Existence and Regularity of Solutions for Unbounded Elliptic Equations with Singular Nonlinearities

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Introduction
Consider the Dirichlet problem for some nonlinear elliptic equations: − div a(x) +|u| q ∇u � f |u| c , x ∈ Ω, u ∈ H 1 0 (Ω), (1) under the following assumptions. e set Ω is a bounded open subset of R N , with N ≥ 3: a: Ω ⟶ R is a measurable function satisfying the following conditions: for almost every x ∈ Ω, where α and β are positive constant, and 0≨f ∈ L m (Ω), with m ≥ 1.
A possible motivation for studying the existence of these types of problems arises from the calculation of variations and stochastic control. For example, if we consider the functional the Euler-Lagrange equation associated to the functional J is − div a(x) +|v| 1− θ ∇v + 1 − θ 2 |∇v| 2 |v| θ sign(v) � f. (6) Several papers deal with existence of solutions to the singular elliptic problems with lower order terms having a quadratic growth with respect to the gradient (for example, [1][2][3][4][5][6][7][8][9]), namely, with the model problem where θ is a positive constant and M: Ω × R ⟶ R is a Carathéodory function. More precisely, existence of positive solutions for (7) was shown in [1][2][3], for M(x, t) � 1 and 0 < θ ≤ 1, and the uniqueness of positive solution, for M(x, t) � 1 and 0 < θ < 1, in [4]. On the contrary, the existence of positive solutions of (7) is shown in [6] for 0 < θ ≤ 1, provided M is a bounded uniformly elliptic matrix and 0≨f ∈ L m (Ω) (m > (2N/N + 2)). Later, in [9], it is proved the existence of solution for (7) with 0 < θ < 1, where M(x, t) � 1 and the data f ∈ L m (Ω) with m > (N/2), and does not satisfy any sign assumption. Recently, a problem introduced by L. Boccardo (see [7,10]) has given a strong impulse to the study of quasilinear problems having the unbounded divergence operator. In particular, in [7], the authors have proved the existence of positive solutions to problem (7) under the assumption that 0 < θ < 1, M(x, t) � 1 + |t| q , and 0≨f ∈ L m (Ω). We refer also that, in [5], the author has shown the same result as in [7], in the case 0 < θ < 1 and without any sign restriction over f. Let us now consider the Dirichlet boundary value problem (7) in the simple case: which is singular on the right-hand side. Let us remark that, in the case of nonnegative f, in [11], the authors considered the elliptic semilinear problems whose model is where c > 0. More precisely, they have shown that the term (f/|u| c ) has a regularizing effect on the solutions u. In [12], the author has shown the existence of solutions to the following elliptic problem with degenerate coercivity: where p, c > 0. e purpose of this paper is to study the same kind of lower order term as in problems (7) and (9) (indeed, (f/|u| c )) in the case of an elliptic operator with unbounded coefficients. e main difficulties posed by this problem were that the principal part of the differential operator div((a(x) + |u| q )∇u) is not well defined on the whole H 1 0 (Ω); the solutions did not belong, in general, to H 1 0 (Ω) and the lower order term has a singularity at u � 0. Despite these difficulties, we prove that, in our case too, the lower order term (f/|u| c ) has a regularizing effect.
Our main existence results are as follows.

Theorem 2.
We suppose that 0≨f ∈ L m (Ω), 1 < m < (N/2) and that (2) and (3) are satisfied. If 0 < q < 1, then, there exists a solution u of (1) in the sense (19), such that Notation: throughout this paper, we fix an integer N ≥ 3. For any p > 1, p ′ � (p/p − 1) will be the Hölder conjugate exponent of p, and if 1 ≤ p < N, we will denote by p * � (Np/N − p) the Sobolev conjugate exponent of p. As usual, let us denote by S the Sobolev constant, i.e.,

S � inf
We denote by P the Poincaré constant given by For all k > 0, we recall the definition of a truncated function T k (s) defined by We also consider As usual, we consider the positive and negative part of a measurable function u(x) 2 International Journal of Differential Equations (18)

The Approximated Problem
To prove our existence results, we will use the following approximating problems: where n ∈ N * , and As in [11], we prove existence of positive solution of the approximated problem.

Lemma 1. Let g be positive function belonging to L ∞ (Ω).
Suppose that (2) and (3) Proof. To prove it, we define the following operator S n : From the results of [13], the operator S n is well defined and w n is bounded by the results of [14]. We take w n as a test function in (19), and we use Hölder's inequality and (3) to deduce that anks to Poincaré's inequality, we deduce Hence, there exists an invariant ball for S n . On the contrary, from the H 1 0 (Ω)↪L 2 (Ω) embedding, it is easily seen that S n is continuous and compact.
e Schauder theorem shows that S n has a fixed point or equivalently, and there exists a solution u n ∈ H 1 0 (Ω) to problems − div a(x) + T n u n q ∇u n � g Moreover, by the maximum principle, it is clear that the sequence u n is nonnegative since g is nonnegative, and we choose G k (u n ) as test function in (25) and use (3) where A k � x ∈ Ω: |u n | > k . By the method of Stampacchia (see [14]), the sequence u n is bounded in L ∞ (Ω). Supposing that u n is bounded by d n in L ∞ (Ω), we have that u n : is a solution of (13). By Lemma 1, it follows the existence of a solution u n ∈ L ∞ (Ω) ∩ H 1 0 (Ω) of (19). Now, we are going to prove that the sequence u n is not 0 in Ω. For this, we are going to prove that it is uniformly away from zero in every compact set in Ω. We will follow a similar technique to that one in [12]. □ Lemma 2. Assume that (2) and (3) hold true. If 0≨f ∈ L 1 (Ω) and u n is the solution of problem (19), then for every n ∈ N * : u n ≤ u n+1 a.e. in Ω. Furthermore, if ω ⊂ ⊂ Ω, then, for every n ∈ N * , there exists c ω > 0 such that u n ≥ c ω > 0 a.e. in ω.
We remark that u 1 is bounded; indeed, |u 1 | ≤ c, for some positive constant c. en, it follows that anks to (3), we have α ≤ a(x) + |u 1 | q ≤ β + c q . us, we infer that u 1 is a supersolution of a linear Dirichlet problem with a strictly positive and bounded, measurable coefficient. e strong maximum principle implies that u 1 > 0. In addition, Harnack's inequality gives the stronger conclusion: for every ω ⊂ ⊂ Ω, there exists c ω such that u 1 ≥ c ω a.e. in ω. Finally, using that the sequence u n is increasing, one deduces that u n ≥ c ω a.e. in ω for every n ∈ N * . □ 2.1. Existence of Bounded Solutions. In this section, we will prove existence of bounded weak solutions for (1).

Lemma 3. Let 0≨f ∈ L m (Ω) with m > (N/2). Suppose that (2) and (3) hold true. Let u n be a sequence solutions of (19)
with f n � f for every n ∈ N * . en, the norm of the sequence u n in L ∞ (Ω) is bounded by a constant which depends on q, m, N, α, c, meas(Ω) and on the norm of f in L m (Ω).

Proof.
e use of G k (u n ) as test function in (19) and (3), implies that where A k � x ∈ Ω: |u n | > k . Hence, we can use eorem 4.1 in [14] and obtain a positive constant, say M, that only depends on the parameters: q, N, α, c, meas(Ω) and ‖f‖ L m (Ω) such that: ‖u n ‖ L ∞ (Ω) ≤ M for all n ∈ N * . □ Lemma 4. We assume that 0≨f ∈ L m (Ω) with m > (N/2), and (2) and (3) are satisfied. Let u n be a sequence solutions of (19) with f n � f for every n ∈ N * . If q < 1 and c ≤ 1 − q, then the sequence u n is uniformly bounded in H 1 0 (Ω).

Proof.
We denote by C a positive constant which may only depend on the parameters of our problem, and its value may vary from line to line. We use (1 + u n ) 1− q − 1 as test function in (19) to obtain and thus (since q ≤ 1), from which the sequence u n is bounded in H 1 0 (Ω).
International Journal of Differential Equations Lemma 5. Let 0≨f ∈ L m (Ω) with m > (N/2), and we suppose that (2) and (3) are satisfied. If q < 1 and c > 1 − q and u n is a solution to problem (19), then u n is uniformly bounded in H 1 loc (Ω).

□
Proof. of eorem 1. We start by proving point (1.i), the rest of the proof of the theorem can be proven similarly. According to Lemmas 3 and 4, there exists a subsequence u n and a function u ∈ H 1 0 (Ω) ∩ L ∞ (Ω) such that u n weakly converges to u in H 1 0 (Ω). Now, we can pass to the limit in the equation satisfied by the approximated solutions u n : where f n (x) � (f(x)/1 + (1/n)f(x)).
For the term of the left-hand side, it is sufficient to observe that ∇u n converge to ∇u weakly in L 2 loc (Ω) and [a(x) + u q n ] a.e. (and weakly − * in L ∞ (Ω) converges towards [a(x) + u q ]. On the contrary, for the limit of the right-hand side of (47), let ω � Suppφ, and one can use Lebesgue's dominated convergence theorem, since Finally, passing to the limit as n goes to infinity in equation (47), we conclude that  (2) and (3) and we assume that holds true.