Existence and concentration of nontrivial solutions for an elastic beam equation with local nonlinearity

1.   Introduction and main results

In this paper, we focus on the following equation

where $\lambda > 0$, $\mu\in \mathbb{R}$, $f:[0, 1]\times\mathbb{R}\rightarrow \mathbb{R}$, $g:\mathbb{R}\rightarrow \mathbb{R}$ is a continuous function, and $h\in C^{1}[0, 1]$ is nonnegative. The problem (1.1) with $h = 0$ describes the static equilibrium of an elastic beam which is fixed at the left end of $x = 0$ and is attached to a bearing device at the right end of $x = 1$, where the corresponding force of the bearing device is given by function $g$, the nonlinear term $f$ is a continuous load which is attached to elastic beam. Moreover, there are many fourth order differential equations which is similar to problem (1.1) in engineering, material mechanics and so on. In recent years, with the development of science and technology, more and more scholars are devoted to the study on the existence and multiplicity of solutions for these fourth order ordinary differential equations. It is well known that different boundary conditions will lead to different physical meanings of these equations. For example, the boundary conditions that $u(0) = 0, u'(0) = 0$, $u(1) = 0$ and $u'(1) = 0$ describe the static equilibrium of an elastic beam fixed at both ends, see ^[1] and references therein. In ^[2], the boundary conditions that $u(0) = \alpha u'(0)-\beta u''(0) = \gamma u(1)+ \delta u''(1) = u'''(1) = 0$ describe that one end of the elastic beam is fixed and the other is sliding when $\alpha = \delta = 1$ and $\beta = \gamma = 0$. In ^[3], the boundary conditions that $u(0) = 0, u''(0) = g(u'(0)), u(1) = 0,$ and $u''(1) = h(u'(1))$ describe that both ends of the elastic beam are attached to fixed torsional represented by two nonlinear functions.

For the boundary value problems like problem (1.1), the existence and multiplicity of solutions have been investigated extensively by some methods, for example, fixed point theory and variational method. In particular, in ^[4], Cabada et al. investigated the problem

where the function $f:[0, 1]\times\mathbb{R}\to\mathbb{R}$ is continuous, $f(x, \mathbb{R_{+}})\subset\mathbb{R_{+}}$ for all $x\in[0, 1]$. They studied the existence, localization and multiplicity of positive solutions by using the critical point theorems in conical shells, Krasnosel'ski\u{\i}'s compression-expansion theorem, and unilateral Harnack type inequalities. In ^[5], Bonanno and Tersian investigated the following problem

where $\lambda > 0$, $\mu > 0$, $f:[0, 1]\times\mathbb{R}\rightarrow \mathbb{R}$ is a $L^{1}$-Carathéodory function, $g:\mathbb{R}\rightarrow \mathbb{R}$ is a continuous function. By using the variational methods, they established the existence result of solutions for problem (1.3). We refer readers to ^[6,7,8,9,10] for more related results of problem (1.3) with $\mu > 0$ or $\mu < 0$, respectively. The problem with perturbed nonlinear term is also an interesting topic. In ^[11], Heidarkhani and Gharehgazlouei investigated the problem

where $\lambda > 0$, $\mu\geq 0$, $f:[0, 1]\times\mathbb{R}\rightarrow \mathbb{R}$ is a $L^{1}$-Carathéodory function, $g:\mathbb{R}\rightarrow \mathbb{R}$ is a nonnegative continuous function, and $k:\mathbb{R}\rightarrow \mathbb{R}$ is a Lipschitz continuous function with a Lipschitz constant $L > 0$, i.e.,

and $k(0) = 0$. They investigated the existence of solution for problem (1.4) by using variational methods. More results about fourth-order boundary value problems with perturbations can be seen in ^{[12,13,14,15,16]} and references therein. Moreover, the multi-point boundary value and integral boundary value problems of fourth order ordinary differential equations have also been studied extensively. We refer readers to ^{[17,18,19,20,21]} and references therein.

In this paper, our work is mainly motivated by ^[22,23]. In ^[23], Costa and Wang considered a class of elliptic problems with a parameter

where $\lambda > 0$, $\Omega$ is a bounded smooth domain in $\mathbb{R}^{N}(N\geq 3)$ and $f\in C^{1}(\mathbb{R}, \mathbb{R})$ has superlinear growth only in neighborhood of $u = 0$. They investigated the existence of both signed and sign-changing solutions for problem (1.5) by using truncation function and minimax method and obtained the existence result when $\lambda$ is large enough. Subsequently, the method was used widely (for example, see ^{[24,25,26,27,28]}). However, the concrete values of lower bound of $\lambda$ were not given in these references. Recently, in ^[22], by using the idea in ^[23], the three authors in this paper, Kang, Liu and Zhang, considered a fractional order Kirchhoff-type system with a parameter

where

$a, b, \lambda > 0, p > 1$ and $1/p < \alpha\leq1$, $u(t) = (u_1(t), \cdots, u_N(t))^\tau\in \mathbb{R}^N$ for a.e. $t\in [0, T]$ and $N$ is a given positive integer, $(\cdot)^\tau$ denote the transpose of a vector, $V(t)\in C([0, T], \mathbb{R})$ with $\mathop{\min}\limits_{t\in[0, T]}V(t) > 0$, $_{0}D^{\alpha}_{T}$ and $_{t}D^{\alpha}_{T}$ are the left and right Riemann-Liouville fractional derivatives, respectively, $\phi_{p}(s): = |s|^{p-2}s$, $F:[0, T]\times\mathbb{R}^N\rightarrow \mathbb{R}$ and $\nabla F(t, x)$ is the gradient of $F$ with respect to $x = (x_1, \cdots, x_N)\in \mathbb{R}^N$, that is, $\nabla F(t, x) = \left(\frac{\partial F}{\partial x_1}, \cdots, \frac{\partial F}{\partial x_N}\right)$. They investigated the existence of solutions for problem (1.6) by using mountain pass theorem when the nonlinear term satisfied the superquadric condition only near the origin. They obtained problem (1.6) has at least one nontrivial solution if $\lambda > \lambda_0$, where $\lambda_0$ is given in detail.

Based on the idea in ^[22,23], in this paper, we investigate the existence of nontrivial weak solutions of problem (1.1) when the nonlinear term $f(x, u)$ with respect to $u$ satisfies the super-quadratic growth condition only near the origin. We obtain a specific lower bound of the parameter $\lambda$ when $\mu > 0$ and $\mu < 0$ respectively, and analyze the relationships between $\lambda$ and $\mu$. Moreover, we also investigate the concentration phenomenon of solutions when $\mu\to 0$. Although the idea origins from ^[22,23], there are still three differences: (1) the model (1.1) is obviously different from (1.5) and (1.6). In particular, (1.5) and (1.6) have only one parameter $\lambda$, while problem (1.1) has two parameters $\lambda$ and $\mu$. Hence, it is necessary to discuss the relationship between these two parameters; (2) we study the concentration phenomenon of solutions when the parameter $\mu\to 0$; (3) the boundary value condition of (1.1) is not Dirichlet boundary condition. In particular, when $\mu\neq 0$, $u'''(1) = \mu g(u(1)) \neq 0$. In the following theorems, we assume that $g$ satisfies the locally subquadratic condition when $\mu > 0$ and locally superquadric condition when $\mu < 0$ with respect to $u$ near the origin. Finally, comparing with those results for the fourth order differential equations like (1.2) and (1.3), we only suppose $f$ and $g$ satisfy the local conditions near the origin, and consider more general model (1.1) since it is easy to see that (1.1) reduces to (1.2) and (1.3) if $h = 0$. Here, we refer to some related references for the case $h\not = 0$, (see ^{[29,30,31,32,33]}) which focus on the existence and multiplicity of solutions for the second order Hamiltonian system with damped term by variational methods. Moreover, as a comparison, we also refer to a recent reference ^[34] in which the Klein-Gordon-Maxwell systems was investigated and the nonlinear term was assumed to satisfy super-quadratic conditions near $\infty$.

Next, we make some assumptions for $F$, and then state our main results.

$(H_{1})$ there exists a constant $\delta > 0$ such that $F(x, u)$ is continuously differentiable in $u\in \mathbb{R}$ with $|u|\le \delta$ for a.e. $x \in [0, 1]$, measurable in $x$ for every $u\in \mathbb{R}$ with $|u|\le \delta$, and there are $a\in C(\mathbb{R}^+, \mathbb{R}^+)$ and $b\in L^1([0, 1];\mathbb{R}^+)$ such that

for all $u\in \mathbb{R}$ with $|u|\le \delta$ and a.e. $x \in [0, 1]$, where $F(x, u) = \int_0^uf(x, s)ds$;

$(H_{2})$ there exist constants $q_{1} > 2$, $q_{2}\in(2, q_{1})$, $M_{1} > 0$ and $M_{2} > 0$ such that

for all $u\in\mathbb{R }$ with $|u|\leq\delta$ and a.e. $x\in[0, 1]$;

$(H_{3})$ there exists a constant $\beta > 2$ such that

for all $u\in\mathbb{R}$ with $|u|\leq\delta$ and a.e. $x\in[0, 1]$.

Theorem 1.1. Suppose that $(H_{1})-(H_{3})$ hold and $G$ satisfies

$(H_{4})$ there exist constants $1 < p_{2} < p_{1} < 2$ and $N_{1}, N_{2} > 0$ such that

for all $u\in\mathbb{R}$ with $|u|\leq\delta$ and a.e. $x\in[0, 1]$, where $G(u) = \int_0^ug(s)ds$;

$(H_{5})$ there exists a constant $\gamma\in(0, 2)$ such that

for all $u\in\mathbb{R}$ with $|u|\leq\delta$ and a.e. $x\in[0, 1]$.

If $\lambda > \lambda^*_{+}: = \max\{\Lambda^{+}_1, \Lambda^{+}_2, \Lambda^{+}_3\}$ and $0 < \mu < \mu_*$, then problem (1.1) has at least a nontrivial solution $u_{\lambda, \mu}$. Moreover,

where

Remark 1.1. It is easy to see that $\Lambda^{+}_{3}\to+\infty$ if $\mu\to\mu_{*}$, and then $\lambda\to+\infty$. Hence $\lim\limits_{\mu\rightarrow \mu_{*}\atop\lambda\rightarrow +\infty}\|u_{\lambda, \mu}\|_{\infty}$ can be simply written as $\lim\limits_{\mu\rightarrow \mu_{*}}\|u_{\lambda, \mu}\|_{\infty}$.

Theorem 1.2. Suppose that $(H_{1})-(H_{3})$ hold and $G$ satisfies

$(H_{6})$ there exist constants $\alpha_{1} > q_{2}$, $\alpha_{2}\in(q_{2}, \alpha_{1})$, $\eta_{1} > 0$ and $\eta_{2} > 0$ such that

for all $u\in\mathbb{R }$ with $|u|\leq\delta$ and a.e. $x\in[0, 1]$;

$(H_{7})$ there exists a constant $\xi > 2$ such that

for all $u\in\mathbb{R}$ with $|u|\leq\delta$ and a.e.$x\in[0, 1]$.

If $\lambda > \lambda^*_{-}: = \max\{\Lambda^{-}_1, \Lambda^{-}_2, \Lambda^{-}_3, \Lambda^{-}_4\}$ and $\mu < 0$, then system (1.1) has at least a nontrivial solution $u_{\lambda, \mu}$. Moreover,

where

In Theorem 1.1 and Theorem 1.2, we investigate problem (1.1) with $\mu > 0$ and $\mu < 0$, respectively. It is natural to ask what happens when $\mu\to 0$. Next, in Theorem 1.3, we show the concentration phenomenon of $\{u_{\lambda, \mu}\}$ as $\mu\to 0$, which means that $u_{\lambda, \mu}\to u_\lambda$ as $\mu\to 0$ for some $u_\lambda\in E$ ($E$ defined by (2.1)) and $u_\lambda$ is a nontrivial solution of the following equation

Theorem 1.3. Suppose that $(H_{1})-(H_{3})$ hold and assume that $(H_{4})-(H_{5})$ hold if $\mu > 0$ and $(H_{6})-(H_{7})$ hold if $\mu < 0$. If $\{u_{\lambda, \mu}\}$ is a family of nontrivial solutions of problem (1.1), which are given in Theorem 1.1 and Theorem 1.2, then problem (1.16) has at least a nontrivial solution $u_{\lambda}$ for all $\lambda > \lambda^*: = \max\{\Lambda_{M_0}, \Lambda_{21}, \Lambda_{22}, \Lambda_3, \Lambda_4^{-}\}$, and for any given $\lambda > \lambda^*$, $u_{\lambda, \mu}\to u_{\lambda}$, as $\mu\to0$. Moreover,

where $M_0$ is any given positive constant,

We organize this paper as follows. In section 2, we present the working space, some conclusions for the working space and a variant of mountain pass theorem. In section 3, we complete the proofs of Theorem 1.1, Theorem 1.2 and Theorem 1.3.

2.   Preliminaries

Consider the space

where

is the Sobolev space. $E$ is a Hilbert space with inner product

and norm

which is equivalent to the usual norm

and

Lemma 2.1. (^[11]) The embedding

is compact and

for all $u\in E$.

Remark 2.1. It is easy to obtain from Lemma 2.1 that

where $H_0 = \min\limits_{s\in [0, 1]} H(s)$.

Lemma 2.2. The embedding

is compact and

for all $u\in E$, where $\|u\|_{2} = \left(\int^{1}_{0}|u(x)|^{2}dx\right)^{\frac{1}{2}}$.

Proof. The compactness of the embedding is easily proved by Lemma 2.1. Next, we prove the embedding inequalities. By H$\ddot{\mbox{o}}$lder's inequality, we have

Similarly,

By Lemma 2.2, it is easy to obtain the following inequalities which is of independent interest.

Lemma 2.3. For all $u\in E$,

Let $X$ be a Banach space. $\chi \in C^{1}(X, \mathbb{R})$ and $c\in\mathbb{R}$. A sequence $\{u_{n}\}\subset X$ is called (PS)$_{c}$ sequence (named after Palais and Smale) if the sequence $\{u_{n}\}$ satisfies

Lemma 2.4. (Mountain Pass Theorem ^[35]) Let $X$ be a Banach space, $\chi \in C^{1}(X, \mathbb{R})$, $\omega\in X$ and $r > 0$ be such that $\|\omega\| > r$ and

Then there exists a (PS)$_{c}$ sequence with

and

3.   Proofs

3.1.   Proof of Theorem 1.1

For each $\lambda > 0, \mu\in \mathbb{R}$, we define the functional $J_{\lambda, \mu}: E\to \mathbb{R}$ as

It is easy to see that the assumption $(H_{1})-(H_{5})$ can not ensure that $J_{\lambda, \mu}$ is well defined on $E$. So we follow the method in ^[23]. Define $m(s)\in C^1(\mathbb{R}, [0, 1])$ as an even cut-off function satisfying $sm'(s)\leq0$ and

Define $\bar{F}:[0, 1]\times\mathbb{R}\to\mathbb{R}$, $\bar{G}:\mathbb{R} \to \mathbb{R}$ as

By $(H_{1})$ and the definition of $\bar{F}$, it is easy to obtain that $\bar{F}$ satisfies $(H_{1})'$$\bar{F}(x, u)$ is continuously differentiable in $u\in \mathbb{R}$ for a.e. $x \in [0, 1]$, measurable in $x$ for every $u\in \mathbb{R}$, and there exist $a\in C(\mathbb{R}^+, \mathbb{R}^+)$ and $b\in L^1([0, 1];\mathbb{R}^+)$ such that

for all $u\in \mathbb{R}$ and a.e. $x \in [0, 1]$, $a_{0} = \max_{s\in[0, \delta]}a(s)$ and $m_{0} = \max_{s\in[\frac{\delta}{2}, \delta]}|m'(s)|$, where $\bar{F}(x, u) = \int_0^{u}\bar{f}(x, s)ds$ (see ^[22]).

Lemma 3.1. (^[22]) Assume that $(H_{2})$ and $(H_{3})$ hold. Then

$(H_{2})'$

$(H_{3})'$

where $\theta = \min\{q_{2}, \beta\}.$

Lemma 3.2. (^[36]) Assume that $(H_{4})$ and $(H_{5})$ hold. Then

$(H_{4})'$

$(H_{5})'$

where $\zeta = \max\{p_{1}, \gamma\}$ and $\bar{g}(u) = \bar{G}'(u).$

Next, we define the variational functional corresponding to $\bar{F}$ and $\bar{G}$ as

for all $u\in E$. It follows from Lemma 3.1 and Lemma 3.2 that $\bar{J}_{\lambda, \mu}\in C^{1}(E, \mathbb{R})$ and

for all $u, v\in E$. Hence, for all $u\in E$, we can get

Lemma 3.3. If $u$ is a critical point of $\bar{J}_{\lambda, \mu}$, then $u$ is a weak solution of the following equation

Proof. If $u$ is a critical point of $\bar{J}_{\lambda, \mu}$, we have

for all $v\in H^{2}([0, 1])$. An integration by parts gives

Define

Then for any $v\in H^{2}[0, 1]\cap H^{2}_{0}[0, 1]$, (3.5) and (3.6) implies that

and then by the arbitrary of $v$, we have

Next, we prove that $u$ satisfies the boundary condition of (3.4). For any $v\in E$, integrating (3.7) by parts, and by (3.5) and (3.6), we can obtain

Then (3.7) and the arbitrary of $v$ imply that the boundary conditions $u''(1) = 0$ and $u'''(1)-\mu \bar{g}(u(1)) = 0$. Therefore, $u$ is a weak solution for problem (3.4).

Lemma 3.4. If $(H_{1})-(H_{5})$ hold. For all $\lambda > \Lambda_{1}^{+}$ and $\mu > 0$, $\bar{J}_{\lambda, \mu}$ satisfies the following conditions:

(i) there exist two positive constants $d_\lambda$ and $\nu_\lambda$ such that $\bar{J}_{\lambda, \mu}|_{\partial B_{\nu_\lambda}}\geq d_\lambda,$ where $B_{r}$ denote a ball with center 0 and radius r;

(ii) there is $\omega\in E/\bar{B}_{\nu_\lambda}$ such that $\bar{J}_{\lambda, \mu}(\omega) < 0$.

Proof. By Lemma 2.1 and 2.2, Lemma 3.1 and 3.2, we have

where $H_{1} = \max\limits_{x\in [0, 1]} H(x)$. For any given $\lambda > 0$, we choose $\nu_{\lambda} = \left(\frac{2e^{\frac{q_2 H_0}{2}}}{\lambda M_{2}e^{H_{1}}}\right)^{\frac{1}{q_{2}-2}}$. Then for all $\|u\|_{H} = \nu_{\lambda}$, we have

Since $\kappa\in E$ with

and

Then for all $\lambda > \Lambda^{+}_{1}$, we have

By (3.9), we have $\|\delta \kappa\|_{\infty}\leq\delta.$ By $(H_{2}), (H_{4})$ and the definitions of $\bar{F}$ and $\bar{G}$, for all $|u|\leq\frac{\delta}{2}$, we have

We also have

and

for all $\frac{\delta}{2} < |u|\le\delta$. Hence by Hölder inequality, for all $\lambda > \Lambda^{+}_{1}$, we can obtain

where $N = \max\{N_{1}, N_{2}\}$. Let $\omega = \delta \kappa$. Then the proof is completed.\qed

Let $\chi = \bar{J}_{\lambda, \mu}$. Then for any given $\lambda > \Lambda^{+}_{1}$ and $\mu > 0$, Lemma 2.4 and Lemma 3.4 imply that $\bar{J}_{\lambda, \mu}$ has a (PS)$_{c_{\lambda, \mu}}$ sequence $\{u_n\}: = \{u_{n, \lambda, \mu}\}$, that is, there exists a sequence $\{u_n\}$ satisfying

where

Lemma 3.5. Suppose that $(H_{1})-(H_{5})$ hold. Then for any given $\lambda > \Lambda^+_1$ and $\mu > 0$, the $(PS)_{c_{\lambda, \mu}}$ sequence $\{u_n\}$ has a convergent subsequence in $E$, that is, there exists a $u_{\lambda, \mu}\in E$ such that $\|u_n-u_{\lambda, \mu}\|_H\to 0$.

Proof. The proof is similar to the argument in ^[37]. Note that $\zeta < \theta$, by Lemma 3.1, Lemma 3.2 and (3.13), there exists a positive constant $M > 0$ such that

So the $(PS)_{c_{\lambda, \mu}}$ sequence $\{u_{n}\}$ is bounded in $E$. Then by Lemma 2.1, there exists a subsequence, denoted by $\{u_{n}\}$, for some $u: = u_{\lambda, \mu}\in E$, such that

By (3.3), we have

So we get

By $(H_{1})'$, the boundedness of $\{u_{n}\}$ in $E$, and $h(x)\in C^{1}([0, 1])$, we have $e^{H(x)}|\bar{f}(x, u_{n})-\bar{f}(x, u)|$ is bounded in $[0, 1]$. Moreover, by $u_{n}\rightarrow u$ in $C^{1}([0, 1])$ and the boundedness of $e^{H(x)}|\bar{g}(u_{n}(1))-\bar{g}(u(1))|$ on $[0, 1]$, we have

and it is easy to see from (3.13) and (3.16) that

Therefore, by (3.18)–(3.20), we get

By the continuity of $\bar{J}_{\lambda, \mu}$, we obtain that $\bar{J}_{\lambda, \mu}(u_{\lambda, \mu}) = c_{\lambda, \mu}$, where $c_{\lambda, \mu}$ is defined by (3.14). Then (3.8) implies that $c_{\lambda, \mu}\geq d_{\lambda} > 0$. Hence, $u_{\lambda, \mu}$ is a nontrivial critical point of $\bar{J}_{\lambda, \mu}$ in $E$ for any given $\lambda > \Lambda^{+}_1$.

Next, we will show that $u_{\lambda, \mu}$ precisely is the nontrivial weak solution of problem (1.1) for any given $\lambda > \lambda^{*}_{+}$. In order to get this, we need to make an estimate for the critical level $c_{\lambda, \mu}$. We introduce the functional $\widetilde{I}_{\lambda, \mu}: E\to \mathbb{R}$ as follows

Lemma 3.6. Suppose $(H_{1})-(H_{5})$ hold. Then for all $\lambda\geq \max\{\Lambda^{+}_{1}, \Lambda^{+}_{2}\}$ and $\mu > 0$, we have

where $C_{*}$ and $C_{**}$ are defined by (1.7) and (1.8), respectively.

Proof. Define $f_i:[0, \infty)\to \mathbb{R}$, $i = 1, 2, 3$, by

where $\kappa_1 = \delta \kappa$ and $\kappa$ is defined in (3.9). Then $f_{1}(s)+f_{2}(s)+f_3(s) = \widetilde{I}_{\lambda, \mu}(s \kappa_1)$. Let

Thus for each given $\lambda > 0$, we have $s = \left(\dfrac{\lambda^{\frac{1}{q_{1}}}\|\kappa_1\|_H^{2}}{\lambda e^{H_{0}}M_1 q_{1} \|\kappa_1\|_{L^{q_{1}}}^{q_{1}}}\right)^{\frac{1}{q_{1}-2}}.$ Then

Obviously, $f_1(0) = 0$ and

So if $\lambda > \Lambda^{+}_{2} = 1$, $f_1(s)$ is decreasing on $s\in [0, 1]$ and then $f_{1}(s) < 0$ for all $s\in [0, 1]$. Moreover, for all $\mu > 0$, we can get

By (3.9), we have

for all $s \in [0, 1]$. So for all $\lambda > \Lambda^{+}_{2}$, by $(3.10)$–$(3.12)$, we have

Proof of Theorem 1.1. Note that $u_{\lambda, \mu}$ is a critical point of $\bar{J}_{\lambda, \mu}$ with critical value $c_{\lambda, \mu}$. Since $\langle \bar{J}'(u_{\lambda, \mu}), u_{\lambda, \mu}\rangle = 0$, similar to the argument in (3.15) and by Lemma 3.6, we have

Note that $\mu\in (0, \mu_*)$, where $\mu_*$ is defined by (1.10). If

then by Remark 2.1 and (3.22), we have

So for all $\lambda > \Lambda^{+}_{3}$ and all $x\in[0, 1]$,

and then

Furthermore, for all $v\in E$, we have

Thus, $u_{\lambda, \mu}$ is precisely the nontrivial weak solution of problem (1.1) when $\lambda > \lambda^{*}_{+}: = \max\{\Lambda^{+}_{1}, \Lambda^{+}_{2}, \Lambda^{+}_{3}\}$.

Next, we discuss the connection between $\mu$, $\lambda$ and $\Lambda^{+}_{i}(i = 1, 2, 3)$.

$(a_{1})$ If $\mu\rightarrow0^{+}$, we have

which shows that if $\mu\to 0^+$, then $\lambda_+^*\to \lambda_*$, where $\lambda_*$ is defined by (1.9). Hence, if $\mu$ is small enough, the range of $\lambda$ can be extended to $(\lambda_*, \infty)$.

$(a_{2})$ If $\mu\rightarrow \mu_{*}^{-}$, we have

and then $\lambda\rightarrow +\infty$, which means that if $\mu$ is close to $\mu_{*}$, problem (1.1) has a nontrivial weak solution when $\lambda$ is sufficiently large.

Finally, by $(a_{1}), (a_{2}), (3.22)$ and $(3.23)$, it is easy to obtain that

3.2.   Proof of Theorem 1.2

By (3.1), we define $\bar{\bar{G}}:\mathbb{R}\rightarrow \mathbb{R}$ as

Similar to Lemma 3.1, we have the following conclusion.

Lemma 3.7. Assume $(H_{6})-(H_{7})$ hold. Then

$(H_{6})'$

$(H_{7})'$

where $\varrho = \min\{\alpha_{2}, \xi\}, \; \; \; \bar{\bar{g}}(u) = \bar{\bar{G}}'(u).$

Next, we define the variational functional corresponding to $\bar{F}$ and $\bar{\bar{G}}$ as

for all $u\in E$. By Lemma 3.1 and Lemma 3.7, we have $\bar{\bar{J}}_{\lambda, \mu}\in C^{1}(E, \mathbb{R})$ and

for all $u, v\in E$. Hence

for all $u\in E.$

Lemma 3.8. If $(H_{1})-(H_{3})$, $(H_{6})$ and $(H_{7})$ hold. If $\lambda > \max\{\Lambda_{1}^{-}, \Lambda_4^{-}\}$ and $\mu < 0$, then $\bar{\bar{J}}_{\lambda, \mu}$ satisfies the following conditions:

(i) there exist two positive constants $d_{\lambda, \mu}$ and $\rho_{\lambda, \mu}$ such that $\bar{\bar{J}}_{\lambda, \mu}|_{\partial B_{\rho_{\lambda, \mu}}}\geq d_{\lambda, \mu};$

(ii) there is $\omega\in E/\bar{B}_{\rho_{\lambda, \mu}}$ such that $\bar{\bar{J}}_{\lambda, \mu}(\omega) < 0$.

Proof. By Lemma 2.1 and 2.2, Remark 2.1 and Lemma 3.7, we have

Note that $\lambda > \Lambda_4^- = \frac{2e^{\frac{q_2 H_0}{2}}}{M_2e^{H_1}}$. If we take $\|u\|_H = \rho_{\lambda, \mu}: = \left(\frac{2e^{\frac{q_{2}H_{0}}{2}}}{\lambda e^{H_{1}}M_{2}-8\mu e^{H(1)}\eta_{2}}\right)^{\frac{1}{q_{2}-2}}$, then $\|u\|_H < 1$. Thus for all $u\in \partial B_{\rho_{\lambda, \mu}}$, we have

We choose $\phi\in E$ such that

Note that

For all $\lambda > \Lambda^{-}_{1}$, we have

By (3.25), we have $\|\delta \phi\|_{\infty}\leq\delta$. By $(H_{2})$, $(H_{4})$ and the definition of $\bar{F}$ and $\bar{\bar{G}}$, for all $|u|\leq\frac{\delta}{2}$, we have

We also have

and

for all $\frac{\delta}{2} < |u| < \delta$. Hence, by Hölder inequality, for any $\lambda > \Lambda^{-}_{1}$, we have

Let $\omega = \delta \phi$. Then the proof is completed.

Let $\chi = \bar{J}_{\lambda, \mu}$. Then for any given $\lambda > \max\{\Lambda^{-}_{1}, \Lambda^{-}_{4}\}$, Lemma 2.4 and Lemma 3.8 imply that $\bar{\bar{J}}_{\lambda, \mu}$ has a (PS)$_{c_{\lambda, \mu}}$ sequence $\{u_n\}: = \{u_{n, \lambda, \mu}\}$, that is, there exists a sequence $\{u_n\}$ satisfying

where

Lemma 3.9. Suppose that $(H_{1})-(H_{3})$, $(H_{6})$ and $(H_{7})$ hold. Then for any given $\lambda > \max\{\Lambda^{-}_{1}, \Lambda^{-}_{4}\}$ and $\mu < 0$, the $(PS)_{c_{\lambda, \mu}}$ sequence $\{u_{n}\}$ has a convergent subsequence in E, that is, there exists a $u_{\lambda, \mu}\in E$ such that $\|u_{n}-u_{\lambda, \mu}\|_{H}\to 0$.

Proof. Note that $\rho = \min\{\theta, \varrho\}$. By Lemma 3.1, Lemma 3.7 and (3.29), there exists a positive constant $M > 0$ such that

So the $(PS)_{c_{\lambda, \mu}}$ sequence $\{u_{n}\}$ is bounded in $E$, when $n\to\infty$. The rest proof is similar to the argument in Lemma 3.5.

By the continuity of $\bar{\bar{J}}_{\lambda, \mu}$, we obtain that $\bar{\bar{J}}_{\lambda, \mu}(u_{\lambda, \mu}) = c_{\lambda, \mu}$, where $c_{\lambda, \mu}$ is defined by (3.30). Then (3.24) implies that $c_{\lambda, \mu}\geq d_{\lambda, \mu} > 0$. Hence $u_{\lambda, \mu}$ is a nontrivial critical point of $\bar{\bar{J}}_{\lambda, \mu}$ in $E$ for any given $\lambda > \max\{\Lambda^{-}_1, \Lambda^{-}_4\}$.

Next, we will show that $u_{\lambda, \mu}$ precisely is the nontrivial weak solution of problem (1.1) for any given $\lambda > \lambda^{*}_{-}$. In order to get this, we need to make an estimate for the critical level $c_{\lambda, \mu}$. We introduce the functional $\bar{I}_{\lambda, \mu}: E\to \mathbb{R}$ as follows

Lemma 3.10.Suppose $(H_{1})-(H_{3})$, $(H_{6})$ and $(H_{7})$ hold. Then for all $\lambda > \max\{\Lambda^{-}_{1}, \Lambda^{-}_{2}, \Lambda^{-}_{4}\}$ and $\mu < 0$, we have

where $D_{*}$ is defined by (1.14).

Proof. Define $k_i:[0, \infty)\to \mathbb{R}$, $i = 1, 2, 3$, by

where $\phi_1 = \delta \phi$ and $\phi$ is defined in (3.25). Then $k_{1}(s)+k_{2}(s)+k_3(s) = \bar{I}_{\lambda, \mu}(s \phi_1)$. Let

Thus for each given $\lambda > 0$, we have $s = \left(\dfrac{\lambda^{\frac{1}{q_{1}}}\|\phi_1\|_H^{2}}{\lambda e^{H_{0}}M_1 q_{1} \|\phi_1\|_{{q_{1}}}^{q_{1}}}\right)^{\frac{1}{q_{1}-2}}.$ Then

Obviously, $f_1(0) = 0$ and

So if $\lambda > \Lambda^{-}_{2} = 1$, $k_1(s)$ is decreasing on $s\in[0, 1]$ and then $k_{1}(s)\le0$. Moreover, obviously,

By (3.25), we have

for all $s \in [0, 1]$. Then for all $\lambda > \max\{\Lambda^{-}_{1}, \Lambda^{-}_{2}, \Lambda^{-}_{4}\}$, by (3.26)–(3.27), we have

Proof of Theorem 1.2. Note that $u_{\lambda, \mu}$ is a critical point of $\bar{\bar{J}}_{\lambda, \mu}$ with critical value $c_{\lambda, \mu}$. Since $\langle \bar{\bar{J}}'(u_{\lambda, \mu}), u_{\lambda, \mu}\rangle = 0$, similar to the argument in (3.31) and by Lemma 3.10, we have

Since

by Remark 2.1 and (3.33), we have

So for all $\lambda > \Lambda^{-}_{3}$, we have

and then

Furthermore, for all $v\in E$, we have

Thus, $u_{\lambda, \mu}$ is precisely the nontrivial weak solution of problem (1.1) when $\lambda > \lambda^{*}_{-}: = \max\{\Lambda^{-}_{1}, \Lambda^{-}_{2}, \Lambda^{-}_{3}, \Lambda^{-}_{4}\}$.

Next, we discuss the connection between $\mu, \lambda$ and $\Lambda^{-}_{i}(i = 1, 2, 3, 4)$. If $\mu\rightarrow0^{-}$, we can get

which shows that if $\mu\to 0^{-}$, then $\lambda^{*}_{-}\to \Lambda_{*}.$ Hence, if $|\mu|$ is small enough, the range of $\lambda$ can be extended to $(\Lambda_{*}, \infty)$, where $\Lambda_{*}$ is defined by (1.15).

Finally, by $(3.33)$ and $(3.34)$, it is easy to see that (1.11)–(1.13) hold.

3.3.   Proof of Theorem 1.3

Let $\{u_n: = u_{\lambda, \mu_n}\}\subset E$ be the critical points of $\bar{J}_{\lambda, \mu_{n}}$ (if $\mu_n > 0$) and $\widetilde{J}_{\lambda, \mu_{n}}$ (if $\mu_n < 0$) with respect to $c_{\lambda, \mu_{n}}$. Then we have

and

for all $v\in E$. Define the functional $\bar{J}_{\lambda}:E\to \mathbb{R}$ as

Obviously, $\bar{J}_{\lambda, \mu_{n}}(u_{n})\to \bar{J}_{\lambda}(u_n)$ as $\mu_n\to 0^+$ and $\widetilde{J}_{\lambda, \mu_{n}}(u_{n})\to \bar{J}_{\lambda}(u_n)$ as $\mu_n\to 0^-$. Moreover, it is easy to see that the critical point of $\bar{J}_\lambda$ is a weak solution of the following elastic beam equation

Next, we prove that $\{u_{n}\}$ is a bounded (PS) sequence of $\bar{J}_{\lambda}$. Since $\mu_n\to 0$, there exists a positive constant $N_0$ such that $-M_0\le\mu_n\le M_0$ for all $n\ge N _0$. Note that $\Lambda^{+}_{1}$ is decreasing to $\Lambda_{11}$ as $\mu_n\to 0^+$ and $q_1 > 2$. Then Lemma 3.6 implies that

for all $n\ge N_0$ and all $\lambda > \max\{\Lambda_{M_0}, \Lambda_{21}\}\ge \max\{\Lambda^{+}_{1}, \Lambda^{+}_{2}\}$, where $\Lambda_{M_0}$ is defined by (1.17). Moreover, since $\Lambda^{-}_{1}$ is increasing to $\Lambda_{22}$ as $\mu_n\to 0^-$, we have

for all $n\ge N_0$. Then Lemma 3.10 implies that

for all $n\ge N_0$ and all $\lambda > \max\{\Lambda_{22}, \Lambda_{21}\}\ge \max\{\Lambda^{-}_{1}, \Lambda^{-}_{2}, \Lambda^{-}_{4}\}$. (3.38) and (3.39) imply that $c_{\lambda, \mu_n}$ is bounded for all $n\ge N_0$ and all $\lambda > \max\{\Lambda_{M_0}, \Lambda_{21}, \Lambda_{22}, \Lambda^{-}_{4}\}$. Then it follows from (3.22) and (3.33) that there exists a positive constant $K_0$ such that $\|u_n\|_H\le K_0$ for all $n\ge N_0$. Thus we have

for all $n$ and all $\lambda > \max\{\Lambda_{M_0}, \Lambda_{21}, \Lambda_{22}, \Lambda^{-}_{4}\}$. By (3.40), (H2)$'$ and Remark 2.1, it is easy to see that $\bar{J}_{\lambda}(u_n)$ is bounded.

If $\mu_n > 0,$ then by (3.35), for any given $v \in E$, we have

Since $\mu_n\to 0^+$ as $n\to \infty$, the continuity of $g$, Remark 2.1, (3.40) and (3.41) imply that

Similarly, if $\mu_n < 0,$ we also have

Hence, $\{u_{n}\}$ is a bounded (PS) sequence of $\bar{J}_{\lambda}$. By make a standard procedure for $\bar{J}_{\lambda}$ (see ^[37]), we can obtain a convergent subsequence, still denoted by $\{u_{n}\}$, such that $u_{n}\to u_{\lambda}$ for some $u_{\lambda}\in E$. Consequently, by (3.42), we have $\bar{J}_{\lambda}'(u_\lambda) = 0$ and $\bar{J}_{\lambda}(u_n)\to\bar{J}_{\lambda}(u_\lambda): = c_\lambda$ as $n\to \infty$, which shows that $u_\lambda$ is a critical point of $\bar{J}_{\lambda}(u)$. Moreover, it follows from (3.8) and (3.24) that

and

which implies that $u_{\lambda}$ is a nontrivial. It follows from Lemma 3.6 and Lemma 3.10 that

Then similar to the argument in (3.22) and (3.33) with $\mu = 0$, we can obtain that

Hence, we have

Then when $\lambda > \lambda^{*} = \max\{\Lambda_{M_0}, \Lambda_{21}, \Lambda_{22}, \Lambda_{3}, \Lambda_{4}^{-}\}$, where $\Lambda_3$ is defined by (1.20) and $K_{*}$ is defined by (1.21), Remark 2.1 implies that

Hence, $\bar{F}(x, u_\lambda) = F(x, u_\lambda)$. Thus $u_{\lambda}$ is a nontrival solution of Eq (1.16) if

and by (3.47) and (3.48), it is easy to see that

The proof is completed.

4.   Conclusions

Some sufficient conditions about existence of a nontrivial solution for Eq (1.1) are obtained. (1.1) is a generalization of (1.2) and (1.3) which can be used to describe the static equilibrium of an elastic beam. The nonlinear terms $F$ and $G$ are assumed to satisfy (H1)$-$(H7) which are some growth conditions only near the origin. The concrete lower bounds of the parameter $\lambda$ are given for the cases $\mu > 0$ and $\mu < 0$, respectively. Finally, in Theorem 1.3, the concentration phenomenon of $\{u_{\lambda, \mu}\}$ is revealed as $\mu\to 0$.

Acknowledgments

This project is supported by Yunnan Ten Thousand Talents Plan Young & Elite Talents Project and Candidate Talents Training Fund of Yunnan Province (No. 2017HB016).

Conflict of interest

The authors declare that they have no conflicts of interest.