A semi-supervised deep neuro-fuzzy iterative learning system for automatic segmentation of hippocampus brain MRI

1.   Introduction

Consider with the following fourth-order elliptic Navier boundary problem

where $\Delta^2: = \Delta(\Delta)$ denotes the biharmonic operator, $\Omega\subset \mathbb{R}^N (N\geq 4)$ is a smooth bounded domain, $c < \lambda_1$ $(\lambda_1$ is the first eigenvalue of $-\Delta$ in $H_0^1(\Omega))$ is a constant, $1 < s < 2, \lambda\geq 0$ is a parameter, $a\in L^{\infty}(\Omega), a(x)\geq 0, a(x)\not \equiv 0,$ and $f\in C(\bar{\Omega}\times \mathbb{R}, \mathbb{R})$. It is well known that some of these fourth order elliptic problems appear in different areas of applied mathematics and physics. In the pioneer paper Lazer and Mckenna ^[13], they modeled nonlinear oscillations for suspensions bridges. It is worth mentioning that problem (1.1) can describe static deflection of an elastic plate in a fluid, see ^[21,22]. The static form change of beam or the motion of rigid body can be described by the same problem. Equations of this type have received more and more attentions in recent years. For the case $\lambda = 0,$ we refer the reader to ^{[3,7,11,14,16,17,20,23,27,29,34,35,36,37]} and the reference therein. In these papers, existence and multiplicity of solutions have been concerned under some assumptions on the nonlinearity $f$. Most of them considered the case $f(x, u) = b[(u+1)^+-1]$ or $f$ having asymptotically linear growth at infinity or $f$ satisfying the famous Ambrosetti-Rabinowitz condition at infinity. Particularly, in the case $\lambda\neq 0$, that is, the combined nonlinearities for the fourth-order elliptic equations, Wei ^[33] obtained some existence and multiplicity by using the variational method. However, the author only considered the case that the nonlinearity $f$ is asymptotically linear. When $\lambda = 1,$ Pu et al. ^[26] did some similar work. There are some latest works for problem (1.1), for example ^[10,18] and the reference therein. In this paper, we study problem (1.1) from two aspects. One is that we will obtain two multiplicity results when the nonlinearity $f$ is superlinear at infinity and has the standard subcritical polynomial growth but not satisfy the Ambrosetti-Rabinowitz condition, the other is we can establish some existence results of multiple solutions when the nonlinearity $f$ has the exponential growth but still not satisfy the Ambrosetti-Rabinowitz condition. In the first case, the standard methods for the verification of the compactness condition will fail, we will overcome it by using the functional analysis methods, i.e., Hahn-Banach Theorem combined the Resonance Theorem. In the last case, although the original version of the mountain pass theorem of Ambrosetti-Rabinowitz ^[1] is not directly applied for our purpose. Therefore, we will use a suitable version of mountain pass theorem and some new techniques to finish our goal.

When $N > 4$, there have been substantial lots of works (such as ^{[3,7,11,16,17,26,34,35,36,37]}) to study the existence of nontrivial solutions or the existence of sign-changing for problem (1.1). Furthermore, almost all of the works involve the nonlinear term $f(x, u)$ of a standard subcritical polynomial growth, say:

(SCP): There exist positive constants $c_1$ and $q\in (1, p^*-1)$ such that

where $p^* = \frac{2N}{N-4}$ expresses the critical Sobolev exponent. In this case, people can deal with problem (1.1) variationally in the Sobolev space $H^2(\Omega)\cap H_0^1(\Omega)$ owing to the some critical point theory, such as, the method of invariant sets of descent flow, mountain pass theorem and symmetric mountain pass theorem. It is worth while to note that since Ambrosetti and Rabinowitz presented the mountain pass theorem in their pioneer paper ^[1], critical point theory has become one of the main tools on looking for solutions to elliptic equation with variational structure. One of the important condition used in many works is the so-called Ambrosetti-Rabinowitz condition:

(AR) There exist $\theta > 2$ and $R > 0$ such that

where $F(x, t) = \int_0^t f(x, s)ds$. A simple computation explains that there exist $c_2, \; c_3 > 0$ such that $F(x, t)\geq c_2|t|^{\theta}-c_3$ for all $(x, t)\in \bar{\Omega}\times \mathbb{R}$ and $f$ is superlinear at infinity, i.e., $\lim\limits_{t\rightarrow \infty}\frac{f(x, t)}{t} = +\infty$ uniformly in $x\in \Omega$. Thus problem (1.1) is called strict superquadratic if the nonlinearity $f$ satisfies the (AR) condition. Notice that (AR) condition plays an important role in ensuring the boundedness of Palais-Smale sequences. However, there are many nonlinearities which are superlinear at infinity but do not satisfy above (AR) condition such as $f(x, t) = tln(1+|t|^2)+|sint|t$.

In the recent years many authors tried to study problem (1.1) with $\lambda = 0$ and the standard Laplacian problem where (AR) is not assumed. Instead, they regard the general superquadratic condition:

(WSQC) The following limit holds

with additional assumptions (see ^{[3,5,7,11,12,15,17,19,24,26,31,37]} and the references therein). In the most of them, there are some kind of monotonicity restrictions on the functions $F(x, t)$ or $\frac{f(x, t)}{t},$ or some convex property for the function $tf(x, t)-2F(x, t)$.

In the case $N = 4$ and $c = 0$, motivated by the Adams inequality, there are a few works devoted to study the existence of nontrivial solutions for problem (1.1) when the nonlinearity $f$ has the exponential growth, for example ^[15] and the references therein.

Now, we begin to state our main results: Let $\mu_1$ be the first eigenvalue of $(\Delta^2-c\Delta, H^2(\Omega)\cap H_0^1(\Omega))$ and suppose that $f(x, t)$ satisfies:

$(H_1)$ $f(x, t)t\geq 0, \; \forall (x, t)\in \Omega \times \mathbb{R}$;

$(H_2)$ $\lim\limits_{t\rightarrow0}\frac{f(x, t)}{t} = f_0$ uniformly for a.e. $x\in \Omega,$ where $f_0\in [0, +\infty);$

$(H_3)$ $\lim\limits_{t\rightarrow \infty}\frac{F(x, t)}{t^2} = +\infty$ uniformly for a.e. $x\in \Omega$, where $F(x, t) = \int_0^t f(x, s)ds.$

In the case of $N > 4$, our results are stated as follows:

Theorem 1.1. Assume that $f$ has the standard subcritical polynomial growth on $\Omega$ (condition $\mathrm{(SCP)}$) and satisfies $(H_1)$–$(H_3)$. If $f_0 < \mu_1$ and $a(x)\geq a_0$ $(a_0$ is a positive constant $)$, then there exists $\Lambda^* > 0$ such that for $\lambda\in (0, \Lambda^*),$ problem (1.1) has five solutions, two positive solutions, two negative solutions and one nontrivial solution.

Theorem 1.2. Assume that $f$ has the standard subcritical polynomial growth on $\Omega$ (condition $\mathrm{(SCP)}$) and satisfies $(H_2)$ and $(H_3)$. If $f(x, t)$ is odd in $t$.

$a)$ For every $\lambda\in R$, problem (1.1) has a sequence of solutions $\{u_k\}$ such that $I_\lambda(u_k)\rightarrow \infty, k\rightarrow \infty,$ definition of the functional $I_\lambda$ will be seen in Section 2.

$b)$ If $f_0 < \mu_1$, for every $\lambda > 0$, problem (1.1) has a sequence of solutions $\{u_k\}$ such that $I_\lambda(u_k) < 0$ and $I_\lambda(u_k)\rightarrow 0, k\rightarrow \infty.$

Remark. Since our the nonlinear term $f(x, u)$ satisfies more weak condition $(H_3)$ compared with the classical condition (AR), our Theorem 1.2 completely contains Theorem 3.20 in ^[32].

In case of $N = 4$, we have $p^* = +\infty$. So it's necessary to introduce the definition of the subcritical exponential growth and critical exponential growth in this case. By the improved Adams inequality (see ^[28] and Lemma 2.2 in Section 2) for the fourth-order derivative, namely,

So, we now define the subcritical exponential growth and critical exponential growth in this case as follows:

(SCE): $f$ satisfies subcritical exponential growth on $\Omega$, i.e., $\lim\limits_{t\rightarrow \infty}\frac{|f(x, t)|}{\exp(\alpha t^2)} = 0$ uniformly on $x\in \Omega$ for all $\alpha > 0$.

(CG): $f$ satisfies critical exponential growth on $\Omega$, i.e., there exists $\alpha_0 > 0$ such that

and

When $N = 4$ and $f$ satisfies the subcritical exponential growth (SCE), our work is still to consider problem (1.1) where the nonlinearity $f$ satisfies the (WSQC)-condition at infinity. As far as we know, this case is rarely studied by other people for problem (1.1) except for ^[24]. Hence, our results are new and our methods are technique since we successfully proved the compactness condition by using the Resonance Theorem combined Adams inequality and the truncated technique. In fact, the new idea derives from our work ^[25]. Our results are as follows:

Theorem 1.3. Assume that $f$ satisfies the subcritical exponential growth on $\Omega$ (condition $\mathrm{(SCE)}$) and satisfies $(H_1)$–$(H_3)$. If $f_0 < \mu_1$ and $a(x)\geq a_0$ $(a_0$ is a positive constant $)$, then there exists $\Lambda^* > 0$ such that for $\lambda\in (0, \Lambda^*),$ problem (1.1) has five solutions, two positive solutions, two negative solutions and one nontrivial solution.

Remark. Let $F(x, t) = t^2e^{\sqrt{|t|}}, \forall (x, t) \in \Omega\times \mathbb{R}.$ Then it satisfies that our conditions $(H_1)$–$(H_3)$ but not satisfy the condition $\mathrm{(AR)}$. It's worth noting that we do not impose any monotonicity condition on $\frac{f(x, t)}{t}$ or some convex property on $tf(x, t)-2F(x, t)$. Hence, our Theorem 1.3 completely extends some results contained in ^[15,24] when $\lambda = 0$ in problem (1.1).

Theorem 1.4. Assume that $f$ satisfies the subcritical exponential growth on $\Omega$ (condition $\mathrm{(SCE)}$) and satisfies $(H_2)$ and $(H_3)$. If $f_0 < \mu_1$ and $f(x, t)$ is odd in $t$.

$a)$ For $\lambda > 0$ small enough, problem (1.1) has a sequence of solutions $\{u_k\}$ such that $I_\lambda(u_k)\rightarrow \infty, k\rightarrow \infty.$

$b)$ For every $\lambda > 0$, problem (1.1) has a sequence of solutions $\{u_k\}$ such that $I_\lambda(u_k) < 0$ and $I_\lambda(u_k)\rightarrow 0, k\rightarrow \infty.$

When $N = 4$ and $f$ satisfies the critical exponential growth $\mathrm{(CG)}$, the study of problem (1.1) becomes more complicated than in the case of subcritical exponential growth. Similar to the case of the critical polynomial growth in $\mathbb{R}^N\; (N\geq 3)$ for the standard Laplacian studied by Brezis and Nirenberg in their pioneering work ^[4], our Euler-Lagrange functional does not satisfy the Palais-Smale condition at all level anymore. For the class standard Laplacian problem, the authors ^[8] used the extremal function sequences related to Moser-Trudinger inequality to complete the verification of compactness of Euler-Lagrange functional at some suitable level. Here, we still adapt the method of choosing the testing functions to study problem (1.1) without (AR) condition. Our result is as follows:

Theorem 1.5. Assume that $f$ has the critical exponential growth on $\Omega$ (condition $\mathrm{(CG)}$) and satisfies $(H_1)$–$(H_3)$. Furthermore, assume that

$(H_4)$ $\lim\limits_{t\rightarrow \infty}f(x, t) \exp(-\alpha_0t^2)t\geq \beta > \frac{64}{\alpha_0r_0^4}$, uniformly in $(x, t),$ where $r_0$ is the inner radius of $\Omega$, i.e., $r_0: =$ radius of the largest open ball $\subset \Omega.$ and

$(H_5)$ $f$ is in the class $(H_0)$, i.e., for any $\{u_n\}$ in $H^2(\Omega)\cap H_0^1(\Omega)$, if $u_n\rightharpoonup 0$ in $H^2(\Omega)\cap H_0^1(\Omega)$ and $f(x, u_n)\rightarrow 0$ in $L^1(\Omega),$ then $F(x, u_n)\rightarrow 0$ in $L^1(\Omega)$ (up to a subsequence).

If $f_0 < \mu_1,$ then there exists $\Lambda^* > 0$ such that for $\lambda\in (0, \Lambda^*),$ problem (1.1) has at least four nontrivial solutions.

Remark. For standard biharmonic problems with Dirichlet boundary condition, Lam and Lu ^[15] have recently established the existence of nontrivial nonnegative solutions when the nonlinearity $f$ has the critical exponential growth of order $\exp(\alpha u^2)$ but without satisfying the Ambrosetti- Rabinowitz condition. However, for problem (1.1) with Navier boundary condition involving critical exponential growth and the concave term, there are few works to study it. Hence our result is new and interesting.

The paper is organized as follows. In Section 2, we present some necessary preliminary knowledge and some important lemmas. In Section 3, we give the proofs for our main results. In Section 4, we give a conclusion.

2.   Preliminaries and some lemmas

We let $\lambda_k$ $(k = 1, 2, \cdot\cdot\cdot)$ denote the eigenvalue of $-\Delta$ in $H_0^1(\Omega)$, then $0 < \mu_1 < \mu_2 < \cdot\cdot\cdot < \mu_k < \cdot\cdot\cdot$ be the eigenvalues of $(\Delta^2-c\Delta, H^2(\Omega)\cap H_0^1(\Omega))$ and $\varphi_k(x)$ be the eigenfunction corresponding to $\mu_k$. Let $X_k$ denote the eigenspace associated to $\mu_k.$ In fact, $\mu_k = \lambda_k(\lambda_k-c).$ Throughout this paper, we denote by $\|\cdot\|_p$ the $L^p(\Omega)$ norm, $c < \lambda_1$ in $\Delta^2-c\Delta$ and the norm of $u$ in $H^2(\Omega)\cap H_0^1(\Omega)$ will be defined by the

We also denote $E = H^2(\Omega)\cap H_0^1(\Omega)$.

Definition 2.1. Let ($\mathbb{E}, ||\cdot||_\mathbb{E})$ be a real Banach space with its dual space $(\mathbb{E}^*, ||\cdot||_{\mathbb{E}^*})$ and $I\in C^1(\mathbb{E}, \mathbb{R})$. For $c^*\in \mathbb{ R},$ we say that $I$ satisfies the $\mathrm{(PS)_{c^*}}$ condition if for any sequence $\{x_n\}\subset \mathbb{E}$ with

there is a subsequence $\{x_{n_k}\}$ such that $\{x_{n_k}\}$ converges strongly in $\mathbb{E}$. Also, we say that $I$ satisfy the $\mathrm{(C)_{c^*}}$ condition if for any sequence $\{x_n\}\subset \mathbb{E}$ with

there exists subsequence $\{ x_{n_k}\}$ such that $\{ x_{n_k}\}$ converges strongly in $\mathbb{E}.$

Definition 2.2. We say that $u\in E$ is the solution of problem (1.1) if the identity

holds for any $\varphi\in E.$

It is obvious that the solutions of problem (1.1) are corresponding with the critical points of the following $C^1$ functional:

Let $u^+ = \max\{u, 0\}, u^- = \min\{u, 0\}.$

Now, we concern the following problem

where

Define the corresponding functional $I_\lambda^+:E\rightarrow \mathbb{R}$ as follows:

where $F^+(x, u) = \int_0^u f^+(x, s)ds.$ Obviously, the condition $\mathrm{(SCP)}$ or $\mathrm{(SCE)}$ ($\mathrm{(CG)}$) ensures that $I_\lambda^+\in C^1(E, \mathbb{R}).$ Let $u$ be a critical point of $I_\lambda^+,$ which means that $u$ is a weak solution of problem (2.1). Furthermore, since the weak maximum principle (see ^[34]), it implies that $u\geq 0$ in $\Omega$. Thus $u$ is also a solution of problem (1.1) and $I_\lambda^+(u) = I_\lambda(u)$.

Similarly, we define

and

where $F^-(x, u) = \int_0^u f^-(x, s)ds.$ Similarly, we also have $I_\lambda^-\in C^1(E, \mathbb{R})$ and if $v$ is a critical point of $I_\lambda^-$ then it is a solution of problem (1.1) and $I_\lambda^-(v) = I_\lambda(v)$.

Prosition 2.1. (^[6,30]). Let $\mathbb{E}$ be a real Banach space and suppose that $I \in C^1(\mathbb{E}, \mathbb{R})$ satisfies the condition

for some $\alpha < \beta,$ $\rho > 0$ and $u_1\in \mathbb{E}$ with $||u_1|| > \rho.$ Let $c^*\geq\beta$ be characterized by

where $\Gamma = \{\gamma\in C([0, 1], \mathbb{E}), \gamma(0) = 0, \gamma(1) = u_1\}$ is the set of continuous paths joining $0$ and $u_1.$ Then, there exists a sequence $\{u_n\}\subset \mathbb{E}$ such that

Lemma 2.1. (^[28]). Let $\Omega\subset \mathbb{R}^4$ be a bounded domain. Then there exists a constant $C > 0$ such that

and this inequality is sharp.

Next, we introduce the following a revision of Adams inequality:

Lemma 2.2. Let $\Omega\subset \mathbb{R}^4$ be a bounded domain. Then there exists a constant $C^* > 0$ such that

and this inequality is also sharp.

Proof. We will give a summarize proof in two different cases. In the case of $c\leq0$ in the definition of $\|.\|$, if $\|u\|\leq 1,$ we can deduce that $\|\Delta u\|_2\leq 1$ and by using Lemma 2.1 combined with the Proposition 6.1 in ^[28], the conclusion holds.

In the case of $0 < c < \lambda_1$ in the definition of $\|.\|$, from Lemma 2.1, the proof and remark of Theorem 1 in ^[2] and the proof of Proposition 6.1 in ^[28], we still can establish this revised Adams inequality.

Lemma 2.3. Assume $(H_{1})$ and $(H_{3})$ hold. If $f$ has the standard subcritical polynomial growth on $\Omega$ (condition $\mathrm{(SCP)}$), then $I_\lambda^+$ ($I_\lambda^-$) satisfies $\mathrm{(C)}_{c^*}$.

Proof. We only prove the case of $I_\lambda^+$. The arguments for the case of $I_\lambda^-$ are similar. Let $\{u_n\}\subset E$ be a $\mathrm{(C)}_{c^*}$ sequence such that

Obviously, (2.3) implies that

Step 1. We claim that $\{u_n\}$ is bounded in $E$. In fact, assume that

Define

Then, $\|v_n\| = 1$, $\forall n\in {\bf N}$ and then, it is possible to extract a subsequence (denoted also by $\{v_n\}$) converges weakly to $v$ in $E$, converges strongly in $L^p(\Omega) (1\leq p < p^*)$ and converges $v$ a.e. $x\in\Omega$.

Dividing both sides of (2.2) by $\|u_n\|^2$, we get

Set

By $(H_3)$, we imply that

If $|\Omega_+|$ is positive, since Fatou's lemma, we get

which contradicts with (2.5). Thus, we have $v\leq 0$. In fact, we have $v = 0$. Indeed, again using (2.3), we get

Thus, we have

by noticing that since $v\leq 0,$ $f^+(x, u_n)v\leq 0 \ \text{a.e.} \ x\in\Omega,$ thus $-\int_\Omega f^+(x, u_n)vdx\geq 0.$ So we get

On the other hand, from $v_n\rightharpoonup v$ in $E$, we have

which implies $v = 0$.

Dividing both sides of (2.4) by $\|u_n\|$, for any $\varphi \in E$, then there exists a positive constant $M(\varphi)$ such that

Set

Thus, by (SCP), we know that $\{{\bf f}_n\}$ is a family bounded linear functionals defined on $E$. Combing (2.7) with the famous Resonance Theorem, we get that $\{|{\bf f}_n|\}$ is bounded, where $|{\bf f}_n|$ denotes the norm of ${\bf f}_n$. It means that

Since $E\subset L^{\frac{p^*}{p^*-q}}(\Omega)$, using the Hahn-Banach Theorem, there exists a continuous functional $\hat{{\bf f}}_n$ defined on $L^{\frac{p^*}{p^*-q}}(\Omega)$ such that $\hat{{\bf f}}_n$ is an extension of ${\bf f}_n$, and

where $\|\hat{{\bf f}}_n\|_{\frac{p^*}{q}}$ denotes the norm of $\hat{{\bf f}}_n(\varphi)$ in $L^{\frac{p^*}{q}}(\Omega)$ which is defined on $L^{\frac{p^*}{p^*-q}}(\Omega)$.

On the other hand, from the definition of the linear functional on $L^{\frac{p^*}{p^*-q}}(\Omega),$ we know that there exists a function $S_n(x)\in L^{\frac{p^*}{q}}(\Omega)$ such that

So, from (2.9) and (2.11), we obtain

which implies that

According to the basic lemma of variational, we can deduce that

Thus, by (2.8) and (2.10), we have

Now, taking $\varphi = v_n-v$ in (2.4), we get

where $A: E\rightarrow E^*$ defined by

By the H$\mathrm{\ddot{o}}$lder inequality and (2.12), we obtain

Then from (2.13), we can conclude that

This leads to a contradiction since $\|v_n\| = 1$ and $v = 0.$ Thus, $\{u_n\}$ is bounded in $E$.

Step 2. We show that $\{u_n\}$ has a convergence subsequence. Without loss of generality, we can suppose that

Now, it follows from $f$ satisfies the condition (SCP) that there exist two positive constants $c_4, c_5 > 0$ such that

then

Similarly, since $u_n\rightharpoonup u$ in $E,$ $\int_\Omega |u_n-u|dx\rightarrow 0$ and $\int_\Omega |u_n-u|^{\frac{p^*}{p^*-q}}dx\rightarrow 0.$

Thus, from (2.4) and the formula above, we obtain

So, we get $\|u_n\|\rightarrow \|u\|$. Thus we have $u_n\rightarrow u$ in $E$ which implies that $I_\lambda^+$ satisfies $\mathrm{(C)}_{c^*}$.

Lemma 2.4. Let $\varphi _{1} > 0$ be a $\mu_{1}$-eigenfunction with $\| \varphi_{1} \| = 1$ and assume that $(H_{1})$–$(H_{3})$ and $\mathrm{(SCP)}$ hold. If $f_{0} < \mu _{1},$ then:

$\mathrm{(i)}\ $ For $\lambda > 0$ small enough, there exist $\rho, \alpha > 0$ such that $I_\lambda^{\pm}(u)\geq \alpha$ for all $u\in E$ with $\| u \| = \rho,$

$\mathrm{(ii)}\ I_\lambda^{\pm}(t\varphi_1)\rightarrow -\infty$ as $t\rightarrow +\infty$.

Proof. Since $\mathrm{(SCP)}$ and $(H_1)$–$(H_3),$ for any $\varepsilon > 0,$ there exist $A = A(\varepsilon),$ $M$ large enough and $B = B(\varepsilon)$ such that for all $(x, s)\in \Omega\times \mathbb{R},$

Choose $\varepsilon > 0$ such that $(f_0+\varepsilon) < \mu_1.$ By (2.14), the Poincaré inequality and the Sobolev embedding, we obtain

where $K, C^{**}$ are constant.

Write

We can prove that there exists $t^*$ such that

In fact, letting $h'(t) = 0$, we get

According to the knowledge of mathematical analysis, $h(t)$ has a minimum at $t = t^*$. Denote

Taking $t^*$ in $h(t)$, we get

where $\Lambda^* = (\frac{\nu}{K\vartheta^{\hat{\vartheta}}+C^{**}\vartheta^{\bar{\vartheta}}})^{\frac{1}{\bar{\vartheta}}}.$ So, part (i) holds if we take $\rho = t^*$.

On the other hand, from (2.15), we get

Similarly, we have

Thus part $\mathrm{(ii)}$ holds.

Lemma 2.5. Let $\varphi _{1} > 0$ be a $\mu_{1}$-eigenfunction with $\| \varphi_{1} \| = 1$ and assume that $(H_{1})$–$(H_{3})$ and $\mathrm{(SCE)}$(or $\mathrm{(CG)}$) hold. If $f_{0} < \mu _{1},$ then:

$\mathrm{(i)}\ $ For $\lambda > 0$ small enough, there exist $\rho, \alpha > 0$ such that $I_\lambda^{\pm}(u)\geq \alpha$ for all $u\in E$ with $\| u \| = \rho,$

$\mathrm{(ii)}\ I_\lambda^{\pm}(t\varphi_1)\rightarrow -\infty$ as $t\rightarrow +\infty$.

Proof. From $\mathrm{(SCE)}$ (or (CG)) and $(H_1)$-$(H_3),$ for any $\varepsilon > 0,$ there exist $A_1 = A_1(\varepsilon),$ $M_1$ large enough, $B_1 = B_1(\varepsilon),$ $\kappa_1 > 0$ and $q_1 > 2$ such that for all $(x, s)\in \Omega\times \mathbb{R},$

Choose $\varepsilon > 0$ such that $(f_0+\varepsilon) < \mu_1.$ By (2.16), the Hölder inequality and the Adams inequality (see Lemma 2.2), we obtain

where $r_1 > 1$ sufficiently close to $1$, $\|u\|\leq \sigma$ and $\kappa_1 r_1\sigma^{2} < 32\pi^2.$ Remained proof is completely similar to the proof of part (ⅰ) of Lemma 2.4, we omit it here. So, part (ⅰ) holds if we take $\|u\| = \rho > 0$ small enough.

On the other hand, from (2.17), we get

Similarly, we have

Thus part $\mathrm{(ii)}$ holds.

Lemma 2.6. Assume $(H_{1})$ and $(H_{3})$ hold. If $f$ has the subcritical exponential growth on $\Omega$ (condition $\mathrm{(SCE)}$), then $I_\lambda^+$ ($I_\lambda^-$) satisfies $\mathrm{(C)}_{c^*}$.

Proof. We only prove the case of $I_\lambda^+$. The arguments for the case of $I_\lambda^-$ are similar. Let $\{u_n\}\subset E$ be a $\mathrm{(C)}_{c^*}$ sequence such that the formulas (2.2)–(2.4) in Lemma 2.3 hold.

Now, according to the previous section of Step 1 of the proof of Lemma 2.3, we also obtain that the formula (2.7) holds. Set

Then from for any $u\in E$, $e^{\alpha u^2}\in L^1(\Omega)$ for all $\alpha > 0$, we can draw a conclusion that $\{{\bf f}_n\}$ is a family bounded linear functionals defined on $E$. Using (2.7) and the famous Resonance Theorem, we know that $\{|{\bf f}_n|\}$ is bounded, where $|{\bf f}_n|$ denotes the norm of ${\bf f}_n$. It means that the formula (2.8) (see the proof of Lemma 2.3) holds.

Since $E\subset L^{q_0}(\Omega)$ for some $q_0 > 1$, using the Hahn-Banach Theorem, there exists a continuous functional $\hat{{\bf f}}_n$ defined on $L^{q_0}(\Omega)$ such that $\hat{{\bf f}}_n$ is an extension of ${\bf f}_n$, and

where $\|\hat{{\bf f}}_n\|_{q_0^*}$ is the norm of $\hat{{\bf f}}_n(\varphi)$ in $L^{q_0^*}(\Omega)$ which is defined on $L^{q_0}(\Omega)$ and $q_0^*$ is the dual number of $q_0$.

By the definition of the linear functional on $L^{q_0}(\Omega),$ we know that there is a function $S_n(x)\in L^{q_0^*}(\Omega)$ such that

Similarly to the last section of the Step 1 of the proof of Lemma 2.3, we can prove that $\mathrm{(C)}_{c^*}$ sequence $\{u_n\}$ is bounded in $E$. Next, we show that $\{u_n\}$ has a convergence subsequence. Without loss of generality, assume that

Since $f$ has the subcritical exponential growth $\mathrm{(SCE)}$ on $\Omega$, we can find a constant $C_{\beta^*} > 0$ such that

Thus, from the revised Adams inequality (see Lemma 2.2),

where $k > 1$ and $k'$ is the dual number of $k$. Similar to the last proof of Lemma 2.3, we have $u_n\rightarrow u$ in $E$ which means that $I_\lambda^+$ satisfies $\mathrm{(C)}_{c^*}$.

Lemma 2.7. Assume $(H_{3})$ holds. If $f$ has the standard subcritical polynomial growth on $\Omega$ (condition $\mathrm{(SCP)}$), then $I_\lambda$ satisfies $\mathrm{(PS)}_{c^*}$.

Proof. Let $\{u_n\}\subset E$ be a $\mathrm{(PS)}_{c^*}$ sequence such that

Step 1. To prove that $\{u_n\}$ has a convergence subsequence, we first need to prove that it is a bounded sequence. To do this, argue by contradiction assuming that for a subsequence, which is still denoted by $\{u_n\}$, we have

Without loss of generality, assume that $\|u_n\|\geq 1$ for all $n\in {\bf N}$ and let

Clearly, $\|v_n\| = 1$, $\forall n\in {\bf N}$ and then, it is possible to extract a subsequence (denoted also by $\{v_n\}$) converges weakly to $v$ in $E$, converges strongly in $L^p(\Omega) (1\leq p < p^*)$ and converges $v$ a.e. $x\in\Omega$.

Dividing both sides of (2.21) by $\|u_n\|^2$, we obtain

Set

By $(H_3)$, we get that

If $|\Omega_0|$ is positive, from Fatou's lemma, we obtain

which contradicts with (2.23).

Dividing both sides of (2.22) by $\|u_n\|$, for any $\varphi \in E$, then there exists a positive constant $M(\varphi)$ such that

Set

Thus, by (SCP), we know that $\{{\bf f}_n\}$ is a family bounded linear functionals defined on $E$. By (2.25) and the famous Resonance Theorem, we get that $\{|{\bf f}_n|\}$ is bounded, where $|{\bf f}_n|$ denotes the norm of ${\bf f}_n$. It means that

Since $E\subset L^{\frac{p^*}{p^*-q}}(\Omega)$, using the Hahn-Banach Theorem, there exists a continuous functional $\hat{{\bf f}}_n$ defined on $L^{\frac{p^*}{p^*-q}}(\Omega)$ such that $\hat{{\bf f}}_n$ is an extension of ${\bf f}_n$, and

where $\|\hat{{\bf f}}_n\|_{\frac{p^*}{q}}$ denotes the norm of $\hat{{\bf f}}_n(\varphi)$ in $L^{\frac{p^*}{q}}(\Omega)$ which is defined on $L^{\frac{p^*}{p^*-q}}(\Omega)$.

Remained proof is completely similar to the last proof of Lemma 2.3, we omit it here.

Lemma 2.8. Assume $(H_{3})$ holds. If $f$ has the subcritical exponential growth on $\Omega$ (condition $\mathrm{(SCE)}$), then $I_\lambda$ satisfies $\mathrm{(PS)}_{c^*}$.

Proof. Combining the previous section of the proof of Lemma 2.7 with slightly modifying the last section of the proof of Lemma 2.6, we can prove it. So we omit it here.

To prove the next Lemma, we firstly introduce a sequence of nonnegative functions as follows. Let $\Phi(t)\in C^\infty[0, 1]$ such that

We let

and $\psi_n(r) = H((ln n)^{-1}ln \frac{1}{r}).$ Notice that $\psi_n(x)\in E$, $B$ the unit ball in $\mathbb{R}^N$, $\psi_n(x) = 1$ for $|x|\leq \frac{1}{n}$ and, as it was proved in ^[2],

where $0\leq \lim\limits_{n\rightarrow \infty} A_n\leq 1.$ Thus, we take $x_0\in \Omega$ and $r_0 > 0$ such that $B(x_0, r)\subset \Omega,$ denote

Lemma 2.9. Assume $(H_{1})$ and $(H_{4})$ hold. If $f$ has the critical exponential growth on $\Omega$ (condition $\mathrm{(CG)}$), then there exists $n$ such that

Proof. We only prove the case of $I_\lambda^+$. The arguments for the case of $I_\lambda^-$ are similar. Assume by contradiction that this is not the case. So, for all $n$, this maximum is larger or equal to $\frac{16\pi^2}{\alpha_0}.$ Let $t_n > 0$ be such that

From $(H_1)$ and (2.29), we conclude that

Also at $t = t_n$, we have

which implies that

Since $(H_4)$, for given $\epsilon > 0$ there exists $R_\epsilon > 0$ such that

So by (2.31), we deduce that, for large $n$

By (2.30), the inequality above is true if, and only if

Set

and break the integral in (2.31) into a sum of integrals over $A_n^*$ and $B_n$. By simple computation, we have

The last integral in (2.34), denote $I_n$ is evaluated as follows:

Thus, finally from (2.34) we get

which means $\beta\leq \frac{64}{\alpha_0r_0^4}.$ This results in a contradiction with $(H_4)$.

To conclude this section we state the Fountain Theorem of Bartsch ^[32].

Define

Lemma 2.10. (Dual Fountain Theorem). Assume that $I_\lambda\in C^1(\mathbb{E}, \mathbb{R})$ satisfies the $\mathrm{(PS)_c^*}$ condition (see ^[32]), $I_\lambda(-u) = I_\lambda(u)$. If for almost every $k\in {\bf N},$ there exist $\rho_k > r_k > 0$ such that

$\mathrm{(i)}$ $a_k: = \inf\limits_{u\in Z_k, \|u\| = \rho_k}I_\lambda(u)\geq 0,$

$\mathrm{(ii)}$ $b_k: = \max\limits_{u\in Y_k, \|u\| = r_k}I_\lambda(u) < 0,$

$\mathrm{(iii)}$ $b_k = \inf\limits_{u\in Z_k, \|u\| = \rho_k}I_\lambda(u)\rightarrow 0, \ \mathit{\text{as}}\ k\rightarrow \infty,$

then $I_\lambda$ has a sequence of negative critical values converging $0$.

3.   Proofs of the main results

Proof of Theorem 1.1. For $I_\lambda^{\pm},$ we first demonstrate that the existence of local minimum $v_{\pm}$ with $I_\lambda^{\pm}(v_{\pm}) < 0$. We only prove the case of $I_\lambda^+.$ The arguments for the case of $I_\lambda^-$ are similar.

For $\rho$ determined in Lemma 2.4, we write

Then $\bar{B}(\rho)$ is a complete metric space with the distance

From Lemma 2.4, we have for $0 < \lambda < \Lambda^*,$

Furthermore, we know that $I_\lambda^+\in C^1(\bar{B}(\rho), \mathbb{R}),$ hence $I_\lambda^+$ is lower semi-continuous and bounded from below on $\bar{B}(\rho)$. Set

Taking $\tilde{\phi}\in C_0^\infty (\Omega)$ with $\tilde{\phi} > 0,$ and for $t > 0$, we get

for all $t > 0$ small enough. Hence, $c_1^* < 0$.

Since Ekeland's variational principle and Lemma 2.4, for any $m > 1$, there exists $u_m$ with $\|u_m\| < \rho$ such that

Hence, there exists a subsequence still denoted by $\{u_m\}$ such that

Thus $v_+$ is a weak solution of problem (1.1) and $I_\lambda^+(v_+) < 0$. In addition, from the maximum principle, we know $v_+ > 0$. By a similar way, we obtain a negative solution $v_-$ with $I_\lambda^-(v_-) < 0$.

On the other hand, from Lemmas 2.3 and 2.4, the functional $I_\lambda^+$ has a mountain pass-type critical point $u_+$ with $I_\lambda^+(u_+) > 0.$ Again using the maximum principle, we have $u_+ > 0$. Hence, $u_+$ is a positive weak solution of problem (1.1). Similarly, we also obtain a negative mountain pass-type critical point $u_-$ for the functional $I_\lambda^-$. Thus, we have proved that problem (1.1) has four different nontrivial solutions. Next, our method to obtain the fifth solution follows the idea developed in ^[33] for problem (1.1). We can assume that $v_+$ and $v_-$ are isolated local minima of $I_\lambda$. Let us denote by $b_\lambda$ the mountain pass critical level of $I_\lambda$ with base points $v_+, v_-:$

where $\Gamma = \{\gamma\in C([0, 1], E), \gamma(0) = v_+, \gamma(1) = v_-\}$. We will show that $b_\lambda < 0$ if $\lambda$ is small enough. To this end, we regard

We claim that there exists $\delta > 0$ such that

If not, we have $t_0\in (0, 1)$ such that $I_\lambda(t_0v_{\pm})\geq 0$ for $\lambda$ small enough. Similarly, we also have $I_\lambda(tv_{\pm}) < 0$ for $t > 0$ small enough. Let $\rho_0 = t_0\|v_{\pm}\|$ and $\check{c}_*^{\pm} = \inf \{I_\lambda^{\pm}(u), u\in \bar{B}(\rho_0)\}.$ Since previous arguments, we obtain a solution $v_{\pm}^*$ such that $I_\lambda(v_{\pm}^*) < 0,$ a contradiction. Hence, (3.1) holds.

Now, let us consider the $2$-dimensional plane $\Pi_2$ containing the straightlines $tv_-$ and $tv_+$, and take $v\in \Pi_2$ with $\|v\| = \epsilon.$ Note that for such $v$ one has $\|v\|_s = c_s\epsilon.$ Then we get

Thus, for small $\epsilon$,

Consider the path $\bar{\gamma}$ obtained gluing together the segments $\{tv_-:\epsilon \|v_-\|^{-1}\leq t\leq 1\}, \{tv_+:\epsilon \|v_+\|^{-1}\leq t\leq 1\}$ and the arc $\{v\in \Pi_2: \|v\| = \epsilon\}$. by (3.1)and (3.2), we get

which verifies the claim. Since the (PS) condition holds because of Lemma 2.3, the level$\{I_\lambda(v) = b_\lambda\}$ carries a critical point $v_3$ of $I_\lambda$, and $v_3$ is different from $v_{\pm}$.

Proof of Theorem 1.2. We first use the symmetric mountain pass theorem to prove the case of $a)$. It follows from our assumptions that the functional $I_\lambda$ is even. Since the condition (SCP), we know that $(I_1')$ of Theorem 9.12 in ^[30] holds. Furthermore, by condition $(H_3)$, we easily verify that $(I_2')$ of Theorem 9.12 also holds. Hence, by Lemma 2.7, our theorem is proved.

Next we use the dual fountain theorem (Lemma 2.10) to prove the case of $b)$. Since Lemma 2.7, we know that the functional $I_\lambda$ satisfies $\mathrm{(PS)_c^*}$ condition. Next, we just need to prove the conditions (ⅰ)-(ⅲ) of Lemma 2.10.

First, we verify (ⅰ) of Lemma 2.10. Define

From the conditions (SCP) and $(H_2)$, we get, for $u\in Z_k, \|u\|\leq R,$

Here, $R$ is a positive constant and $\epsilon > 0$ small enough. We take $\rho_k = (4\mu_1\lambda \beta_k^s/[(\mu_1-f_0-\epsilon)s])^{\frac{1}{2-s}}.$ Since $\beta_k\rightarrow 0, k\rightarrow \infty,$ it follows that $\rho_k\rightarrow 0, k\rightarrow \infty.$ There exists $k_0$ such that $\rho_k\leq R$ when $k\geq k_0$. Thus, for $k\geq k_0, u\in Z_k$ and $\|u\| = \rho_k,$ we have $I_\lambda(u)\geq 0$ and (ⅰ) holds. The verification of (ⅱ) and (ⅲ) is standard, we omit it here.

Proof of Theorem 1.3. According to our assumptions, similar to previous section of the proof of Theorem 1.1, we obtain that the existence of local minimum $v_{\pm}$ with $I_\lambda^{\pm}(v_{\pm}) < 0$. In addition, by Lemmas 2.5 and 2.6, for $I_\lambda^{\pm}$, we obtain two mountain pass type critical points $u_+$ and $u_-$ with positive energy. Similar to the last section of the proof of Theorem 1.1, we can also get another solution $u_3$, which is different from $v_{\pm}$ and $u_{\pm}$. Thus, this proof is completed.

Proof of Theorem 1.4. We first use the symmetric mountain pass theorem to prove the case of $a)$. It follows from our assumptions that the functional $I_\lambda$ is even. Since the condition (SCE), we know that $(I_1')$ of Theorem 9.12 in ^[30] holds. In fact, similar to the proof of (ⅰ) of Lemma 2.5, we can conclude it. Furthermore, by condition $(H_3)$, we easily verify that $(I_2')$ of Theorem 9.12 also holds. Hence, by Lemma 2.8, our theorem is proved.

Next we use the dual fountain theorem (Lemma 2.10) to prove the case of $b)$. Since Lemma 2.8, we know that the functional $I_\lambda$ satisfies $\mathrm{(PS)_c^*}$ condition. Next, we just need to prove the conditions (ⅰ)-(ⅲ) of Lemma 2.10.

First, we verify (ⅰ) of Lemma 2.10. Define

From the conditions (SCE), $(H_2)$ and Lemma 2.2, we get, for $u\in Z_k, \|u\|\leq R,$

Here, $R$ is a positive constant small enough and $\epsilon > 0$ small enough. We take $\rho_k = (4\mu_1\lambda \beta_k^s/[(\mu_1-f_0-\epsilon)s])^{\frac{1}{2-s}}.$ Since $\beta_k\rightarrow 0, k\rightarrow \infty,$ it follows that $\rho_k\rightarrow 0, k\rightarrow \infty.$ There exists $k_0$ such that $\rho_k\leq R$ when $k\geq k_0$. Thus, for $k\geq k_0, u\in Z_k$ and $\|u\| = \rho_k,$ we have $I_\lambda(u)\geq 0$ and (ⅰ) holds. The verification of (ⅱ) and (ⅲ) is standard, we omit it here.

Proof of Theorem 1.5. According to our assumptions, similar to previous section of the proof of Theorem 1.1, we obtain that the existence of local minimum $v_{\pm}$ with $I_\lambda^{\pm}(v_{\pm}) < 0$. Now, we show that $I_\lambda^+$ has a positive mountain pass type critical point. Since Lemmas 2.5 and 2.9, then there exists a $\mathrm{(C)_{c_M}}$ sequence $\{u_n\}$ at the level $0 < c_M\leq \frac{16\pi^2}{\alpha_0}$. Similar to previous section of the proof of Lemma 2.6, we can prove that $\mathrm{(C)_{c_M}}$ sequence $\{u_n\}$ is bounded in $E$. Without loss of generality, we can suppose that

Following the proof of Lemma 4 in ^[9], we can imply that $u_+$ is weak of problem (1.1). So the theorem is proved if $u_+$ is not trivial. However, we can get this due to our technical assumption $(H_5)$. Indeed, assume $u_+ = 0$, similarly as in ^[9], we obtain $f^+(x, u_n)\rightarrow 0$ in $L^1(\Omega)$. Since $(H_5),$ $F^+(x, u_n)\rightarrow 0$ in $L^1(\Omega)$ and we get

and again following the proof in ^[9], we get a contradiction.

We claim that $v_+$ and $u_+$ are distinct. Since the previous proof, we know that there exist sequence $\{u_n\}$ and $\{v_n\}$ in $E$ such that

and

Now, argue by contradiction that $v_+ = u_+.$ Since we also have $v_n\rightharpoonup v_+$ in $E$, up to subsequence, $\lim\limits_{n\rightarrow \infty}\|v_n\|\geq \|v_+\| > 0.$ Setting

we know that $\|w_n\| = 1$ and $w_n\rightharpoonup w_0$ in $E$.

Now, we consider two possibilities:

If (ⅰ) happens, we have $v_n\rightarrow v_+$ in $E$, so that $I_\lambda^+(v_n)\rightarrow I_\lambda^+(v_+) = c_*^+.$ This is a contradiction with (3.5) and (3.6).

Now, suppose that (ⅱ) happens. We claim that there exists $\delta > 0$ such that

for $n$ large enough. In fact, by the proof of $v_+$ and Lemma 2.9, we get

Thus, we can choose $h > 1$ sufficiently close to $1$ and $\delta > 0$ such that

Since $v_n\rightharpoonup v_+,$ by condition $(H_5)$, up to a subsequence, we conclude that

Thus, for $n$ sufficiently large we get

Thus, from (3.9) and the definition of $w_0$, (3.10) implies (3.7) for $n$ large enough.

Now, taking $\tilde{h} = (h+\epsilon)\alpha_0\|v_n\|^2,$ it follows from (3.7) and a revised Adams inequality (see ^[28]), we have

for $\epsilon > 0$ small enough. Thus, from our assumptions and the Hölder inequality we get $v_n\rightarrow v_+$ and this is absurd.

Similarly, we can find a negative mountain pass type critical point $u_-$ which is different that $v_-$. Thus, the proof is completed.

4.   Conclusions

In this research, we mainly studied the existence and multiplicity of nontrivial solutions for the fourth-order elliptic Navier boundary problems with exponential growth. Our method is based on the variational methods, Resonance Theorem together with a revised Adams inequality.

Acknowledgements

The authors would like to thank the referees for valuable comments and suggestions in improving this article. This research is supported by the NSFC (Nos. 11661070, 11764035 and 12161077), the NSF of Gansu Province (No. 22JR11RE193) and the Nonlinear mathematical physics Equation Innovation Team (No. TDJ2022-03).

Conflict of interest

There is no conflict of interest.