Abstract
This paper is devoted to modifying the Schrödinger-type identity related to singular boundary value problem in (Zhang et al. in Bound. Value Probl. 2018:135, 2018). We also present some mathematical consequences of the method, including a stability result. The main technical tools used to develop the mathematical analysis are local and global bifurcation, monotonicity techniques, fixed point theory in b-metric spaces in (Liu et al. in Bull. Aust. Math. Soc. 94(1):121–130, 2016) and the maximum principle approach with respect to the Schrödinger operator in (Fan et al. in Math. Appl. 31(1):42–48, 2018). As an application, the uniqueness of solutions for singular boundary value problem for the Schrödinger equation is proved.
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1 Introduction
In this paper we consider a singular boundary value problem with mixed boundary conditions and spatial heterogeneities given by (see [4,5,6])
where:
-
(i)
\(\gimel =(0,W)\times (0,w)\) is a bounded rectangular domain in \(\mathbb{R}^{2}\), ℷ represents a porous medium, with Lipschitz boundary \(\partial \gimel =\daleth _{1}\cup \daleth _{2}\) where
$$ \daleth _{2}=\bigl(\{0\}\times [0,w]\bigr)\cup \bigl([0,W]\times \{w\} \bigr)\cup \bigl(\{W\} \times [0,w]\bigr) $$is the part in contact with air or covered by fluid and
$$ \daleth _{1}= [0,W]\times \{0\} $$is the impervious part of ∂ℷ. Let \(P=\gimel \times (0,M)\), where \(M>0\);
-
(ii)
−Δ stands for the minus Laplacian operator, χ is a function of the variable t satisfying
$$ c_{1} \leq \chi (t)\leq c_{2} \quad \text{a.e. } t\in (0,W) $$(2)for two positive constants \(c_{1}\) and \(c_{2}\) and \(\omega (t)\) satisfies
$$ 0\leq \omega (t)\leq 1\quad \text{a.e. } t\in \gimel ; $$(3) -
(iii)
the spatial heterogeneities on the boundary come given by the potentials \(V, b \in {\mathcal{C}}(\daleth _{2})\), where \(b>0\) on \(\daleth _{2}\) and V possesses arbitrary sign in each point \(x \in \daleth _{2}\);
-
(iv)
\(\partial f (x)\) stands for the outer normal derivative of f at \(t \in \daleth _{2}\).
In 2008, Polidoro and Ragusa in [7] proved a Harnack inequality for the positive solutions of ultraparabolic equations of the type \(Lf+Vf=0\), where L is a linear second order hypoelliptic operator and V belongs to a class of functions of Stummel-Kato type. They also obtained the existence of a Green function and an uniqueness result for the Cauchy–Dirichlet problem. In 2016, Guariglia and Silvestrov in [8] described a wavelet expansion theory for positive definite distributions over the real line and define a fractional derivative operator for complex functions in the distribution sense. In 2017, Goubet and Hamraoui in [9] investigated both numerically and theoretically the influence of a defect on the blow-up of radial solutions to a cubic NLS equation in dimension 2. Colorado in [10] showed the existence of positive bound and ground states for a system of coupled nonlinear Schrödinger–Korteweg–de Vries equations. In 2018, Khader and Adel in [11] introduced a study of the convergence analysis and error estimation of the obtained approximation solution. The FLDE is reduced to a system of algebraic equations with the help of the properties of wavelets polynomials. Rybalko in [12] studied an initial value problem for the one-dimensional non-stationary linear Schrödinger equation with a point singular potential. Zhang and Gu in [13] considered a three components system of nonlinear Schrödinger equations related to the Raman amplification in a plasma. In 2019, Scapellato in [14] showed some regularity properties of solutions to the elliptic equations on Herz spaces with two variable exponents. Meng in [15, 16] discussed the application of the new criteria for minimally thin sets with respect to the Schrödinger operator to an approximate solution of singular Schrödinger-type boundary value problems.
The classical Randon transform (see [17]) is defined by the Cauchy principal value of the singular integral. In 2018, Zhang et al. in [1] established the Schrödinger-type identity and applied it to a Schrödinger integral equation and then gave three examples where the kernel function is a Green’s function for a two-order system of Schrödinger integral equations.
In this paper, we shall use this identity to study the problem (1) with a power nonlinearity and an exponential nonlinearity both of which are singular as the solutions approaches 0. The results from previous applications are generalized and extended. In our approach, the problem is considered as a system of coupled Schrödinger boundary value problems on two half-lines. This system can exhibit different qualitative behaviors depending on the value of the vaccination-isolation reproductive number. Following the ideas of this method, we studied the existence and the behavior of periodic solutions of a generalized model with general heterogeneous coefficients, by using a continuation theorem based on coincidence degree theory. Regarding existence of a solution of the problem (1) with respect to the Schrödinger operator was obtained in [2] by applying the maximum principle approach for a two-order system of Schrödinger operator in [18,19,20,21,22]. The regularity of the solution of the problem (1) with respect to a two-order system of Schrödinger operator was also discussed in [23], where it was proved that \(\omega \in C^{0}([0,T]; L^{p}( \gimel ))\) for all \(p\in [1,+\infty )\) in both classes of free boundary problem of types (see [6])
and
and that \(f\in C^{0}([0,M]; L^{p}(\gimel ))\) for all \(p\in [1,2]\) in the second class.
2 A modified Schrödinger-type identity
For simplicity we shall denote the solution of (1) as \((f,\omega )\).
Lemma 2.1
Let
and
-
(i)
Let \(\epsilon >0\), \(k\geq 0\) and \(\varsigma \in \mathrm{D}( \mathbb{R}^{2}\times (0,M))\) such that \(\varsigma \geq 0\), \(\varsigma =0\) on \(\varSigma _{3}\). Then
$$ \int _{P} \chi (t_{1}) (f_{t_{2}}+\omega ) \biggl(\min \biggl(\frac{ {(f-k)^{+}}}{ \epsilon },\varsigma \biggr) \biggr)_{t_{2}} \,dt \,ds=0. $$(4) -
(ii)
Let \(\varsigma =0\) on \(\varSigma _{2}\). Then
$$ \int _{P} \chi (t_{1}) (f_{t_{2}}+\omega ) \biggl(\min \biggl(\frac{{(k-u)^{+}}}{ \epsilon },\varsigma \biggr)-\min \biggl( \frac{k}{\epsilon },\varsigma \biggr) \biggr)_{t_{2}} \,dt \,ds=0. $$(5)
Proof
Let ψ be a measure function satisfying
and
Then we know that
vanishes on \(\varSigma _{2}\) for any \(\kappa \in (-\kappa _{0},\kappa _{0})\), where \(\kappa _{0}>0\).
So
which yields
Equation (6) still holds for any \(\psi \in L^{2}(0,T; H^{1}( \gimel ))\), where \(\psi =0\) on \(\varSigma _{2}\) and \(\psi =0\) on \(\gimel \times ((0,\kappa _{0})\cup (M-\kappa _{0},M))\).
So
which shows that \(\varsigma \geq 0\), \(\varsigma =0\) on \(\varSigma _{3}\).
Set
Then (6) also gives
for any \(\kappa \in (-\kappa _{0},\kappa _{0})\), where
So
for any \(\kappa \in (-\kappa _{0},\kappa _{0})\).
By applying (7), we know that (4) holds for \(\varsigma \in \mathrm{D}(\mathbb{R}^{2}\times (0,M))\) such that \(\varsigma \geq 0\), \(\varsigma =0\) on \(\varSigma _{3}\).
If we put \(\varsigma =0\) on \(\varSigma _{2}\) and set
then (6) also yields
for any \(\kappa \in (-\kappa _{0},\kappa _{0})\), where
So
for all \(\kappa \in (-\kappa _{0},\kappa _{0})\).
By applying (8), we know that (5) holds for \(\varsigma \in \mathrm{D}(\mathbb{R}^{2}\times (0,M))\) such that \(\varsigma \geq 0\), \(\varsigma =0\) on \(\varSigma _{2}\). □
3 Uniqueness of the solution
In this section, we shall state and prove our main result: the solution of problem (1) is unique. Let us assume that
Now, we can state our uniqueness theorem.
Theorem 3.1
The solution of the problem (1) associated with the initial data \(\omega _{0}\) is unique.
Proof
Let \((f_{1}, \omega _{1})\) and \((f_{2}, \omega _{2})\) be two solutions of the problem (1) satisfying
a.e. in ℷ.
Set
and
It follows that
from Lemma 2.1.
Let
and
Note that the function χ admits an extension to \(\mathbb{R}\) from (9), still denoted by χ, such that \(\chi \in C^{1}( \mathbb{R},\mathbb{R})\) (see [24]).
It follows that
from Fubini’s theorem, where \(\varepsilon \in (0,\frac{\varepsilon _{0}}{2})\), \(\rho _{\varepsilon }\in \mathrm{D}(\mathbb{R}^{2})\) with
and \(f_{\varepsilon }=\rho _{\varepsilon }\ast f\).
Note that
is nonnegative for all \(y\in B(0,\varepsilon )\) and belongs to \(C_{0}^{1}(\gimel )\).
Since (10) holds for \(0\leq \varphi \in C_{0}^{1}(\gimel )\), we have
which yields
Put
and suppose that there exists \(t_{0}\in \mathcal{A}_{\varepsilon _{0}} \cap \gimel \) and \(\varepsilon _{1}\in (0,\frac{\varepsilon _{0}}{2})\) such that \(\alpha _{\varepsilon _{1}}(t_{0})>0\).
Since \(\alpha _{\varepsilon _{1}}\) is continuous and \(\mathcal{A}_{ \varepsilon _{0}}\cap \gimel \) is an open set,
and \(\alpha _{\varepsilon _{1}}(t)>0\) for all \(t\in \overline{B(t_{0},r)}\), where \(r>0\).
It follows that
in \(\overline{B(t_{0},r)}\) from (2).
Let us consider the following Dirichlet problem associated to a linear second order partial differential equation:
which yields
where
Note that \((a_{\varepsilon _{1}ij}(t))_{ij}\) is strictly elliptic in \(B(t_{0},r)\) with a positive constant
and the coefficients \(\frac{1}{\alpha _{\varepsilon _{1}}}\), \(\beta _{\epsilon _{1}}\) are in \(C^{1}(\overline{B(t_{0},r)})\). So, by the regularity theory (see [25] for example), the problem (12) has a unique solution \(\hat{\varsigma }\in C^{2}(\overline{B(t _{0},r)})\).
Moreover, since the function in the right side of the first equation of (12) satisfies \(\frac{1}{\alpha _{\varepsilon _{1}}}>0\) in \(B(t_{0},r)\), we have \(\hat{\varsigma }\geq 0\) in \(\overline{B(t_{0},r)}\) from the maximum principle (see [26]).
So (11) yields
where \(\varepsilon \in (0,\frac{\varepsilon _{0}}{2} )\).
It follows that
and
from (12), which gives that
So
where \(\varepsilon \in (0,\frac{\varepsilon _{0}}{2} )\) and \(t\in \mathcal{A}_{\varepsilon _{0}}\cap \gimel \), which yields
for any \(\varepsilon \in (0,\frac{\varepsilon _{0}}{2} )\).
By passing to the limit as \(\varepsilon \to 0\), we obtain
and
Notice that
for all \(0\leq \iota \in \mathrm{D}(0,M)\), which gives that
Next we prove
Put
where \(\sigma >0\) and \(s\in (0,M]\).
Note that \(\iota \in C^{1}([0,s])\) and
Note that \(\varsigma \iota ^{2}\in H^{1}(P)\), \(\varsigma \iota ^{2}=0\) on \(\partial \gimel \times (0,M)\) and
Choosing \(\pm \varsigma \iota ^{2}\) as test functions for (1) (see [27] for example), we obtain
It follows from (16) and (15) that
By using the Lebesgue theorem to \(\mathcal{R}_{\sigma }^{1}\) (see [28]), we have
We now estimate \(\mathcal{R}_{\sigma }^{2}\). We know that
from the definition of \(\iota '\), where
We have
(see [29, Proposition 2.1]), \(\iota \in C^{0}([0,s])\), \(\iota (0)=0\) and ι is uniformly bounded independently of σ.
Notice that
is right-continuous and vanishes at 0.
So
Similarly,
By letting \(\sigma \to 0\) in (19) and applying (20)–(21), we obtain
Combining (17), (18) and (22), we obtain
where \(s\in [0,M]\).
Since
is continuous on \([0,M]\) (see [30]), we deduce that
for all \(s\in [0,M]\) from (23).
So
which yields
If we choose
as the test functions in (25), then we have
By applying the Lebesgue theorem to \(S_{\sigma }^{1}\), we obtain
and
which yield
Put
and let \((\delta _{0},\omega _{0})\) and \((\delta ^{\varrho }, \omega _{\varrho })\) be the principal eigenvalue associated to −Δ operator in the domains ℷ and \(\gimel _{\varrho }\), respectively (see [31]).
It is obvious that
By the definitions of \(\omega _{0}\) and \(\omega _{\varrho }\), we obtain
where \(t \in \partial \gimel =\daleth _{0} \cup \daleth _{1}\), and
where \(t \in \daleth _{1}\).
Put
Put
and
respectively.
It is well known that
where \(\varrho >0\).
So
Combining (31), (35) and (36), there exists \(\varrho _{0}>0\) satisfying
where \(\varrho \in (0,\varrho _{0}]\).
Thanks to the regularity of \(\omega _{\varrho }\) and \(\omega _{0}\), there exists \(\varrho _{1} \in (0,\varrho _{0}]\) satisfying
where \(\varrho \in (0,\varrho _{1}]\) and \(t \in \daleth _{1}\).
Let \(\bar{x}_{\varrho }\in \daleth _{1}\) satisfy
It follows that there exists \(\bar{z} \in \daleth _{1}\) satisfying
Put
where
There exists \(\varrho _{2} \in (0,\varrho _{1}]\) such that
is a positive strict subsolution of (1) for each \(\varrho \in (0,\varrho _{2}]\) and \(\lambda \in (\delta ^{\varrho },\delta _{0})\).
It follows that
for each \(\varrho \in (0,\varrho _{1}]\) and \(\lambda \in (\delta ^{ \varrho },\delta _{0})\).
It follows that
from (30), (31), (32), (38), (39), (40), (41) and (43), which, together with (34), shows that there exists \(\varrho _{2} \in (0,\varrho _{1}]\) such that
for each \(\varrho \in (0,\varrho _{2}]\) and \(t \in \daleth _{1}\).
And hence
where \(t\in \daleth _{1}\) and \(\varrho \in (0,\varrho _{2}]\).
So
Combining (44), (45) and (46), we have
for \(\varrho \in (0,\varrho _{2}]\) and \(\lambda \in (\delta ^{\varrho }, \delta _{0})\).
It follows from (29) that there exists \(\tilde{y}_{\varrho } \in \daleth _{1}\) satisfying
for each \(\varrho \in (0,\varrho _{2}]\) and there exists \(\underline{z} \in \daleth _{1}\) satisfying
Combining (48) and (49), we obtain
for each \(t \in \daleth _{1}\) and \(\varrho \in (0,\varrho _{2}]\), which yields
uniformly on \(\daleth _{1}\).
Passing to the limit as \(\delta \to 0\) in (26) and using (27)–(28), we have
If we choose \(t_{2}\varsigma \) as a test function in (50), then we have
Define the following functions h and g in \((t_{1}^{1},t_{1}^{2})\) (resp. \((t_{2}^{1},t_{2}^{2})\)):
and
respectively (see [25]), where \(t_{1}^{1},t_{1}^{2}\in (0,W)\) and \(t_{2}^{1},t_{2}^{2}\in (0,w) \) satisfying that \(t_{1}^{1}< t_{1} ^{2}\), \(t_{2}^{1}< t_{2}^{2}\),
and δ is a positive real number.
We have
and \(\mathrm{h},\mathrm{g}\geq 0\).
If we set
then we see that \(\mathrm{h}\mathrm{g}^{2}\in C^{2}(\overline{\gimel _{1,2}})\) and \(\mathrm{h}\mathrm{g}^{2}\geq 0\).
Let us extend \(\mathrm{h}\mathrm{g}^{2}\) outside \(\gimel _{\varepsilon _{0}}\) by 0 and still denote by \(\mathrm{h} \mathrm{g}^{2}\) this function. We obtain
by choosing \(\varsigma =\mathrm{h}\mathrm{g}^{2}\) as a test function in (51).
We have
and
where
Since the function
is right-continuous and vanishes at \(t_{2}^{1}\), uniformly bounded independently of δ and \(|g'|\sim \frac{1}{\delta }\), we know that
Similarly
It follows from (55)–(56) that
by letting \(\delta \to 0\) in (54).
Similarly, it follows from (53) and (57) that
by passing to the limit as \(\delta \to 0\) in (52), where \(t\in [0,M]\).
Finally, it follows that
by letting \(\varepsilon \to 0\) in (58), where \(t\in [0,M]\).
So
where \(t\in [0,M]\), which yields
for all \(t\in [0,M]\).
Thanks to (2), we deduce that
for all \(t\in [0,M]\) (see [32]).
So
By exchanging the roles of \(\omega _{1}\) and \(\omega _{2}\), we obtain
We conclude that
Hence, (14) holds. If we combine (13) and (14), we see that the solution of problem (1) associated with the initial data \(\omega _{0}\) is unique. □
4 Examples
In this section, two boundary value problems involving nonlocal integral boundary conditions will be tested by using the present method.
Example 4.1
Consider the boundary value problem
Here, \(\delta =1/2\), \(\sigma =3/5\), \(\xi =1/3\), \(p_{2}=1\), \(p_{1}=3\), \(p_{0}=2\), \(\lambda =1\), A is a positive constant and
Clearly the constants \(p_{2}\), \(p_{1}\), and \(p_{0}\) satisfy the condition of Lemma 2.1, and
where \(\ell =A/7\). Using the given values, we find \(\alpha \approx 0.44269\) and \(\alpha _{1}\approx 0.21725\), It is easy to check that
and \(\ell \alpha _{1}<1\) when \(A<32.22094\). As all the conditions of Theorem 3.1 are satisfied the problem (59)–(60) has at least one solution on \([0,1]\). On the other hand, \(\ell \alpha <1\) whenever \(A<15.81242\) and thus there exists a unique solution for the problem (59)–(60) on \([0,1]\) by Theorem 3.1.
Example 4.2
Consider the boundary value problem
Here, \(\delta =1/2\), \(\sigma =3/5\), \(\xi =1/3\), \(p^{2}_{1}-4p_{2}p _{0}=1>0\), \(\lambda =1\), and
Clearly
where \(g(t)=1\), \(\psi (\|x\|)=\frac{1}{2}\|x\|+1\).
Then, by using the condition (A4), we find that \(K> 0.56853\) (we have used \(\alpha = 0.44269\)). Thus, the conclusion of Theorem 3.1 applies to problem (61)–(62).
5 Conclusion
This paper was devoted to modifying the Schrödinger-type identity related to the singular boundary value problem. We also presented some mathematical consequences of the method including a stability result. The main technical tools used to develop the mathematical analysis are local and global bifurcation, monotonicity techniques, fixed point theory in b-metric spaces and the maximum principle approach with respect to the Schrödinger operator. As an application, the uniqueness of solutions for singular boundary value problem for the Schrödinger equation was proved.
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He, H., Pang, Z. A modified Schrödinger-type identity: uniqueness of solutions for singular boundary value problem for the Schrödinger equation. Bound Value Probl 2019, 147 (2019). https://doi.org/10.1186/s13661-019-1261-6
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DOI: https://doi.org/10.1186/s13661-019-1261-6