Positive solutions of nonlinear Dirichlet BVPs in ODEs with time and space singularities

In the present work, we analyze the existence of positive solutions to the singular Dirichlet BVP,

(1a)

(1b)

The differential operator on the left-hand side of Eq. (1a) can be equivalently written as ${(t^{-a}{(t^{a}u)}^{\prime })}^{\prime }$ and, after the substitution $x(t)=t^{a}u(t)$, it takes the form ${(t^{-a}{x}^{\prime })}^{\prime }$, which arises in numerous important applications. Operators of such type were studied in phase transitions of Van der Waals fluids [1–4], in population genetics, especially in models for the spatial distribution of the genetic composition of a population [5, 6], in the homogeneous nucleation theory [7], in relativistic cosmology for description of particles which can be treated as domains in the universe [8], and in the nonlinear field theory [9], in particular, when describing bubbles generated by scalar fields of Higgs type in the Minkowski spaces [10].

The aim of this paper is to study the case $a\in (-1,0)$ which is fundamentally different from the case $a\in (-\mathrm{\infty },-1)$. The latter setting was studied in [11, 12], where the structure and properties of the set of all positive solutions to (1a) and (1b) were investigated (the cardinality of this set is a continuum).

In the sequel, we introduce the basic notation and state the preliminary results required in the analysis of problem (1a) and (1b). Here, we focus our attention on the case $a\in (-1,0)$ and prove the existence of at least one positive solution of (1a) and (1b). In contrast to [11, 12], we consider the more general situation in which f may be also singular at $y=0$. This means that we have to deal with the following additional difficulties.

Let u be a positive solution of problem (1a) and (1b), where $f(t,x,y)$ has a singularity at $y=0$. Then there exists $t_0\in (0,T)$ such that $u(t_0)>0$, $u^{\prime }(t_0)=0$ and hence f is unbounded in a neighborhood of the point $(t_0,u(t_0),u^{\prime }(t_0))$. Unfortunately, we do not know the exact position of $t_0$ and therefore, it is not possible to construct a universal Lebesgue integrable majorant for all functions $f(t,u_n(t),u_n^{\prime }(t))$, where $u_n$ are positive solutions of a sequence of auxiliary regular problems. Consequently, the Lebesgue dominated convergence theorem is not applicable and we have to use arguments based on the Vitali convergence theorem instead; see Lemma 2. Another tool used in the proofs is a combination of regularization and sequential techniques with the Leray-Schauder nonlinear alternative.

The investigation of singular Dirichlet BVPs has a long history and a lot of methods for their analysis are available. One of the most important ones is the topological degree method providing various fixed point theorems and existence alternative theorems; see, e.g., Lemma 1. For more information on the topological degree method and its application to numerous BVPs, including Dirichlet problems, we refer the reader to the monographs by Mawhin [13–15].

Throughout this paper, we work with the following conditions on the function f in (1a):

(H₁) $f\in Car([0,T]\times \mathcal{D})$, where $\mathcal{D}={\mathbb{R}}_{+}\times {\mathbb{R}}_0$.

holds, where $\phi \in L^1[0,T]$, $h\in C([0,\mathrm{\infty })\times [0,\mathrm{\infty }))$, $g,r\in C({\mathbb{R}}_{+})$ are positive, h is nondecreasing in both its arguments, g and r are nonincreasing, and

we denote the norms in $C[0,T]$ and $L^1[0,T]$, respectively. $A{C}^1[0,T]$ denotes the set of functions whose first derivative is absolutely continuous on $[0,T]$, while $A{C}_{\mathrm{loc}}^1(0,T]$ is the set of functions having absolutely continuous first derivative on each compact subinterval of $(0,T]$. We use the symbol $meas(\mathcal{M})$ to denote the Lebesgue measure of ℳ.

Definition 1 We say that a function $u\in A{C}^1[0,T]$ is a positive solution of problem (1a) and (1b) if $u>0$ on $(0,T)$, u satisfies the boundary conditions (1b) and (1a) holds for a.e. $t\in [0,T]$.

respectively. Then it follows from (H₁) that $f_n\in Car([0,T]\times {\mathbb{R}}^2)$ and (H₂) and (H₃) yield

(3)

(4)

As a first step in the analysis, we investigate auxiliary regular BVPs of the form

(7a)

(7b)

To show the solvability of problem (7a) and (7b), we use the following alternative of Leray-Schauder type which follows from [[16], Theorem 5.1].

Lemma 1 Let E be a Banach space, U be an open subset of E and $\ell \in U$. Assume that $\mathcal{F}:\overline{U}\to E$ is a compact operator. Then either

A1: ℱ has a fixed point in $\overline{U}$, or

A2: there exists an element $u\in \partial U$ and $\lambda \in (0,1)$ with $u=\lambda \mathcal{F}(u)+(1-\lambda )\ell$.

In limit processes, we apply the following Vitali convergence theorem; cf. [17–19].

The paper is organized as follows. In Section 2, we collect auxiliary results used in the subsequent analysis. Section 3 is devoted to the study of limit properties of solutions to Eq. (7a). In Section 4, we investigate auxiliary regular problems associated with the singular problem (1a) and (1b). We show their solvability and describe properties of their solutions. An existence result for the singular problem (1a) and (1b) is given in Section 5. Finally, in Section 6, we illustrate the theoretical findings by means of numerical experiments.

Throughout the paper $a\in (-1,0)$.

In this section, auxiliary statements necessary for the subsequent analysis are formulated.

Lemma 3 Let $\rho \in L^1[0,T]$ and

holds for a.e. $t\in [0,T]$.

Then $p\in C(0,T]$ and $H(t)=tp(t)$ for $t\in (0,T]$. We now show that p can be extended on $[0,T]$ in such a way that $p\in C[0,T]$. Integrating by parts yields

(10)

and therefore, $H^{″}\in L^1[0,T]$. Consequently, $H\in A{C}^1[0,T]$. Finally, it follows from $H^{\prime }(t)=p(t)-r(t)$ and $p(t)=\frac{H(t)}{t}$ that $r(t)=\frac{H(t)}{t}-H^{\prime }(t)$. Since, by (11), $\frac{a}{t}r(t)=H^{″}(t)+\rho (t)$, we see that equality (8) is satisfied for a.e. $t\in [0,T]$ which completes the proof. □

we conclude from the relation

The last equality contradicts (13). Consequently, (12) holds and the result follows. □

for $t\in [0,T]$, estimate (14) holds. □

Here, we investigate asymptotic properties of solutions of (7a). We also provide a related integral equation this solution satisfies.

holds for $t\in [0,T]$.

Then ${\rho }_m\in L^1[0,T]$ and ${lim}_{m\to \mathrm{\infty }}{\rho }_m(t)=\rho (t)$ for a.e. $t\in [0,T]$. Moreover, $|{\rho }_m(t)|\le \mu (t)$ for a.e. $t\in [0,T]$ and all $m\in \mathbb{N}$, where $\mu (t)=sup\{|f_n(t,x,y)|:|x|\le L,|y|\le L\}\in L^1[0,T]$. Consequently, by the Lebesgue dominated convergence theorem, $\rho \in L^1[0,T]$.

Let H be the function given in Lemma 3. By Lemma 3(ii), H can be extended on $[0,T]$ with $H\in A{C}^1[0,T]$ and −H satisfies (16) for a.e. $t\in [0,T]$. Therefore, each function $v\in A{C}_{\mathrm{loc}}^1(0,T]$ which satisfies Eq. (16) a.e. on $[0,T]$ has the form $v(t)=c^{\ast }t+d^{\ast }{t}^{-a}-H(t)$ for $t\in (0,T]$, with some $c^{\ast },d^{\ast }\in \mathbb{R}$. By assumption we know that $u\in A{C}_{\mathrm{loc}}^1(0,T]$ satisfies (16) a.e. on $[0,T]$, and therefore there exist $c,d\in \mathbb{R}$ such that $u(t)=ct+d{t}^{-a}-H(t)$, $t\in (0,T]$. Since by assumption $|u^{\prime }|\le L$ on $(0,T]$, we have $d=0$. Consequently, the function u can be extended on the interval $[0,T]$ in the class $A{C}^1[0,T]$ and (15) holds on $[0,T]$. □

Corollary 1 Let (H₁) hold. Let $u\in A{C}^1[0,T]$ be a solution of Eq. (7a). Then there exists a constant $c\in \mathbb{R}$ such that equality (15) is satisfied for $t\in [0,T]$.

Proof The result holds by Lemma 6 with $sup\{|u(t)|+|u^{\prime }(t)|:t\in [0,T]\}<\mathrm{\infty }$. □

Remark 2 Corollary 1 says that the set of all solutions $u\in A{C}^1[0,T]$ of Eq. (7a) depends on one parameter $c\in \mathbb{R}$ and $u(0)=0$.

In order to prove the solvability of problem (7a) and (7b), we first have to investigate the problem

(17a)

(17b)

depending on the parameter λ. Here, $\epsilon >0$ is from (H₂) and $n\in \mathbb{N}$.

The following result shows that the solvability of problem (17a) and (17b) is equivalent to the solvability of an integral equation in the set $C^1[0,T]$.

in the set $C^1[0,T]$.

holds for $t\in [0,T]$. Hence, $u(0)=0$ and $u(T)=0$ if and only if $c=0$. Consequently, if u is a solution of problem (17a) and (17b), then u is a solution of Eq. (18) in $C^1[0,T]$.

Let u be a solution of Eq. (18) in $C^1[0,T]$. Then $f_n(t,u(t),u^{\prime }(t))\in L^1[0,T]$. Hence, Lemma 3(ii) (with ρ replaced by $-\lambda f_n+(1-\lambda )\epsilon$) guarantees that $u\in A{C}^1[0,T]$ and u is a solution of Eq. (17a). Moreover, $u(0)=u(T)=0$. Consequently, u is a solution of problem (17a) and (17b) which completes the proof. □

The following results provide bounds for solutions of problem (17a) and (17b).

Lemma 8 Let (H₁)-(H₃) hold. Then there exists a positive constant S (independent of $n\in \mathbb{N}$ and $\lambda \in [0,1]$) such that for all solutions u of problem (17a) and (17b), the estimates

(19)

(20)

It is clear that L is independent of the choice of solution u to problem (17a) and (17b) and independent of $n\in \mathbb{N}$, $\lambda \in [0,1]$.

Hence, $u^{\prime }$ is decreasing on $[0,T]$, and therefore $u^{\prime }$ vanishes at a unique point $\xi \in (0,T)$ due to $u(0)=u(T)=0$. The inequality (21) now follows from the relations

Consequently, there exists $S>0$ such that $Kh(1+Tz,1+z)+M<z$ for $z\ge S$. Now, due to (26), ${\parallel u^{\prime }\parallel }_{\mathrm{\infty }}<S$, and therefore, by (25), ${\parallel u\parallel }_{\mathrm{\infty }}<ST$. □

We are now in the position to prove the existence result for problem (7a) and (7b).

Lemma 9 Let (H₁)-(H₃) hold. Then for each $n\in \mathbb{N}$, problem (7a) and (7b) has a solution u satisfying inequalities (19)-(21), where S is a positive constant independent of n.

It follows from Lemma 5 and (9) that

holds for a.e. $t\in [0,T]$ and all $x\in \overline{\mathrm{\Omega }}$ (cf. (9)). Consequently, the set $\{\mathcal{K}{(1,x)}^{\prime }:x\in \overline{\mathrm{\Omega }}\}$ is equicontinuous on $[0,T]$. Hence, the set $\mathcal{K}(1,\overline{\mathrm{\Omega }})$ is relatively compact in $C^1[0,T]$ by the Arzelà-Ascoli theorem. As a result, $\mathcal{K}(1,\cdot )$ is a compact operator and the condition (i) follows.

Due to the fact that by Lemma 7 any fixed point u of the operator $\mathcal{K}(\lambda ,\cdot )$ is a solution of problem (17a) and (17b), Lemma 8 guarantees that u satisfies inequality (20). Therefore, $\mathcal{K}$ has property (ii). Consequently, by Lemmas 1 and 8, for each $n\in \mathbb{N}$, problem (7a) and (7b) has a solution u satisfying estimates (19)-(21). □

Let $u_n$ be a solution of problem (7a) and (7b) for $n\in \mathbb{N}$. The following property of the sequence $\{|f_n(t,u_n(t),u_n^{\prime }(t))|\}$ is an important prerequisite for solving problem (1a) and (1b).

Lemma 10 Let (H₁)-(H₃) hold. Let $u_n$ be a solution of problem (7a) and (7b) for $n\in \mathbb{N}$. Then the sequence $\{|f_n(t,u_n(t),u_n^{\prime }(t))|\}$ is uniformly integrable on $[0,T]$.