1 Introduction

In this paper, we further consider the following Schrödinger problem (see [1]):

$$ iz_{t}=-\Delta z+W(x)z-a(x)h\bigl( \vert z \vert ^{2}\bigr)z-k\Delta l\bigl( \vert z \vert ^{2} \bigr)l'\bigl( \vert z \vert ^{2}\bigr)z, $$
(1)

where \(x \in \mathbb{R}^{n}\), \(z:\mathbb{R}\times \mathbb{R}^{n} \to \mathbb{C}\), \(a,W:\mathbb{R}^{n}\to \mathbb{R}\) is a given potential, k is real constant, and l and h are real functions. The above quasilinear equations have been accepted as models of several physical phenomena corresponding to various types of l; we refer to [2] and the references given therein for physical applications of these problems. Specifically, we would like to mention that the superfluid film equation in plasma physics has this structure for \(l(s)=s\) (see e.g. [3, 4]), while in the case \(l(s)=(1 +s)^{1/2}\), (1) models the self-channeling of a high-power ultrashort laser in matter (see e.g. [5, 6]).

The standing waves solutions of (1); that is, solutions of the type \(z(t,x)=\exp (-iEt)u(x)\) where \(E \in \mathbb{R}\) and \(u>0\) is a real function. Inserting z into (1), with \(l(s)=s\) and \(l(s)=(1 +s^{2})^{1/2}\), turns, respectively, the following equations (see e.g. [7]):

$$\begin{aligned}& -\Delta u + V_{\infty }u-k\Delta \bigl(u^{2}\bigr))u = a(x)h(u), \\ & -\Delta u + V_{\infty }u-k\Delta \bigl(\bigl(1+u^{2} \bigr)^{1/2}\bigr) \frac{u}{(1+u^{2})^{1/2}}= a(x)h(u), \end{aligned}$$

where \(x \in \mathbb{R}^{n}\) and \(V_{\infty }=W-E\).

It is well known that an unknown Borel probability measure on \(W= S\times T\) controls the sampling process, where \(T=\mathbb{R}\) and S is a compact metric space in \(\mathbb{R}^{n}\). As in [8], the exact weak solutions of (1) can be defined by \(g_{\varrho }(s)= \int _{T} y \,d\varrho (t|s)\), where \(\varrho (\cdot |s)\) is the conditional probability measure induced by ϱ on T given \(s\in S\).

To our knowledge, the criteria for minimally thin sets with respect to the Schrödinger operator (1) was introduced for the first time in the context of the stationary Schrödinger equations in [9, 10]. In 2018, Jiang, Zhang and Li (see [11]) further improved this complex method and applied to study meromorphic solutions for the linear differential equations with analytic coefficients and obtain some applications. Recently, Zhang (see [12, 13]) defined a new type of minimal thinness with respect to the stationary Schrödinger operator, established new criteria for it and applied the result to study growth properties at infinity of the maximum modulus with respect to the Schrödinger operator.

In this paper, we will continue to apply new criteria for solutions for a kind of quasilinear Schrödinger equations. Although we are motivated here by [9,10,11,12,13], there were substantial difficulties to adapt the above approach to the present situation. Let \(\mathfrak{H}_{E}\) be the completion of the linear span of the set of functions \(\{E_{s} :=E(s,\cdot ) : s \in S \}\) equipped with (see [8, 14])

$$ \Biggl\langle \sum_{i=1}^{n} \xi _{i} E_{s_{i}}, \sum_{l=1}^{m} \o _{j} E_{t_{j}} \Biggr\rangle _{E} := \sum _{i=1}^{n} \sum_{l=1}^{m} \xi _{i} \o _{j} E(s_{i} , t_{j}). $$

Let \(s\in S\) and \(g\in \mathfrak{H}_{E}\). Define (see [15, Remark 2.3])

$$\begin{aligned} g(s)=\langle g, \,E_{s}\rangle _{E}. \end{aligned}$$
(2)

It follows from (2) that (see [16])

$$\begin{aligned} \Vert g \Vert _{\infty }\leq \kappa \Vert g \Vert _{E}, \end{aligned}$$
(3)

where

$$ \kappa := \sup_{t, s\in S} \bigl\vert E(s,t) \bigr\vert < \infty . $$

Define (see [17])

$$\begin{aligned}& g_{\mathbf{w},\chi }(s)=g_{\mathbf{w},\zeta ,\chi ,s}(s)=g_{ \mathbf{w},\zeta ,\chi ,s}(u)|_{u=s}, \\& g_{\mathbf{w},\zeta ,\chi ,s} :=\arg \min_{f\in \mathfrak{H}_{E}} \Biggl\{ \frac{1}{m}\sum_{i=1}^{m}\varPhi \biggl(\frac{s}{\zeta },\frac{s _{i}}{\zeta } \biggr) \bigl(t_{i}-g(s_{i}) \bigr)^{2}+\chi \Vert g \Vert _{E}^{2} \Biggr\} , \end{aligned}$$
(4)

where

$$\begin{aligned} & \varPhi (s,t)\leq 1,\quad \forall s,t \in \mathbb{R}^{n}, \end{aligned}$$
(5)
$$\begin{aligned} & \varPhi (s,t)\geq c_{q},\quad \forall \vert s-t \vert \leq 1. \end{aligned}$$
(6)

Scheme (4) yields (see [18, 19])

$$\begin{aligned}& g_{\mathbf{w},\varsigma }(s)=g_{\mathbf{w},\zeta ,\varsigma ,s}(s)=g _{\mathbf{w},\zeta ,\varsigma ,s}(u)|_{u=s}, \\& g_{\mathbf{w},\zeta ,\varsigma ,s}=\arg \min_{f\in \mathfrak{H}_{E,\,\mathbf{w}} } \Biggl\{ \frac{1}{m}\sum _{i=1} ^{m}\varPsi \biggl(\frac{s}{\zeta }, \frac{s_{i}}{\zeta } \biggr) \bigl(g(s_{i})-t _{i} \bigr)^{2}+\varsigma \sum_{i=1}^{m} \vert \xi _{i} \vert ^{q} \Biggr\} , \end{aligned}$$

and

$$\begin{aligned} \mathfrak{H}_{E,\,\mathbf{w}}= \Biggl\{ g(s)=\sum_{i=1}^{m} \xi _{i}E(s,s _{i}):\xi =(\xi _{1},\ldots ,\xi _{m})\in \mathbb{R}^{m},m\in \mathbb{N} \Biggr\} . \end{aligned}$$

In order to study the boundary behaviors of \(g_{\mathbf{w},\varsigma }\), we derive

$$ \Vert g_{\mathbf{w},\varsigma }-g_{\varrho } \Vert _{\varrho _{S}} $$

with (see [20,21,22,23] for more details)

$$ \bigl\Vert g(\cdot ) \bigr\Vert _{\varrho _{S}}:=\biggl( \int _{S} \bigl\vert g(\cdot ) \bigr\vert ^{2}\,d{ \varrho _{S}}\biggr)^{ \frac{1}{2}}. $$

The remainder of this paper is organized as follows. In Sect. 2, we will provide the main results. In Sect. 3, some basic but important estimates and properties are summarized. The proofs of main results will be given in Sect. 4. Section 5 contains the conclusions of the paper.

2 Main results

The integral operator \(L_{E}:L_{\varrho _{S}} ^{2}(S)\rightarrow L_{ \varrho _{S}} ^{2}(S)\) is defined by

$$ (L_{E} g) (s) = \int _{S} E(s,t)g(t)\,d\varrho _{S}(t). $$

Let \(\{\mu _{i}\} \) be the eigenvalues of \(L_{E}\) and \(\{e_{i}\}\) be the corresponding eigenfunctions. Then we define

$$ L_{E}^{r}(g)=\sum_{i=1}^{\infty } \mu _{i}^{r}\langle g,e_{i} \rangle _{L_{\varrho _{S}} ^{2}}e_{i} $$

for \(g\in L_{\varrho _{S}} ^{2}(S)\). We assume that \(g_{\varrho }\) satisfies \(L_{E}^{-r}g_{\varrho }\in L^{2}_{\varrho _{S}}\), where r is a positive constant depending on the size of the initial data in a suitable norm.

Let \(c_{p}~(0< p<2)\) be a positive constant. Define (see [24])

$$\begin{aligned} \log \mathfrak{N}_{2}(B_{1}, \epsilon )\leq c_{p}\epsilon ^{-p}, \end{aligned}$$
(7)

where

$$ B_{1}= \bigl\{ f\in \mathfrak{H}_{E,\,\mathbf{w}}: \Vert g \Vert _{E}\leq 1 \bigr\} . $$

Now we are in a position to obtain the existence of solutions for the problem (1).

Theorem 1

Suppose \(L_{E}^{-r}g_{\varrho }\in L^{2}_{\varrho _{S}}\) with \(r>0\), (7) with \(0< p<2\). Then there exist solutions for the problem (1), which can be defined by

$$ \mathfrak{H}(\mathbf{w},\chi ,\varsigma )= \int _{S}\bigl(\mathfrak{E}_{ \mathbf{w},s}\bigl(\gamma _{M}(g_{\mathbf{w}, \zeta ,\varsigma ,s})\bigr)+\varsigma \varOmega _{\mathbf{w}}(g_{\mathbf{w}, \zeta ,\varsigma ,s}) \bigr)\,d\varrho _{S}(s) $$

and

$$ \mathfrak{H}(\mathbf{w},\chi ,\varsigma )\leq \frac{m\varsigma M^{2}}{(m \chi )^{q}}. $$

For the further application of Theorem 1, we have the following result. Similar results for solutions of the stationary Schrödinger equations, we refer the reader to the papers (see [13, 25]).

Proposition 1

Let \(L_{E}^{-r}g_{\varrho }\in L^{2}_{\varrho _{S}}\), where \(r>0\). Then

$$\begin{aligned} \mathfrak{D}(\chi )\leq C_{1}\chi ^{\min \{2r,1\}}. \end{aligned}$$
(8)

It follows from Theorem 1 that we can decompose solutions for the problem (1) into two parts, \(\mathfrak{H}_{1}( \mathbf{w},\varsigma )+\mathfrak{H}_{2}(\mathbf{w},\chi )\), where

$$ \int _{S} \bigl\{ \mathfrak{E}_{s}\bigl(\gamma _{M}(g_{\mathbf{w}, \zeta , \varsigma ,s})\bigr)-\mathfrak{E}_{s}(g_{\varrho })- \mathfrak{E}_{ \mathbf{w},s}\bigl(\gamma _{M}(g_{\mathbf{w}, \zeta ,\varsigma ,s})\bigr)+ \mathfrak{E}_{\mathbf{w},s}(g_{\varrho }) \bigr\} \,d\varrho _{S}(s) $$

and

$$ \int _{S} \bigl\{ \mathfrak{E}_{\mathbf{w},s}(g_{\chi })- \mathfrak{E} _{\mathbf{w},s}(g_{\varrho })-\mathfrak{E}_{s}(g_{\chi })+ \mathfrak{E}_{s}(g_{\varrho }) \bigr\} \,d\varrho _{S}(s). $$

Finally, we further study the boundary behaviors for solutions for the problem (1).

Theorem 2

Let the assumptions of Theorem 1 hold. Then

$$\begin{aligned} \mathfrak{H}_{2}(\mathbf{w},\chi )\leq \frac{\mathfrak{D}(\chi )}{2}+\frac{7 (3M+\kappa \sqrt{\frac{\mathfrak{D}(\chi )}{\chi }} )^{2} \log (2/\delta )}{3m}, \end{aligned}$$
(9)

where \(0<\delta <1\).

Theorem 3

Let the assumptions of Theorem 1 hold. Then

$$\begin{aligned} \mathfrak{H}_{1}(\mathbf{w},\varsigma )\leq {}& \frac{1}{2} \int _{S} \bigl\{ \mathfrak{E}_{s}\bigl(\gamma _{M}(g_{\mathbf{w}, \zeta ,\varsigma ,s})\bigr)- \mathfrak{E}_{s}(g_{\varrho }) \bigr\} \,d\varrho _{S}(s) \\ &{}+\frac{176M^{2}}{m}\log \biggl(\frac{2}{\delta } \biggr)+C_{p,M}R_{ \varsigma }^{\frac{2p}{2+p}}m^{-\frac{2}{2+p}}, \end{aligned}$$
(10)

where \(0<\delta <1\) and

$$ R_{\varsigma }=\kappa m^{1-\frac{1}{q}} \biggl(\frac{M^{2}}{\varsigma } \biggr)^{\frac{1}{q}}. $$

3 Lemmas

Some basic but important estimates are needed in this section. The following lemma indicates that the natural and weakened total energies are conserved in time.

Lemma 1

We have the following estimates:

$$\begin{aligned}& \mathfrak{E}_{\tau ,g}(t) = \mathfrak{E}_{\tau ,g}(0), \quad \forall t \in [0,\tau ], \end{aligned}$$
(11)
$$\begin{aligned}& \widetilde{E}_{\tau ,g}(t) = \widetilde{E}_{\tau ,g}(0), \quad \forall t \in [0,\tau ]. \end{aligned}$$
(12)

Proof

Multiplying the first equation by \(g_{\varrho }'\), we obtain

$$\bigl\langle g_{\varrho }''(t) - \partial _{g}^{2} g_{\varrho }(t) + \delta g_{\chi }(t) , g_{\varrho }'(t) \bigr\rangle _{\mathbb{R}^{N},g} =0. $$

It follows that

$$\begin{aligned} &\bigl\langle g_{\varrho }''(t), g_{\varrho }'(t) \bigr\rangle _{\mathbb{R}^{N},g} + \bigl\langle \bigl(-\partial _{g}^{2}\bigr)^{1/2} g_{\varrho }(t) , \bigl(-\partial _{g} ^{2} \bigr)^{1/2} g_{\varrho }'(t) \bigr\rangle _{\mathbb{R}^{N},g} + \delta \bigl\langle g_{\chi }(t), g_{\varrho }'(t) \bigr\rangle _{\mathbb{R}^{N},g} =0. \end{aligned}$$

Therefore

$$ \frac{d}{dt} \gamma _{M}(g_{\varrho };t) + \delta \bigl\langle g_{\chi }(t), g_{\varrho }'(t) \bigr\rangle _{\mathbb{R}^{N},g} =0, $$
(13)

which leads to

$$ \frac{d}{dt} \gamma _{M}(g_{\chi };t) + \delta \bigl\langle g_{\varrho }(t), g_{\chi }'(t) \bigr\rangle _{\mathbb{R}^{N},g} =0. $$
(14)

Adding (13) and (14), we can write

$$\frac{d}{dt} \mathfrak{E}_{\tau ,g}(t) =0, $$

which is equivalent to (11).

By taking the sum of the resulting two identities we obtain

$$\begin{aligned} &\frac{d}{dt} \widetilde{\mathfrak{E}}_{g}(g_{\varrho };t) + \frac{d}{dt} \widetilde{\mathfrak{E}}_{g}(g_{\chi };t) + \delta \bigl\langle g_{\chi }(t),\bigl(-\partial _{g}^{2} \bigr)^{-1} g_{\varrho }'(t) \bigr\rangle _{\mathbb{R}^{N},g} + \delta \bigl\langle g_{\varrho }(t),\bigl(-\partial _{g}^{2} \bigr)^{-1} g_{ \chi }'(t) \bigr\rangle _{\mathbb{R}^{N},g} =0, \end{aligned}$$

using the symmetry of the matrix \((-\partial _{g}^{2} )^{-1}\) we obtain

$$\frac{d}{dt} \widetilde{\mathfrak{E}}_{\tau ,g}(t) =0. $$

 □

From Lemma 1, we deduce the following result.

Lemma 2

Let \(0 \leq \delta \leq \frac{\delta _{0}}{3}\). Then

$$ \int _{S} \bigl( \gamma _{M}(g_{\varrho };t) + \widetilde{\mathfrak{E}}_{g}(g _{\chi };t) \bigr) \,dt \geq \frac{C\tau }{2} \bigl( \widetilde{\mathfrak{E}}_{g}(g _{\varrho };0) + \widetilde{\mathfrak{E}}_{g}(g_{\chi };0) \bigr) $$
(15)

for a positive constant Cτ depending only on τ.

Proof

We recall

$$\gamma _{M}(g_{\varrho };t) = \frac{1}{2} \bigl\Vert g_{\varrho }'(t) \bigr\Vert _{\mathbb{R}^{N},g}^{2} + \frac{1}{2} \bigl\Vert \bigl(-\partial _{g}^{2} \bigr)^{1/2} g_{\varrho }(t) \bigr\Vert _{\mathbb{R}^{N},g}^{2}, $$

and we can write

$$\gamma _{M}(g_{\varrho };t) \geq \frac{\delta _{0}}{2} \bigl\Vert \bigl(-\partial _{g}^{2}\bigr)^{-1/2} g_{\varrho }'(t) \bigr\Vert _{\mathbb{R}^{N},g}^{2} + \frac{ \delta _{0}}{2} \bigl\Vert g_{\varrho }(t) \bigr\Vert _{\mathbb{R}^{N},g}^{2} = \delta _{0} \widetilde{ \mathfrak{E}}_{g}(g_{\varrho };t). $$

It follows from Lemma 1 that

$$ \int _{S} \bigl( \gamma _{M}(g_{\varrho };t) + \widetilde{\mathfrak{E}}_{g}(g _{\chi };t) \bigr)\,dt \geq C \int _{S} \bigl( \widetilde{\mathfrak{E}}_{g}(g_{ \varrho };t) + \widetilde{\mathfrak{E}}_{g}(g_{\chi };t) \bigr)\,dt. $$
(16)

On the other hand

$$\bigl\vert \widetilde{\mathfrak{E}}_{\tau ,g}(t) - \bigl( \widetilde{ \mathfrak{E}} _{g}(g_{\varrho };t) + \widetilde{ \mathfrak{E}}_{g}(g_{\chi };t)\bigr) \bigr\vert = \bigl\vert \delta \bigl\langle \bigl(-\partial _{g}^{2} \bigr)^{-1} g_{\varrho }(t),g_{ \chi }(t) \bigr\rangle _{\mathbb{R}^{N},g} \bigr\vert , $$

and thanks to Lemma 1 and [26, Theorem 2.1], one has

$$ \bigl\vert \widetilde{\mathfrak{E}}_{\tau ,g}(t) - \bigl( \widetilde{\mathfrak{E}} _{g}(g_{\varrho };t) + \widetilde{ \mathfrak{E}}_{g}(g_{\chi };t)\bigr) \bigr\vert \leq \frac{\delta }{\delta _{0}} \bigl( \widetilde{\mathfrak{E}}_{g}(g_{ \varrho };t) + \widetilde{\mathfrak{E}}_{g}(g_{\chi };t) \bigr). $$
(17)

Hence

$$\widetilde{\mathfrak{E}}_{g}(g_{\varrho };t) + \widetilde{ \mathfrak{E}}_{g}(g_{\chi };t) \geq \frac{\delta _{0}}{ \delta _{0} + \delta } \widetilde{\mathfrak{E}}_{\tau ,g}(t). $$

Integrating this last inequality over \(t \in [0,\tau ]\) and using the fact that the energy \(\widetilde{\mathfrak{E}}_{\tau ,g}(t)\) is conservative, we deduce that

$$ \int _{S} \bigl( \widetilde{\mathfrak{E}}_{g}(g_{\varrho };t) + \widetilde{\mathfrak{E}}_{g}(g_{\chi };t) \bigr) \,dt \geq \frac{\delta _{0} \tau }{\delta _{0} + \delta }\widetilde{\mathfrak{E}}_{\tau ,g}(0). $$
(18)

Moreover, thanks to inequality (17), we have

$$\widetilde{\mathfrak{E}}_{\tau ,g}(0) \geq \frac{\delta _{0} - \delta }{\delta _{0}} \bigl( \widetilde{\mathfrak{E}}_{g}(g_{\varrho };0) + \widetilde{ \mathfrak{E}}_{g}(g_{\chi };0) \bigr), $$

and inserting this last equation into (18) yields

$$ \int _{S} \bigl( \widetilde{\mathfrak{E}}_{g}(g_{\varrho };t) + \widetilde{\mathfrak{E}}_{g}(g_{\chi };t) \bigr) \,dt \geq \frac{\delta _{0} - \delta }{\delta _{0} + \delta }\tau \bigl( \widetilde{\mathfrak{E}}_{g}(g _{\varrho };0) + \widetilde{\mathfrak{E}}_{g}(g_{\chi };0) \bigr). $$
(19)

However, since

$$\frac{\delta _{0} - \delta }{\delta _{0} + \delta } \geq \frac{1}{2} $$

for all \(\delta \leq \frac{\delta _{0}}{3}\), we deduce from (19) that

$$\int _{S} \bigl( \widetilde{\mathfrak{E}}_{g}(g_{\varrho };t) + \widetilde{\mathfrak{E}}_{g}(g_{\chi };t) \bigr) \,dt \geq \frac{\tau }{2} \bigl( \widetilde{\mathfrak{E}}_{g}(g_{\varrho };0) + \widetilde{\mathfrak{E}}_{g}(g_{\chi };0) \bigr). $$

Inserting this inequality into (16), the desired estimate (15) is obtained. □

We complete this subsection with the following lemma.

Lemma 3

We have

$$\begin{aligned}& \begin{aligned}[b] \int _{S} \widetilde{\mathfrak{E}}_{g}(g_{\chi };t) \,dt \leq {}&\frac{C}{ \delta (\sqrt{\delta _{0}}-\delta )} \bigl( \gamma _{M}(g_{\varrho };0) + \widetilde{\mathfrak{E}}_{g}(g_{\chi };0) \bigr) \\ &{} + \frac{C}{(\sqrt{\delta _{0}}-\delta )^{2}} \int _{S} \gamma _{M}(g_{\varrho };t) \,dt, \end{aligned} \end{aligned}$$
(20)
$$\begin{aligned}& \begin{aligned}[b] \int _{S} \bigl\Vert g_{\chi }(t) \bigr\Vert _{\mathbb{R}^{N},g}^{2} \,dt \leq {}&\frac{C}{\delta (\sqrt{\delta _{0}}-\delta )} \bigl( \gamma _{M}(g _{\varrho };0) + \widetilde{\mathfrak{E}}_{g}(g_{\chi };0) \bigr) \\ &{}+ \frac{C}{(\sqrt{\delta _{0}}-\delta )^{2}} \int _{S} \gamma _{M}(g_{\varrho };t) \,dt, \end{aligned} \end{aligned}$$
(21)
$$\begin{aligned}& \begin{aligned}[b] \widetilde{\mathfrak{E}}_{g}(g_{\chi }; \tau ) + \widetilde{\mathfrak{E}}_{g}(g_{\chi }; 0) \leq{}& \frac{C}{\sqrt{ \delta _{0}}-\delta } \bigl( \gamma _{M}(g_{\varrho };0) + \widetilde{ \mathfrak{E}}_{g}(g_{\chi };0) \bigr) \\ &{}+ \frac{C \delta }{(\sqrt{\delta _{0}}-\delta )^{2}} \int _{S} \gamma _{M}(g_{\varrho };t) \,dt, \end{aligned} \end{aligned}$$
(22)

where \(0 \leq \delta \leq \min ( \delta _{0}, \sqrt{\delta _{0}} )\).

Proof

First, we recall the following estimates:

$$\begin{aligned}& \begin{aligned}[b] \int _{S} \bigl\Vert g_{\chi }(t) \bigr\Vert _{\mathbb{R}^{N},g}^{2} \,dt \leq{}& \frac{C}{\delta ( \sqrt{\delta _{0}}-\delta )} \bigl( \gamma _{M}(g _{\varrho };0) + \widetilde{\mathfrak{E}}_{g}(g_{\chi };0) \bigr) \\ &{} + \frac{C}{(\sqrt{\delta _{0}}-\delta )^{2}} \int _{S} \bigl( \bigl\Vert g_{\varrho }(t) \bigr\Vert _{\mathbb{R}^{N},g}^{2} + \bigl\Vert g_{ \varrho }'(t) \bigr\Vert _{\mathbb{R}^{N},g}^{2} \bigr)\,dt, \end{aligned} \\& \begin{aligned} \int _{S} \bigl\Vert \bigl(-\partial _{g}^{2} \bigr)^{-1/2}g_{\chi }'(t) \bigr\Vert _{ \mathbb{R}^{N},g}^{2} \,dt \leq{}& \frac{C}{\delta (\sqrt{\delta _{0}}- \delta )} \bigl( \gamma _{M}(g_{\varrho };0) + \widetilde{\mathfrak{E}}_{g}(g _{\chi };0) \bigr) \\ &{} + \frac{C}{(\sqrt{\delta _{0}}-\delta )^{2}} \int _{S} \bigl( \bigl\Vert g_{\varrho }(t) \bigr\Vert _{\mathbb{R}^{N},g}^{2} + \bigl\Vert g_{ \varrho }'(t) \bigr\Vert _{\mathbb{R}^{N},g}^{2} \bigr)\,dt, \end{aligned} \end{aligned}$$
(23)

from the proof of Lemma 2.

Taking the sum of these two inequalities, we obtain

$$ \begin{aligned}[b] \int _{S} \widetilde{\mathfrak{E}}_{g}(g_{\chi };t) \,dt\leq{}& \frac{C}{ \delta (\sqrt{\delta _{0}}-\delta )} \bigl( \gamma _{M}(g_{\varrho };0) + \widetilde{\mathfrak{E}}_{g}(g_{\chi };0) \bigr) \\ &{} + \frac{C}{(\sqrt{\delta _{0}}-\delta )^{2}} \int _{S} \bigl( \bigl\Vert g_{\varrho }(t) \bigr\Vert _{\mathbb{R}^{N},g}^{2} + \bigl\Vert g_{ \varrho }'(t) \bigr\Vert _{\mathbb{R}^{N},g}^{2} \bigr)\,dt. \end{aligned} $$
(24)

And thanks to Lemma 2, we improve (24) as follows:

$$\begin{aligned} \int _{S} \widetilde{\mathfrak{E}}_{g}(g_{\chi };t) \, dt\leq{}& \frac{C}{ \delta (\sqrt{\delta _{0}}-\delta )} \bigl( \gamma _{M}(g_{\varrho };0) + \widetilde{\mathfrak{E}}_{g}(g_{\chi };0) \bigr) \\ &{} + \frac{C}{(\sqrt{\delta _{0}}-\delta )^{2}} \int _{S} \gamma _{M}(g_{\varrho };t) \, dt, \end{aligned}$$

which proves the inequality (20).

The other estimates (21) and (22), are obtained easily from equations (23), (24) and the relation

$$\int _{S} \bigl( \bigl\Vert g_{\varrho }(t) \bigr\Vert _{\mathbb{R}^{N},g}^{2} + \bigl\Vert g_{\varrho }'(t) \bigr\Vert _{\mathbb{R}^{N},g}^{2} \bigr)\,dt \leq \max \biggl( \frac{1}{\delta _{0}},1 \biggr) \int _{S} \gamma _{M}(g_{\varrho };t) \,dt. $$

 □

4 Proofs of main results

Now we derive the learning rates.

Proof of Theorem 1

Let \(\mathbf{{y}}=(t_{1}, t_{2}, t_{3},\ldots , t_{m})^{ \tau }\), \(K[{\mathbf{{s}}}]=(E(s_{i},s_{j}))_{i,j=1}^{m}\) and \(\mathbf{{a}}^{\mathbf{w}}=(a_{1}^{\mathbf{w}},\ldots ,a _{m}^{\mathbf{w}})\) be the coefficient of \(g_{\mathbf{w},\varsigma }\). It follows from the representation theorem (see [27, 28]) that

$$\begin{aligned} a_{i}^{\mathbf{w}}=\frac{1}{\chi m}\varPsi \biggl( \frac{s}{\zeta },\frac{s _{i}}{\zeta } \biggr) \bigl(t_{i}-g_{\mathbf{w}, \zeta ,\chi ,s}(s_{i}) \bigr) \end{aligned}$$

for \(i=1,2,\ldots ,m\).

By the Hölder inequality, we have

$$\begin{aligned} \sum_{i=1}^{m} \bigl\vert a_{i}^{\mathbf{w}} \bigr\vert ^{q} ={}&\frac{1}{(\chi m)^{q}} \sum_{i=1}^{m} \biggl\vert \varPsi \biggl( \frac{s}{\zeta },\frac{s_{i}}{\zeta } \biggr) \bigl(t _{i}-g_{\mathbf{w}, \zeta ,\chi ,s}(s_{i}) \bigr) \biggr\vert ^{q} \\ \leq {}&\frac{1}{(\chi m)^{q}} \Biggl(\sum_{i=1}^{m} \varPsi \biggl(\frac{s}{ \zeta },\frac{s_{i}}{\zeta } \biggr)^{\frac{1}{2-q}} \Biggr)^{1- \frac{q}{2}} \\ &{} \times \Biggl(\sum_{i=1}^{m} \biggl( \frac{s}{\zeta },\frac{s_{i}}{ \zeta } \biggr) \bigl(t_{i}-g_{\mathbf{w}, \zeta ,\chi ,s}(s_{i}) \bigr)^{2} \Biggr)^{ \frac{q}{2}}. \end{aligned}$$

It follows that

$$\begin{aligned} \sum_{i=1}^{m} \bigl\vert a_{i}^{\mathbf{w}} \bigr\vert ^{q}\leq \frac{m}{(\chi m)^{q}} \bigl(\mathfrak{E}_{\mathbf{w},s}(g_{\mathbf{w}, \zeta ,\chi ,s}) \bigr)^{\frac{q}{2}} \end{aligned}$$

from (5).

Thus

$$\begin{aligned} & \mathfrak{E}_{\mathbf{w},s}\bigl(\gamma _{M}(g_{\mathbf{w}, \zeta , \varsigma ,s}) \bigr)+\varsigma \varOmega _{\mathbf{w}}(g_{\mathbf{w}, \zeta , \varsigma ,s}) \\ &\quad \leq \mathfrak{E}_{\mathbf{w},s}(g_{\mathbf{w}, \zeta ,\varsigma ,s})+ \varsigma \varOmega _{\mathbf{w}}(g_{\mathbf{w}, \zeta ,\varsigma ,s}) \\ &\quad \leq \mathfrak{E}_{\mathbf{w},s}(g_{\mathbf{w}, \zeta ,\chi ,s})+ \varsigma \varOmega _{\mathbf{w}}(g_{\mathbf{w}, \zeta ,\chi ,s}) \\ &\quad \leq \mathfrak{E}_{\mathbf{w},s}(g_{\mathbf{w}, \zeta ,\chi ,s})+\frac{m \varsigma }{(\chi m)^{q}} \bigl( \mathfrak{E}_{\mathbf{w},s}(g_{ \mathbf{w}, \zeta ,\chi ,s}) \bigr)^{\frac{q}{2}} \\ &\quad \leq \mathfrak{E}_{\mathbf{w},s}(g_{\mathbf{w}, \zeta ,\chi ,s})+ \chi \Vert g_{\mathbf{w}, \zeta ,\chi ,s} \Vert _{E}^{2} \\ &\qquad{} +\frac{m\varsigma }{(\chi m)^{q}} \bigl(\mathfrak{E}_{\mathbf{w},s}(g _{\mathbf{w}, \zeta ,\chi ,s})+ \chi \Vert g_{\mathbf{w}, \zeta ,\chi ,s} \Vert _{E}^{2} \bigr)^{\frac{q}{2}}. \end{aligned}$$

Since

$$\begin{aligned} \mathfrak{E}_{\mathbf{w},s}(g_{\mathbf{w}, \zeta ,\chi ,s})+\chi \Vert g _{\mathbf{w}, \zeta ,\chi ,s} \Vert _{E}^{2}\leq \mathfrak{E}_{\mathbf{w},s}(0)+ \chi \Vert 0 \Vert _{E}^{2}, \end{aligned}$$

we get

$$\begin{aligned} & \mathfrak{E}_{\mathbf{w},s}\bigl(\gamma _{M}(g_{\mathbf{w}, \zeta , \varsigma ,s}) \bigr)+\varsigma \varOmega _{\mathbf{w}}(g_{\mathbf{w}, \zeta , \varsigma ,s}) \\ &\quad\leq \mathfrak{E}_{\mathbf{w},s}(g_{\mathbf{w}, \zeta ,\chi ,s})+ \chi \Vert g_{\mathbf{w}, \zeta ,\chi ,s} \Vert _{E}^{2}+\frac{m\varsigma M ^{2}}{(\chi m)^{q}}. \end{aligned}$$

This yields our desired estimation. □

Proof of Theorem 2

Let

$$ h(u,t)= \int _{S}\varPsi \biggl(\frac{s}{\zeta },\frac{u}{\zeta } \biggr)\bigl[\bigl(t-g _{\chi }(u)\bigr)^{2}- \bigl(t-g_{\varrho }(u)\bigr)^{2}\bigr]\,d\varrho _{S}(s) $$

for any \(z=(u,t)\in Z\). Then

$$\begin{aligned} & \int _{Z}h\,d\varrho = \int _{S} \bigl\{ \mathfrak{E}_{s}(g_{\chi })- \mathfrak{E}_{s}(g_{\varrho }) \bigr\} \,d\varrho _{S}(s); \\ &\frac{1}{m}\sum_{i=1}^{m}h(w_{i})= \int _{S} \bigl\{ \mathfrak{E}_{ \mathbf{w},s}(g_{\chi })- \mathfrak{E}_{\mathbf{w},s}(g_{\varrho }) \bigr\} \,d\varrho _{S}(s). \end{aligned}$$

By (3) we have

$$\begin{aligned} \Vert g_{\chi } \Vert _{\infty }\leq \kappa \Vert g_{\chi } \Vert _{E}\leq \kappa \sqrt{\frac{ \mathfrak{D}(\chi )}{\chi }}. \end{aligned}$$

Combining with (5), we have

$$\begin{aligned} \bigl\vert h(u,t) \bigr\vert &\leq \bigl( \Vert g_{\chi } \Vert _{\infty }+M\bigr) \bigl(3M+ \Vert g_{\chi } \Vert _{\infty } \bigr) \\ &\leq \biggl(3M+\kappa \sqrt{\frac{\mathfrak{D}(\chi )}{\chi }} \biggr)^{2}:=B_{\chi }. \end{aligned}$$

Therefore

$$ \biggl\Vert h(u,t)- \int _{Z}h\,d\varrho \biggr\Vert _{\infty }\leq 2B_{\chi } $$

and

$$\begin{aligned} \zeta ^{2}(h) \leq& \int _{Z}h^{2}\,d\varrho \\ =& \int _{Z} \biggl( \int _{S}\varPsi \biggl(\frac{s}{\zeta },\frac{u}{\zeta } \biggr)\,d\varrho _{S}(s) \biggr)^{2} \bigl(g_{\chi }(u)-g_{\varrho }(u) \bigr)^{2} \\ &{} \times \bigl(g_{\chi }(u)+g_{\varrho }(u)-2y \bigr)^{2}\,d\varrho (u,t) \\ \leq& \bigl(3M+ \Vert g_{\chi } \Vert _{\infty } \bigr)^{2} \Vert g_{\chi }-g_{\varrho } \Vert _{ \varrho _{S}}^{2} \\ \leq& B_{\chi }\mathfrak{D}(\chi ). \end{aligned}$$

By Lemma 1,

$$\begin{aligned} \frac{1}{m}\sum_{i=1}^{m}h(w_{i})- \int _{Z}h\,d\varrho \leq \frac{ \mathfrak{D}(\chi )}{2}+\frac{7B_{\chi }\log (2/\delta )}{3m}. \end{aligned}$$
(25)

 □

Proof of Theorem 3

Consider the set of functions

$$\begin{aligned} \mathfrak{G}_{R}= \biggl\{ &h(u,t)= \int _{S}\varPsi \biggl(\frac{s}{\zeta },\frac{u}{ \zeta } \biggr) \bigl(\bigl(t-\gamma _{M}(g) (u)\bigr)^{2} -\bigl(t-g_{\varrho }(u)\bigr)^{2} \bigr)\,d\varrho _{S}(s):f\in B_{R} \biggr\} . \end{aligned}$$

We have

$$\begin{aligned} \bigl\vert h(u,t) \bigr\vert &\leq \int _{S}\varPsi \biggl(\frac{s}{\zeta },\frac{u}{\zeta } \biggr) \bigl\vert \bigl(\gamma _{M}(g) (u)-g_{\varrho }(u) \bigr) \times \bigl(\gamma _{M}(g) (u)+g_{\varrho }(u)-2y \bigr) \bigr\vert \,d\varrho _{S}(s) \\ &\leq 8M^{2} \end{aligned}$$

from (5), which yields

$$\begin{aligned} \bigl\vert h(u,t) \bigr\vert ^{2} &= \biggl\vert \int _{S}\varPsi \biggl(\frac{s}{\zeta },\frac{u}{ \zeta } \biggr) \bigl(\gamma _{M}(g) (u)-g_{\varrho }(u) \bigr) \times \bigl(\gamma _{M}(g) (u)+g_{\varrho }(u)-2y \bigr)\,d\varrho _{S}(s) \biggr\vert ^{2} \\ &\leq 16M^{2} \int _{S}\varPsi \biggl(\frac{s}{\zeta },\frac{u}{\zeta } \biggr) \bigl(\gamma _{M}(g) (u)-g_{\varrho }(u) \bigr)^{2}\,d\varrho _{S}(s) \int _{S}\varPsi \biggl(\frac{s}{\zeta },\frac{u}{\zeta } \biggr)\,d\varrho _{S}(s). \end{aligned}$$

So

$$\begin{aligned} \mathfrak{E}\bigl(h^{2}\bigr)\leq 16M^{2} \int _{S} \biggl( & \int _{S}\varPsi \biggl(\frac{s}{ \zeta },\frac{u}{\zeta } \biggr) \bigl(\gamma _{M}(g) (u)-g_{\varrho }(u) \bigr)^{2}\,d\varrho _{S}(u) \biggr)\,d\varrho _{S}(s). \end{aligned}$$

It has been proved in [13, 29] that

$$\begin{aligned} & \int _{S}\varPsi \biggl(\frac{s}{\zeta },\frac{u}{\zeta } \biggr) \bigl(g(u)-g _{\varrho }(u) \bigr)^{2}\,d\varrho _{S}(u) \\ &\quad= \int _{Z}\varPsi \biggl(\frac{s}{\zeta },\frac{u}{\zeta } \biggr)\bigl[\bigl(g(u)-t\bigr)^{2}-\bigl(g _{\varrho }(u)-t \bigr)^{2}\bigr]\,d\varrho (u,t), \end{aligned}$$

which implies that

$$\begin{aligned} \mathfrak{E}\bigl(h^{2}\bigr)\leq{}& 16M^{2} \int _{S} \biggl( \int _{Z}\varPsi \biggl(\frac{s}{ \zeta },\frac{u}{\zeta } \biggr)\bigl[\bigl(\gamma _{M}(g) (u)-t\bigr)^{2} \\ &{} -\bigl(g_{\varrho }(u)-t\bigr)^{2}\bigr]\,d\varrho (u,t) \biggr)\,d\varrho _{S}(s) \\ ={}&16M^{2} \int _{Z} \biggl( \int _{S}\varPsi \biggl(\frac{s}{\zeta },\frac{u}{ \zeta } \biggr)\bigl[\bigl(\gamma _{M}(g) (u)-t\bigr)^{2} \\ & {} -\bigl(g_{\varrho }(u)-t\bigr)^{2}\bigr]\,d\varrho _{S}(s) \biggr)\,d\varrho (u,t) \\ ={}&16M^{2}\mathfrak{E}(h). \end{aligned}$$

Then we get

$$\begin{aligned} & \bigl\vert h_{1}(u,t)-h_{2}(u,t) \bigr\vert \\ &\quad= \biggl\vert \int _{S}\varPsi \biggl(\frac{s}{\zeta },\frac{u}{\zeta } \biggr) \bigl(\bigl(\gamma _{M}(g_{1}) (u)-t \bigr)^{2}-\bigl(\gamma _{M}(g_{2}) (u)-t \bigr)^{2} \bigr)\,d \varrho _{S}(s) \biggr\vert \\ &\quad \leq \biggl| \int _{S}\varPsi \biggl(\frac{s}{\zeta },\frac{u}{\zeta } \biggr) \bigl(\gamma _{M}(g_{1}) (u)\bigr)-\gamma _{M}(g_{2}) (u)) \\ & \qquad{} \times \bigl(\gamma _{M}(g_{1}) (u)+\gamma _{M}(g_{2}) (u)-2t\bigr)\,d\varrho _{S}(s) \biggr| \\ &\quad \leq 4M \bigl\vert g_{1}(u)-g_{2}(u) \bigr\vert \end{aligned}$$
(26)

for any \(h_{1}\), \(h_{2}\in \mathfrak{G}_{R}\), which yields

$$\begin{aligned} \mathfrak{N}_{2}(\mathfrak{G}_{R},\varepsilon )\leq \mathfrak{N}_{2} \biggl(B_{R},\frac{\varepsilon }{4M} \biggr)= \mathfrak{N}_{2} \biggl(B_{1},\frac{ \varepsilon }{4MR} \biggr). \end{aligned}$$

It follows from the capacity condition (7) that

$$\begin{aligned} \log \mathfrak{N}_{2}(\mathfrak{G}_{R},\epsilon )\leq c_{p}(4M)^{p}R ^{p}\epsilon ^{-p}. \end{aligned}$$

By applying Lemma 2 to \(\mathscr{G}\) with \(Q=8M^{2}\) we have

$$\begin{aligned} \mathfrak{E}g-\frac{1}{m}\sum_{i=1}^{m}h(w_{i}) &\leq \frac{ \mathfrak{E}g}{2}+\frac{176M^{2}}{m}\log \biggl(\frac{2}{\delta } \biggr)+C_{p,M}R^{\frac{2p}{2+p}}m^{-\frac{2}{2+p}} \end{aligned}$$

for any \(0<\delta <1\), where

$$ C_{p,M}=c_{p}'(4M)^{\frac{4}{2+p}}c_{p}^{\frac{2}{2+p}}. $$

Moreover, we take \(f=g_{\mathbf{w},\zeta ,\varsigma ,s}\) and derive the following bound of \(g_{\mathbf{w},\zeta ,\varsigma ,s}\) by using the same method in [9, Lemma 3] and (5):

$$\begin{aligned} \Vert g_{\mathbf{w},\varsigma } \Vert _{E}\leq \kappa m^{1-\frac{1}{q}} \biggl(\frac{M ^{2}}{\varsigma } \biggr)^{\frac{1}{q}}. \end{aligned}$$

If we take

$$ R=R_{\varsigma }=\kappa m^{1-\frac{1}{q}} \biggl(\frac{M^{2}}{\varsigma } \biggr)^{\frac{1}{q}}, $$

then we can complete the proof of Theorem 3. □

5 Conclusion

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 were discussed in this study. The method was based on approximating functions and their derivatives by using the natural and weakened total energies. This study showed that the new criteria were very effective and powerful tools in solving such problems. At the end of the paper, we were also concerned with the boundary behaviors of solutions for a kind of quasilinear Schrödinger equation.