# Singular Nonlocal Problem Involving Measure Data

- 193 Downloads

## Abstract

*f*is a nonnegative function over \(\Omega \).

## Keywords

Elliptic PDE Fractional Sobolev space Schauder fixed point theorem## Mathematics Subject Classification

35J35 35J60## 1 Introduction

*m*refers to the Lebesgue measure. Certainly, for bounded \(\Omega \) we have \({M}^q\subset {M}^{\bar{q}}\) if \(q\ge \bar{q}\), for some fixed positive \(\bar{q}\). We bring to mind that the following continuous embeddings hold

### Definition 1.1

*f*.

## 2 Results on Existence of Solutions

*h*is a non-increasing continuous function that may blow up at zero.

*u*is affected by the behavior of

*h*at infinity. So, we need to assume the following

### Definition 2.1

We define the space \(W_0^{s,2}(\Omega )\) as \(W_0^{s,2}(\Omega )=\{u\in W^{s,2}(\Omega ):u=0~\text {in}~\mathbb {R}^N{\setminus }\Omega \}\).

### Definition 2.2

For both the cases, \(\gamma \le 1\) and \(\gamma >1\), we will show the existence of weak solutions for the problem (2.1) in the subsequent Subsects. 2.1 and 2.2.

*n*.

### Lemma 2.3

Problem (2.7) admits a nonnegative weak solution \(u_n\in W_0^{s,2} (\Omega )\cap L^{\infty }(\Omega )\).

### Proof

*w*to the following problem

*w*as a test function in the weak formulation (2.8) with the test function space \(W_0^{s,2}(\Omega )\). Thus

*v*. We now prove that the map

*G*is continuous on \(L^2(\Omega )\). Consider a sequence (\(v_k\)) that converges to

*v*in \(L^2(\Omega )\). Then by the dominated convergence theorem,

*G*is continuous over \(L^2(\Omega )\).

Now, applying the Schauder fixed point theorem to obtain that *G* has a fixed point \(u_n\in L^2(\Omega )\) that is a solution to Eq. (2.7) in \(W_0^{s,2}(\Omega )\). Furthermore, \(u_n\) belongs to \(L^{\infty }(\Omega )\) by Canino et al. (2017).

Since, \(\left( h_n\left( u_n+\frac{1}{n}\right) f_n+{\mu }_n\right) \ge 0\) then by the maximum principle \(u_n\ge 0\) and this concludes the proof. \(\square \)

The next step is to prove that (\(u_n\)) is uniformly bounded from below on compact subsets of \(\Omega \).

### Lemma 2.4

The sequence (\(u_n\)) is such that for every \(K\subset \subset \Omega \) there exists \(C_K\) (independent of n) such that \(u_n(x)\ge C_K >0\), a.e. in *K*, and for every \(n\in \mathbb {N}\).

### Proof

*x*in

*K*and \(C_K\) is independent of

*n*. Before establishing this we will show that the sequence (\(w_n\)) is monotonically increasing. Towards this we consider

*K*of \(\Omega \), there exists a constant \(L_K\) such that \(w_1\ge L_K>0\).

*K*, for if not, i.e. there exists a subset of

*K*which is of non zero Lebesgue measure on which \(u_n<w_n\). On taking \((u_n-w_n)^-\) as a test function in the Eq. (2.19), we get

*K*, and so \(\forall K \subset \subset \Omega \) there exists \(C_K\) such that \(u_n \ge C_K >0\) a.e. in

*K*. \(\square \)

We are now in a position to prove the existence of a solutions to the problem (2.1). In order to do this we differentiate between the following two cases.

### 2.1 When \(\gamma \le 1\)

In this subsection, we consider the problem in Eq. (2.7) for the case of \(\gamma \le 1\).

### Lemma 2.5

Let \(u_n\) be a solution of Eq. (2.7), where *h* satisfy Eqs. (2.3) and (2.4), with \(\gamma \le 1\). Then \((u_n)\) is bounded in \(W_0^{s,q}(\Omega )\) for every \(q<\frac{N}{N-s}\).

### Proof

*m*we get

### Theorem 2.6

Let \(\gamma \le 1\). Then there exists a weak solution *u* of the problem (2.1) in \(W_0^{s,q}(\Omega )\) for every \(q< \frac{N}{N-s}\).

### Proof

*u*such that the sequence \((u_n)\) converges weakly to

*u*in \(W_0^{s,q}(\Omega )\) for every \(q< \frac{N}{N-s}\). This implies that for \(\varphi \) in \(C_c^1(\Omega )\)

*u*both strongly in \(L^1(\Omega )\) and a.e. in \(\Omega \). Thus, taking \(\varphi \) in \(C_c^1(\Omega )\), we have

*K*is the set \(\{x\in \Omega : \varphi (x)\ne 0\}\). This is sufficient to apply the dominated convergence theorem to obtain

### 2.2 When \(\gamma >1\)

This case corresponds to a strongly singular case, which is why we can produce some local estimates on \(u_n\) in the fractional Sobolev space. We shall give global estimates on \(T_k^{\frac{\gamma +1}{2}}(u_n)\) in \(W_0^{s,2}(\Omega )\) with the aim of giving sense, at least in a weak sense, to the boundary values of *u*.

### Lemma 2.7

Let \(u_n\) be a weak solution of the problem in Eq. (2.7) with \(\gamma >1\). Then \(T_k^{\frac{\gamma +1}{2}}(u_n)\) is bounded in \(W_0^{s,2}(\Omega )\) for every fixed \(k>0\).

### Proof

Now, in order to pass to the limit \(n\rightarrow \infty \) in the weak formulation (2.8), we require to prove some local estimates on \(u_n\). We first prove the following.

### Lemma 2.8

Let \(u_n\) be a weak solution of the problem (2.7) with \(\gamma >1\). Then (\(u_n\)) is bounded in \(W_{loc}^{s,q}(\Omega )\) for every \(q<\frac{N}{N-s}\).

### Proof

We follow Benilan et al. (1995) to prove this lemma. We prove the lemma in two steps.

\({\textit{Step 1}}\) We claim that the sequence \(\left( G_1(u_n)\right) \) is bounded in \(W_0^{s,q}(\Omega )\) for every \(q<\frac{N}{N-s}\).

We can see that \(G_1(u_n)=0\) when \(0\le u_n\le 1\), \(G_1(u_n)=u_n-1\), otherwise i.e when \(u_n>1\). So \((-\Delta )^{s/2} G_1(u_n)=(-\Delta )^{s/2} u_n\) for \(u_n>1\).

*m*we have

*Step 2* We claim that \(T_1(u_n)\) is bounded in \(W_{loc}^{s,q}(\Omega )\) for every \(q<\frac{N}{N-s}\).

*n*. We want to show that for every \(K\subset \subset \Omega \),

*K*in Lemma 2.4. We again use the Proposition 3.3 due to Canino et al. (2017) to obtain

The proof is complete since \(u_n=T_1(u_n)+G_1(u_n)\) and hence the sequence (\(u_n\)) is bounded in \(W_{loc}^{s,q}(\Omega )\) for every \(q<\frac{N}{N-s}\). \(\square \)

As a consequence we have the following existence result.

### Theorem 2.9

Let \(\gamma >1\). Then there exists a weak solution *u* of Eq. (2.1) in \(W_{loc}^{s,q}(\Omega )\) for every \(q<\frac{N}{N-s}\).

The proof of this theorem is a straightforward application of the Theorem 2.6 and using the results in Lemmas 2.7 and 2.8.

## 3 Further Analysis of the Case \(\gamma <1\)

*very weak*’ solution which is defined as follows.

### Definition 3.1

We will show existence of a nonnegative very weak solution to the problem (3.1).

### Definition 3.2

### Theorem 3.3

Let \(\underline{u}\) is a subsolution and \(\bar{u}\) is a supersolution to the problem (3.1) with \(\underline{u}\le \bar{u}\) in \(\Omega \), then there exists a solution *u* to Eq. (3.1) according to the Definition 3.1 such that \(\underline{u}\le u\le \bar{u}\).

### Proof

*u*satisfies

*u*is a very weak solution to Eq. (3.1).

We only show that \(u\le \bar{u}\) in \(\Omega \). The proof of the other side of the inequality, \(\underline{u}\le u\), follows similarly.

*u*is a solution to Eq. (3.5) and \(\bar{u}\) is a supersolution to Eq. (3.1). Subtracting Eq. (3.6) from Eq. (3.4) we have, for every \(\varphi \in C_0^1(\bar{\Omega })\) such that \(\varphi \ge 0\) and \((-\Delta )^s\phi \in L^{\infty }(\Omega )\),

*u*to the following linear problem

*G*is well-defined since a unique solution exists to the problem in Eq. (3.7) due to Petitta (2016). We need to show that this map is continuous in \(L^1(\Omega )\). Let us choose a sequence \((v_n)\) converging to some function

*v*in \(L^1(\Omega )\), then as

*h*is a non-increasing continuous function we get

*G*is continuous.

*G*(

*v*) is bounded in \(W_0^{s,q}(\Omega )\) for every \(q<\frac{N}{N-s}\) and therefore, by Rellich-Kondrachov theorem we get \(G(L^1(\Omega ))\) is bounded and hence relatively compact in \(L^1(\Omega )\).

We can now apply the Schauder fixed point theorem to see that *G* has a fixed point \(u\in L^1(\Omega )\). According to the result from Step 1, we conclude that *u* is a very weak solution to Eq. (3.1) such that \(\underline{u}\le u\le \bar{u}\). \(\square \)

### Proof

We want to find both a subsolution and a supersolution to the problem (3.1) in the sense of Definition 3.2. Then we will use the result of Theorem 3.3 to prove the existence of a solution to the problem (3.1) in the sense of Definition 3.1.

*h*and (3)

*v*being a solution to Eq. (3.8). Hence we have \(v > 0\) in \(\Omega \).

*h*given in Eqs. (2.3) and (2.4), we can see that \(h(\varphi _1)\in L^1(\Omega )\) if and only if \(\gamma <1\) by using arguments as in Lazer and McKenna (1991). We see that the first eigenfunction of the Laplacian, denoted by \(\phi _1\), is a subsolution to the problem (3.8). It is proved in Lazer and McKenna (1991) that there exists a solution \(v\in L^1(\Omega )\) to the problem (3.8) for \(h(s)=\frac{1}{s^{\gamma }}\). On using the conditions over

*h*in Eqs. (2.3) and (2.4) and the result in Lazer and McKenna (1991), we guarantee the existence of a solution to Eq. (3.8). Hence we have \(v > 0\) in \(\Omega \), \(h(v)f \in L^1(\Omega )\), and

*v*is a subsolution to the problem (3.1).

*w*be the solution of

*v*is a very weak solution to Eq. (3.8). Then we get

*w*is nonnegative, then we have \(0<h(Z)\le h(v)\). Thus, we can say

*Z*is a positive function in \(L^1(\Omega )\) such that \(h(Z)\le h(v)\in L^1(\Omega )\) and

*Z*is a supersolution to Eq. (3.1). We can now apply Theorem 3.3 to get the conclusion that there exists a solution

*u*to problem (3.1) in the sense of Definition 3.1. \(\square \)

### 3.1 When \(f\in L^1(\Omega )\cap L^{\infty }(\Omega _\delta )\).

We proved Theorem 3.4 by assuming a strong regularity on *f* i.e. *f* belongs to \(C^{\beta }(\bar{\Omega })\) for some \(0<\beta < 1\). In this section we do some relaxation on our assumption on *f* in order to prove the existence of solution.

For a fix \(\delta >0\), let us define \(\Omega _\delta =\{x\in \Omega :\text {dist}(x,\partial \Omega )<\delta \}\), and let *f* be an a.e. positive function in \(L^1(\Omega )\cap L^\infty (\Omega _\delta )\).

### Theorem 3.5

Let \(f\in L^1(\Omega )\cap L^{\infty }(\Omega _\delta )\) such that \(f>0\) a.e. in \(\Omega \) for some fixed \(\delta >0\). Then there exists a solution for problem (3.1) in the sense of Definition 3.1.

### Proof

*x*in \(\Omega \)

*x*from \(\partial \Omega \). Thus, we have

## 4 Appendix

## Notes

### Acknowledgements

Two of the authors, Sekhar Ghosh and Ratan Kr. Giri, thanks the financial assistantship received from the Council of Scientic and Industrial Research (CSIR), India and the Ministry of Human Resource Development (MHRD), Govt. of India respectively.

## References

- Arcoya, D., Moreno-Mérida, L., Leonori, T., Martínez-Aparicio, P.J., Orsina, L., Petitta, F.: Existence and nonexistence results. J. Differ. Equ.
**246**(10), 4006–4042 (2009)CrossRefzbMATHGoogle Scholar - Benilan, P., Boccardo, L., Gallouët, T., Gariepy, R., Pierre, M., Vazquez, J.L.: An \(L^1\) theory of existence and uniqueness of nonlinear elliptic equations. Ann. Scuola Norm. Sup. Pisa
**22**, 240–273 (1995)zbMATHGoogle Scholar - Boccardo, L., Orsina, L.: Semilinear elliptic equations with singular nonlinearities. Calc. Var. PDEs
**37**(3–4), 363–380 (2010)MathSciNetCrossRefzbMATHGoogle Scholar - Brezis, H., Cabré, X.: Some simple nonlinear PDE’s without solutions. Boll. della Unione Mat. Ital. Ser. 8
**1–B**(2), 223–262 (1998)MathSciNetzbMATHGoogle Scholar - Canino, A., Montoro, L., Sciunzi, B., Squassina, M.: Nonlocal problems with singular nonlinearity. Bull. Sci. Math.
**141**(3), 223–250 (2017)MathSciNetCrossRefzbMATHGoogle Scholar - Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math.
**136**(5), 521–573 (2012)Google Scholar - Evans, L.C.: Partial Differential Equations. Graduate Studies in Mathematics, vol. 19. American Mathematical Society, Providence (2010)Google Scholar
- Folland, G.B.: Real analysis: modern techniques and their applications, 2nd edn. Wiley, Hoboken (2013)zbMATHGoogle Scholar
- Gatica, J.A., Oliker, V., Waltman, P.: Singular nonlinear boundary-value problems for second-order ordinary differential equations. J. Differ. Equ.
**79**, 62–78 (1989)MathSciNetCrossRefzbMATHGoogle Scholar - Ghanmi, A., Saoudi, K.: A multiplicity results for a singular problem involving the fractional \(p\)-Laplacian operator. Complex Var. Elliptic Equ.
**61**(9), 1199–1216 (2016)MathSciNetCrossRefzbMATHGoogle Scholar - Giachetti, D., Martinez-Aparicio, P.J., Murat, F.: Definition, existence, stability and uniqueness of the solution to a semilinear elliptic problem with a strong singularity at \(u = 0\). Annali della Scuola Normale Superiore di Pisa (2017a) (hal-01348682v2)Google Scholar
- Giachetti, D., Martínez-Aparicio, P.J., Murat, F.: A semilinear elliptic equation with a mild singularity at \(u=0\): existence and homogenization. J. Math. Pures Appl.
**107**(1), 41–77 (2017b)Google Scholar - Lazer, A.C., McKenna, P.J.: On a singular nonlinear elliptic boundary-value problem. Proc. Am. Math. Soc.
**111**(3), 721–730 (1991)MathSciNetCrossRefzbMATHGoogle Scholar - Leray, J., Lions, J.L.: Quelques résultates de višik sur les problémes elliptiques semilinéaires par les méthodes de Minty et Browder. Bull. Soc. Math. Fr.
**93**, 97–107 (1965)CrossRefzbMATHGoogle Scholar - Marcus, M., Véron, L.: Nonlinear second order elliptic equations involving measures. De Gruyter Series in Nonlinear Analysis and Applications. De Gruyter, Berlin (2013)Google Scholar
- Montenegro, M., Ponce, A.C.: The sub-supersolution method for weak solutions. Proc. Am. Math. Soc.
**136**(7), 2429–2438 (2008)MathSciNetCrossRefzbMATHGoogle Scholar - Oliva, F., Petitta, F.: On singular elliptic equations with measure sources. ESAIM Control Optim. Calc. Var.
**22**, 289308 (2016)MathSciNetCrossRefzbMATHGoogle Scholar - Oliva, F., Petitta, F.: Finite and infinite energy solutions of singular elliptic problems: existence and uniqueness. J. Differ. Equ.
**264**(1), 311–340 (2018)MathSciNetCrossRefzbMATHGoogle Scholar - Panda, A., Ghosh, S., Choudhuri, D.: Elliptic partial differential equation involving singularity, arXiv:1709.00905 [math.AP] (2017)
- Petitta, F.: Some remarks on the duality method for integro-differential equations with measure data. Adv. Nonlinear Stud.
**16**(1), 115124 (2016)MathSciNetCrossRefzbMATHGoogle Scholar - Taliaferro, S.: A nonlinear singular boundary value problem. Nonlinear Anal.
**3**, 897–904 (1979)MathSciNetCrossRefzbMATHGoogle Scholar