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A new application of Schrödinger-type identity to singular boundary value problem for the Schrödinger equation

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Abstract

In this paper, we present a modified Schrödinger-type identity related to the Schrödinger-type boundary value problem with mixed boundary conditions and spatial heterogeneities. This identity can be regarded as an \(L^{1}\)-version of Fisher–Riesz’s theorem and has a broad range of applications. Using it and fixed point theory in \(L^{1}\)-metric spaces, we prove that there exists a unique solution for the singular boundary value problem with mixed boundary conditions and spatial heterogeneities. We finally provide two examples, which show the effectiveness of the Schrödinger-type identity method.

Introduction

In this paper, we consider a singular boundary value problem with mixed boundary conditions and spatial heterogeneities given by (see [1,2,3,4,5,6,7])

$$ \begin{aligned} &-\Delta f=\chi f \quad \text{in } \gimel , \\ &f=0 \quad \text{on } \daleth _{1} , \\ &\partial f+V(t)f=\chi \omega (t)u^{q} \quad \text{on } \daleth _{2} , q>1 , \end{aligned} $$
(1.1)

where:

  1. (i)

    \(\gimel =(0,W)\times (0,w)\) is a bounded rectangular domain, 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 . Let \(P=\gimel \times (0,M)\), where \(M>0\);

  2. (ii)

    −Δ stands for the minus Laplacian operator, χ is a function of the variable t satisfying

    $$ c_{1} \leq \chi (t)\leq c_{2} \quad \text{for a.e. } t\in (0,W) $$
    (1.2)

    for two positive constants \(c_{1}\) and \(c_{2}\), and \(\omega (t)\) satisfies

    $$ 0\leq \omega (t)\leq 1\quad \text{for a.e. } t\in \gimel . $$
    (1.3)
  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 at each point \(x \in \daleth _{2}\);

  4. (iv)

    \(\partial f (x)\) stands for the outer normal derivative of f at \(t \in \daleth _{2}\).

Our goal in this paper is to analyze the Schrödinger-type identity for (1.1). In the Schrödinger-type identity the continuous part of the corresponding Schrödinger operator is unchanged, and only the discrete part of the spectrum is changed by adding or removing a finite number of discrete eigenvalues to the spectrum. We can view the process of adding or removing discrete eigenvalues as changing the “unperturbed” potential and the “unperturbed” wavefunction into the “perturbed” potential and the “perturbed” wavefunction, respectively. Hence our goal is to present a Schrödinger-type identity at the potential and wavefunction levels by expressing the change in the potential and wavefunction in terms of quantities related to the perturbation and the unperturbed quantities.

The singular boundary value problem arises in many areas of applied mathematics and physics, and only its positive solution is significant in practice (see [8,9,10,11,12]). In recent years the study of positive solutions for ordinary elliptic systems and of positive radial solutions for elliptic systems in annular domains has received considerable attention; see [13,14,15,16,17] and the references therein. These references discussed mainly (1.1) for the particular case \(\omega (t)=1\) and \(V(t)=0\) and established some interesting results by applying the fixed point theorems of cone compression type, the lower and upper solutions method, and the fixed point index theory in cones, and especially extended the relevant results on the scalar second-order ordinary differential equations. For instance, Huang [18] has developed the Randon transform of the singular integral, where they have considered a linear stochastic Schrödinger equation in terms of local quantum Bernoulli noise. Subsequently, Sun [19] obtained new applications of the above identity for obtaining transmutations via the fixed point index for nonlinear integral equations. It is possible to derive a wide range of transmutation operators by this method. Zhang et al. [20] introduced a Schrödinger-type identity for a Schrödinger free boundary problem in \(\mathbb{R}^{n}\) and established necessary and sufficient conditions for the product of some distributional functions with uniformly sublinear term. Bahrouni et al. [21] obtained qualitative properties of entire solutions to a Schrödinger equation with sublinear nonlinearity and sign-changing potentials. Their analysis considered three distinct cases, and they established sufficient conditions for the existence of infinitely many solutions. In 2019, they [22] also considered the bound state solutions of sublinear Schrödinger equations with lack of compactness. Using variational methods, they proved the existence of two solutions with negative and positive energies, one of these solutions being nonnegative. Rybalko [23] studied an initial value problem for a one-dimensional nonstationary linear Schrödinger equation with a point singular potential. Xiang et al. [24] considered the existence and multiplicity of solutions for the Schrödinger–Kirchhof-type problems involving the fractional p-Laplacian and critical exponent. Xue and Tang [25] established the existence of bound state solutions for a class of quasilinear Schrödinger equations whose nonlinear term is asymptotically linear.

Recently, there have been also many extensive attentions (see [26, 27] and references therein) for singular Schrödinger-type boundary value problems under a general sublinear condition or a general superlinear condition involving the principal eigenvalue of the Schrödinger operator, and in some sense their conditions are optimal.

Our paper is organized as follows. In Sect. 2, we present a modified Schrödinger-type identity when a bound state is added to the spectrum of the Schrödinger operator. Applying it, in Sect. 3, we prove that there exists a unique solution for the singular boundary value problem with mixed boundary conditions and spatial heterogeneities. Finally, in Sect. 4, we present some illustrative examples for better understanding of the results introduced.

A modified Schrödinger-type identity

In this section, we introduce the following modified Schrödinger-type identity for the solution of (1.1). As for the classical Schrödinger-type identity, we refer the reader to [19] for more detail.

Lemma 2.1

Let

$$ \varSigma _{1}=\daleth _{1}\times (0,M),\qquad \varSigma _{2}=\daleth _{2}\times (0,M),\qquad \varSigma _{3}=\varSigma _{2}\cap \{\phi >0\} $$

and

$$ \varSigma _{4}=\varSigma _{2}\cap \{\phi =0\}. $$
  1. (i)

    Let \(\epsilon >0\), \(k\geq 0\), and \(\varsigma \in \mathfrak{D}(\mathbb{R}^{2}\times (0,M))\) be such that \(\varsigma \geq 0\) and \(\varsigma =0\) on \(\varSigma _{3}\). Then

    $$ \int _{P} \chi (t) (f_{t}+\omega ) \biggl(\min \biggl(\frac{ {(f-k)^{+}}}{ \epsilon },\varsigma \biggr) \biggr)_{t}\,dt \,ds=0. $$
    (2.1)
  2. (ii)

    Let \(\varsigma =0\) on \(\varSigma _{2}\). Then

    $$ \int _{P} \chi (t) (f_{t}+\omega ) \biggl(\min \biggl(\frac{{(k-u)^{+}}}{ \epsilon },\varsigma \biggr)-\min \biggl(\frac{k}{\epsilon }, \varsigma \biggr) \biggr)_{t}\,dt \,ds=0. $$
    (2.2)

Proof

Let ψ be a measure function satisfying

$$ d\bigl(\operatorname{supp}(\psi ), \varSigma _{2}\bigr)>0 $$

and

$$ \operatorname{supp}(\psi )\subset \mathbb{R}^{2}\times (0,M). $$

Then we have that

$$ (t,s)\mapsto \pm \psi (t,t-\kappa ) $$

vanishes on \(\varSigma _{2}\) and in \(\gimel \times \{0,M\}\) for any \(\kappa \in (-\kappa _{0},\kappa _{0})\), where \(\kappa _{0}\) is a positive constant.

Note that there exist two constants \(d_{1}>0\) and \(d_{2}>0\) such that

$$ d_{1} \Vert f_{t} \Vert _{\ast }\leq \Vert f_{t} \Vert _{\mathcal{H}^{1}}\leq d_{2} \Vert f_{t} \Vert _{\ast }\quad \text{for all } f_{t}\in \varSigma _{1}. $$
(2.3)

It follows from (1.2), (1.3), and (2.3) that

$$ \bigl\Vert \mathcal{F} _{l}(f_{t})-\mathcal{F} _{l}(\bar{f} _{t}) \bigr\Vert _{\ast } \leq \frac{d_{2}}{d_{1}}\exp \biggl( \biggl( \frac{1}{2}R \tilde{K}_{R}-1\biggr) W\biggr) \Vert f _{t}- \bar{f}_{t} \Vert _{\ast } $$
(2.4)

for all \(f_{t}\), \(\bar{f}_{t}\in \varSigma _{1}\).

So

$$ \begin{aligned}[b] & \bigl\Vert f_{l}(y) \bigr\Vert _{\mathcal{H}^{1}}^{2}+ \bigl\Vert f_{lt}(y) \bigr\Vert _{\mathcal{H} ^{1}}^{2}+2(1-\kappa ) \int _{0}^{t}\bigl( \bigl\Vert f_{l}(y) \bigr\Vert _{\mathcal{H}^{1}} ^{2}+ \bigl\Vert f_{lt}(y) \bigr\Vert _{\mathcal{H}^{1}}^{2}\bigr)\,dy \\ &\quad \leq \mathcal{R} ^{2}+\frac{1}{2\kappa } \int _{0}^{M} \bigl\Vert w(y) \bigr\Vert _{\mathcal{H}^{1}}^{2}\,dy \leq \mathcal{D}_{M}. \end{aligned} $$
(2.5)

On the other hand, we obtain that

$$ \begin{aligned}[b] &2 \int _{0}^{M} \bigl\Vert f_{l}'(y) \bigr\Vert _{\mathcal{H}^{1}}^{2}\,dy+ \int _{0}^{M} \frac{d}{dy} \biggl[ \bigl\Vert f_{l}(y) \bigr\Vert _{\mathcal{H}^{1}}^{2}+2 \int _{0}^{1}\tilde{V }\bigl(f_{lt}(t,y) \bigr)\,dx \biggr]\,dy \\ &\quad =2 \int _{0}^{M}\bigl\langle w(y),f_{l}'(y) \bigr\rangle \,dy. \end{aligned} $$
(2.6)

It follows from (2.5) and (2.6) that

$$ \begin{aligned}[b] & \int _{0}^{M}\frac{d}{dy} \biggl[ \bigl\Vert f_{l}(y) \bigr\Vert _{\mathcal{H}^{1}}^{2}+2 \int _{0}^{1}\tilde{V} \bigl(f_{lt}(t,y) \bigr)\,dx \biggr]\,dy \\ &\quad = \bigl\Vert f_{l}(W) \bigr\Vert _{\mathcal{H}^{1}}^{2}- \bigl\Vert f_{l}(0) \bigr\Vert _{\mathcal{H}^{1}}^{2}+2 \int _{0}^{1} \bigl[ \tilde{V} \bigl(f_{lt}(t,W)\bigr)-\tilde{ V}\bigl(f_{lt}(t,0)\bigr) \bigr]\,dx=0. \end{aligned} $$
(2.7)

Putting \(\iota _{R}=\sup_{|t| \leq \sqrt{2}R} \iota (t)\), we obtain that

$$ \begin{aligned}[b] &2 \int _{0}^{M}\bigl\langle \iota \bigl(f_{lt}(y)\bigr)f_{l}(y),f_{l}'(y) \bigr\rangle \,dy \\ &\quad \leq 2\iota _{R} \int _{0}^{M} \bigl\Vert f_{l}(y) \bigr\Vert \bigl\Vert f_{l}'( y) \bigr\Vert \,dy \\ &\quad \leq 2R\iota _{R} \int _{0}^{M} \bigl\Vert f_{l}'( y) \bigr\Vert \,dy \\ &\quad \leq 2W\mathcal{R} ^{2}\iota _{R}^{2}+ \frac{1}{2} \int _{0}^{M} \bigl\Vert f_{l}'(y) \bigr\Vert ^{2}\,dy. \end{aligned} $$
(2.8)

From (2.6), (2.7), and (2.8) it follows that

$$ \int _{0}^{M} \bigl\Vert f_{l}'(y) \bigr\Vert _{\mathcal{H}^{1}}^{2}\,dy \leq 2W\mathcal{R} ^{2}\iota _{R}^{2} +2 \int _{0}^{M} \bigl\Vert w(y) \bigr\Vert ^{2}\,dy\leq \mathcal{D}_{W} $$
(2.9)

for all \(m\in \mathbb{N}\) and \(t\in [0,M]\).

For fixed i and j, from (2.9) we deduce that

$$ \begin{aligned} &\int _{0}^{M}\bigl\langle f_{l}'(y)+f_{l}(y), \chi _{j}\zeta _{i}(y)\bigr\rangle \,dy \to \int _{0}^{M}\bigl\langle f'(y)+f(y), \chi _{j}\zeta _{i}(y)\bigr\rangle \,dy, \\ &\int _{0}^{M}\bigl\langle f_{lt}'(y)+f_{lt}(y), \chi _{jt}\zeta _{i}(y)\bigr\rangle \,dy\to \int _{0}^{M}\bigl\langle f_{t}'(y)+f_{t}(y), \chi _{jt}\zeta _{i}(y) \bigr\rangle \,dy. \end{aligned} $$
(2.10)

It follows from (2.8) that

$$ \begin{gathered} \int _{0}^{M}\bigl\langle \bar{V} \bigl(f_{lt}(y)\bigr),\chi _{jt}\zeta _{i}(y) \bigr\rangle \,dy\to \int _{0}^{M}\bigl\langle \bar{V} \bigl(f_{t}(y)\bigr),\chi _{jt}\zeta _{i}(y) \bigr\rangle \,dy, \\ \int _{0}^{M}\bigl\langle \iota \bigl(f_{lt}(y)\bigr)f_{l}(y),\chi _{j}\zeta _{i}(y) \bigr\rangle \,dy\to \int _{0}^{M}\bigl\langle \iota \bigl(f_{t}(y)\bigr)f(y),\chi _{j}\zeta _{i}(y)\bigr\rangle \,dy. \end{gathered} $$
(2.11)

So

$$ \begin{aligned}[b] & \int _{0}^{M}\bigl\langle f'(y)+f(y), \chi _{j}\zeta _{i}(y)\bigr\rangle \,dy+ \int _{0}^{M}\bigl\langle f_{t}'(y) +f_{t}(y),\chi _{jt}\zeta _{i}(y)\bigr\rangle \,dy \\ &\qquad {}+ \int _{0}^{M}\bigl\langle \bar{V} \bigl(f_{t}(y)\bigr),\chi _{jt}\zeta _{i}(y) \bigr\rangle \,dy+ \int _{0}^{M}\bigl\langle \iota \bigl(f_{t}(y)\bigr)f(y),\chi _{j}\zeta _{i}(y)\bigr\rangle \,dy \\ &\quad = \int _{0}^{M}\bigl\langle w(y),\chi _{j}\zeta _{i}(y) \bigr\rangle \,dy, \end{aligned} $$
(2.12)

which yields the equation

$$ \begin{aligned}[b] & \int _{0}^{M}\bigl\langle f'(y)+f(y), \chi (y)\bigr\rangle \,dy+ \int _{0}^{M} \bigl\langle f_{t}'(y)+f_{t}(y), \chi _{t}(y)\bigr\rangle \,dy \\ &\qquad {}+ \int _{0}^{M} \bigl\langle \bar{V} \bigl(f_{t}(y)\bigr),\chi _{t}(y) \bigr\rangle \,dy+ \int _{0}^{M}\bigl\langle \iota \bigl(f_{t}(y)\bigr)f(y),\chi (y)\bigr\rangle \,dy \\ &\quad = \int _{0}^{M}\bigl\langle w(y), \chi (y)\bigr\rangle \,dy\quad \text{for all }w\in L^{2}\bigl(0,W;\mathcal{H} _{0}^{1}\bigr). \end{aligned} $$
(2.13)

It follows from (2.13) that modified Schrödinger-type identities (2.1) and (2.2) hold. □

Uniqueness of the solution

In this section, we obtain our main result that a solution of problem (1.1) is unique. We assume that

$$ \chi \in C^{1}\bigl([0,W]\bigr). $$
(3.1)

Now we can state our uniqueness theorem.

Theorem 3.1

The solution of problem (1.1) associated with the initial data \(\omega _{0}\) is unique and satisfies

$$ f\in L^{\infty }\bigl(0,W;\mathcal{H}_{0}^{1}\cap \mathcal{H}^{2}\bigr),\qquad f' \in L^{2} \bigl(0,W;\mathcal{H}_{0}^{1}\bigr). $$
(3.2)

Furthermore, we have the estimate

$$ \Vert f \Vert _{L^{\infty }(\gimel )}\leq \max \bigl\{ \Vert \tilde{f} _{0} \Vert _{L^{ \infty }(\varSigma _{2} )}, \Vert f \Vert _{L^{\infty }(\gimel )} \bigr\} . $$
(3.3)

Proof

Consider a special orthonormal basis \(\{\chi _{j}\}\) on \(\mathcal{H} _{0}^{1}:\chi _{j}(t)=\sqrt{2}\sin (j\pi x)\), \(j\in \mathbb{N}\),

$$ -\triangle \chi _{j}=\tau _{j}\chi _{j},\quad \chi _{j}\in C^{\infty }\bigl([0,1]\bigr), \tau _{j}=(j\pi )^{2},\quad j=1,2,\dots . $$

Put (see [28])

$$ f_{l}( s) =\sum_{j=1}^{l}d_{lj}( s) \chi _{j}, $$
(3.4)

where

$$ \begin{aligned} &\bigl\langle f_{l}'( s) ,\chi _{j}\bigr\rangle +\bigl\langle f_{lt}'( s) ,\chi _{jt} \bigr\rangle +\bigl\langle f_{lt}( s) +\bar{V}\bigl(f_{lt}(s) \bigr),\chi _{jt}\bigr\rangle \\ &\quad {} +\bigl\langle \bigl( 1+\iota \bigl(f_{lt}(s)\bigr) \bigr)f_{l}(s),\chi _{j}\bigr\rangle =\bigl\langle w(s), \chi _{j}\bigr\rangle , \quad 1\leq j\leq l, \\ &f_{l}(0)=f_{0l}, \end{aligned} $$
(3.5)

in which

$$ f_{0l}=\sum_{j=1}^{l}\beta _{lj}\chi _{j}\to \tilde{f}_{0}\quad \text{strongly in }\mathcal{H}_{0}^{1}\cap \mathcal{H}^{2}. $$
(3.6)

Equality (3.5) yields that

$$ \begin{aligned} &d_{li}'( s) +d_{li}( s) +\frac{1}{1+\tau _{i}} \bigl[ \bigl\langle \bar{V} \bigl(f _{lt}(s)\bigr),\chi _{ix}\bigr\rangle +\bigl\langle \iota \bigl(f_{lt}(s)\bigr)f_{l}(s),\chi _{i}\bigr\rangle \bigr] \\ &\quad =\frac{1}{1+\tau _{i}}\bigl\langle w(s),\chi _{i} \bigr\rangle , \\ &d_{li}( 0) =\beta _{li}, \quad 1\leq i\leq l. \end{aligned} $$
(3.7)

Multiplying the jth equation of (3.7) by \(d_{lj}(s)\) and summing up with respect to j, we obtain that

$$ \mathcal{S}_{l}(s)=\mathcal{S}_{l}(0)+2 \int _{0}^{t}\bigl\langle w(y),f_{l}(y) \bigr\rangle \,dy, $$
(3.8)

where

$$ \begin{aligned}[b] \mathcal{S}_{l}(s) ={}& \bigl\Vert f_{l}(s) \bigr\Vert _{\mathcal{H}^{1}}^{2}+2 \int _{0} ^{t} \bigl\Vert f_{l}(y) \bigr\Vert _{\mathcal{H}^{1}}^{2}\,dy \\ &{} +2 \int _{0}^{t} \bigl\langle \bar{V} \bigl(f_{lt}(y)\bigr),f_{lt}(y)\bigr\rangle \,dy+2 \int _{0}^{t}\bigl\langle \iota \bigl(f_{lt}(y)\bigr),f_{l}^{2}(y)\bigr\rangle \,dy. \end{aligned} $$
(3.9)

So

$$ \mathcal{S}_{l}(0)= \Vert f_{0l} \Vert _{\mathcal{H}^{1}}^{2}\leq \bar{S}_{0} $$
(3.10)

for \(m\in \mathbb{N}\), where \(f_{0l}\to \tilde{f}_{0}\) strongly in \(\mathcal{H}_{0}^{1}\cap \mathcal{H}^{2}\).

Consider

$$ y\bar{V}(y)=y \int _{0}^{y}V (r)\,dr\geq 0 $$

for \(y\in \mathbb{R}\).

So

$$ \begin{aligned}[b] 2 \int _{0}^{t}\bigl\langle w(y),f_{l}(y) \bigr\rangle \,dy &\leq \int _{0}^{t} \bigl\Vert w(y) \bigr\Vert ^{2}\,dy+ \int _{0}^{t} \bigl\Vert f_{l}(y) \bigr\Vert ^{2}\,dy \\ &\leq \int _{0}^{W} \bigl\Vert w(y) \bigr\Vert ^{2}\,dy+\frac{1}{2}\mathcal{S}_{l}(s), \end{aligned} $$
(3.11)

which yields that

$$ \mathcal{S}_{l}(s)\leq 2\bar{S}_{0}+2 \int _{0}^{W} \bigl\Vert w(y) \bigr\Vert ^{2}\,dy \leq D_{W}^{(1)}. $$
(3.12)

Further, we obtain that

$$ \begin{aligned}[b] &\bigl\langle f_{lt}'( s) ,\chi _{jt}\bigr\rangle +\bigl\langle \Delta f_{l}'( s) , \Delta \chi _{j}\bigr\rangle +\bigl\langle \Delta f_{l}( s) ,\Delta \chi _{j} \bigr\rangle \\ &\qquad {}+\bigl\langle f_{lt}(s),\chi _{jt}\bigr\rangle +\bigl\langle V \bigl(f_{lt}(s)\bigr) \Delta f_{l}( s) ,\Delta \chi _{j}\bigr\rangle \\ &\qquad {}+\bigl\langle \iota '\bigl(f_{lt}(s)\bigr)f _{l}(s)\Delta f_{l}( s) +\iota \bigl(f_{lt}(s) \bigr)f_{lt}(s),\chi _{jt}\bigr\rangle \\ &\quad =\bigl\langle f_{t}(s),\chi _{jt}\bigr\rangle ,\quad 1\leq j\leq l. \end{aligned} $$
(3.13)

Similarly,

$$ \begin{aligned}[b] \mathcal{P}_{l}(s) ={}& \mathcal{P}_{l}(0)-2 \int _{0}^{t} \bigl[ \bigl\langle \iota '\bigl(f_{lt}(y)\bigr)f_{l}(y) \Delta f_{l}( s) ,f_{lt}(y)\bigr\rangle \\ & {}+\bigl\langle \iota \bigl(f_{lt}(y)\bigr), \bigl\vert f_{lt}(y) \bigr\vert ^{2}\bigr\rangle \bigr]\,dy +2 \int _{0}^{t}\bigl\langle f_{t}(y),f_{lt}( s) \bigr\rangle \,dy \\ ={}&\mathcal{P} _{l}(0)+\mathcal{I}_{1}+ \mathcal{I}_{2}, \end{aligned} $$
(3.14)

where

$$ \begin{aligned}[b] \mathcal{P}_{l}(s) ={}& \bigl\Vert f_{lt}(s) \bigr\Vert ^{2}+ \bigl\Vert \Delta f_{l}(s) \bigr\Vert ^{2}+2 \int _{0}^{t}\bigl( \bigl\Vert f_{lt}(y) \bigr\Vert ^{2}+ \bigl\Vert \Delta f_{l}(y) \bigr\Vert ^{2}\bigr)\,dy \\ & {}+2 \int _{0}^{t}\bigl\langle V \bigl(f_{lt}(y)\bigr), \bigl\vert \Delta f_{l}( s) \bigr\vert ^{2} \bigr\rangle \,dy. \end{aligned} $$
(3.15)

On the other hand, we have

$$ \mathcal{P}_{l}(0)= \bigl\Vert f_{lt}(0) \bigr\Vert ^{2}+ \bigl\Vert \triangle f_{l}( 0) \bigr\Vert ^{2} = \Vert f_{0mx} \Vert ^{2}+ \Vert \triangle f_{0l} \Vert ^{2}\leq \bar{P}_{0} $$
(3.16)

for any \(m\in \mathbb{N}\), where \(\bar{P}_{0}\) always indicates a constant depending on \(\tilde{f}_{0}\) (see [29]).

It follows that

$$ \begin{aligned}[b] \mathcal{I}_{1} ={}&{-}2 \int _{0}^{t} \bigl[ \bigl\langle \iota '\bigl(f_{lt}(y)\bigr)f_{l}(y) \Delta f_{l}( s) , f_{lt}(y)\bigr\rangle \\ &{} +\bigl\langle \iota \bigl(f_{lt}(y)\bigr), \bigl\vert f_{lt}(y) \bigr\vert ^{2}\bigr\rangle \bigr]\,dy \leq 0 \end{aligned} $$
(3.17)

and

$$ \begin{aligned}[b] \mathcal{I}_{2} &=2 \int _{0}^{t}\bigl\langle f_{t}(y),f_{lt}( s) \bigr\rangle \,dy \leq \int _{0}^{W} \bigl\Vert f_{t}(y) \bigr\Vert \bigl\Vert f_{lt}(y) \bigr\Vert \,dy \\ &\leq \int _{0}^{W} \bigl\Vert f_{t}(y) \bigr\Vert \sqrt{\mathcal{S}_{l}(y)}\,dy \leq \sqrt{D_{W}^{(1)}} \int _{0}^{W} \bigl\Vert f_{t}(y) \bigr\Vert \,dy \end{aligned} $$
(3.18)

from (3.14), (3.15), and (3.16).

So from (3.14), (3.16), (3.17), and (3.18) we have

$$ \mathcal{P}_{l}(s)\leq \bar{P}_{0}+ \sqrt{D_{W}^{(1)}} \int _{0}^{W} \bigl\Vert f_{t}(y) \bigr\Vert \,dy\leq D_{W}^{(2)}. $$
(3.19)

Define

$$ f_{l}(s)= \textstyle\begin{cases} 2(\frac{s}{\sigma })^{2} &\text{if } s\in [0,\frac{\sigma }{2}], \\ 1-2(1-\frac{s}{\sigma })^{2} &\text{if } s\in (\frac{\sigma }{2}, \sigma ], \\ 1 &\text{if } s\in (\sigma ,M-\sigma ], \\ 1-2(1-\frac{M-t}{\sigma })^{2} &\text{if } t\in (M-\sigma ,M-\frac{ \sigma }{2}], \\ 2(\frac{M-t}{\sigma })^{2} &\text{if } t\in (M-\frac{\sigma }{2},M], \end{cases} $$

where \(M\in (0,W]\), and σ is a positive real number.

Note that \(f_{l} \in C^{1}([0,W])\) and

$$ f_{l}'(s)= \textstyle\begin{cases} 4\frac{s}{\sigma ^{2}} &\text{if } s\in [0,\frac{\sigma }{2}], \\ \frac{4}{\sigma }(1-\frac{s}{\sigma }) &\text{if } s\in (\frac{ \sigma }{2},\sigma ], \\ 0 &\text{if } s\in (\sigma ,M-\sigma ], \\ -\frac{4}{\sigma }(1-\frac{M-s}{\sigma }) &\text{ if } t\in (M- \sigma ,M-\frac{\sigma }{2}], \\ -\frac{4}{\sigma }(\frac{M-s}{\sigma }) &\text{if } s\in (M-\frac{ \sigma }{2},M]. \end{cases} $$

Let \(\varSigma _{1}\) be the linear space generated by \(\chi _{1},\chi _{2}, \dots ,\chi _{l}\). We consider the following problem: Find a function \(f_{l}(s)\) in the form (3.4) satisfying system (3.5) and the W-periodic condition (see [30])

$$ f_{l}(0)=f_{l}(W). $$
(3.20)

We consider the initial value problem given by (3.5), where \(f_{0l}\) is given in \(\varSigma _{1}\).

It follows that

$$ \begin{aligned}[b] &\frac{d}{ds} \bigl\Vert f_{l}(s) \bigr\Vert _{\mathcal{H}^{1}}^{2}+2 \bigl\Vert f_{l}(s) \bigr\Vert _{ \mathcal{H}^{1}}^{2}+2 \bigl\langle \bar{V} \bigl(f_{lt}(s)\bigr),f_{lt}(s)\bigr\rangle \\ &\qquad {}+2 \bigl\Vert \sqrt{\iota \bigl(f_{lt}(s)\bigr)}f_{l}(s) \bigr\Vert ^{2} \\ &\quad =2\bigl\langle w(s),f _{l}(s)\bigr\rangle . \end{aligned} $$
(3.21)

So we have the following inequality:

$$ 2\bigl\langle w(s),f_{l}(s)\bigr\rangle \leq \frac{1}{2\delta _{1}} \bigl\Vert w(s) \bigr\Vert ^{2}+2\kappa \bigl\Vert f_{l}(s) \bigr\Vert ^{2} \leq \frac{1}{2\delta _{1}} \bigl\Vert w(s) \bigr\Vert ^{2}+2\kappa \bigl\Vert f_{l}(s) \bigr\Vert _{\mathcal{H}^{1}}^{2} $$
(3.22)

for all \(0<\delta _{1}<1\).

From (3.21) and (3.22) it follows that

$$ \begin{aligned}[b] &\frac{d}{ds} \bigl\Vert f_{l}(s) \bigr\Vert _{\mathcal{H}^{1}}^{2}+2(1-\delta _{1}) \bigl\Vert f _{l}(s) \bigr\Vert _{\mathcal{H}^{1}}^{2}+2\bigl\langle \bar{V}\bigl(f_{lt}(s) \bigr),f_{lt}(s) \bigr\rangle \\ &\qquad {}+2 \bigl\Vert \sqrt{\iota \bigl(f_{lt}(s)\bigr)}f_{l}(s) \bigr\Vert ^{2} \\ &\quad \leq \frac{1}{2 \delta _{1}} \bigl\Vert w(s) \bigr\Vert ^{2}. \end{aligned} $$
(3.23)

So

$$ \begin{aligned}[b] &\frac{d}{ds} \bigl\Vert f_{lt}(s) \bigr\Vert _{\mathcal{H}^{1}}^{2}+2 \bigl\Vert f_{lt}(s) \bigr\Vert _{\mathcal{H}^{1}}^{2}+2 \bigl\Vert \sqrt{ V \bigl(f_{lt}(s)\bigr)}\triangle f_{l}(s) \bigr\Vert ^{2} \\ &\qquad {}+2\bigl\langle \iota '\bigl(f_{lt}(s) \bigr)f_{l}(s)\Delta f_{l}( s) + \iota \bigl(f_{lt}(s)\bigr)f_{lt}(s),f_{lt}(s)\bigr\rangle \\ &\quad =2\bigl\langle f_{t}(s),f _{lt}(s)\bigr\rangle . \end{aligned} $$
(3.24)

Similarly,

$$ \begin{aligned}[b] &2\bigl\langle \iota ' \bigl(f_{lt}(s)\bigr)f_{l}(s)\Delta f_{l}( s) +\iota \bigl(f_{lt}(s)\bigr)f _{lt}(s),f_{lt}(s) \bigr\rangle \\ &\quad =2 \int _{0}^{1}f_{l}(t,s)f_{lt}(t,s) \iota '\bigl(f_{lt}(t,s)\bigr)\Delta f_{l}( x,s)\,dx \\ &\qquad {} +2 \int _{0}^{1}f _{lt}^{2}(t,s) \iota \bigl(f_{lt}(t,s)\bigr)\,dx \\ &\quad =2 \int _{0}^{1}f_{l}(t,s) \frac{ \partial }{\partial t} \biggl( \int _{0}^{f_{lt}(t,s)}y\iota '(y) \biggr)\,dx +2 \int _{0}^{1}f_{lt}^{2}(t,s) \iota \bigl(f_{lt}(t,s)\bigr)\,dx \\ &\quad =-2 \int _{0} ^{1}f_{lt}(t,s) \biggl( \int _{0}^{f_{lt}(t,s)}y\iota '(y) \biggr)\,dx +2 \int _{0}^{1}f_{lt}^{2}(t,s) \iota \bigl(f_{lt}(t,s)\bigr)\,dx \\ &\quad =2 \int _{0}^{1} \biggl[ f_{lt}^{2}(t,s) \iota \bigl(f_{lt}(t,s)\bigr)-f_{lt}(t,s) \biggl( \int _{0} ^{f_{lt}(t,s)}y\iota '(y) \biggr) \biggr]\,dx\geq 0, \end{aligned} $$
(3.25)

which implies

$$ \begin{aligned}[b] &\frac{d}{ds} \bigl\Vert f_{lt}(s) \bigr\Vert _{\mathcal{H}^{1}}^{2}+2(1-\delta _{1}) \bigl\Vert f _{lt}(s) \bigr\Vert _{\mathcal{H}^{1}}^{2}+2 \bigl\Vert \sqrt{V \bigl(f_{lt}(s) \bigr)}\triangle f_{l}(s) \bigr\Vert ^{2} \\ &\quad \leq \frac{1}{2\delta _{1}} \bigl\Vert f_{t}(s) \bigr\Vert ^{2}. \end{aligned} $$
(3.26)

From (3.23) and (3.26) it follows that

$$ \begin{aligned}[b] &\frac{d}{ds} \bigl[ \bigl\Vert f_{l}(s) \bigr\Vert _{\mathcal{H}^{1}}^{2}+ \bigl\Vert f_{lt}(s) \bigr\Vert _{\mathcal{H}^{1}}^{2} \bigr] +2(1-\delta _{1}) \bigl( \bigl\Vert f_{l}(s) \bigr\Vert _{\mathcal{H}^{1}}^{2}+ \bigl\Vert f_{lt}(s) \bigr\Vert _{\mathcal{H}^{1}}^{2}\bigr) \\ &\quad \leq \frac{1}{2\delta _{1}} \bigl\Vert w(s) \bigr\Vert _{\mathcal{H}^{1}}^{2}, \end{aligned} $$
(3.27)

which, together with (3.27), gives

$$ \begin{aligned}[b] & \bigl\Vert f_{l}(s) \bigr\Vert _{\mathcal{H}^{1}}^{2}+ \bigl\Vert f_{lt}(s) \bigr\Vert _{\mathcal{H} ^{1}}^{2} \\ &\quad \leq \bigl( \Vert f_{0l} \Vert _{\mathcal{H}^{1}}^{2}+ \Vert f_{0mx} \Vert _{\mathcal{H}^{1}}^{2}- \mathcal{R}^{2} \bigr)e^{-2(1-\delta _{1})t} \\ &\qquad {} + \biggl( \mathcal{R}^{2}+\frac{1}{2\delta _{1}} \int _{0}^{t}e ^{2(1-\delta _{1})s} \bigl\Vert w(y) \bigr\Vert _{\mathcal{H}^{1}}^{2}\,dy \biggr) e^{-2(1- \delta _{1})t} \\ &\quad \leq \bigl( \Vert f_{0l} \Vert _{\mathcal{H}^{1}}^{2}+ \Vert f _{0mx} \Vert _{\mathcal{H}^{1}}^{2}- \mathcal{R}^{2} \bigr) e^{-2(1-\delta _{1})t}+\mathcal{R}^{2}, \end{aligned} $$
(3.28)

where \(\mathcal{R}^{2}=\underset{0\leq t\leq W}{\sup }\mathcal{R}_{1}(s)\),

$$ \mathcal{R}_{1}(s)= \textstyle\begin{cases} \frac{1}{2\delta _{1}}\frac{1}{e^{2(1-\delta _{1})t}-1} \int _{0}^{t}e ^{2(1-\delta _{1})s} \Vert w(y) \Vert _{\mathcal{H}^{1}}^{2}\,dy, & 0< s\leq W, \\ \frac{1}{4\delta _{1}(1-\delta _{1})} \Vert w(0) \Vert _{\mathcal{H}^{1}}^{2}, & s=0. \end{cases} $$
(3.29)

Note that \(\| f_{0l}\| _{\mathcal{H}^{1}}^{2}+\| f_{0lx}\| _{ \mathcal{H}^{1}}^{2}\leq \mathcal{R}^{2}\). It follows from (3.28) that

$$ \bigl\Vert f_{l}(s) \bigr\Vert _{\mathcal{H}^{1}}^{2}+ \bigl\Vert f_{lt}(s) \bigr\Vert _{\mathcal{H}^{1}} ^{2}\leq \mathcal{R}^{2},\quad \text{that is, $\varSigma _{1}=W$ for all }l. $$
(3.30)

Let \(\bar{B}_{l}(0,R)\) be a closed ball in the space \(\varSigma _{1}\) of linear combinations of the functions \(\chi _{1}\), \(\chi _{2},\ldots \) , \(\chi _{l}\). Put

$$ \begin{gathered} \mathcal{F} _{l}:\bar{B}_{l}(0,R) \to \bar{B}_{l}(0,R), \\ f_{0l}\mapsto f_{l}(W). \end{gathered} $$
(3.31)

It is obvious that \(y_{l}(s)\) satisfies

$$ \begin{aligned}[b] &\bigl\langle y_{l}'( s) +y_{l}(s),\chi _{j}\bigr\rangle +\bigl\langle y_{lt}'( s) +y _{lt}( s) ,\chi _{jt}\bigr\rangle \\ &\quad {}+\bigl\langle \bar{V}\bigl(f_{lt}(s)\bigr)-\bar{V}\bigl( \bar{f}_{lt}(s)\bigr),\chi _{jt}\bigr\rangle \\ &\quad {}+\bigl\langle \iota \bigl(f_{lt}(s)\bigr)f_{l}(s)- \iota \bigl(\bar{f}_{lt}(s)\bigr) \bar{f}_{l}(s),\chi _{j}\bigr\rangle =0. \end{aligned} $$
(3.32)

Similarly,

$$ \begin{aligned}[b] &\frac{d}{ds} \bigl\Vert y_{l}(s) \bigr\Vert _{\mathcal{H}^{1}}^{2} +2 \bigl\Vert y_{l}(s) \bigr\Vert _{ \mathcal{H}^{1}}^{2}+2 \bigl\langle \bar{V}\bigl(f_{lt}(s)\bigr)-\bar{V}\bigl( \bar{f}_{lt}(s)\bigr),y _{lt}(s)\bigr\rangle \\ &\quad {} +2\bigl\langle \iota \bigl(f_{lt}(s)\bigr)f_{l}(s) - \iota \bigl( \bar{f}_{lt}(s)\bigr)\bar{f}_{l}(s),y_{l}(s) \bigr\rangle =0. \end{aligned} $$
(3.33)

From (3.33) it follows that

$$ \frac{d}{ds} \bigl\Vert y_{l}(s) \bigr\Vert _{\mathcal{H}^{1}}^{2}+(2-R\tilde{K} _{R}) \bigl\Vert y_{l}(s) \bigr\Vert _{\mathcal{H}^{1}}^{2}\leq 0, $$
(3.34)

which yields that

$$\begin{aligned}& \begin{aligned}&\bigl\Vert y_{l}(W) \bigr\Vert _{\mathcal{H}^{1}}^{2} \leq e^{(R\tilde{K} _{R}-2)W} \Vert f _{0l}-\bar{f}_{t} \Vert _{\mathcal{H}^{1}}^{2}, \\ &\bigl\Vert \mathcal{F} _{l}(f_{0l})-\mathcal{F} _{l}(\bar{f}_{t}) \bigr\Vert _{ \mathcal{H}^{1}} \leq \exp \biggl( \biggl( \frac{1}{2}R\tilde{K}_{R}-1 \biggr) W \biggr) \Vert f_{0l}-\bar{f}_{t} \Vert _{\mathcal{H}^{1}}. \end{aligned} \end{aligned}$$
(3.35)

We obtain that

$$ \begin{aligned}[b] \mathcal{Q}_{l}(s) &= \mathcal{Q}_{l}(0)-2 \int _{0}^{t}\bigl\langle \iota \bigl(f _{lt}(y)\bigr)f_{l}(y),f_{l}'(y) \bigr\rangle \,dy+2 \int _{0}^{t}\bigl\langle w(y),f_{l}'(y) \bigr\rangle \,dy \\ &=\mathcal{Q}_{l}(0)+\mathcal{J}_{1}+ \mathcal{J}_{2} \end{aligned} $$
(3.36)

by multiplying the jth equation of (3.35), where

$$\begin{aligned}& \begin{aligned}&\mathcal{Q}_{l}(s)= \bigl\Vert f_{l}(s) \bigr\Vert _{\mathcal{H}^{1}}^{2}+2 \int _{0}^{t} \bigl\Vert f_{l}'(y) \bigr\Vert _{\mathcal{H}^{1}}^{2}\,dy +2 \int _{0}^{1}\tilde{V}\bigl(f_{lt}(t,s) \bigr)\,dx, \\ &\tilde{V}(x)= \int _{0}^{z}\bar{V}(y)\,dy\geq 0\quad \forall z \in \mathbb{R}. \end{aligned} \end{aligned}$$
(3.37)

It is obvious that there exists a positive constant \(\bar{Q}_{0} \) independent of m such that

$$ \mathcal{Q}_{l}(0)= \Vert f_{0l} \Vert _{\mathcal{H}^{1}}^{2}+2 \int _{0}^{1} \tilde{V}\bigl(f_{0mx}(t) \bigr)\,dx\leq \bar{Q}_{0}\quad \forall m\in \mathbb{N} $$
(3.38)

since \(f_{0l}\to \tilde{f}_{0}\) strongly in \(\mathcal{H}_{0}^{1} \cap \mathcal{H}^{2}\).

From (3.19) it follows that

$$\begin{aligned} \bigl\vert f_{lt}(t,s) \bigr\vert &\leq \bigl\Vert f_{lt}(y) \bigr\Vert _{C^{0}([0,1])}\leq \sqrt{2} \bigl\Vert f _{lt}(y) \bigr\Vert _{\mathcal{H}^{1}} \\ &\leq \sqrt{2}\sqrt{ \bigl\Vert f_{lt}(y) \bigr\Vert ^{2}+ \bigl\Vert \Delta f_{l}(y) \bigr\Vert ^{2}}\leq \sqrt{2}\sqrt{2 \bigl\Vert \Delta f_{l}(y) \bigr\Vert ^{2}} \\ &\leq 2 \bigl\Vert \Delta f_{l}(y) \bigr\Vert \leq 2\sqrt{ \mathcal{P}_{l}(y)}\leq 2 \sqrt{D _{W}^{(2)}}, \end{aligned}$$

which yields that

$$ \begin{aligned}[b] \mathcal{J}_{1} &=-2 \int _{0}^{t}\bigl\langle \iota \bigl(f_{lt}(y)\bigr)f_{l}(y),f _{l}'(y) \bigr\rangle \,dy \\ &\leq 2\sup_{ \vert x \vert \leq 2\sqrt{D_{W}^{(2)}}} \iota (x) \int _{0}^{t} \bigl\Vert f_{l}(y) \bigr\Vert \bigl\Vert f_{l}'(y) \bigr\Vert \,dy \\ &\leq 2 \sup_{ \vert x \vert \leq 2\sqrt{D_{W}^{(2)}}} \iota (x) \int _{0}^{t}\sqrt{ \mathcal{S}_{l}(y)} \bigl\Vert f_{l}'(y) \bigr\Vert \,dy \\ &\leq 2\sqrt{D_{W}^{(1)}} \sup_{ \vert x \vert \leq 2\sqrt{D_{W}^{(2)}}} \iota (x) \int _{0}^{t} \bigl\Vert f_{l}'(y) \bigr\Vert \,dy \\ &\leq 2WD_{W}^{(1)}\sup_{ \vert x \vert \leq 2\sqrt{D_{W}^{(2)}}} \iota ^{2}(x)+\frac{1}{2} \int _{0}^{t} \bigl\Vert f_{l}'(y) \bigr\Vert ^{2}\,dy \\ &\leq 2WD _{W}^{(1)}\sup_{ \vert x \vert \leq 2\sqrt{D_{W}^{(2)}}} \iota ^{2}(x)+ \frac{1}{4}\mathcal{Q}_{l}(s) \end{aligned} $$
(3.39)

and

$$ \begin{aligned}[b] \mathcal{J}_{2} &=2 \int _{0}^{t}\bigl\langle w(y),f_{l}'(y) \bigr\rangle \,dy \\ & \leq 2 \int _{0}^{W} \bigl\Vert w(y) \bigr\Vert ^{2}\,dy+\frac{1}{2} \int _{0}^{t} \bigl\Vert f_{l}'(y) \bigr\Vert ^{2}\,dy \\ &\leq 2 \int _{0}^{W} \bigl\Vert w(y) \bigr\Vert ^{2}\,dy+\frac{1}{4} \mathcal{Q}_{l}(s). \end{aligned} $$
(3.40)

Combining (3.36) and (3.38)–(3.40), we have

$$ \mathcal{Q}_{l}(s)\leq 2 \biggl( \bar{Q}_{0}+2WD_{W}^{(1)} \underset{ \vert x \vert \leq 2\sqrt{D_{W}^{(2)}}}{ \sup }\iota ^{2}(x)+2 \int _{0}^{W} \bigl\Vert w(y) \bigr\Vert ^{2}\,dy \biggr) \leq D_{W}^{(3)}. $$
(3.41)

It follows from (3.12), (3.19), and (3.41) that there exists a subsequence of \(\{f_{l}\}\), still denoted by \(\{f_{l}\}\), such that

$$ \begin{gathered} f_{l}\to u \quad \text{in } L^{\infty }\bigl(0,W;\mathcal{H}_{0}^{1}\cap \mathcal{H}^{2}\bigr) \text{ weakly*,} \\ f_{l}'\to f' \quad \text{in } L^{2}\bigl(0,W;\mathcal{H}_{0}^{1}\bigr) \text{ weakly.} \end{gathered} $$
(3.42)

Applying the modified Schrödinger-type identity, by Lemma 2.1 there exists a subsequence of \(\{f_{l}\}\) such that

$$ \begin{gathered} f_{l}\to f \quad \text{strongly in $L^{2}(0,W;\mathcal{H}_{0}^{1})$ and a.e. in } \gimel , \\ f_{lt}\to f_{t} \quad \text{strongly in $L^{2}(\gimel )$ and a.e. in $\gimel $}. \end{gathered} $$
(3.43)

It follows from (3.43) that

$$ \begin{gathered} \bar{V}\bigl(f_{lt}(t,s)\bigr)\to \bar{V}\bigl(f_{t}(t,s)\bigr)\quad \text{for a.e. $(t,s)$ in } \gimel , \\ \iota \bigl(f_{lt}(t,s)\bigr)f_{l}(t,s)\to \iota \bigl(f_{t}(t,s)\bigr)f(t,s) \quad \text{for a.e. $(t,s)$ in } \gimel . \end{gathered} $$
(3.44)

Inequalities (3.19) yield that

$$ \begin{gathered} \begin{aligned} \bigl\vert f_{lt}(t,s) \bigr\vert &\leq \bigl\Vert f_{lt}(s) \bigr\Vert _{C^{0}([0,1])} \leq \sqrt{2} \bigl\Vert f _{lt}(s) \bigr\Vert _{\mathcal{H}^{1}} \\ &\leq 2 \bigl\Vert \triangle f_{l}( s) \bigr\Vert \leq 2\sqrt{ \mathcal{P}_{l}(s)}\leq 2\sqrt{D_{W}^{(2)}}; \end{aligned} \\ \bigl\vert \bar{V}\bigl(f_{lt}(t,s)\bigr) \bigr\vert \leq \sup _{ \vert x \vert \leq 2\sqrt{D_{W}^{(2)}}} \bigl\vert \bar{V}(x) \bigr\vert \leq D_{W}; \\ \begin{aligned} \bigl\vert \iota \bigl(f_{lt}(t,s) \bigr)f_{l}(t,s) \bigr\vert &\leq \bigl\Vert f_{lt}(s) \bigr\Vert \bigl\vert \iota \bigl(f_{lt}(t,s)\bigr) \bigr\vert \\ &\leq \sqrt{D_{W}^{(2)}}\underset{ \vert x \vert \leq 2\sqrt{ D_{W}^{(2)}}}{\sup } \bigl\vert \iota (x) \bigr\vert \leq D_{W}. \end{aligned} \end{gathered} $$
(3.45)

It follows from (3.44) and (3.45) that

$$ \begin{gathered} \bar{V}(f_{lt})\to \bar{V}(f_{t}) \quad \text{strongly in } L^{2}( \gimel ), \\ \iota (f_{lt})f_{l}\to \iota (f_{t})u\quad \text{strongly in } L^{2}( \gimel ). \end{gathered} $$
(3.46)

So

$$ \begin{aligned} &\bigl\langle f_{s}(s),w\bigr\rangle +\bigl\langle f_{xs}(s),\chi _{t}\bigr\rangle +\bigl\langle f_{t}(s)+ \bar{V} \bigl(f_{t}(s)\bigr),\chi _{t}\bigr\rangle +\bigl\langle \bigl( 1+\iota \bigl(f_{t}(s)\bigr)\bigr) f(s),w\bigr\rangle \\ &\quad =\bigl\langle w(s),w\bigr\rangle , \quad \forall w\in \mathcal{H}_{0}^{1}, \\ &f(0)=\tilde{f}_{0}. \end{aligned}$$
(3.47)

Furthermore,

$$ f\in L^{\infty }\bigl(0,W;\mathcal{H}_{0}^{1}\cap \mathcal{H}^{2}\bigr),\qquad f' \in L^{2} \bigl(0,W;\mathcal{H}_{0}^{1}\bigr). $$

Let f and v be two weak solutions of (1.1) such that

$$ f, v\in L^{\infty }\bigl(0,W;\mathcal{H}_{0}^{1} \cap \mathcal{H}^{2}\bigr), \qquad f', v' \in L^{2}\bigl(0,W;\mathcal{H}_{0}^{1}\bigr). $$
(3.48)

Put \(\chi =f-v\), which satisfies

$$ \begin{aligned} &\bigl\langle \chi _{s}(s),y\bigr\rangle +\bigl\langle \chi _{xs}(s),y_{t} \bigr\rangle + \bigl\langle \chi _{t}(s),y_{t}\bigr\rangle +\bigl\langle \chi (s),y\bigr\rangle \\ &\quad {}+ \bigl\langle \bar{V}\bigl(f_{t}(s)\bigr)-\bar{V} \bigl(v_{t}(s)\bigr),y_{t}\bigr\rangle +\bigl\langle \iota \bigl(f_{t}(s)\bigr)u-\iota \bigl(v_{t}(s)\bigr)v,y\bigr\rangle =0, \quad \forall y \in \mathcal{H}_{0}^{1}, \\ &\chi (0)=0, \\ &u,v,\chi \in L^{\infty }\bigl(0,W;\mathcal{H}_{0}^{1} \cap \mathcal{H}^{2}\bigr), \qquad f_{s},v_{s}, \chi _{s}\in L^{2}\bigl(0,W;\mathcal{H}_{0}^{1} \bigr). \end{aligned}$$
(3.49)

Define the following functions \(\varrho _{1}\) and \(\varrho _{2}\) of \((s_{1}^{1},s_{1}^{2})\) (resp. \((s_{2}^{1},s_{2}^{2})\)) by

$$ \varrho _{1}(s_{1})= \textstyle\begin{cases} 2(\frac{s_{1}-s_{1}^{1}}{\delta })^{2} &\text{if } s_{1}\in [t_{1} ^{1},s_{1}^{1}+\frac{\delta }{2}], \\ 1-2(1-\frac{s_{1}-s_{1}^{1}}{\delta })^{2} &\text{if } s_{1}\in (s _{1}^{1}+\frac{\delta }{2},s_{1}^{1}+\delta ], \\ 1 &\text{if } s_{1}\in (s_{1}^{1}+\delta ,s_{1}^{2}-\delta ], \\ 1-2(1-\frac{s_{1}^{2}-s_{1}}{\delta })^{2} &\text{if } s_{1}\in (s _{1}^{2}-\delta ,s_{1}^{2}-\frac{\delta }{2}], \\ 2(\frac{s_{1}^{2}-s_{1}}{\delta })^{2} &\text{if } s_{1}\in (s_{1} ^{2}-\frac{\delta }{2},s_{1}^{2}], \end{cases} $$

and

$$ \varrho _{2}(s_{2})= \textstyle\begin{cases} 2(\frac{s_{2}-s_{2}^{1}}{\delta })^{2} &\text{if } s_{2}\in [s_{2} ^{1},s_{2}^{1}+\frac{\delta }{2}], \\ 1-2(1-\frac{s_{2}-s_{2}^{1}}{\delta })^{2} &\text{if } s_{2}\in (s _{2}^{1}+\frac{\delta }{2},s_{2}^{1}+\delta ], \\ 1 &\text{if } s_{2}\in (s_{2}^{1}+\delta ,s_{2}^{2}-\delta ], \\ 1-2(1-\frac{s_{2}^{2}-s_{2}}{\delta })^{2} &\text{if } s_{2}\in (s _{2}^{2}-\delta ,s_{2}^{2}-\frac{\delta }{2}], \\ 2(\frac{s_{2}^{2}-s_{2}}{\delta })^{2} &\text{if } s_{2}\in (s_{2} ^{2}-\frac{\delta }{2},s_{2}^{2}]. \end{cases} $$

Putting \(y=\chi =u-v\) in (3.49) and integrating with respect to t, we have

$$ \begin{aligned}[b] \varrho (s) ={}&-2 \int _{0}^{t}\bigl\langle \bar{V} \bigl(f_{t}(y)\bigr)-\bar{V} \bigl(v_{t}(y)\bigr), \chi _{t}(y)\bigr\rangle \,dy \\ &{} -2 \int _{0}^{t}\bigl\langle \iota \bigl(f_{t}(y)\bigr)f(y) -\iota \bigl(v_{t}(y)\bigr)v(y), \chi (y)\bigr\rangle \,dy \\ ={}&\varrho _{1}(s)+\varrho _{2}(s), \end{aligned} $$
(3.50)

where

$$ \varrho (s)= \bigl\Vert \chi (s) \bigr\Vert _{\mathcal{H}^{1}}^{2}+2 \int _{0}^{t} \bigl\Vert \chi (y) \bigr\Vert _{\mathcal{H}^{1}}^{2}\,dy. $$
(3.51)

Noting the monotonicity of the function \(z\mapsto \bar{V}(x)\), we have

$$ \varrho _{1}(s)=-2 \int _{0}^{t}\bigl\langle \bar{V} \bigl(f_{t}(y)\bigr)-\bar{V} \bigl(v_{t}(y)\bigr), \chi _{t}(y)\bigr\rangle \,dy\leq 0. $$
(3.52)

Furthermore,

$$ \begin{aligned}[b]\bigl[ \iota (f_{t})u-\iota (v_{t})v\bigr] w &=\bigl[ \iota (f_{t})w+\bigl( \iota (f_{t})-\iota (v_{t})\bigr) v\bigr] w \\ &=\iota (f _{t})w^{2}+\bigl( \iota (f_{t})- \iota (v_{t})\bigr) vw \\ &\geq \bigl( \iota (f_{t})- \iota (v_{t})\bigr) vw, \end{aligned} $$
(3.53)

which implies that

$$ \begin{aligned}[b] \varrho _{2}(s) &=-2 \int _{0}^{t}\bigl\langle \iota \bigl(f_{t}(y)\bigr)f(y)-\iota \bigl(v _{t}(y)\bigr)v(y), \chi (y)\bigr\rangle \,dy \\ &\leq -2 \int _{0}^{t}\bigl\langle \bigl[ \iota \bigl(f_{t}(y)\bigr)-\iota \bigl(v_{t}(y)\bigr) \bigr] v(y),\chi (y)\bigr\rangle \,dy \\ & \leq 2 \int _{0}^{t} \bigl\Vert \bigl[ \iota \bigl(f_{t}(y)\bigr)-\iota \bigl(v_{t}(y)\bigr) \bigr] v(y) \bigr\Vert \bigl\Vert \chi (y) \bigr\Vert \,dy \\ &\leq 2 \int _{0}^{t} \bigl\Vert \iota \bigl(f_{t}(y)\bigr)- \iota \bigl(v_{t}(y)\bigr) \bigr\Vert \bigl\Vert v_{t}(y) \bigr\Vert \bigl\Vert \chi (y) \bigr\Vert \,dy. \end{aligned} $$
(3.54)

Putting

$$ M= \Vert u \Vert _{L^{\infty }(0,W;\mathcal{H}_{0}^{1}\cap \mathcal{H}^{2})} + \Vert v \Vert _{L^{\infty }(0,W;\mathcal{H}_{0}^{1}\cap \mathcal{H}^{2})} $$

and

$$ L_{M}=\sup_{ \vert x \vert \leq M} \bigl\vert \iota '(x) \bigr\vert , $$

we obtain that

$$ \bigl\vert \iota (f_{t})-\iota (v_{t}) \bigr\vert \leq L_{M} \vert \chi _{t} \vert . $$
(3.55)

So

$$ \begin{aligned}[b] \varrho _{2}(s) &\leq 2L_{M} \int _{0}^{t} \bigl\Vert \chi _{t}(y) \bigr\Vert \bigl\Vert v_{t}(y) \bigr\Vert \bigl\Vert \chi (y) \bigr\Vert \,dy \\ &\leq 2ML_{M} \int _{0}^{t} \bigl\Vert \chi _{t}(y) \bigr\Vert \bigl\Vert \chi (y) \bigr\Vert \,dy \\ &\leq ML_{M} \int _{0}^{t}\varrho (y)\,dy. \end{aligned} $$
(3.56)

Then from (3.50), (3.52), and (3.56) it follows that

$$ \varrho (s)\leq ML_{M} \int _{0}^{t}\varrho (y)\,dy, $$
(3.57)

which leads to \(\varrho (s)=0\), that is, \(\chi =f-v=0\).

Let us assume that

$$ f_{0}(t)\leq M\quad \text{for a.e. }t\in \varSigma _{2}\quad \mbox{and}\quad \max \bigl\{ \Vert \tilde{f}_{0} \Vert _{L^{\infty }}, \Vert f \Vert _{L^{\infty }(\gimel )}\bigr\} \leq M. $$
(3.58)

Then \(\omega =f-M\) satisfies

$$ \begin{aligned} &\omega _{s}-\omega _{tts}-\frac{\partial }{\partial t}\bigl( \omega _{t}+ \bar{V} (x_{t})\bigr) +z+(x+M)\iota (x_{t}) \\ &\quad =w(t,s)-M, \quad 0< t< 1, 0< s< W, \\ &z(0,s)=z(1,s)=-M, \\ &z(t,0)=\tilde{f}_{0}(t)-M. \end{aligned}$$
(3.59)

So

$$ \begin{aligned}[b] &\bigl\langle \omega _{s}(s),v\bigr\rangle +\bigl\langle \omega _{xs}(s),v_{t}\bigr\rangle + \bigl\langle \omega _{t}(s)+\bar{V}\bigl(x_{t}(s) \bigr),v_{t}\bigr\rangle \\ &\qquad{} +\bigl\langle z(s)+\bigl(x(s)+M\bigr) \iota \bigl(x_{t}(s)\bigr),v \bigr\rangle \\ &\quad =\bigl\langle w(s)-M,v\bigr\rangle \quad \text{for all }v\in \mathcal{H}_{0}^{1}. \end{aligned} $$
(3.60)

We deduce that the solution of the singular boundary value problem with mixed boundary conditions and spatial heterogeneities (1.1) satisfies \(f\in L^{\infty }(0,W;\mathcal{H}_{0}^{1}\cap \mathcal{H} ^{2})\), \(f'\in L^{2}(0,W;\mathcal{H}_{0}^{1})\), so that we are allowed to take \(v=\omega ^{+}=\frac{1}{2}(| x| +z)\) in (3.60).

So

$$ \begin{aligned}[b] &\bigl\langle \omega _{s}(s),\omega ^{+}(s)\bigr\rangle +\bigl\langle \omega _{xs}(s), \omega _{t}^{+}(s)\bigr\rangle +\bigl\langle \omega _{t}(s)+\bar{V} \bigl(x_{t}(s)\bigr), \omega _{t}^{+}(s)\bigr\rangle \\ &\qquad {}+\bigl\langle z(s)+\bigl(x(s)+M\bigr)\iota \bigl(x_{t}(s)\bigr), \omega ^{+}(s)\bigr\rangle \\ &\quad =\bigl\langle w(s)-M,\omega ^{+}(s)\bigr\rangle , \end{aligned} $$
(3.61)

which yields that

$$ \begin{aligned}[b] &\frac{1}{2}\frac{d}{dt}\bigl( \bigl\Vert \omega ^{+}(s) \bigr\Vert ^{2}+ \bigl\Vert \omega _{t}^{+}(s) \bigr\Vert ^{2} \bigr) + \bigl\Vert \omega _{t}^{+}(s) \bigr\Vert ^{2}+ \bigl\Vert \omega ^{+}(s) \bigr\Vert ^{2} \\ &\quad =- \bigl\langle \bar{V}\bigl(x_{t}^{+}(s)\bigr),\omega _{t}^{+}(s)\bigr\rangle -\bigl\langle \bigl(x ^{+}(s)+M\bigr)\iota \bigl(x_{t}^{+}(s) \bigr),\omega ^{+}(s)\bigr\rangle \\ &\qquad {} +\bigl\langle w(s)-M,\omega ^{+}(s)\bigr\rangle \leq 0 \end{aligned} $$
(3.62)

and

$$ \begin{aligned}[b] \bigl\langle \omega _{s}(s),\omega ^{+}(s)\bigr\rangle &= \int _{0}^{1}\omega _{s}(t,s) \omega ^{+}(t,s)\,dx \\ & = \int _{0,z>0}^{1}\bigl(x^{+}(t,s) \bigr)_{t }\omega ^{+}(t,s)\,dx \\ & =\frac{1}{2}\frac{d}{dt} \int _{0, z>0}^{1} \bigl\vert \omega ^{+}(t,s) \bigr\vert ^{2}\,dx \\ &=\frac{1}{2} \frac{d}{dt} \int _{0}^{1} \bigl\vert \omega ^{+}(t,s) \bigr\vert ^{2}\,dx \\ & =\frac{1}{2}\frac{d}{dt} \bigl\Vert \omega ^{+}(s) \bigr\Vert ^{2}, \end{aligned} $$
(3.63)

and on the domain \(z>0\), we have \(\omega ^{+}=z\), \(\omega _{t}=(x^{+})_{t}\), and \(\omega _{s}=(x^{+})_{s}\).

It follows from (3.62) that

$$ \bigl\Vert \omega ^{+}(s) \bigr\Vert ^{2}+ \bigl\Vert \omega _{t}^{+}(s) \bigr\Vert ^{2} \leq \bigl\Vert \omega ^{+}(0) \bigr\Vert ^{2}+ \bigl\Vert \omega _{t}^{+}(0) \bigr\Vert ^{2}. $$
(3.64)

Since

$$\begin{aligned}& \omega ^{+}(t,0)=\bigl(f(t,0)-M\bigr)^{+}=\bigl( \tilde{f}_{0}(t)-M\bigr)^{+}=0, \\& \omega _{t}^{+}(t,0)=0, \end{aligned}$$

we obtain that \(\| \omega ^{+}(s)\| ^{2}+\|\omega _{t}^{+}(s)\| ^{2}=0\). Thus \(\omega ^{+}=0\) and \(f(t,s)\leq M\) for a.e. \((t,s)\in \gimel \).

The case \(-M\leq f_{0}(t)\) for a.e. \(t\in \varSigma _{2} \) and

$$ M\geq \max \bigl\{ \Vert \tilde{f}_{0} \Vert _{L^{\infty }}, \Vert f \Vert _{L^{\infty }( \gimel )}\bigr\} $$

can be dealt with by considering \(\omega =u+M\) and \(\omega ^{-}= \frac{1}{2}(|x| -z)\); we also have \(\omega ^{-}=0\), and hence \(f(t,s)\geq -M\) for a.e. \((t,s)\in \gimel \).

Furthermore, we obtain that \(| f(t,s)| \leq M\) for a.e. \((t,s)\in \gimel \), that is,

$$ \Vert u \Vert _{L^{\infty }(\gimel )}\leq M $$
(3.65)

for all

$$ M\geq \max \bigl\{ \Vert \tilde{f}_{0} \Vert _{L^{\infty }}, \Vert f \Vert _{L^{\infty }( \gimel )}\bigr\} , $$

which implies (3.3). The proof is complete. □

Examples

In this section, we will test two singular boundary value problems with mixed boundary conditions and spatial heterogeneities by using the presented method.

Example 4.1

Consider the singular boundary value problem with mixed boundary conditions and spatial heterogeneities

$$\begin{aligned}& \bigl(4^{c}\mathcal{D}^{1/4}+3^{c} \mathcal{D}^{2/3}+2^{c}\mathcal{D}^{3/4} \bigr)s(y) =\frac{M}{\sqrt{t^{2}+81}} \bigl(\cos x + \cot ^{-1}t \bigr), \quad 0< y< 1, \end{aligned}$$
(4.1)
$$\begin{aligned}& s(0)=0, \qquad s(1/4)=0,\qquad s(1)= \int _{0}^{1/5}s(r)\,dr. \end{aligned}$$
(4.2)

Here \(M>0\), and

$$ f(y,s)=\frac{M}{\sqrt{t^{2}+81}} \bigl(\cos x + \cot ^{-1}t \bigr). $$

Put \(p_{0}=2\) and \(p_{1}=p_{2}=3\). It is easy to see that they satisfy the conditions of Lemma 2.1 and

$$ \bigl\vert f(y,s)-f(y,t) \bigr\vert \leq \frac{2}{9}M \vert s-t \vert . $$

Using the given values, we know that \(\phi \approx 0.44269\) and \(\phi _{1}\approx 0.21725\). So

$$ \bigl\vert f(y,s) \bigr\vert \leq \frac{M(3+2\pi )}{3\sqrt{t^{3}+81}}=\theta (y) $$

and \(\ell \phi _{1}<1\) when \(M<41.32901\).

On the one hand, all conditions of Theorem 3.1 hold. So problem (4.1)–(4.2) has at least one weak solution in \([0,W]\). On the other hand, \(\ell \phi <1\) whenever \(M<17.28439\). So it follows from Theorem 3.1 that there exists a unique weak solution for problem (4.1)–(4.2) in \([0,1]\).

Example 4.2

Consider the singular boundary value problem with mixed boundary conditions and spatial heterogeneities

$$\begin{aligned}& \bigl(3^{c}\mathcal{D}^{1/4}+3^{c} \mathcal{D}^{2/3}+2^{c}\mathcal{D}^{3/4} \bigr)s(y) =\frac{1}{4\pi }\cos {(2\pi s)}+\frac{ \vert s \vert ^{2}}{1+ \vert s \vert ^{2}}, \quad 0< y< 1, \end{aligned}$$
(4.3)
$$\begin{aligned}& s(0)=0, \qquad s(1/4)=0,\qquad s(1)= \int _{0}^{3/4}s(r)\,dr. \end{aligned}$$
(4.4)

Here

$$ f(y,s)=\frac{2}{3\pi }\cos {(3\pi s)}+\frac{ \vert s \vert ^{2}}{1+ \vert s \vert ^{2}}. $$

Similarly to Example 4.1, we obtain that

$$ \bigl\vert f(y,s) \bigr\vert \leq \biggl\vert \frac{2}{3\pi }\cos {(3\pi s)}+\frac{ \vert s \vert ^{2}}{1+ \vert s \vert ^{2}} \biggr\vert \leq \frac{2}{3} \Vert s \Vert +3, $$

\(g(y)=1\), and \(\psi (\|s\|)=\frac{1}{2}\|s\|+1\).

It is clear that \(M> 0.23971\) (we have used \(\phi = 0.38471\)). Thus the conclusion of Theorem 3.1 applies to problem (4.3)–(4.4).

Conclusions

In this paper, we presented a modified Schrödinger-type identity related to the Schrödinger-type boundary value problem with mixed boundary conditions and spatial heterogeneities. This identity can be regarded as an \(L^{1}\)-version of Fisher–Riesz’s theorem, and it had a broad range of applications. Using it and fixed point theory in \(L^{1}\)-metric spaces, we proved that there exists a unique solution for the singular boundary value problem with mixed boundary conditions and spatial heterogeneities. We finally provided two examples, which show the effectiveness of the Schrödinger-type identity method.

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This work was supported by the Post-Doctoral Applied Research Projects of Qingdao (no. 2015122) and the Scientific Research Foundation of Shandong University of Science and Technology for Recruited Talents (no. 2014RCJJ032).

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Meng, B. A new application of Schrödinger-type identity to singular boundary value problem for the Schrödinger equation. Bound Value Probl 2019, 161 (2019) doi:10.1186/s13661-019-1279-9

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Keywords

  • Boundary value problem
  • Schrödinger-type identity
  • Uniqueness