Regularizing Effects for Some Nonlocal Elliptic Problems

Abstract

We study the regularizing effect of the zero order term in some elliptic problems, which present in the principal part a nonlocal operator, i.e., the fractional Laplacian operator. We prove the existence of bounded energy solutions, even if the data are assumed to be very irregular.

1. Introduction and Main Results

In this paper, we study existence and regularity results for the solution to the following nonlocal elliptic problem

$$\left\{\begin{array}{cc}{(-\Delta )}^{s}u+b(x)u=f\hfill & \mathrm{in} \Omega ,\hfill \\ u=0,\hfill & \mathrm{in} {\mathbb{R}}^N\setminus \Omega ,\hfill \end{array}\right.$$

(1)

where $\Omega$ is a bounded open set of ${\mathbb{R}}^N,N\ge 2$ and b, and f satisfy the following assumptions:

$$b(x)\ge 0,b(x)\in L^1(\Omega ),$$

(2)

$$f(x)\ge 0,f(x)\in L^1(\Omega ),$$

(3)

and there exists a constant $\beta >0$ such that

$$f(x)\le \beta b(x).$$

(4)

The operator which appears on the left-hand side is the fractional Laplace operator defined, up to a normalization factor, by the Riesz potential as

$$-{(-\Delta )}^{s}u(x):=P.V.{\int }_{{\mathbb{R}}^N}{\displaystyle \frac{u(x+y)+u(x-y)-2u(x)}{y^{N+2s}}}dy,x\in {\mathbb{R}}^N,$$

when $s\in (0,1)$ is a fixed parameter (for more details see [1,2,3]).

We are interested to investigate the relation between the zero order term $b(x)u(x)$ and the source term $f(x)$ and in what way the presence of the lower order term influences the regularity of the solution to Problem (1).

This kind of study has been analyzed in the local context by Boccardo and Arcoya in [4]. Taking in the principal part of Problem (1), an operator in divergence form like the Laplacian operator, and assuming (4) with the same regularity hypothesis on the data $b(x)$ and $f(x)$, i.e., (2) and (3), they proved the existence of bounded solutions. This fact was already surprising in the local context. Indeed, if we consider the following simple model problem in the absence of a zero order term

$$\left\{\begin{array}{cc}-\Delta u=f\hfill & \mathrm{in}\Omega ,\hfill \\ u=0,\hfill & \mathrm{on}\partial \Omega ,\hfill \end{array}\right.$$

(5)

we can obtain $L^{\infty }(\Omega )$ solutions requiring that f belongs to $L^m(\Omega )$ with $m>N/2$ (see for example [5]). An analogous result can be obtained in the nonlocal context if we take into account the following nonlocal elliptic problem

$$\left\{\begin{array}{cc}{(-\Delta )}^{s}u=f\hfill & \mathrm{in}\Omega ,\hfill \\ u=0,\hfill & \mathrm{in}{\mathbb{R}}^N\setminus \Omega .\hfill \end{array}\right.$$

(6)

In [6], the authors proved that Problem (6) admits bounded solutions with the assumption that f belongs to the space $L^m(\Omega )$ with $m>N/2s$ (see Theorem 13 of [6]). Once again, also in the nonlocal context, it is required for the source term f to be regular enough.

Similarly to what happens in the local context, when $b(x)$ is not equivalent to zero, assuming a relation of Equation (4) between $b(x)$ and $f(x)$ and requiring that the data are only summable functions (so they could also be unbounded functions), we are able to prove the existence of an energy-bounded solution (see Theorem 1). This is a surprising fact that shows in what way the presence of the zero order term $b(x)u(x)$ can have a regularizing effect on the solution of Problem (1), also in the nonlocal context, where the situation could be more difficult to handle, since we have to take into account not only what happens locally but also in all the space ${\mathbb{R}}^N$.

We introduce now our notion of the solution to Problem (1).

Definition 1.

We say that $u\in X_0^s(\Omega )$ is an energy solution to Problem (1) if it satisfies

$${\displaystyle \frac{\gamma (N,s)}{2}}{\int \int }_{{\mathbb{R}}^{2N}}{\displaystyle \frac{(u(x)-u(y))(\phi (x)-\phi (y))}{{|x-y|}^{N+2s}}}dxdy+{\int }_{\Omega }b(x)u(x)\phi (x)dx={\int }_{\Omega }f(x)\phi (x)dx,$$

(7)

for every $\phi \in X_0^s(\Omega )$.

For the definition of the space $X_0^s(\Omega )$ (see (9)) and other preliminaries, we direct the reader to Section 2.

The main result of the paper is the following.

Theorem 1.

Let us assume (2)–(4). Then, there exists a nonnegative solution u to Problem (1) in the sense of Definition 1. Moreover, it results in $u\in L^{\infty }(\Omega )$ and the following estimate holds true

$${\parallel u\parallel }_{L^{\infty }(\Omega )}\le \beta ,$$

(8)

where β is defined in (4).

This paper is organized as follows: In Section 2, we introduce preliminaries and definitions useful to prove our results; in Section 3, we present a sequence of approximating problems (see (15)) and we prove the existence of a solution $u_n\in L^{\infty }(\Omega )\cap X_0^s(\Omega )$ to these problems; in Section 4, we obtain some a priori estimates useful to pass to the limit as $n\to +\infty$ in Section 5 in order to obtain the results of Theorem 1. In Section 6, we summarize the results obtained mentioning some open problems.

2. Preliminaries and Functional Setting

In this section, we introduce the functional spaces in which we set our problem and other preliminaries useful to prove our results.

For $0<s<1$ we, introduce the fractional Sobolev space of order s

$$H^s({\mathbb{R}}^N)=\{u\in L^2({\mathbb{R}}^N){:|\xi |}^s\mathcal{F}(u)(\xi )\in L^2({\mathbb{R}}^N)\},$$

where $\mathcal{F}(u)$ is the Fourier transform of u and ${|\xi |}^s$ the Fourier multiplier. Since we are interested in dealing with Dirichlet problems in bounded domains, we introduce the following space

$$X_0^s(\Omega )=\{u\in L^2({\mathbb{R}}^N)\mathrm{with}u=0\mathrm{a}.\mathrm{e}.\mathrm{in}{\mathbb{R}}^N\setminus \Omega \},$$

(9)

endowed with the norm

$${\parallel u\parallel }_{X_0^s(\Omega )}={\left({\int \int }_D{\displaystyle \frac{{|u(x)-u(y)|}^2}{{|x-y|}^{N+2s}}}\right)}^{{\displaystyle \frac{1}{2}}},$$

where $D={\mathbb{R}}^{2N}\setminus (\mathcal{C}\Omega \times \mathcal{C}\Omega )$. The pair $(X_0^s(\Omega ),\parallel ·{\parallel }_{X_0^s(\Omega )})$ yields a Hilbert space (see, for instance, Lemma 7 of [7]).

Moreover, the fractional Laplacian ${(-\Delta )}^s:X_0^s(\Omega )\to X^{-s}(\Omega )$ is a continuous operator, with $X^{-s}(\Omega )$ the dual space of $X_0^s(\Omega )$.

In the following, we will use the relation between the norm in the energy space $X_0^s(\Omega )$ and the $L^2$-norm of the fractional Laplacian (for more details, see Proposition 3.6 of [1])

$${\parallel u\parallel }_{X_0^s(\Omega )}^2={\displaystyle \frac{2}{\gamma (N,s)}}{\parallel {(-\Delta )}^{s/2}u\parallel }_{L^2({\mathbb{R}}^N)}^2,$$

(10)

where

$$\gamma (N,s)={\displaystyle \frac{4^s\Gamma (\frac{N}{2}+s)}{-{\pi }^{\frac{N}{2}}\Gamma (-s)}}.$$

(11)

Another important theorem, i.e., Sobolev’s theorem in the nonlocal context, that we will use in the proof of our results is the following.

Theorem 2

(Theorem 6.5 of [1]). Let $s\in (0,1)$ and $N>2s.$ There exists a constant $S(N,s)$ such that, for any measurable and compactly supported function $f:{\mathbb{R}}^N\to \mathbb{R}$, we have

$${\parallel f\parallel }_{L^{2_s^{*}}({\mathbb{R}}^N)}^2\le S(N,s){\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}{\displaystyle \frac{|f(x)-f(y)|}{{|x-y|}^{N+2s}}}dxdy,$$

where $2_s^{*}={\displaystyle \frac{2N}{N-2s}}$ is the Sobolev fractional critical exponent.

Moreover, if u and $\phi$ are smooth enough, with vanishing condition outside $\Omega$, we have the following duality product,

$${\displaystyle \frac{2}{\gamma (N,s)}}{\int }_{{\mathbb{R}}^N}u{(-\Delta )}^s\phi dx={\int \int }_D{\displaystyle \frac{(u(x)-u(y))(\phi (x)-\phi (y))}{{|x-y|}^{N+2s}}}dxdy,$$

which means that ${(-\Delta )}^s$ is self-adjoined in $X_0^s(\Omega )$.

We recall here some definitions of functions that we will use in the following.

For any measurable function v, we will denote by $v^{+}$ and $v^{-}$, respectively, the positive part and the negative part of the function v as follows

$$v^{+}=max(v,0),v^{-}=max(-v,0).$$

(12)

For $k\ge 0$, we introduce the functions $G_k(r)$ and $T_k(r)$ defined, respectively, as

$$G_k(r)={(|r|-k)}^{+}\mathrm{sign}(r),$$

(13)

$$T_k(r)=max\{-k;min\{k,r\}\}.$$

(14)

3. Approximating Problems

In order to prove the existence of an energy-bounded solution to Problem (1), we reason by approximation. In particular, we consider the following sequence of approximating problems

$$\left\{\begin{array}{cc}{(-\Delta )}^s{u}_n+b_n{u}_n=f_n,\hfill & \mathrm{in}\Omega ,\hfill \\ u_n=0,\hfill & \mathrm{on}{\mathbb{R}}^N\setminus \Omega ,\hfill \end{array}\right.$$

(15)

where $s\in (0,1)$, $N\ge 2$, $b_n$ and $f_n$ are bounded functions defined as follows

$$f_n(x)={\displaystyle \frac{f(x)}{1+\frac{1}{n}f(x)}},b_n(x)={\displaystyle \frac{b(x)}{1+\frac{\beta }{n}b(x)}}.$$

(16)

Notice that, since the function $w(r)={\displaystyle \frac{r}{1+\frac{r}{n}}}$ is an increasing function, by (4) we deduce that

$$0\le f_n(x)={\displaystyle \frac{f(x)}{1+\frac{1}{n}f(x)}}\le {\displaystyle \frac{\beta b(x)}{1+\frac{\beta }{n}b(x)}}=\beta b_n(x).$$

(17)

The existence of a solution to Problem (15) is established in the next theorem.

Theorem 3.

Let us assume (2), (3) and (17). Then, there exists a nonnegative solution $u_n\in X_0^s(\Omega )\cap L^{\infty }(\Omega )$ of (15), which satisfies

$${\displaystyle \frac{\gamma (N,s)}{2}}{\int \int }_D{\displaystyle \frac{(u_n(x)-u_n(y))(\phi (x)-\phi (y))}{{|x-y|}^{N+2s}}}dxdy+{\int }_{\Omega }{b}_n(x)u_n(x)\phi (x)dx={\int }_{\Omega }{f}_n(x)\phi (x)dx.$$

(18)

for every $\phi \in X_0^s(\Omega )$.

Proof. 

Let us fix $n\in \mathbb{N}$. Let v be a function in $L^2(\Omega )$ and we define $w=\mathcal{T}(v)$ as the unique solution in $X_0^s(\Omega )$ to the following problem

$$\left\{\begin{array}{cc}-{(\Delta )}^{s}w+b_{n}w=f_n,\hfill & \mathrm{in}\Omega ,\hfill \\ w=0,\hfill & \mathrm{in}{\mathbb{R}}^N\setminus \Omega .\hfill \end{array}\right.$$

(19)

We observe that w belongs to $L^{\infty }(\Omega )$ since $b_n$ and $f_n$ are bounded functions.

Using w as a test function in (19), we find

$${\displaystyle \frac{\gamma (N,s)}{2}}{\int \int }_D{\displaystyle \frac{{(w(x)-w(y))}^2}{{|x-y|}^{N+2s}}}dxdy+{\int }_{\Omega }{b}_n{w}^{2}dx\le {\int }_{\Omega }{f}_n|w|\le n{\int }_{\Omega }|w|={n\parallel w\parallel }_{L^1(\Omega )}.$$

Dropping the nonnegative integral on the left-hand side, it follows, by Sobolev’s embedding (see Theorem 2), that

$${\parallel w\parallel }_{X_0^s(\Omega )}\le Cn,$$

with $C=C(N,s,\Omega )$ and the ball of radius $Cn$ is invariant under $\mathcal{T}$ in the space $X_0^s(\Omega )$.

Furthermore, once again, by Sobolev’s embedding, $\mathcal{T}$ is a continuous and compact operator from $X_0^s(\Omega )$ to $X_0^s(\Omega )$. Using Schauder’s Fixed-Point Theorem, there exists a solution $u_n\in X_0^s(\Omega )\cap L^{\infty }(\Omega )$ to Problem (15). In order to establish the nonnegativity of the function $u_n$ in $\Omega$, we take $-u_n^{-}$ as a test function in (15) (see (12)) and we have

$$\begin{gathered} {\displaystyle \frac{\gamma (N,s)}{2}}{\int \int }_D{\displaystyle \frac{(u_n(x)-u_n(y))(-u_n^{-}(x)+u_n^{-}(y))}{{|x-y|}^{N+2s}}}dxdy+{\int }_{\Omega }{b}_n{u}_n(-u_n^{-})dx= \\ {\int }_{\Omega }{f}_n(-u_n^{-})dx\le 0. \end{gathered}$$

(20)

Decomposing $u_n=u_n^{+}-u_n^{-}$, we observe that the integrand function in the first integral of (20) can be estimated in the following way

$$\begin{gathered} (u_n(x)-u_n(y))(-u_n^{-}(x)+u_n^{-}(y))=u_n(x)(-u_n^{-}(x)+u_n^{-}(y))-u_n(y)(-u_n^{-}(x)+u_n^{-}(y))= \\ {(u_n^{-}(x))}^2+u_n(x)u_n^{-}(y)+u_n(y)u_n^{-}(x)+{(u_n^{-}(y))}^2= \\ {(u_n^{-}(x))}^2+{(u_n^{-}(y))}^2-2u_n^{-}(x)u_n^{-}(y)+2u_n^{-}(x)u_n^{-}(y)+u_n(x)u_n^{-}(y)+u_n(y)u_n^{-}(x)= \\ {(u_n^{-}(x)-u_n^{-}(y))}^2+2u_n^{-}(x)u_n^{-}(y)+u_n(x)u_n^{-}(y)+u_n(y)u_n^{-}(x)= \\ {(u_n^{-}(x)-u_n^{-}(y))}^2+2u_n^{-}(x)u_n^{-}(y)+(u_n^{+}(x)-u_n^{-}(x))u_n^{-}(y)+(u_n^{+}(y)-u_n^{-}(y))u_n^{-}(x)= \\ {(u_n^{-}(x)-u_n^{-}(y))}^2+2u_n^{-}(x)u_n^{-}(y)+u_n^{+}(x)u_n^{-}(y)-u_n^{-}(x)u_n^{-}(y)+u_n^{+}(y)u_n^{-}(x)-u_n^{-}(y)u_n^{-}(x)= \\ {(u_n^{-}(x)-u_n^{-}(y))}^2+2u_n^{-}(x)u_n^{-}(y)-2u_n^{-}(x)u_n^{-}(y)+u_n^{+}(x)u_n^{-}(y)+u_n^{+}(y)u_n^{-}(x)\ge {(u_n^{-}(x)-u_n^{-}(y))}^2. \end{gathered}$$

Consequently, since $b_n\ge 0$ and

$${\int }_{\Omega }{b}_n{u}_n(-u_n^{-})dx={\int }_{\Omega }{b}_n(u_n^{+}-u_n^{-})(-u_n^{-})dx={\int }_{\Omega }{b}_n{(u_n^{-})}^{2}dx\ge 0,$$

Inequality (20) becomes

$$0\le {\displaystyle \frac{\gamma (N,s)}{2}}{\parallel u_n^{-}(x)-u_n^{-}(y)\parallel }_{X_0^s(\Omega )}^2\le {\int \int }_D{\displaystyle \frac{(u_n(x)-u_n(y))(-u_n^{-}(x)+u_n^{-}(y))}{{|x-y|}^{N+2s}}}dxdy\le 0,$$

which means

$${\parallel u_n^{-}\parallel }_{X_0^s(\Omega )}^2=0$$

that implies that $u_n^{-}=0$ a.e. in $\Omega$, i.e., $u_n\ge 0$ a.e. in $\Omega$. □

4. A Priori Estimates

In this section, we prove some a priori estimates on the solution $u_n$ of the approximating Problem (15) that are useful to pass to the limit in the distributional formulation (18).

Proposition 1.

Assume (2), (3) and (17). Then, $u_n$ is uniformly bounded in $L^{\infty }(\Omega ),$ i.e.,

$$||u_n{||}_{L^{\infty }(\Omega )}\le \beta ,$$

(21)

where β is as in (4).

Proof. 

Let $k\ge \beta$. Taking $G_k(u_n)\in L^{\infty }(\Omega )$ (see (13)) as a test function in Problem (15), using (17), we obtain

$$\begin{gathered} {\displaystyle \frac{\gamma (N,s)}{2}}{\int \int }_D{\displaystyle \frac{(u_n(x)-u_n(y))(G_k(u_n(x))-G_k(u_n(y)))}{{|x-y|}^{N+2s}}}dxdy+{\int }_{\Omega }{b}_n(x)u_n(x)G_k(u_n(x))dx= \\ {\int }_{\Omega }{f}_n(x)G_k(u_n(x))dx\le \beta {\int }_{\Omega }{b}_n(x)G_k(u_n(x))dx, \end{gathered}$$

(22)

from which, observing that

$${\int }_{\Omega }{b}_n(x)u_n(x)G_k(u_n(x))dx\ge k{\int }_{\Omega }{b}_n(x)G_k(u_n(x))dx,$$

Inequality (22) becomes

$$\begin{gathered} {\displaystyle \frac{\gamma (N,s)}{2}}{\int \int }_D{\displaystyle \frac{(u_n(x)-u_n(y))(G_k(u_n(x))-G_k(u_n(y)))}{{|x-y|}^{N+2s}}}dxdy+ \\ k{\int }_{\Omega }{b}_n(x)G_k(u_n(x))dx-\beta {\int }_{\Omega }{b}_n(x)G_k(u_n(x))dx\le 0. \end{gathered}$$

(23)

Since $k\ge \beta$, we have that

$$k{\int }_{\Omega }{b}_n(x)G_k(u_n(x))dx-\beta {\int }_{\Omega }{b}_n(x)G_k(u_n(x))dx\ge 0.$$

(24)

Moreover, we observe that, for any $r\in \mathbb{R}$, $r=T_k(r)+G_k(r)$ (see definitions (14) and (13)), so we have that

$$\begin{gathered} (u_n(x)-u_n(y))(G_k(u_n(x))-G_k(u_n(y)))= \\ \left[T_k(u_n(x))+G_k(u_n(x))-T_k(u_n(y))-G_k(u_n(y))\right](G_k(u_n(x))-G_k(u_n(y)))= \\ (G_k(u_n(x))-G_k(u_n(y)))(G_k(u_n(x))-G_k(u_n(y)))+ \\ (T_k(u_n(x))-T_k(u_n(y)))(G_k(u_n(x))-G_k(u_n(y)))= \\ {(G_k(u_n(x))-G_k(u_n(y)))}^2+(T_k(u_n(x))-T_k(u_n(y)))(G_k(u_n(x))-G_k(u_n(y))). \end{gathered}$$

(25)

We have that

$$(T_k(u_n(x))-T_k(u_n(y)))(G_k(u_n(x))-G_k(u_n(y)))\ge 0.$$

(26)

Indeed, if we set

$$R_1=\{(x,y)\in \Omega :u(x)\ge k,u(y)\ge k\},$$

$$R_2=\{(x,y)\in \Omega :u(x)\le k,u(y)\le k\},$$

$$R_3=\{(x,y)\in \Omega :u(x)\le k,u(y)\ge k\},$$

$$R_4=\{(x,y)\in \Omega :u(x)\ge k,u(y)\le k\},$$

remembering the definitions of $T_k(s)$ and $G_k(s)$ (see (14) and (13)), we obtain

$$\begin{gathered} (T_k(u_n(x))-T_k(u_n(y)))(G_k(u_n(x))-G_k(u_n(y))){\chi }_{R_1}=(k-k)(u(x)-k-u(y)+k){\chi }_{R_1}=0, \\ (T_k(u_n(x))-T_k(u_n(y)))(G_k(u_n(x))-G_k(u_n(y))){\chi }_{R_2}=0, \\ (T_k(u_n(x))-T_k(u_n(y)))(G_k(u_n(x))-G_k(u_n(y))){\chi }_{R_3}=(u(x)-k)(k-u(y)){\chi }_{R_3}\ge 0, \end{gathered}$$

and

$$(T_k(u_n(x))-T_k(u_n(y)))(G_k(u_n(x))-G_k(u_n(y))){\chi }_{R_4}=(k-u(y))(u(x)-k){\chi }_{R_4}\ge 0.$$

It follows that (26) holds true. Using (25) and (26), we have found that

$${\int \int }_D{\displaystyle \frac{(u_n(x)-u_n(y))(G_k(u_n(x))-G_k(u_n(y)))}{{|x-y|}^{N+2s}}}dxdy\ge {\int \int }_D{\displaystyle \frac{{(G_k(u_n(x))-G_k(u_n(y)))}^2}{{|x-y|}^{N+2s}}}dxdy.$$

(27)

Using (23) and (27), it follows that

$${\parallel G_k(u_n(x))-G_k(u_n(y))\parallel }_{X_0^s(\Omega )}^2\le 0,$$

which implies $G_k(u_n)=0$, a.e. in $\Omega$, i.e., $u_n\le k$; hence, (21) holds, choosing $k=\beta$. □

Thanks to the fundamental estimate (21), we can obtain the following energy estimate for the solution $u_n$ to Problem (15).

Proposition 2.

Assume (2), (3) and (17). Then, $u_n$ is uniformly bounded in $X_0^s(\Omega )$, i.e.,

$$||u_n{||}_{X_0^s(\Omega )}\le C,$$

(28)

where $C=C(\gamma (N,s),\beta ,\parallel f{\parallel }_{L^1(\Omega )})$ is a positive constant independent of n.

Proof. 

Choosing $u_n$ as a test function in (18), using (21), we obtain

$${\displaystyle \frac{\gamma (N,s)}{2}}\parallel u_n{\parallel }_{X_0^s(\Omega )}^2+{\int }_{\Omega }{b}_n(x)u_n^2(x)dx={\int }_{\Omega }f(x)u_n(x)dx\le \beta {\int }_{\Omega }f\le {C(\beta ,\parallel f\parallel }_{L^1(\Omega )}).$$

(29)

Dropping the second integral on the left-hand side of (29) since it is nonnegative, it follows that (28) holds true with $C=C(\gamma (N,s),\beta ,\parallel f{\parallel }_{L^1(\Omega )})$. □

5. Proof of Theorem 1

In order to prove Theorem 1, we observe that, by Energy Estimate (28), the solution $u_n$ is bounded in the space $X_0^s(\Omega )$; hence, $u_n$ weakly converges to a function u in $X_0^s(\Omega )$. Moreover, using Estimate (21), we deduce, thanks to Fatou’s lemma, that $|u|\le \beta$, so (8) holds true.

In this way, thanks to the fact that $u_n$ weakly converges to u in $X_0^s(\Omega )$, we are able to pass to the limit in the first integral of (18), obtaining that

$$\begin{gathered} \underset{n\to +\infty }{lim}{\displaystyle \frac{\gamma (N,s)}{2}}{\int \int }_D{\displaystyle \frac{(u_n(x)-u_n(y))(\phi (x)-\phi (y))}{{|x-y|}^{N+2s}}}dxdy= \\ {\displaystyle \frac{\gamma (N,s)}{2}}{\int \int }_D{\displaystyle \frac{(u(x)-u(y))(\phi (x)-\phi (y))}{{|x-y|}^{N+2s}}}dxdy \end{gathered}$$

for every $\phi \in X_0^s(\Omega ).$ Thanks to the fact that the sequence $b_n(x)u_n(x)\phi (x)$$L^1$-converges to $b(x)u(x)\phi (x)$ and $f_n(x)\phi (x)$$L^1$-converges to $f(x)\phi (x)$, we can also pass to the limit in the other integrals of Formula (18), so we obtain that u satisfies (7) and, consequently, the proof of Theorem 1 is concluded.

6. Conclusions and Open Problems

Theorem 1 proves the existence of a nonnegative bounded solution u to Problem (1) in the sense of Definition 1. The boundedness of the solution in the space $L^{\infty }(\Omega )$ is a surprising fact, under the assumption that f is only $L^1(\Omega )$. A key role is played by the zero order term $b(x)u(x)$ that gives a strong regularizing effect.

According to the open problems, it could be interesting to consider the evolutive problem related to Problem (1) and investigate if we can obtain the same regularizing effect for the solution thanks to the presence of the zero order term, under weak assumptions on the data.