 Research
 Open Access
 Published:
Multiple solutions for biharmonic equations with improved subcritical polynomial growth and subcritical exponential growth
Boundary Value Problems volume 2015, Article number: 115 (2015)
Abstract
The main purpose of this paper is to establish the existence of three nontrivial solutions for a class of fourthorder elliptic equations with subcritical polynomial growth and subcritical exponential growth by using the minimax method and Morse theory.
Introduction
Consider the following Navier boundary value problem:
where \(\Delta^{2}\) is the biharmonic operator and Ω is a bounded smooth domain in \({\mathbb{R}}^{N}\) (\(N\geq4\)).
In problem (1), let \(f(x,u)=b[(u+1)^{+}1]\), then we get the following Dirichlet problem:
where \(u^{+}=\max\{u,0\}\) and \(b\in\mathbb{R}\). We let \(\lambda_{k}\) (\(k=1,2,\ldots\)) denote the eigenvalues of −Δ in \(H_{0}^{1}(\Omega)\).
Fourthorder problems of this class with \(N>4\) have been studied by many authors. In [1], Lazer and Mckenna pointed out that this type of nonlinearity provides a model to study traveling waves in suspension bridges. Since then, more general nonlinear fourthorder elliptic boundary value problems have been studied. For problem (2), Lazer and Mckenna [2] proved the existence of \(2k1\) solutions when \(N=1\), and \(b>\lambda_{k}(\lambda_{k}c)\) by the global bifurcation method. In [3], Tarantello found a negative solution when \(b\geq \lambda_{1}(\lambda_{1}c)\) by a degree argument. For problem (1) when \(f(x,u)=bg(x,u)\), Micheletti and Pistoia [4] proved that there exist two or three solutions for a more general nonlinearity g by the variational method. Xu and Zhang [5] discussed the problem when f satisfies the local superlinearity and sublinearity. Zhang [6] proved the existence of solutions for a more general nonlinearity \(f(x,u)\) under some weaker assumptions. Zhang and Li [7] proved the existence of multiple nontrivial solutions by means of Morse theory and local linking. An and Liu [8] and Liu and Wang [9] also obtained the existence result for nontrivial solutions when f is asymptotically linear at positive infinity. In [10], Zhang and Wei obtained the existence of infinitely many solutions when the nonlinearity involves a combination of superlinear and asymptotically linear terms. As far as the problem (1) is concerned, existence results of signchanging solutions were also obtained (see, e.g., [11, 12]).
We notice that almost all the works (see [4–12]) mentioned above involve the nonlinear term \(f(x,u)\) of a subcritical (polynomial) growth, say,

(SCP):
there exist positive constants \(c_{1}\) and \(c_{2}\) and \(q_{0}\in (1, p^{*}1)\) such that
$$\bigl\vert f(x,t)\bigr\vert \leq c_{1}+ c_{2}t^{q_{0}} \quad \mbox{for all } t\in\mathbb{R} \mbox{ and } x\in\Omega, $$
where \(p^{*}=2N/(N4)\) denotes the critical Sobolev exponent. One of the main reasons to assume this condition (SCP) is to use the Sobolev compact embedding \(H^{2}(\Omega) \cap H_{0}^{1}(\Omega)\hookrightarrow L^{q}(\Omega)\) (\(1\leq q< p^{*}\)). In that case, it is easy to see that seeking weak solutions of problem (1) is equivalent to finding nonzero critical points of the following functional on \(H^{2}(\Omega)\cap H_{0}^{1}(\Omega)\):
where \(F(x,u)=\int_{0}^{u}f(x,t)\, dt\). In this paper, inspired by Lam and Lu [13], our first main result will be to study problem (1) in the improved subcritical polynomial growth
which is weaker than (SCP). Note that, in this case, we do not have the Sobolev compact embedding anymore. Our work is to study problem (1) when nonlinearity f does not satisfy the AmbrosettiRabinowitz condition, i.e., for some \(\theta>2\) and \(\gamma>0\),
In fact, this condition was studied by Liu and Wang in [14] in the case of Laplacian by the Nehari manifold approach. However, we will use a suitable version of the mountain pass theorem and Morse theory to get three nontrivial solutions for problem (1) in the general case \(N>4\). Indeed, in this case, we have obtained two nontrivial solutions for problem (1) in [15] via the mountain pass theorem and a truncated technique.
Let us now state our results. In this paper, we always assume that \(f(x,t) \in C^{1}(\bar{\Omega}\times\mathbb{R})\). The conditions imposed on \(f(x,t)\) are as follows:
 (H_{1}):

\(f(x,t)t\geq0\) for all \(x\in\Omega\), \(t\in{\mathbb{R}}\);
 (H_{2}):

\(\lim_{t\rightarrow0}\frac{f(x,t)}{t}=f_{0}\) uniformly for \(x \in\Omega\), where \(f_{0}\) is a constant;
 (H_{3}):

\(\lim_{t\rightarrow\infty}\frac{f(x,t)}{t} =+\infty\) uniformly for \(x \in\Omega\);
 (H_{4}):

\(\frac{f(x,t)}{t}\) is nondecreasing in \(t\in \mathbb{R}\) for any \(x\in\Omega\).
Let \(0<\mu_{1}\) be the first eigenvalue of \((\Delta^{2}c\Delta ,H^{2}(\Omega)\cap H_{0}^{1}(\Omega))\) (\(c<\lambda_{1}\)) and \(\varphi_{1}(x)>0\) be the eigenfunction corresponding to \(\mu_{1}\). Throughout this paper, we denote by \(\cdot_{p}\) the \(L^{p}(\Omega)\) norm and the norm of u in \(H^{2}(\Omega)\cap H_{0}^{1}(\Omega)\) will be defined by
We also define \(E=H^{2}(\Omega)\cap H_{0}^{1}(\Omega)\). In fact, the norm \(\\cdot\\) is equivalent to another norm \(\\cdot\_{E} \) defined by
on E, i.e., there exist two positive constants \(C^{*}\), \(C^{**}\) such that
Theorem 1.1
Let \(N>4\) and assume that f has the improved subcritical polynomial growth on Ω (condition (SCPI)) and satisfies (H_{1})(H_{4}). If \(f_{0}<\mu_{1}\), then problem (1) has at least three nontrivial solutions.
In the case of \(N=4\), we have \(p^{*}=+\infty\). So it is necessary to introduce the definition of the subcritical (exponential) growth in this case. By the improved Adams inequality (see [16]) for the fourthorder derivative, we have
where C is a positive constant (see Lemma 2.2). So, we now define the subcritical (exponential) growth in this case as follows:

(SCE):
f has subcritical (exponential) growth on Ω, i.e., \(\lim_{t\rightarrow\infty}\frac{f(x,t)}{\exp(\alpha t^{2})} =0 \) uniformly on \(x\in\Omega\) for all \(\alpha>0\).
When \(N=4\) and f has the subcritical (exponential) growth (SCE), our work is still to study problem (1) without the (AR) condition. Our result is as follows.
Theorem 1.2
Let \(N=4\) and assume that f has the subcritical exponential growth on Ω (condition (SCE)) and satisfies (H_{1})(H_{4}). If \(f_{0}<\mu_{1}\), then problem (1) has at least three nontrivial solutions.
Remark 1.1
Indeed, in this case we have already obtained two nontrivial solutions for problem (1) in [15] via the mountain pass theorem and a truncated technique. So, this paper is a completion for our previous work (see [15]).
Preliminaries and auxiliary lemmas
Definition 2.1
Let \((E,\\cdot\_{E})\) be a real Banach space with its dual space \((E^{*},\\cdot\_{E^{*}})\) and \(I\in C^{1}(E,\mathbb{R})\). For \(c\in \mathbb{R}\), we say that I satisfies the \((C)_{c}\) condition (Cerami condition) if for any sequence \(\{x_{n}\}\subset E\) with
there is a subsequence \(\{ x_{n_{k}}\}\) such that \(\{ x_{n_{k}}\}\) converges strongly in E.
We have the following version of the mountain pass theorem (see [17]).
Proposition 2.1
Let E be a real Banach space and suppose that \(I \in C^{1}(E,R)\) satisfies the condition
for some \(\alpha<\beta\), \(\rho>0\) and \(u_{1}\in E\) with \(\u_{1}\>\rho\). Let \(c\geq\beta\) be characterized by
where \(\Gamma=\{\gamma\in C([0,1],E), \gamma(0)=0, \gamma(1)=u_{1}\}\) is the set of continuous paths joining 0 and \(u_{1}\). Then there exists a sequence \(\{u_{n}\}\subset E\) such that
Consider the following problem:
where
Define a functional \(I_{+}:E\rightarrow{\mathbb{R}}\) by
where \(F_{+}(x,t)=\int_{0}^{t} f_{+}(x,s)\, ds\); then \(I_{+}\in C^{1}(E,{\mathbb{R}})\).
Lemma 2.1
Let \(N>4 \) and \(\varphi_{1}> 0\) be a \(\mu_{1}\)eigenfunction with \(\\varphi_{1} \=1\) and assume that (H_{2}), (H_{3}) and (SCPI) hold. If \(f_{0} <\mu_{1}\), then

(i)
there exist \(\rho, \alpha>0\) such that \(I_{+}(u)\geq\alpha\) for all \(u\in E \) with \(\ u \=\rho\),

(ii)
\(I_{+}(t\varphi_{1} )\rightarrow\infty\) as \(t\rightarrow +\infty\).
Proof
By (SCPI), (H_{2}) and (H_{3}), for any \(\varepsilon>0\), there exist \(A_{1}=A_{1}(\varepsilon)\), \(B_{1}=B_{1}(\varepsilon)\) and \(l>2\mu_{1}\) such that for all \((x,s)\in \Omega\times\mathbb{R}\),
Choose \(\varepsilon>0\) such that \((f_{0}+\varepsilon)<\mu_{1}\). By (4), the Poincaré inequality and the Sobolev inequality \(u_{p^{*}}^{p^{*}}\leq K\u\^{p^{*}}\), we get
So, part (i) is proved if we choose \(\u\=\rho>0\) small enough.
On the other hand, from (5) we have
Thus part (ii) is proved. □
Lemma 2.2
(see [16])
Let \(\Omega\subset\mathbb{R}^{4}\) be a bounded domain. Then there exists a constant \(C>0\) such that
where \(\u\_{E}=(\int_{\Omega}\Delta u^{2}\, dx)^{\frac{1}{2}}\). This inequality is sharp.
Lemma 2.3
Let \(N=4\) and \(\varphi_{1}> 0\) be a \(\mu_{1}\)eigenfunction with \(\\varphi_{1} \=1\) and assume that (H_{2}), (H_{3}) and (SCE) hold. If \(f_{0} <\mu_{1}\), then

(i)
there exist \(\rho, \alpha>0\) such that \(I_{+}(u)\geq\alpha\) for all \(u\in E \) with \(\ u\=\rho\),

(ii)
\(I_{+}(t\varphi_{1} )\rightarrow\infty\) as \(t\rightarrow +\infty\).
Proof
By (SCE), (H_{2}) and (H_{3}), for any \(\varepsilon>0\), there exist \(A_{1}=A_{1}(\varepsilon)\), \(B_{1}=B_{1}(\varepsilon)\), \(\kappa>0\), \(q>2\) and \(l>2\mu_{1}\) such that for all \((x,s)\in \Omega\times\mathbb{R}\),
Choose \(\varepsilon>0\) such that \((f_{0}+\varepsilon)<\mu_{1}\). By (6), the Hölder inequality and Lemma 2.2, we get
where \(r>1\) is sufficiently close to 1, \(\u\_{E}\leq\sigma\) and \(\kappa r\sigma^{2}<32\pi^{2}\). So, part (i) is proved if we choose \(\u\=\rho>0\) small enough since \(\u\_{E}\leq C^{*}\u\\).
On the other hand, from (7) we have
Thus part (ii) is proved. □
Lemma 2.4
For the functional \(I_{+}\), if condition (H_{4}) holds, and for any \(\{u_{n}\} \in E\) with
then there is a subsequence, still denoted by \(\{u_{n}\}\), such that
Proof
This lemma is essentially due to [18]. We omit the proof. □
Lemma 2.5
Under the assumptions of Theorem 1.1, then \(I_{+}\) and I satisfy the \((C)_{c}\) condition.
Proof
We first do the proof for \(I_{+}\). Let \(\{u_{n}\}\subset E\) be a \((C)_{c}\) (\(c\in\mathbb{R}\)) sequence such that for every \(n\in\mathbb{N}\),
Clearly, (9) implies that
To complete our proof, we first need to verify that \(\{u_{n}\}\) is bounded in E. Assume \(\u_{n}\\rightarrow+\infty\) as \(n\rightarrow\infty\). Let
Since \(\{w_{n}\}\) is bounded in E, it is possible to extract a subsequence (denoted also by \(\{w_{n}\}\)) such that
where \(w_{n}^{+}= \max\{ w_{n},0\}\), \(w_{0}\in E\) and \(h\in L^{2}(\Omega)\).
We claim that if \(\u_{n}\\rightarrow+\infty\) as \(n\rightarrow +\infty\), then \(w^{+}(x)\equiv0\). In fact, we set \(\Omega_{1}=\{ x\in \Omega: w^{+}=0\}\), \(\Omega_{2}=\{ x\in\Omega: w^{+}>0\}\). Obviously, by (11), \(u_{n}^{+}\rightarrow+\infty\) a.e. in \(\Omega_{2}\); noticing condition (H_{3}), then, for any given \(K>0\), we have
From (10), (11), and (12), we obtain
Noticing that \(w^{+}>0\) in \(\Omega_{2}\) and \(K>0\) can be chosen large enough, so \(\Omega_{2}=0\) and \(w^{+}\equiv0\) in Ω. However, if \(w^{+}\equiv0\), then \(\lim_{n\rightarrow +\infty}\int_{\Omega}F_{+}(x,w_{n}^{+})\, dx=0\) and consequently
By \(\u_{n}\\rightarrow+\infty\) as \(n\rightarrow+\infty\) and in view of (11), we observe that \(s_{n}\rightarrow0\), then it follows from Lemma 2.4 and (8) that
Clearly, (13) and (14) are contradictory. So \(\{u_{n}\}\) is bounded in E.
Next, we prove that \(\{u_{n}\}\) has a convergent subsequence. In fact, we can suppose that
Now, since f has the improved subcritical growth on Ω, for every \(\varepsilon>0\), we can find a constant \(C(\varepsilon)>0\) such that
then
Similarly, since \(u_{n}\rightharpoonup u\) in E, \(\int_{\Omega}u_{n}u\, dx\rightarrow0\). Since \(\varepsilon>0\) is arbitrary, we can conclude that
By (10), we have
So we have \(u_{n}\rightarrow u\) in E, which means that \(I_{+}\) satisfies \((C)_{c}\).
Next we prove that I satisfies \((C)_{c}\). In fact, by (H_{4}), we have
for \((x,t)\in\Omega\times\mathbb{R}\) and \(s\in[0,1]\). By the proof of Lemma 2.1 in [19], we can similarly prove that \((C)_{c}\) sequence \(\{u_{n}\}\) is bounded. The other part of the proof is similar to the case already proved and is omitted. □
Lemma 2.6
Under the assumptions of Theorem 1.2, \(I_{+}\) and I satisfy the \((C)_{c}\) condition.
Proof
We only do the proof for \(I_{+}\). Similar to the first part in the proof of Lemma 2.5, we easily know that \((C)_{c}\) sequence \(\{u_{n}\}\) is bounded in E. Next, we prove that \(\{u_{n}\}\) has a convergent subsequence. Without loss of generality, suppose that
By the equivalence of the norm on E, we have
Let \(\beta=C^{*}\beta_{0}\). Now, since \(f_{+}\) has the subcritical exponential growth (SCE) on Ω, we can find a constant \(C_{\beta}>0\) such that
Thus, by the Adamstype inequality (see Lemma 2.2),
Similar to the last part in the proof of Lemma 2.5, we have \(u_{n}\rightarrow u\) in E, which means that \(I_{+}\) satisfies \((C)_{c}\). □
Computation of the critical groups
It is well known that critical groups and Morse theory are the main tools in solving elliptic partial differential equations. Let us recall some results which will be used later. We refer the readers to the book [20] for more information on Morse theory.
Let H be a Hilbert space and \(I\in C^{1}(H,{\mathbb{R}})\) be a functional satisfying the (PS) condition (or \((C)_{c}\) condition), and \(H_{q}(X,Y)\) be the qth singular relative homology group with integer coefficients. Let \(u_{0}\) be an isolated critical point of I with \(I(u_{0})=c\), \(c\in{\mathbb{R}}\), and let U be a neighborhood of \(u_{0}\). The group
is said to be the qth critical group of I at \(u_{0}\), where \(I^{c}= \{ u\in H:I(u)\leq c\}\).
Let \(K:=\{u\in H:I'(u)=0\}\) be the set of critical points of I and \(a<\inf I(K)\), the critical groups of I at infinity are formally defined by (see [21])
For the convenience of our proof, we firstly prove two important propositions.
Proposition 3.1
If the assumptions of Theorem 1.1 (or Theorem 1.2) hold, then
Proof
Let \(S^{\infty}=\{ u\in E : \u\=1\}\) be the unit sphere in E and \(B^{\infty}=\{u\in E : \u\\leq1\}\). By (H_{3}), for any \(M>0\) there exists \(c>0\), such that \(F(x,t)\geq Mt^{2}c\), for \((x,t)\in\Omega\times\mathbb{R}\), which implies
for any \(u\in S^{\infty}\). Using (H_{4}), we have
Choose
Then, for any \(u\in S^{\infty}\), there exists \(t>1\) such that \(I(tu)\leq a\), that is,
which implies
The rest of the proof is similar to the proof of Lemma 2.6 in [19]. □
Proposition 3.2
If the assumptions of Theorem 1.1 (or Theorem 1.2) hold, then
Proof
By Lemma 2.1 (Lemma 2.3), \(u=0\) is a local minimizer of I. So we have
□
Proof of the main results
Proofs of Theorem 1.1 and Theorem 1.2
By Lemma 2.1 (Lemma 2.3), Lemma 2.5 (Lemma 2.6), and Proposition 2.1, the functional \(I_{+}\) has a critical point \(u_{1}\) satisfying \(I_{+}(u_{1})\geq\beta\). Since \(I_{+}(0)=0\), \(u_{1}\neq0\) and by the maximum principle, we get \(u_{1}>0\). Hence \(u_{1}\) is a positive solution of the problem (1) and satisfies
Using Lemma 3.1 in [22], we obtain
Similarly, we can obtain another negative critical point \(u_{2}\) of I satisfying
Now, from Proposition 3.2, we have
On the other hand, from Proposition 3.1, we have
Then from (19), (20), (21), (22), and the Morse relation, we have
with \(Q(t)=\sum_{q\geq0}d_{q}t^{q}\) where \(d_{q}\in \mathbb{N}=\{0,1,2,\ldots,n,\ldots\}\) for all \(q\geq0\). Then \(d_{0}=d_{1}=1\) and so the righthand side has a term \(t^{2}\), a contradiction. This means that there is one more critical point of I, \(u_{3}\notin\{0,u_{1},u_{2}\}\). Then \(u_{3}\in E\) is a nontrivial solution of problem (1). □
References
 1.
Lazer, AC, Mckenna, PJ: Large amplitude periodic oscillation in suspension bridges: some new connections with nonlinear analysis. SIAM Rev. 32, 537578 (1990)
 2.
Lazer, AC, Mckenna, PJ: Global bifurcation and a theorem of Tarantello. J. Math. Anal. Appl. 181, 648655 (1994)
 3.
Tarantello, G: A note on a semilinear elliptic problem. Differ. Integral Equ. 5, 561566 (1992)
 4.
Micheletti, AM, Pistoia, A: Multiplicity solutions for a fourth order semilinear elliptic problems. Nonlinear Anal. TMA 31, 895908 (1998)
 5.
Xu, GX, Zhang, JH: Existence results for some fourthorder nonlinear elliptic problems of local superlinearity and sublinearity. J. Math. Anal. Appl. 281, 633640 (2003)
 6.
Zhang, JH: Existence results for some fourthorder nonlinear elliptic problems. Nonlinear Anal. TMA 45, 2936 (2001)
 7.
Zhang, JH, Li, SJ: Multiple nontrivial solutions for some fourthorder semilinear elliptic problems. Nonlinear Anal. TMA 60, 221230 (2005)
 8.
An, YK, Liu, RY: Existence of nontrivial solutions of an asymptotically linear fourthorder elliptic equation. Nonlinear Anal. TMA 68, 33253331 (2008)
 9.
Liu, Y, Wang, ZP: Biharmonic equations with asymptotically linear nonlinearities. Acta Math. Sci. 27, 549560 (2007)
 10.
Zhang, J, Wei, ZL: Infinitely many nontrivial solutions for a class of biharmonic equations via fountain theorems. Nonlinear Anal. TMA 74, 74747485 (2011)
 11.
Liu, XQ, Huang, YS: On signchanging solution for a fourthorder asymptotically linear elliptic problem. Nonlinear Anal. TMA 72, 22712276 (2010)
 12.
Zhou, JW, Wu, X: Signchanging solutions for some fourthorder nonlinear elliptic problems. J. Math. Anal. Appl. 342, 542558 (2008)
 13.
Lam, N, Lu, GZ: NLaplacian equations in \(\mathbb{R}^{N}\) with subcritical and critical growth without the AmbrosettiRabinowitz condition. Adv. Nonlinear Stud. 13, 289308 (2013)
 14.
Liu, ZL, Wang, ZQ: On the AmbrosettiRabinowitz superlinear condition. Adv. Nonlinear Stud. 4, 563574 (2004)
 15.
Pei, RC, Zhang, JH: Biharmonic equations with improved subcritical polynomial growth and subcritical exponential growth. Bound. Value Probl. 2014, 162 (2014)
 16.
Ruf, B, Sani, F: Sharp Adamstype inequalities in \(\mathbb{R}^{N}\). Trans. Am. Math. Soc. 365, 645670 (2013)
 17.
Costa, DG, Miyagaki, OH: Nontrivial solutions for perturbations of the pLaplacian on unbounded domains. J. Math. Anal. Appl. 193, 737755 (1995)
 18.
Zhou, HS: Existence of asymptotically linear Dirichlet problem. Nonlinear Anal. TMA 44, 909918 (2001)
 19.
Sun, JJ, Ma, SW: Nontrivial solutions for Kirchhoff type equations via Morse theory. Discrete Contin. Dyn. Syst. 13, 483494 (2014)
 20.
Chang, KC: Infinite Dimensional Morse Theory and Multiple Solutions Problems. Birkhäuser, Boston (1993)
 21.
Bartsch, T, Li, SJ: Critical point theory for asymptotically quadratic functionals and applications to problems with resonance. Nonlinear Anal. 28, 419441 (1997)
 22.
Qian, AX, Li, SJ: Multiple solutions for a fourthorder asymptotically linear elliptic problem. Acta Math. Sin. 22, 11211126 (2006)
Acknowledgements
This study was supported by the National NSF (Grant No. 11101319) of China and Planned Projects for Postdoctoral Research Funds of Jiangsu Province (Grant No. 1301038C). The authors are very grateful for the reviewers’ valuable comments and suggestions in improving this paper.
Author information
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Received
Accepted
Published
DOI
Keywords
 Morse theory
 Adamstype inequality
 subcritical polynomial growth
 subcritical exponential growth