Singular elliptic problem involving a Hardy potential and lower order term

Abstract

We consider the following non-linear singular elliptic problem $\begin{matrix} (1) & \{\begin{matrix} - div (M (x) | \nabla u |^{p - 2} \nabla u) + b | u |^{r - 2} u = a \frac{u^{p - 1}}{| x |^{p}} + \frac{f}{u^{γ}} & in Ω \\ u > 0 & in Ω \\ u = 0 & on \partial Ω, \end{matrix} \end{matrix}$ where $1 < p < N$ ; $Ω \subset R^{N}$ is a bounded regular domain containing the origin and $0 < γ < 1$ , $a ⩾ 0, b > 0, 0 ⩽ f \in L^{m} (Ω) and 1 < m < \frac{N}{p}$ . The main goal of this work is to study the existence and regularizing effect of some lower order terms in Dirichlet problems despite the presence of Hardy the potentials and the singular term in the right hand side.

Keywords

Singular non-linearity Hardy Potential Lower Order Terms Regularizing Effect

1. Introduction and main results

In the paper [1], Abdellaoui et al proved an existence and summability result on the solutions of the Dirichlet problem $\begin{matrix} (2) & \{\begin{matrix} - Δ_{p} u = λ \frac{u^{p - 1}}{| x |^{p}} + \frac{h (x)}{u^{γ}} & in Ω \\ u = 0 & on \partial Ω, \end{matrix} \end{matrix}$ where $1 < p < N, Ω \subset R^{N}$ is a bounded regular domain containing the origin and $γ > 0$ . h is a nonnegative measurable function with suitable hypotheses.

Problems of the form (2) in the case $γ = 0$ are introduced as models for several physical phenomena related to the equilibrium of anisotropic continuous media which possibly are somewhere ‘perfect’ insulators (see [13]). For more results let us refer to papers [2,4,8,9,20] and the references therein.

When $λ = 0$ , the equations in the form of (2) have been widely studied in the last few decades. We refer to the papers [14,15] and the references therein. For $λ > 0$ and $p = 2$ , J. Tyagi studied in [30] the existence and regularity of solutions to the following semilinear elliptic problem with a singular nonlinearity $\begin{matrix} (3) & \{\begin{matrix} - div (M (x) \nabla u) - λ \frac{u}{| x |^{2}} = \frac{f (x)}{u^{θ}} & in Ω \\ u > 0 & in Ω \\ u = 0 & on \partial Ω, \end{matrix} \end{matrix}$ where $θ > 0, 0 ⩽ f \in L^{m} (Ω), 1 < m < \frac{N}{2}, 0 < λ < {(\frac{N - 2}{2})}^{2}$ , and M is a bounded elliptic matrix; i.e., there exist $α, β > 0$ such that $\begin{matrix} α | ξ |^{2} ⩽ M (x) ξ \cdot ξ, | M (x) | ⩽ β . \end{matrix}$

On the other hand, L. M. De Cave and F. Oliva in their leading work [16], obtained existence and regularity of the solution to the problem $\begin{matrix} (4) & \{\begin{matrix} - Δ u + u^{s} = \frac{f (x)}{u^{γ}} & in Ω \\ u = 0 & on \partial Ω . \end{matrix} \end{matrix}$ The problem (4) was then generalized by F.Oliva in the case of the p-laplacian in [25], when the author studied the following singular problem $\begin{matrix} \{\begin{matrix} - Δ_{p} u + u^{q} = \frac{f}{u^{γ}} & in Ω \\ u > 0 & in Ω \\ u = 0 & on \partial Ω, \end{matrix} \end{matrix}$ where Ω is an open bounded subset of $R^{N} (N ⩾ 2), Δ_{p} u$ is the p-Laplacian operator for $1 ⩽ p < N, q > 0, γ ⩾ 0$ and f is a nonnegative function in $L^{m} (Ω)$ for some $m ⩾ 1$ . We refer the readers to Refs. [5,7,10–12,17–19,21,22,26–29]. It is clear that problem (1) is related to the classical Hardy–Sobolev inequality $\begin{matrix} Λ_{N, p} \int_{Ω} \frac{| ϕ |^{p}}{| x |^{p}} d x ⩽ \int_{Ω} | \nabla ϕ |^{p} d x for all ϕ \in W_{0}^{1, p} (Ω), \end{matrix}$ where $Λ_{N, p} = {(\frac{N - p}{p})}^{p}$ is optimal and not achieved, we refer to [23] for more details about this constant. In the case $0 < γ < 1$ , Abdellaoui et al obtained in [1] the existence and regularity of a solution in a suitable Sobolev space for all $λ ⩽ Λ_{N, p}$ .

Based on the above cited results, the main objective of this work is to explain the combined effects of lower order term $b | u |^{r - 2} u$ and Hardy potential $a (u^{p - 1} / | x |^{p})$ on the existence and regularity of solution to problem (1), for every $a ⩾ 0$ (and not only for a smaller than the Hardy constant).

Let us consider the following singular elliptic problem $\begin{matrix} (5) & \{\begin{matrix} - div (M (x) | \nabla u |^{p - 2} \nabla u) + b | u |^{r - 2} u = a \frac{u^{p - 1}}{| x |^{p}} + \frac{f}{u^{γ}} & in Ω \\ u > 0 & in Ω \\ u = 0 & on \partial Ω, \end{matrix} \end{matrix}$ where $1 < p < N$ ; $Ω \subset R^{N}$ is a bounded regular domain containing the origin and $0 < γ < 1$ . We assume that M: $Ω \to R$ , is a Lipschitz continuous function such that for some positive constants α and β $\begin{matrix} (6) & α ⩽ M (x) ⩽ β for a.e. x \in Ω . \end{matrix}$ and $\begin{array}{l} (7) & r > p^{*} and a ⩾ 0, b > 0 \\ (8) & 0 ⩽ f \in L^{m} (Ω), 1 < m < \frac{N}{p} . \end{array}$ Now, we give our definition of solution for problem (5)

Definition 1.1.
We say that $u \in W_{0}^{1, 1} (Ω)$ is a distributional solution to problem (5) if $\begin{matrix} (9) & | \nabla u |^{p - 1} \in L_{loc}^{1} (Ω), a \frac{u^{p - 1}}{| x |^{p}} + \frac{f}{u^{γ}} \in L_{loc}^{1} (Ω), b | u |^{r - 2} u \in L_{loc}^{1} (Ω) \end{matrix}$ and for all $φ \in C_{c}^{1} (Ω)$ , we have $\begin{matrix} (10) & \int_{Ω} M (x) | \nabla u |^{p - 2} \nabla u \cdot \nabla φ + b \int_{Ω} | u |^{r - 2} u φ = \int_{Ω} (a \frac{u^{p - 1}}{| x |^{p}} + \frac{f}{u^{γ}}) φ . \end{matrix}$

The main results of the paper are the following:
Theorem 1.1.
Assume that ( 6 ), ( 7 ) and $0 < γ < 1$ hold true. Let f be a nonnegative function in $L^{m} (Ω)$ with $\begin{matrix} \frac{r}{r - 1 + γ} ⩽ m < \frac{N}{p} \frac{r - p}{r - 1 + γ} . \end{matrix}$ Then there exists a distributional solution u of ( 5 ), which belongs to $W_{0}^{1, p} (Ω) \cap L^{m (r - 1 + γ)} (Ω)$ .
Remark 1.1.
Observe that the interval $[\frac{r}{r - 1 + γ}, \frac{N}{p} \frac{r - p}{r - 1 + γ})$ is not empty if $r > p^{}$ .
Remark 1.2.
Thanks to the presence of the lower order term $b | u |^{r - 2} u$ , the result of Theorem 1 improves that of [30] (where $b = 0$ and $p = 2$ ) in several directions. First of all, if $\begin{matrix} \frac{r}{r - 1 + γ} ⩽ m < \frac{2 N}{N + 2 + γ (N - 2)}, \end{matrix}$ we have finite energy solutions (instead of infinite energy ones). Furthermore, we have that solutions exist for every $a > 0$ (and not only for a smaller than the Hardy constant). Finally, the summability in Lebesgue spaces, $m (r - 1 + γ)$ , is better than the summability $(1 + γ) m^{ *}$ obtained in [30].
Theorem 1.2.
Assume that ( 6 ), ( 7 ), $0 < γ < 1$ hold true. Let f be a nonnegative function in $L^{m} (Ω)$ with $\begin{matrix} 1 < m < \frac{r}{r - 1 + γ} . \end{matrix}$ Then there exists a distributional solution u of ( 5 ) which belongs to $L^{m (r - 1 + γ)} (Ω)$ and $| \nabla u |^{q} \in L^{1} (Ω)$ , with $q = p m \frac{r - 1 + γ}{r}$ . Moreover, if p and m satisfy one of the following assumptions
$\frac{N + (1 - γ) N}{N + 1 - γ} ⩽ p < N$ and $1 < m < \frac{r}{r - 1 + γ}$ ,

$1 < p < \frac{N + (1 - γ) N}{N + 1 - γ}$ and $\frac{r}{p (r - 1 + γ)} < m < \frac{r}{r - 1 + γ}$
then $u \in W_{0}^{1, q} (Ω)$ .
Remark 1.3.
In the case $p = 2$ we can observe also that the result of Theorem 1.2 improves the result of [30]. Once again we have solutions for every $a > 0$ , and the summability of the gradients in Lebesgue spaces, i.e., $\frac{2 m (r - 1 + γ)}{r}$ , is better than the summability $q = \frac{N m (θ + 1)}{m (1 + θ) + (N - 2 m)}$ obtained in [30], since $m < \frac{N}{2} \frac{r - 2}{r - 1 + γ}$ . Note that $q < 2$ because $m < \frac{r}{r - 1 + γ}$ .
Remark 1.4.
If $p = 2$ and $γ \to 0^{+}$ , the result of Theorem 1.1 and Theorem 1.2 coincides with regularity results for elliptic equation problems involving Hardy potential (see [3, Theorem 2.1 and Theorem 3.1]).

We organize the work as follows. In Section 2, we introduce an approximation of problem (5), we prove the uniform positivity of the approximating solutions. In Section 3, we give the a priori estimates valid for the case of finite and infinite energy solutions. Finally, in Section 4, we prove Theorems 1.1 and 1.2.

Notations. If no otherwise specified, we will denote by C several constants whose value may change from line to line and, sometimes, on the same line. These values will only depend on the data (for instance C can depend on $Ω, γ, N, a, b, α$ and β) but they will never depend on the indexes of the sequences we will often introduce. For the sake of simplicity we will often use the simplified notation $\begin{matrix} \int_{Ω} f : = \int_{Ω} f (x) d x, \end{matrix}$ when referring to integrals when no ambiguity on the variable of integration is possible. For fixed $k > 0$ we will made use of the truncation functions $T_{k}$ and $G_{k}$ defined as $\begin{matrix} T_{k} (s) = max (- k, min (s, k)), \forall s \in R . \end{matrix}$
2. The approximation scheme

To prove our existence results, we work with an approximation of (5). Let $n \in N$ , let $f_{n} (x) : = T_{n} (f)$ . Let us consider the approximate problem: $\begin{matrix} (11) & \{\begin{matrix} - div (M (x) | \nabla u_{n} |^{p - 2} \nabla u_{n}) + b | u_{n} |^{r - 2} u_{n} = a \frac{{(T_{n} (| u_{n} |))}^{p - 1}}{(| x |^{p} + \frac{1}{n})} + \frac{f_{n} (x)}{{(| u_{n} | + \frac{1}{n})}^{γ}} & in Ω \\ u_{n} = 0 & on \partial Ω . \end{matrix} \end{matrix}$

Lemma 2.1.
For each integer $n \in N$ , the problem ( 11 ) admits a non-negative solution in $W_{0}^{1, p} (Ω)$ for all $1 < p < N$ .
Proof.
Let $n \in N$ be fixed and let $v \in L^{p} (Ω)$ . We define the map $\begin{matrix} S : L^{p} (Ω) & \to L^{p} (Ω) \\ v & \mapsto S (v), \end{matrix}$ where $w = S (v)$ is the weak solution to the following problem $\begin{matrix} \{\begin{matrix} - div (M (x) | \nabla w |^{p - 2} \nabla w) + b | w |^{r - 2} w = a \frac{{(T_{n} (| v |))}^{p - 1}}{(| x |^{p} + \frac{1}{n})} + \frac{f_{n} (x)}{{(| v | + \frac{1}{n})}^{γ}} & in Ω \\ w = 0 & on \partial Ω . \end{matrix} \end{matrix}$ The existence of a solution $w \in W_{0}^{1, p} (Ω)$ follows from the classical results of [24]. Let us take w as a test function and by (6), we get $\begin{matrix} \int_{Ω} M (x) | \nabla w |^{p - 2} \nabla w \cdot \nabla w + b \int_{Ω} | w |^{r} = \int_{Ω} a \frac{{(T_{n} (| v |))}^{p - 1} w}{(| x |^{p} + \frac{1}{n})} + \int_{Ω} \frac{f_{n} (x) w}{{(| v | + \frac{1}{n})}^{γ}}, \end{matrix}$ dropping the second positive order term, we obtain $\begin{matrix} α \int_{Ω} | \nabla w |^{p} d x ⩽ \int_{Ω} a \frac{{(T_{n} (| v |))}^{p - 1} w}{(| x |^{p} + \frac{1}{n})} + \int_{Ω} \frac{f_{n} (x) w}{{(| v | + \frac{1}{n})}^{γ}} . \end{matrix}$ Therefore, using the Sobolev inequality on the left hand side and the Hölder inequality on the right hand side one has $\begin{matrix} {[\int_{Ω} | w |^{p^{}}]}^{p / p^{}} ⩽ C (a, n, α, γ) {(\int_{Ω} | w |^{p^{}})}^{\frac{1}{p^{}}}, \end{matrix}$ for some constant C independent on v. This implies $\begin{matrix} ‖ w ‖_{L^{p^{}} (Ω)} ⩽ C (a, n, α, γ) = R, \end{matrix}$ so that the ball of $L^{p^{}} (Ω)$ of radius R is invariant for S. It is easy to prove, using the Sobolev embedding, that S is both continuous and compact on $L^{p^{}} (Ω)$ , so that by Schauder’s fixed point Theorem there exists $u_{n} \in W_{0}^{1, p} (Ω)$ such that $u_{n} = T (u_{n})$ , i.e., $u_{n}$ solves $\begin{matrix} \{\begin{matrix} - div (M (x) | \nabla u_{n} |^{p - 2} \nabla u_{n}) + b | u_{n} |^{r - 2} u_{n} = a \frac{{(T_{n} (| u_{n} |))}^{p - 1}}{(| x |^{p} + \frac{1}{n})} + \frac{f_{n} (x)}{{(| u_{n} | + \frac{1}{n})}^{γ}} & in Ω \\ u_{n} = 0 & on \partial Ω . \end{matrix} \end{matrix}$ Moreover, since $\frac{f_{n}}{{(| u_{n} | + \frac{1}{n})}^{γ}} ⩾ 0$ , taking $u_{n}^{-} = min (u_{n}, 0)$ test function in (11) and using (6), then we get $\begin{matrix} α \int_{Ω} {| \nabla u_{n}^{-} |}^{p} + \int_{Ω} | u_{n} |^{r - 2} {(u_{n}^{-})}^{2} = a \int_{Ω} \frac{{(T_{n} (| u_{n} |))}^{p - 1}}{(| x |^{p} + \frac{1}{n})} u_{n}^{-} + \int_{Ω} \frac{f_{n}}{{(| u_{n} | + \frac{1}{n})}^{γ}} u_{n}^{-} ⩽ 0, \end{matrix}$ we obtain $\begin{matrix} α \int_{Ω} {| \nabla u_{n}^{-} |}^{p} ⩽ 0, \end{matrix}$ so that $u_{n} ⩾ 0$ almost everywhere in Ω. □

The next step consists in the proof that $u_{n}$ is uniformly bounded from below on the compact subsets of Ω. Lemma 2.2.
Let* $u_{n}$ be a solution of ( 11 ). Then for every subset $ω \subset \subset Ω$ there exists a positive constant $c_{ω}$ , independent on n, such that $\begin{matrix} u_{n} (x) ⩾ c_{ω} > 0, for every x \in ω and for every n \in N . \end{matrix}$
Proof.
Since $u_{n}$ solution of (11), then $\begin{matrix} - div (M (x) | \nabla u_{n} |^{p - 2} \nabla u_{n}) + b u_{n}^{r - 1} = a \frac{{(T_{n} (u_{n}))}^{p - 1}}{| x |^{p} + \frac{1}{n}} + \frac{f_{n}}{{(u_{n} + \frac{1}{n})}^{γ}}, \end{matrix}$ as $a ⩾ 0$ , then we obtain $\begin{matrix} - div (M (x) | \nabla u_{n} |^{p - 2} \nabla u_{n}) + b u_{n}^{r - 1} ⩾ \frac{f_{n}}{{(u_{n} + \frac{1}{n})}^{γ}}, \end{matrix}$ this implies that the sequence $u_{n}$ is a supersolution to problem $\begin{matrix} \{\begin{matrix} - div (M (x) | \nabla v |^{p - 2} \nabla v) + b | v |^{r - 2} v = \frac{f_{n}}{{(| v | + \frac{1}{n})}^{γ}} & in Ω \\ v > 0 & in Ω \\ v = 0 & on \partial Ω . \end{matrix} \end{matrix}$ Thanks to Lemma 2.2 in [5], $\exists c_{ω} > 0$ (independent of n) such that $v ⩾ c_{w}$ in $ω, \forall n \in N, \forall ω \subset \subset Ω$ , since $u_{n} ⩾ v$ , so $\begin{matrix} u_{n} ⩾ c_{ω} in ω, \forall n \in N, \forall ω \subset \subset Ω . \end{matrix}$ □

3. A priori estimates

In order to prove the existence of solutions for problem (5), we first need some a priori estimates on $u_{n}$ . We start by proving the following lemma

Lemma 3.1.
Let $u_{n}$ be the solution of problem ( 11 ), with $0 < γ < 1$ . Assume that f be a nonnegative function in $L^{m} (Ω)$ with $\frac{r}{r - 1 + γ} ⩽ m < \frac{N}{p} \frac{r - p}{r - 1 + γ}$ and suppose that ( 6 ) and ( 7 ) hold. Then, $u_{n}$ is bounded in $W_{0}^{1, p} (Ω) \cap L^{m (r - 1 + γ)} (Ω)$ .
Proof.
Let $u_{n}$ be a solution of (11). We use $u_{n}^{λ + 1}$ with $λ = (m - 1) (r - 1) - 1 + γ m$ ( $λ ⩾ 0$ , since $m ⩾ \frac{r}{r - 1 + γ}$ ) as test function in (11) and using (6), we get $\begin{matrix} (12) & α (λ + 1) \int_{Ω} | \nabla u_{n} |^{p} u_{n}^{λ} + b \int_{Ω} u_{n}^{λ + r} ⩽ a \int_{Ω} \frac{u_{n}^{λ + p}}{| x |^{p} + \frac{1}{n}} + \int_{Ω} f u_{n}^{λ + 1 - γ}, \end{matrix}$ dropping the positive first term, then the last inequality becomes $\begin{matrix} (13) & b \int_{Ω} u_{n}^{λ + r} ⩽ a \int_{Ω} \frac{u_{n}^{λ + p}}{| x |^{p} + \frac{1}{n}} + \int_{Ω} f u_{n}^{λ + 1 - γ} . \end{matrix}$ In addition, using the Hölder inequality with exponent m and taking into account that $(λ + 1 - γ) m^{'} = λ + r$ , we arrive at $\begin{matrix} \int_{Ω} f u_{n}^{λ + 1 - γ} ⩽ ‖ f ‖_{L^{m} (Ω)} {(\int_{Ω} u_{n}^{(λ + 1 - γ) m^{'}})}^{\frac{1}{m^{'}}} = ‖ f ‖_{L^{m} (Ω)} {(\int_{Ω} u_{n}^{λ + r})}^{\frac{1}{m^{'}}} . \end{matrix}$ Under condition, $r > p^{}$ we have $λ + p < λ + r$ . Thus, since $\begin{matrix} {(\frac{λ + r}{λ + p})}^{'} = \frac{λ + r}{r - p}, \end{matrix}$ we can write $\begin{matrix} \int_{Ω} \frac{u_{n}^{λ + p}}{| x |^{p} + \frac{1}{n}} ⩽ {(\int_{Ω} u_{n}^{λ + r})}^{\frac{λ + p}{λ + r}} {(\int_{Ω} \frac{1}{| x |^{p \frac{λ + r}{r - p}}})}^{\frac{r - p}{λ + r}} . \end{matrix}$ Since $\begin{matrix} p \frac{λ + r}{r - p} < N, \end{matrix}$ gives $\begin{matrix} m < \frac{N}{p} \frac{r - p}{r - 1 + γ} . \end{matrix}$ Therefore, we deduce that $\begin{matrix} \int_{Ω} \frac{u_{n}^{λ + p}}{| x |^{p} + \frac{1}{n}} ⩽ C {(\int_{Ω} u_{n}^{λ + r})}^{\frac{λ + p}{λ + r}} . \end{matrix}$ By the fact that $λ + r = m (r - 1 + γ)$ and removing the positive first term of (12), we find that $\begin{matrix} b \int_{Ω} u_{n}^{m (r - 1 + γ)} ⩽ a C {(\int_{Ω} u_{n}^{m (r - 1 + γ)})}^{\frac{λ + p}{λ + r}} + ‖ f ‖_{L^{m} (Ω)} {(\int_{Ω} u_{n}^{m (r - 1 + γ)})}^{\frac{1}{m^{'}}}, \end{matrix}$ the above estimate implies $\begin{matrix} (14) & \int_{Ω} u_{n}^{m (r - 1 + γ)} ⩽ C . \end{matrix}$ By (14) and going to back to (12) we conclude that $\begin{matrix} \int_{Ω} | \nabla u_{n} |^{p} u_{n}^{λ} ⩽ C . \end{matrix}$ Since $λ ⩾ 0$ , we obtain $\begin{matrix} (15) & \int_{{u_{n} ⩾ 1}} | \nabla u_{n} |^{p} ⩽ C . \end{matrix}$ Observe now that, since ${u_{n}}$ is bounded in $L^{m (r - 1 + γ)} (Ω)$ , and since $\frac{1}{| x |^{p}}$ belongs to $L^{ρ} (Ω)$ for every $ρ < \frac{N}{p}$ , the sequence ${{(T_{n} (u_{n}))}^{p - 1} / (| x |^{p} + \frac{1}{n})}$ is bounded in $L^{s} (Ω)$ for every s such that $\begin{matrix} \frac{1}{s} > \frac{p}{N} + \frac{p - 1}{m (r - 1 + γ)}, \end{matrix}$ that, implies $\begin{matrix} s < \frac{N m (r - 1 + γ)}{N (p - 1) + p m (r - 1 + γ)} . \end{matrix}$ Taking into consideration that $\begin{matrix} \frac{N m (r - 1 + γ)}{N (p - 1) + p m (r - 1 + γ)} > 1, \end{matrix}$ is equivalent to $\begin{matrix} m > \frac{N}{N - p} \frac{p - 1}{r - 1 + γ}, \end{matrix}$ which is true, since $\frac{N}{N - p} \frac{p - 1}{r - 1 + γ} < 1$ , by the assumption $r > p^{}$ . Therefore, $\begin{matrix} (16) & the sequence {{(T_{n} (u_{n}))}^{p - 1} / (| x |^{p} + \frac{1}{n})} is bounded in L^{1} (Ω) . \end{matrix}$

On other hand choosing $T_{k} (u_{n})$ as test function in (11), we obtain, dropping a positive term, $\begin{matrix} (17) & α \int_{Ω} {| \nabla T_{k} (u_{n}) |}^{p} ⩽ C, \end{matrix}$ which, together with (15), yields that ${u_{n}}$ is bounded in $W_{0}^{1, p} (Ω)$ for every $a ⩾ 0$ . □

We now deal with the case of f belonging to $L^{m} (Ω), 1 < m < \frac{r}{r - 1 + γ}$ . In this case, one cannot expect to have solutions in $W_{0}^{1, p} (Ω)$ , but in a larger space. Lemma 3.2.
Let $u_{n}$ be the solution of problem ( 11 ), with $0 < γ < 1$ . Assume that f be a nonnegative function in $L^{m} (Ω)$ with $1 < m < \frac{r}{r - 1 + γ}$ and suppose that ( 6 ) and ( 7 ) hold. Then, $u_{n}$ is bounded in $L^{m (r - 1 + γ)} (Ω)$ and $| \nabla u_{n} |^{q}$ is bounded in $L^{1} (Ω)$ , with $q = p m \frac{r - 1 + γ}{r}$ . Moreover, if p and m satisfy one of the following assumptions
$\frac{N + (1 - γ) N}{N + 1 - γ} ⩽ p < N$ and $1 < m < \frac{r}{r - 1 + γ}$ ,

$1 < p < \frac{N + (1 - γ) N}{N + 1 - γ}$ and $\frac{r}{p (r - 1 + γ)} < m < \frac{r}{r - 1 + γ}$
then, $u_{n}$ is bounded in $W_{0}^{1, q} (Ω)$ .
Proof.
As in the proof of Lemma 3.1, we consider the approximate problems (11). Let $ε > 0$ , since $1 < m < \frac{r}{r - 1 + γ}$ , we have $0 < μ : = 1 - γ m - (m - 1) (r - 1) < 1$ and define $\begin{matrix} (18) & v = \frac{u_{n}}{{(u_{n} + ε)}^{μ}}, \end{matrix}$ then, we can write $\begin{matrix} \nabla v = \frac{\nabla u_{n}}{{(u_{n} + ε)}^{μ}} - μ \frac{\nabla u_{n} u_{n}}{{(u_{n} + ε)}^{μ + 1}} = \frac{(1 - μ) u_{n} + ε}{{(u_{n} + ε)}^{μ + 1}} \nabla u_{n}, \end{matrix}$ by using (6), we deduce that $\begin{matrix} (19) & M (x) | \nabla u_{n} |^{p - 2} \nabla u_{n} \cdot \nabla v ⩾ α \frac{(1 - μ) u_{n} + ε}{{(u_{n} + ε)}^{μ + 1}} | \nabla u_{n} |^{p} ⩾ α (1 - μ) \frac{| \nabla u_{n} |^{p}}{{(u_{n} + ε)}^{μ}} . \end{matrix}$ Now testing (11) by (18) and using (19), we obtain $\begin{matrix} (20) & α (1 - μ) \int_{Ω} \frac{| \nabla u_{n} |^{p}}{{(u_{n} + ε)}^{μ}} + b \int_{Ω} \frac{u_{n}^{r}}{{(u_{n} + ε)}^{μ}} ⩽ a \int_{Ω} \frac{u_{n}^{p - μ}}{| x |^{p} + \frac{1}{n}} + \int_{Ω} f u_{n}^{1 - μ - γ} . \end{matrix}$

Dropping the positive first term, and then letting ε tend to zero, we thus have $\begin{matrix} b \int_{Ω} u_{n}^{r - μ} ⩽ a \int_{Ω} \frac{u_{n}^{p - μ}}{| x |^{p} + \frac{1}{n}} + \int_{Ω} f u_{n}^{1 - μ - γ}, \end{matrix}$ which is nothing but (13), since $μ = - λ$ . Starting from this inequality, and working as in the proof of Lemma 3.1, we prove the boundedness of ${u_{n}}$ in $L^{m (r - 1 + γ)} (Ω)$ . Using this fact and (20), we obtain $\begin{matrix} (21) & \int_{Ω} \frac{| \nabla u_{n} |^{p}}{{(u_{n} + ε)}^{μ}} ⩽ C . \end{matrix}$ Let $q < p$ . Applying Hölder inequality and by (21), we find $\begin{matrix} \int_{Ω} | \nabla u_{n} |^{q} = \int_{Ω} \frac{| \nabla u_{n} |^{q}}{{(u_{n} + ε)}^{\frac{μ q}{p}}} {(u_{n} + ε)}^{\frac{μ q}{p}} ⩽ C {(\int_{Ω} {(u_{n} + ε)}^{\frac{μ q}{p - q}})}^{\frac{p - q}{p}} . \end{matrix}$ Finally we choose q such that $\begin{matrix} \frac{μ q}{p - q} = m (r - 1 + γ), \end{matrix}$ it is easy to verify that $\begin{matrix} q = p m \frac{r - 1 + γ}{r}, \end{matrix}$ then we obtain $\begin{matrix} (22) & \int_{Ω} | \nabla u_{n} |^{q} ⩽ C . \end{matrix}$ Now, under the assumption $m < \frac{r}{r - 1 + γ}$ , we find that $q < p$ . On other hand $\begin{matrix} q > 1 ⟺ m > \frac{r}{p (r - 1 + γ)}, \end{matrix}$ where $\frac{r}{p (r - 1 + γ)} < \frac{r}{(r - 1 + γ)}$ , and that $\begin{matrix} \frac{r}{p (r - 1 + γ)} ⩽ 1 ⟺ p ⩾ \frac{N + (1 - γ) N}{N + 1 - γ}, \end{matrix}$ with $\frac{N + (1 - γ) N}{N + 1 - γ} > 1$ . Then, if the parameters p and m satisfy the assumption (i) or (ii) of the statement, we have that $q > 1$ and so, by (22), we can conclude that ${u_{n}}$ is bounded in $W_{0}^{1, q} (Ω)$ , with $q = p m \frac{r - 1 + γ}{r}$ . □

4. Proofs of Theorems 1.1 and 1.2

We are ready to prove the existence of at least a solution of (5) in the sense of Definition 1.1

Proof of Theorem 1.1.

Thanks to Lemma 3.1, the sequence ${u_{n}}$ is bounded in $W_{0}^{1, p} (Ω)$ . Therefore, there exists a function $u \in W_{0}^{1, p} (Ω)$ such that (up to a subsequence) $\begin{matrix} (23) & \{\begin{matrix} u_{n} ⇀ u & in W_{0}^{1, p} (Ω) \\ u_{n} \to u & a.e. in Ω . \end{matrix} \end{matrix}$ We use the fact that, thanks to (16), (14)) and Lemma 2.2, we have that the right-hand side $\begin{matrix} a \frac{{(T_{n} (| u_{n} |))}^{p - 1}}{(| x |^{p} + \frac{1}{n})} + \frac{f_{n} (x)}{{(| u_{n} | + \frac{1}{n})}^{γ}} - b | u_{n} |^{r - 2} u_{n} is bounded in L_{loc}^{1} (Ω) . \end{matrix}$ Therefore, thanks to Remark 2.2 after Theorem 2.1 of [6], we have that $\begin{matrix} (24) & \nabla u_{n} \to \nabla u a.e. in Ω . \end{matrix}$ For the first term, by (24) we have that $\begin{matrix} M (x) | \nabla u_{n} |^{p - 2} \nabla u_{n} \to M (x) | \nabla u |^{p - 2} \nabla u a.e. in Ω, \end{matrix}$ furthermore $M (x) | \nabla u_{n} |^{p - 2} \nabla u_{n}$ is majorette by $β | \nabla u_{n} |^{p - 1}$ and by Vitali’s Theorem, we have $\begin{matrix} lim_{n \to \infty} \int_{Ω} M (x) | \nabla u_{n} |^{p - 2} \nabla u_{n} \cdot \nabla φ = \int_{Ω} M (x) | \nabla u |^{p - 2} \nabla u \cdot \nabla φ . \end{matrix}$

On other hand, by (23) we can see also that, the sequence ${u_{n}}$ converges to u strongly in $L^{p} (Ω)$ and almost everywhere in Ω. As for the sequence ${T_{n} {(u_{n})}^{p - 1} / (| x |^{p} + \frac{1}{n})}$ , since it is bounded in $L^{s} (Ω)$ for some $s > 1$ , it strongly converges to $\frac{u^{p - 1}}{| x |^{p}}$ in $L^{1} (Ω)$ . Since ${u_{n}}$ is bounded in $L^{m (r - 1 + γ)} (Ω)$ and $m > 1$ , we also have that $\begin{matrix} (25) & | u_{n} |^{r - 2} u_{n} strongly converges to | u |^{r - 2} u in L^{1} (Ω) . \end{matrix}$ Next, let $ω = {φ \neq 0}$ then by Lemma 2.2, one has, for every $φ \in C_{c}^{1} (Ω)$ $\begin{matrix} | \frac{f_{n} φ}{{(u_{n} + \frac{1}{n})}^{γ}} | ⩽ \frac{‖ φ ‖_{L^{\infty} (Ω)}}{c_{ω}^{γ}} f, \end{matrix}$ then from the later estimate, (23) and applying Lebesgue Theorem, we obtain $\begin{matrix} (26) & lim_{n \to \infty} \int_{Ω} \frac{f_{n} φ}{{(u_{n} + \frac{1}{n})}^{γ}} = \int_{Ω} \frac{f φ}{u^{γ}} . \end{matrix}$ Therefore, if φ belongs to $C_{c}^{1} (Ω)$ , we can pass to the limit in the identities $\begin{matrix} (27) & \begin{matrix} \int_{Ω} M (x) | \nabla u_{n} |^{p - 2} \nabla u_{n} \cdot \nabla φ + b \int_{Ω} | u_{n} |^{r - 2} u_{n} φ \\ = a \int_{Ω} \frac{{(T_{n} (u_{n}))}^{p - 1}}{| x |^{p} + \frac{1}{n}} φ + \int_{Ω} \frac{f_{n}}{{(u_{n} + \frac{1}{n})}^{γ}} φ . \end{matrix} \end{matrix}$ Hence, we conclude that the solution u satisfes the conditions (9) and (10) of Definition 1.1, so that the proof of Theorem 1.1 is now completed. □

Proof of Theorem 1.2.

As a consequence of the Lemma 3.2, we have that $| \nabla u_{n} |^{q}$ is bounded in $L^{1} (Ω)$ and so we can use the almost everywhere convergence of $\nabla u_{n}$ to $\nabla u$ and Fatou Lemma to conclude that $| \nabla u |^{q} \in L^{1} (Ω)$ . On other hand if the assumptions (i) or (ii) of the Lemma 3.2 are satisfied, then the sequence of approximated solutions ${u_{n}}$ is bounded in $W_{0}^{1, q} (Ω)$ , with $q = p m \frac{r - 1 + γ}{r}$ , so that it weakly converges (up to sub-sequences) to a function u in the same space. Thanks to the almost everywhere convergence of the gradients in (24), we infer that $M (x) | \nabla u_{n} |^{p - 2} \nabla u_{n}$ converges almost everywhere in Ω to $M (x) | \nabla u |^{p - 2} \nabla u$ . Furthermore, this term is majorised by $β | \nabla u_{n} |^{p - 1}$ . Since $p - 1 < q$ , using Vitali’s Theorem, we deduce that $\begin{matrix} M (x) | \nabla u_{n} |^{p - 2} \nabla u_{n} \to M (x) | \nabla u |^{p - 2} \nabla u in L^{1} (Ω), \end{matrix}$ then, we can pass to the limit, as $n \to \infty$ , in the first term in the left hand side of (27) for every $φ \in C_{c}^{1} (Ω)$ . Finally we can repeat the same argument used in the proof of Theorem 1.1 in order to pass to the limit in the second term in the left hand side and the terms situated in the right hand side of (27). Thus, we conclude that the solution u satisfies the conditions (9) and (10) of Definition 1.1. □

References

Abdellaoui and

Attar, Quasilinear elliptic problem with Hardy potential and singular term, Comm. On Pure and App. Analy. 12(3) (2013), 1363–1380.

Abdellaoui and

Peral, Existence and nonexistence results for quasilinear elliptic problems involving the p-Laplacian, Ann. Mat. Pura Appl. 182(3) (2003), 247–270. doi:10.1007/s10231-002-0064-y.

Adimurth,

Boccardo,

G.R.

Cirmi and

Orsina, The regularizing effect of lower order terms in elliptic problems involving Hardy potential, Adv. Nonlinear Stud. 17(2) (2017), 311–317. doi:10.1515/ans-2017-0001.

Bénilan,

Boccardo,

Gallouët,

Gariepy,

Pierre and

J.L.

Vázquez, An

L^{1}

-theory of existence and uniqueness of solutions of nonlinear elliptic equations, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 22(2) (1995), 241–273.

Boccardo and

Croce, The impact of a lower order term in a Dirichlet problem with a singular nonlinearity, Port. Math. 76(3) (2019), 407–415. doi:10.4171/PM/2041.

Boccardo and

Murat, Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations, Nonlinear Anal. 19(6) (1992), 581–597. doi:10.1016/0362-546X(92)90023-8.

Boccardo,

Murat and

J.P.

Puel, Existence of bounded solutions for nonlinear unilateral problems, Ann. Mat. Pura Appl. 152 (1988), 183–196. doi:10.1007/BF01766148.

Boccardo,

Orsina and

Peral, A remark on existence and optimal summability of solutions of elliptic problems involving Hardy potential, Discrete Contin. Dyn. Syst. 16(5) (2006), 513–523. doi:10.3934/dcds.2006.16.513.

Brézis and

Cabré, Some simple nonlinear PDE’s without solutions, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. 8 (1998), 223–262.

10.

Canino,

Sciunzi and

Trombetta, Existence and uniqueness for p-Laplace equations involving singular nonlinearities, NoDEA Nonlinear Differential Equations Appl. 23(2) (2016), Art. 8.

11.

Chipot, On some singular nonlinear problems for monotone elliptic operators, Rend. Lincei Mat. Appl. 30 (2019), 295–316.

12.

Chipot and

L.M.

De Cave, New techniques for solving some class of singular elliptic equations, Rend. Lincei Mat. Appl. 29(3) (2018), 487–510.

13.

Dautray and

J.L.

Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Physical Origins and Classical Methods, Vol. 1, Springer, Berlin, 1990.

14.

De Cave,

Durastanti and

Oliva, Existence and uniqueness results for possibly singular nonlinear elliptic equations with measure data, Nonlinear Differential Equations and Applications 25 (2018), 18.

15.

L.M.

De Cave, Nonlinear elliptic equation with singular nonlinearities, Asymptotic Analysis 84 (2013), 181–195. doi:10.3233/ASY-131173.

16.

L.M.

De Cave and

Oliva, On the regularizing effect of some absorption and singular lower order terms in classical Dirichlet problems with

L^{1}

data, J. Elliptic Parabol. Equ. 2(1–2) (2016), 73–85. doi:10.1007/BF03377393.

17.

El Hadfi,

Benkirane and

Youssfi, Existence and regularity results for parabolic equations with degenerate coercivity, Complex Variables and Elliptic Equations 63(5) (2018), 715–729. doi:10.1080/17476933.2017.1332596.

18.

El Hadfi,

El Ouardy,

Ifzarne and

Sbai, On nonlinear parabolic equations with singular lower order term, J. Elliptic Parabol. Equ. 8 (2022), 49–75. doi:10.1007/s41808-021-00138-5.

19.

El Ouardy and

El Hadfi, Some nonlinear parabolic problems with singular natural growth term, Results in Mathematics, Results Math 77 (2022), 95. doi:10.1007/s00025-022-01631-6.

20.

El Ouardy,

El Hadfi and

Sbai, Existence of positive solutions to nonlinear singular parabolic equations with Hardy potential, J. Pseudo-Differ. Oper. Appl. 13 (2022), 28. doi:10.1007/s11868-022-00457-8.

21.

Esposito and

Sciunzi, On the Hopf boundary lemma for quasilinear problems involving singular nonlinearities and applications, J. Funct. Anal. 278(4) (2020), 108346.

22.

Ferone,

Mercaldo and

S.A.

Segura de Leon, Singular elliptic equation and a related functional, ESAIM Control Optim. Calc. Var. 27 (2021), 39.

23.

Garcia Azorero and

Peral, Hardy inequalities and some critical elliptic and parabolic problems, J. Differential Equations 144 (1998), 441–476. doi:10.1006/jdeq.1997.3375.

24.

Leray and

J.L.

Lions, Quelques résultats de Visik sur les problemes elliptiques semi-linéaires par les méthodes de Minty et Browder, Bull. Soc. Math. France 93 (1965), 97–107. doi:10.24033/bsmf.1617.

25.

Oliva, Regularizing effect of absorption terms in singular problems, Journal of Mathematical Analysis and Applications 1(472) (2019), 1136–1166. doi:10.1016/j.jmaa.2018.11.069.

26.

Oliva and

Petitta, On singular elliptic equations with measure sources, ESAIM Control Optim. Calc. Var. 22(1) (2016), 289–308. doi:10.1051/cocv/2015004.

27.

Sbai and

El Hadfi, Regularizing effect of absorption terms in singular and degenerate elliptic problems, Journal: Boletim da Sociedade Paranaense de Matemática. (2020), arXiv preprint, arXiv:2008.03597.

28.

Sbai and

El Hadfi, Degenerate elliptic problem with a singular nonlinearity, Complex Variables and Elliptic Equations. doi:10.1080/17476933.2021.2014458.

29.

Sbai,

El Hadfi and

Zeng, Nonlinear singular elliptic equations of p-Laplace type with superlinear growth in the gradient, Mediterr. J. Math. 20 (2023), 32. doi:10.1007/s00009-022-02244-7.

30.

Tyagi, An existence of positive solution to singular elliptic equations, Boll. Union Mat. Ital. 7 (2014), 45–53. doi:10.1007/s40574-014-0003-z.