω). The main result of this section is (1.6) is satisfied e let u λ ∈ H 1 0 (Ω) the solution from (P λ ), which there exists by Theorem 1.1. There holds the following: a) u λ ≥ u 0 for all λ > 0, b) u λ −→ u 0 in H 1 0 (Ω) when λ −→ 0. To prove Theorem 5.1 firstly we will consider the dual problem associated to (P λ ), that is for λ ≥ 0, 0 < p < 1 and 0 b ∈ L ∞ (Ω) consider the following family of problems dual to (P λ ) and let Φ λ the energy functional associated to (D λ ). For each λ ≥ 0 let us denote by the constrained set associated to (D λ ). The proof of Theorem 5.1 is a consequence of the following lemma.  Proof. First we will to prove a).
Let us prove c). To do this, note that v λ ≥ v 0 for all λ > 0 and by the item b) v λ −→ v 0 in H 1 0 (Ω). Thus we can proceed as in the proof of item b) and apply the Lebesgue dominated convergence theorem for get that lim λ→0 Φ λ (v λ ) = Φ 0 (v 0 ).
By the Lemma 5.1 we have the following picture Now we will prove the Theorem 5.1.
Proof. Theorem 5.1. Firstly we prove the item a). Let v λ and v 0 as in the Lemma 5.1. Then u λ = g(v λ ), u 0 = g(v 0 ) and v λ ≥ v 0 by Lemma 5.1 a). So because the function g(t) is increasing for t ≥ 0 (see Lemma 2.1 (9)).