Probabilistic machine learning for breast cancer classification

1.   Introduction

Let $\Omega\subseteq\mathbb{R}^N$ be a bounded domain with a $C^2$-boundary $\partial\Omega$. In this paper, we study the following quasilinear Robin problem

where $\frac{\partial u}{\partial n} = \nabla u\cdot n$ with $n(x)$ being the outward unit normed vector to $\partial\Omega$ at its point $x$, $a(x)\in L^\infty(\Omega)$ satisfying ess $\inf\limits_{x\in\Omega}a(x) > 0$, $\beta(x)\in C^{0, \tau}(\partial\Omega, \mathbb{R}_0^+)$ for some $\tau\in(0, 1)$ and $\beta(x)\neq 0$. $\lambda > 0$ is a real parameter.

The Robin boundary problems with Laplacian operator have been widely investigated in recent years (^{[1,2,3,4,5,6]}). For example, Papageorgiou et al. in ^[4] considered the Robin problems driven by negative Laplacian with a superlinear reaction and proved the existence and multiplicity theorems by variational methods. However, the Robin boundary problems with quasilinear Schrödinger operator have not been dealt with yet. Based on the above issues, this paper aims to study the existence and multiplicity of solutions to problem (1.1). In our work, one of the main difficulties is that there is no suitable space on which the energy functional is well defined. To the best of our knowledge, the first existence result for the equation of the following form

is due to Poppenberg, Schmitt and Wang(^[7]), and their approach to the problem is the constrained minimization argument. Since then, some ideas and approaches were developed to overcome these difficulties. Liu and Wang, in ^[8], reduced the quasilinear equation to a semilinear one by the change of variable. In ^[9], using the same methods in ^[8], Colin and Jeanjean considered the problem (1.2) on the usual Sobolev space. In this paper, we will use the change of variable to overcome the main difficulties.

Let $f:\Omega\times\mathbb{R}\to\mathbb{R}$ be a Carathéodory function such that $f(x, 0) = 0$ for a.e. $x\in\Omega$ and let

We first posit the following assumptions on $f$.

$({\rm{f}}1) \; \; |f(x, t)|\le c_0(1+|t|^{r-1})$ for a.e. $x\in\Omega$, where $c_0 > 0$, $4 < r < 2\cdot2^*$;

$({\rm{f}}2) \; \; \lim\limits_{t\to+\infty}\frac{F(x, t)}{t^4} = 0$ and $\lim\limits_{t\to+\infty}F(x, t) = -\infty$ uniformly for $x\in\Omega$;

$({\rm{f}}3)$ There exists $\tilde{u}\in L^r(\Omega)$ such that $\int_{\Omega}F(x, \tilde{u})dx > 0$;

$({\rm{f}}4)$ There is a constant $\delta > 0$ such that $f(x, t)\le 0$ for a.e. $x\in\Omega$ and $t\in(0, \delta)$;

$({\rm{f}}5)$ For every $\rho > 0$, there exists $\mu_\rho > 0$ such that $t\mapsto\frac{f(x, t)}{\sqrt{1+2t^2}}+\mu_\rho t$ is nondecreasing on $[0, \rho]$;

Remark 1.1. The following function satisfies hypotheses (f1)-(f5), for simplicity we drop the $x$-dependence,

where $3 < k_1 < k_2 < 2k_1 < +\infty$ and $1 < k_3 < k_4 < 4$.

The theorem below is the first result of our paper.

Theorem 1.1. If hypotheses (f1)-(f5) hold, then there exists a critical parameter value $\lambda_* > 0$ such that for all $\lambda > \lambda_*$, problem (1.1) has at least two positive solutions $v_0, v_1\in {\rm int}(C_+)$ with $v_0\le v_1$ in $\Omega$.

To study further the multiplicity of solutions for problem (1.1) under the assumption (f1), we give some other conditions on $f$ as follows:

$({\rm{f}}6) \; \; \lim\limits_{|t|\to+\infty}\frac{F(x, t)}{t^4} = +\infty$ uniformly with respect to $x\in\Omega$, and there exist $\alpha\in(\max\{1, (r-4)\frac{N}{4}\}, 2^*)$, $\hat{c}_0 > 0$ such that

uniformly for $x\in\Omega$.

$({\rm{f}}7) \; \; f(x, -t) = -f(x, t)$.

Remark 1.2. The following function satisfies hypotheses (f1), (f6) and (f7):

Theorem 1.2. If hypotheses (f1), (f6), (f7) hold and $\lambda > 1$, then problem (1.1) has infinitely many high energy solutions in $H^1(\Omega)$.

The paper is organized as follows. In Section 2, we give some preliminary results, which will be used in this paper. We prove Theorem 1.1 and Theorem 1.2 in Section 3 and Section 4 respectively.

2.   Preliminaries and variational setting

In this paper, the main working spaces are $H^1(\Omega)$, $C^1(\bar{\Omega})$ and $L^q(\partial\Omega)$, $1\le q\le+\infty$. We denote the norm of $L^q(\Omega)$ and $H^1(\Omega)$ by

The Banach space $C^1(\bar{\Omega})$ is an ordered Banach space with positive (order) cone

This cone has a nonempty interior given by

On $\partial\Omega$ we will employ the $(N-1)$-dimensional Hausdorff measure $\sigma$. By applying this measure we can define the Lebesgue spaces $L^q(\partial\Omega)$ ($1\le q\le+\infty$). The Trace Theorem says that there exists a unique continuous linear map $\gamma_0:H^1(\Omega)\to L^2(\partial\Omega)$, known as "trace map", such that

Consequently, the trace map extends the notion of boundary values to any Sobolev function, and

Moreover, the trace map $\gamma_0$ is compact from $H^1(\Omega)$ into $L^q(\partial\Omega)$ with $q\in [1, \frac{2(N-1)}{N-2})$ if $N\ge3$ and $q\in [1, +\infty)$ if $N = 1, \ 2$. In the sequel, for the sake of notational simplicity, the use of the trace map $\gamma_0$ will be dropped. The restrictions of all Sobolev functions on the boundary $\partial\Omega$ are understood in the sense of traces.

We know that (1.1) is the Euler-Lagrange equation associated to the energy functional

Unfortunately, the functional $I$ could not be well defined in $H^1(\Omega)$ for $N\ge3$. To overcome this difficulty, we use the argument developed in ^[9]. More precisely, we make the change of variable $u = g(v)$, which is defined by

Now we present some important results about the change of variable $u = g(v)$.

Lemma 2.1. (^[10]) The function $g$ and its derivative satisfy the following properties:

(ⅰ) $g$ is uniquely defined, $C^2$ and invertible;

(ⅱ) $|g'(t)|\le1$ for all $t\in\mathbb{R}$;

(ⅲ) $|g(t)|\le t$ for all $t\in\mathbb{R}$;

(ⅳ) $\frac{g(t)}{t}\to1$, as $t\to0$;

(ⅴ) $|g(t)|\le 2^{1/4}|t|^{1/2}$ for all $t\in\mathbb{R}$;

(ⅵ) $g(t)/2 < tg'(t) < g(t)$ for all $t > 0$, and the reverse inequalities hold for $t < 0$;

(ⅶ) $\frac{g(t)}{\sqrt{t}}\to a_1 > 0$, as $t\to+\infty$;

(ⅷ) $|g(t)g'(t)|\le1/\sqrt{2}$ for all $t\in\mathbb{R}$;

(ⅸ) There exists a positive constant $C_1$ such that

Therefore, after the change of variable, the functional $I(u)$ can be rewritten in the following way

From Lemma 2.1, by a standard argument, we see that $J$ is well defined in $H^1(\Omega)$ and $J\in C^1$. In addition,

for all $v, \varphi \in H^1(\Omega)$.

It is easy to see that the critical points of $J$ correspond exactly to the weak solutions of the semilinear problem

Hence, to prove our main results, we shall look for solutions of problem (2.3), i.e., the critical points of the functional $J$.

Let $X$ be a Banach space and $X^*$ be its topological dual. By $\langle\cdot, \cdot\rangle$ we denote the duality brackets for the pair $(X, X^*)$. We now give the definitions of the $\rm(PS)_c$ condition and Cerami condition in $X$ as follows.

Definition 2.2. Let $\psi\in C^1(X, \mathbb{R})$ and $c\in\mathbb{R}$. We say that $\psi$ satisfies the $(PS)_c$ condition, if every sequence $\{u_n\}\subseteq X$ such that

admits a strongly convergent subsequence.

Definition 2.3. Let $\psi\in C^1(X, \mathbb{R})$ and $c\in\mathbb{R}$. We say that $\psi$ satisfies the Cerami condition, if every sequence $\{u_n\}\subseteq X$ such that

admits a strongly convergent subsequence.

The following two lemmas are known as Mountain Pass Theorem and Fountain Theorem.

Lemma 2.4. (Mountain Pass Theorem, ^[11]) If $\psi\in C^1(X, \mathbb{R})$ satisfies the $(PS)_c$-condition, there are $u_0, \ u_1\in X\ with\ \|u_1-u_0\|_X > \varrho > 0$,

and $c = \inf\limits_{\gamma\in\Gamma}\max\limits_{0\le \tau\le1}\psi(\Gamma(\tau))$ where $\Gamma = \{\gamma\in C([0, 1], \ X):\gamma(0) = u_0, \ \gamma(1) = u_1\}$, then $c\ge m_\varrho$ and $c$ is a critical value of $\psi$.

If $X$ is a reflexive and separable Banach space, then there are $e_j\in X$ and $e_j^*\in X^*$ such that

For convenience, we write

Lemma 2.5. (Fountain Theorem, ^[12]) Let $\psi\in C^1(X, \mathbb{R})$ be an even functional. If, for every $k\in\mathbb{N}$, there exists $\rho_k > \gamma_k > 0$ such that

$\rm{(A1)} \; \; a_k: = \sup\limits_{u\in Y_k, \|u\|_X = \rho_k}\psi(u)\le0$,

$\rm{(A2)} \; \; b_k: = \inf\limits_{u\in Z_k, \|u\|_X = \gamma_k}\psi(u)\to\infty$ as $k\to\infty$,

$\rm{(A3)} \; \; \psi$ satisfies the Cerami condition for every $c > 0$.

Then $\psi$ has an unbounded sequence of critical values.

3.   Existence of positive solutions

In this section we prove the existence of positive solutions of problem (1.1). For simplicity, we may take $f(x, t) = 0$ for a.e. $x\in\Omega$, all $t\le0$.

Proposition 1. If (f1)–(f3) and (f5) hold, then problem (2.3) admits a solution $v_0\in\mathit{{\rm int}}(C_+)$.

Proof. We consider the $C^1$-functional $J_+:H^1(\Omega)\to\mathbb{R}$ defined by

We claim that $J_+$ is coercive. Arguing by contradiction, assume that we can find a sequence $\{v_n\}_{n\ge1}\subseteq H^1(\Omega)$ and a constant $M$ such that

Then from (3.2) we have

for all $n\ge1$. Hypotheses (f1) and (ⅴ) of Lemma 2.1 imply that there exist $c_2, \ c_3 > 0$ such that

Combining this with (3.2), (3.3), we get

Let $w_n = \frac{v_n^+}{\|v_n^+\|}$, $n\ge1$. Then $\|w_n\| = 1$ for all $n\ge1$, and so we may assume that

From (3.3) we have

Invoking (f1), (f2) and (ⅴ) of Lemma 2.1, we can find $c_{\epsilon} > 0$ such that

which yields, for $n\ge1$ large enough,

Passing to the limsup as $n\to\infty$ in (3.5), and using (3.6), we have

In addition, from (3.4) we have $\|\nabla w\|_2^2\le\liminf\limits_{n\to+\infty}\|\nabla w_n\|_2^2$. Combining this with (3.7), we obtain that $\nabla w_n\to\nabla w = 0$ in $L^2(\Omega)$. Hence,

and so $\|w\| = 1, \ w\ge0$. Since $\nabla w = 0$, we have $w = 1/\sqrt{|\Omega|_N}$, and $v_n^+(x)\to+\infty$ for a.e. $x\in\Omega$.

On the other hand, combining with (f1), (f2), (3.3), (ⅶ) of Lemma 2.1 and Fatou's lemma, we get

which is impossible. Thus, $J_+$ is coercive.

Furthermore, from Lemma 2.1, Sobolev embedding theorem and trace theorem, we see that $J_+$ is sequentially weak lower semicontinuous. By Weierstrass-Tonelli theorem we can find $v_0\in H^1(\Omega)$ such that

Consider the integral functional $J_F:L^r(\Omega)\to\mathbb{R}$ defined by

From hypothesis (f1), (ⅰ) of Lemma 2.1 and the dominated convergence theorem, we see that $J_F$ is continuous. By hypothesis (f3) and (ⅸ) of Lemma 2.1, there exists $\tilde{v}\in L^r(\Omega)$ such that

Exploiting the density of $H^1(\Omega)$ in $L^r(\Omega)$, we can find $\bar{v}\in H^1(\Omega)$ such that

Therefore, from (3.1) and (3.9), there exists $\lambda_* > 0$ such that

for all $\lambda > \lambda_*$. This ensure that $v_0\neq0$. From (3.8), we have $J_+'(v_0) = 0$, that is,

for all $\varphi\in H^1(\Omega)$. In (3.10) choosing $\varphi = -v_0^-$, we obtain

which implies $v_0\ge0$ a.e. in $\Omega$. Hence,

for all $\varphi\in H^1(\Omega)$. That is to say $v_0$ is a weak solution of problem (2.3). By Theorem 4.1 in ^[13], we know that $v_0\in L^\infty(\Omega)$. Furthermore, applying Theorem 2 in ^[14], we have $v_0\in C_+\backslash\{0\}$. Taking $\rho = \|g(v_0)\|_\infty$, from (f5), there exists $\mu_\rho > 0$ such that

From (2.3), (3.12) and (ⅲ) of Lemma 2.1 we obtain

Therefore, $v_0\in {\rm int}(C_+)$ by the strong maximum principle(see ^[15]).

To look for another positive solution to problem (2.3), we consider the functional $\hat{J}_+\in C^1(H^1(\Omega), \mathbb{R})$ defined by

where $v_0$ is given in Proposition 1, $\hat{F}(x, t) = \int_{0}^{t}\hat{f}(x, s)ds$, $\hat{K}(x, t) = \int_{0}^{t}\hat{k}(x, s)ds$, and

for all $(x, s)\in\Omega\times\mathbb{R}$.

for all $(x, s)\in\partial\Omega\times\mathbb{R}$. It is easy to see that $\hat{f}(x, s)$ and $\hat{k}(x, s)$ are Carathéodory functions.

Proposition 2. If (f1), (f5) hold and $v$ is a critical point of $\hat{J}_+(v)$, then $v$ is a solution of problem (2.3), and $v\in[0, v_0]\cap int(C_+)$.

Proof. The assumption yields the following equality:

for all $\varphi\in H^1(\Omega)$. Taking $\varphi = -v^-$ in (3.16), we obtain (arguing as in Proposition 1) $v\ge0$ a.e. in $\Omega$.

On the one hand, letting $\varphi = (v-v_0)^+$ in (3.16) again, we have

On the other hand, the fact that $v_0 > 0$ is a critical point of ${J}_+$ yields

Then from (3.17) and (3.18), we obtain

and $(v-v_0)^+ = 0$. Therefore $v\in[0, v_0]$ a.e. in $\Omega$. Now, (3.16) becomes

for all $\varphi\in H^1(\Omega)$. So $v$ is a weak solution of problem (2.3). Reasoning as in the proof of Proposition 1, we have $v\in {\rm int}(C_+)$.

Lemma 3.1. If hypotheses (f1)–(f5) hold, then $\hat{J}_+$ satisfies the $(PS)_c$ condition.

Proof. We first check that $\hat{J}_+$ is coercive. Since $v_0\in C^1\bar{(\Omega)}$, using (f1), (ⅲ) of Lemma 2.1 and (3.14), we can find a constant $c_4 > 0$ such that

Because of (3.15) we have

Moreover, for every $v\in H^1(\Omega)$ we see that

From (3.13), (3.19), (3.20) and (3.21) we derive

It is easy to see that $\hat{J}_+$ is coercive.

Now let $\{v_n\}_{n\ge1}$ be a sequence such that

The coercivity of $\hat{J}_+$ and (3.22) imply that $\{v_n\}_{n\ge1}$ is bounded in $H^1(\Omega)$. Hence, there is $v\in H^1(\Omega)$ such that, along a relabeled subsequence $\{v_n\}_{n\ge1}$,

The second part of (3.22) yields

${\rm as}\ n\to\infty, \ {\rm for\ all}\ \varphi\in H^1(\Omega).$ In (3.24) choosing $\varphi = v_n-v$, using (f1), (3.23) and Lemma 2.1, we find

Therefore $\|v_n\|^2\to\|v\|^2$ as $n\to\infty$. Recalling that the Hilbert space $H^1(\Omega)$ is locally uniformly convex, we obtain $v_n\to v$ in $H^1(\Omega)$. This shows that $\hat{J}_+$ satisfies the $(PS)_c$ condition.

In the next lemma, we state a consequence which depends on the regularity results in ^[13] and Theorem 2 in ^[14]. The proof is similar to Proposition 3 in ^[2].

Lemma 3.2. If (f1) holds and $v\in H^1(\Omega)$ is a local $C^1(\bar{\Omega})$-minimizer of $\hat{J}_+$, then $v\in C^{1, \tau}(\bar{\Omega})$ and it is a local $H^1(\Omega)$-minimizer of $\hat{J}_+$.

Proposition 3. If (f1)-(f5) hold, then $\hat{J}_+$ admits a critical point $v_1\in H^1(\Omega)$ different from $0$ and $v_0$.

Proof. Let $v\in C^1(\bar{\Omega})$ with $0 < \|v\|_{C^1(\bar{\Omega})}\le\delta$. From (f4) and (ⅳ) of Lemma 2.1, we get $\hat{F}(x, g(v))\le0$ for a.e. $x\in\Omega$ which implies

This shows that 0 is a local $C^1(\bar{\Omega})$-minimizer of $\hat{J}_+$. Applying Lemma 3.2, we derive that 0 is a local $H^1(\Omega)$-minimizer of $\hat{J}_+$.

If 0 is not a strict local minimizer of $\hat{J}_+$, the result is obvious because any neighborhood of 0 in $H^1(\Omega)$ contains another critical point of $\hat{J}_+$.

We next only need to discuss that 0 is a strict local minimizer of $\hat{J}_+$. In such a condition, we can find a sufficiently small $\varrho\in(0, 1)$ such that

Note that

This fact, together with (3.25) and Lemma 3.1 permit the use of the Mountain Pass Theorem, which yields a critical point $v_1$ of $\hat{J}_+$ different from 0 and $v_0$.

Theorem 1.1 follows immediately from Proposition 1, Proposition 2 and Proposition 3.

4.   Existence of infinitely many solutions

Lemma 4.1. If (f1) and (f6) hold, then, for any $c > 0$, the functional $J$ satisfies the Cerami condition.

Proof. Let $\{v_n\}_{n\ge1}\subseteq H^1(\Omega)$ be a Cerami sequence of $J$, that is,

First we prove the boundedness of $\{v_n\}_{n\ge1}$. By (4.1) we have

and

for all $\varphi \in H^1(\Omega)$ with $\epsilon_n\ll 1$. Choosing $\varphi = \varphi_n = -\frac{g(v_n)}{g'(v_n)}$, we obtain

Because of Lemma 2.1(ⅵ, ⅷ), $\varphi_n\in H^1(\Omega)$. Therefore,

Now using (f6) and (ⅷ) of Lemma 2.1, it follows from (4.2) and (4.4) that

On the other hand, from (f1) and (f6) we can find $\tilde{c}_0\in(0, \hat{c}_0)$ and $c_5 > 0$ such that

By (4.5) and (4.6), we conclude that $\{g(v_n)\}_{n\ge1}$ is bounded in $L^{2\alpha}(\Omega)$. Furthermore, due to (ix) of Lemma 2.1,

Suppose that $N\neq2$. By (f6), without loss of generality, we may assume that $\alpha < \frac{r}{2} < 2^*$. So, we can find $z\in(0, 1)$ such that

Invoking the interpolation inequality, Sobolev embedding theorem and (4.7) we have

for some $c_6 > 0$. Taking $\varphi = -\varphi_n = \frac{g(v_n)}{g'(v_n)}$ in (4.3), we have

By use of (f1) and (ⅴ) of Lemma 2.1, we can obtain that

From (f1), we assume that $r$ is close to ${2\cdot2^*}$, hence $\alpha\ge2$. Then $\{v_n\}_{n\ge1}$ is bounded in $L^2(\Omega)$. Combining this with (4.9), (4.10) and (4.11), we have

Hypothesis (f6) and (4.8) lead to $zr < 4$, and consequently, $\{v_n\}_{n\ge1}$ is bounded in $H^1(\Omega)$.

If $N = 2$, then $2^* = +\infty$ and the Sobolev embedding theorem says that $H^1(\Omega)\hookrightarrow L^\eta(\Omega)$ for all $\eta\in\lbrack1, +\infty)$. Let $\eta > \frac{r}{2} > \alpha$ and $z\in(0, 1)$ such that

or $zr = \frac{\eta(r-2\alpha)}{\eta-\alpha}$. Note that

Repeating the previous proof method, for $\eta > 1$ large enough, we again obtain that

Next, we show that $\{v_n\}_{n\ge1}$ is strongly convergent in $H^1(\Omega)$. Since $\{v_n\}_{n\ge1}$ is bounded, up to a subsequence(which we still denote by $\{v_n\}_{n\ge1}$), we assume that there exists $v\in H^1(\Omega)$ such that

Choosing $\varphi = v_n-v\in H^1(\Omega)$ in (4.3) and combining (f1) with Lemma 2.1(ⅱ, ⅲ, ⅸ), we obtain $\lim\limits_{n\to\infty}\int_{\Omega}\nabla v_n\nabla(v_n-v)dx = 0$. Hence $\|v_n\|^2\to\|v\|^2$ as $n\to\infty$. Recalling that the Hilbert space $H^1(\Omega)$ is locally uniformly convex, we get $v_n\to v$ in $H^1(\Omega)$.

Lemma 4.2. (^[16]) Let $X = H^1(\Omega)$ and define $Y_k, \ Z_k$ as in (2.4). If $1\le q < 2^*$, then

Proof of Theorem 1.2. Since $g(t)$ and $f(x, t)$ are odd respect to $t$, functional $J\in C^1(H^1(\Omega), \mathbb{R})$ is even obviously. Further, from Lemma 4.1, we know that $J$ satisfies the Cerami condition. So we need only to verify $J$ satisfying the conditions (A1) and (A2) in Lemma 2.5.

Since $Y_k$ is finite-dimensional and all norms are equivalent on $Y_k$, for $v\in Y_k$, we have

From the assumption (f6) and (ⅸ) of Lemma 2.1, there exists $R > 0$ such that

For $v\in Y_k$, $\lambda > 1$, we have

So (A1) is satisfied for sufficiently large $\|v\|$.

From (f1) and (ⅴ) of Lemma 2.1, there exists constant $C_4 > 0$ such that

Let us define

For $v\in Z_k$, using (4.15), we get

Choosing ${\gamma_k = (\frac{1}{2}C_5 r d_k^{\frac{r}{2}})^{\frac{2}{4-r}}}$, if $\|v\| = \gamma_k$, we obtain

By Lemma 4.2, $d_k\to0\ {\rm and}\ \gamma_k\to\infty\ {\rm as}\ k\to\infty$, the condition (A2) is verified. In summary, all conditions of Fountain Theorem are satisfied. Thus $J$ has an unbounded sequence of critical values. Theorem 1.2 is proved.

5.   Conclusions

In this paper, we have studied a class of quasilinear Schrödinger equation in a bounded domain with Robin boundary. By giving different conditions on the reaction, we obtained two existence results of solutions to the equation. The Mountain Pass Theorem and Fountain Theorem were also employed in this study.

Acknowledgments

The authors wish to thank the referees and the editor for their valuable comments and suggestions. This work was supported by the National Natural Science Foundation of China (11171220).

Conflict of interest

All authors declare no conflicts of interest in this paper.