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Biharmonic equations with improved subcritical polynomial growth and subcritical exponential growth
Boundary Value Problems volume 2014, Article number: 162 (2014)
Abstract
The main purpose of this paper is to establish the existence of two nontrivial solutions and the existence of infinitely many solutions for a class of fourthorder elliptic equations with subcritical polynomial growth and subcritical exponential growth by using a suitable version of the mountain pass theorem and the symmetric mountain pass theorem.
Introduction
Consider the following Navier boundary value problem:
where {\mathrm{\u25b3}}^{2} is the biharmonic operator and Ω is a bounded smooth domain in {\mathbb{R}}^{N} (N\ge 4).
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,\dots) denote the eigenvalues of −△ in {H}_{0}^{1}(\mathrm{\Omega}).
Thus, fourthorder problems with N>4 have been studied by many authors. In [[1]], Lazer and McKenna pointed out that this type of nonlinearity furnishes 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\ge {\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.
We noticed that almost all of works (see [[4]–[9]]) 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}^{\ast}1) such that
where {p}^{\ast}=2N/(N4) denotes the critical Sobolev exponent. One of the main reasons to assume this condition (SCP) is that they can use the Sobolev compact embedding {H}^{2}(\mathrm{\Omega})\cap {H}_{0}^{1}(\mathrm{\Omega})\hookrightarrow {L}^{q}(\mathrm{\Omega}) (1\le q<{p}^{\ast}). At that time, it is easy to see that seeking a weak solution of problem (1) is equivalent to finding a nonzero critical points of the following functional on {H}^{2}(\mathrm{\Omega})\cap {H}_{0}^{1}(\mathrm{\Omega}):
In this paper, stimulated by Lam and Lu [[10]], our first main results will be to study problem (1) in the improved subcritical polynomial growth
which is much 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 (AR) condition, i.e., for some \theta >2 and \gamma >0,
In fact, this condition was studied by Liu and Wang in [[11]] in the case of Laplacian by the Nehari manifold approach. However, we will use a suitable version of the mountain pass theorem to get the nontrivial solution to problem (1) in the general case N>4. We will also use the symmetric mountain pass theorem to get infinitely many solutions for problem (1) in the general case N>4 when nonlinearity f is odd.
Let us now state our results. In this paper, we always assume that f(x,t)\in C(\overline{\mathrm{\Omega}}\times \mathbb{R}). The conditions imposed on f(x,t) are as follows:
(H_{1}): f(x,t)t\ge 0 for all x\in \mathrm{\Omega}, t\in \mathbb{R};
(H_{2}): {lim}_{t\to 0}\frac{f(x,t)}{t}={f}_{0} uniformly for x\in \mathrm{\Omega}, where {f}_{0} is a constant;
(H_{3}): {lim}_{t\to \mathrm{\infty}}\frac{f(x,t)}{t}=+\mathrm{\infty} uniformly for x\in \mathrm{\Omega};
(H_{4}): \frac{f(x,t)}{t} is nondecreasing in t\in \mathbb{R} for any x\in \mathrm{\Omega}.
Let 0<{\mu}_{1}<{\mu}_{2}<\cdots <{\mu}_{k}<\cdots be the eigenvalues of ({\mathrm{\u25b3}}^{2}c\mathrm{\u25b3},{H}^{2}(\mathrm{\Omega})\cap {H}_{0}^{1}(\mathrm{\Omega})) and {\phi}_{1}(x)>0 be the eigenfunction corresponding to {\mu}_{1}. Let {E}_{{\mu}_{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}(\mathrm{\Omega}) norm, c<{\lambda}_{1} in {\mathrm{\u25b3}}^{2}c\mathrm{\u25b3} and the norm of u in {H}^{2}(\mathrm{\Omega})\cap {H}_{0}^{1}(\mathrm{\Omega}) will be defined by
We also define E={H}^{2}(\mathrm{\Omega})\cap {H}_{0}^{1}(\mathrm{\Omega}).
Theorem 1.1
LetN>4and 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 two nontrivial solutions.
Theorem 1.2
LetN>4and assume that f has the improved subcritical polynomial growth on Ω (condition (SCPI)), is odd in t and satisfies (H_{3}) and (H_{4}). Iff(x,0)=0, then problem (1) has infinitely many nontrivial solutions.
In the case of N=4, we have {p}^{\ast}=+\mathrm{\infty}. So it is necessary to introduce the definition of the subcritical (exponential) growth in this case. By the improved Adams inequality (see [[12]]) for the fourthorder derivative, namely,
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\to \mathrm{\infty}}\frac{f(x,t)}{exp(\alpha {t}^{2})}=0 uniformly on x\in \mathrm{\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 results are as follows.
Theorem 1.3
LetN=4and 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 two nontrivial solutions.
Theorem 1.4
LetN=4and assume that f has the subcritical exponential growth on Ω (condition (SCE)), is odd in t and satisfies (H_{3}) and (H_{4}). Iff(x,0)=0, then problem (1) has infinitely many nontrivial solutions.
Preliminaries and auxiliary lemmas
Definition 2.1
Let (E,{\parallel \cdot \parallel}_{E}) be a real Banach space with its dual space ({E}^{\ast},{\parallel \cdot \parallel}_{{E}^{\ast}}) and I\in {C}^{1}(E,\mathbb{R}). For {c}^{\ast}\in \mathbb{R}, we say that I satisfies the {(\mathit{PS})}_{{c}^{\ast}} 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. Also, we say that I satisfies the {(C)}_{{c}^{\ast}} 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 [[13]]).
Proposition 2.1
Let E be a real Banach space and suppose thatI\in {C}^{1}(E,R)satisfies the condition
for some\alpha <\beta, \rho >0and{u}_{1}\in Ewith\parallel {u}_{1}\parallel >\rho. Let{c}^{\ast}\ge \betabe characterized by
where\mathrm{\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 Esuch that
Consider the following problem:
where
Define a functional {I}_{+}:E\to \mathbb{R} by
where {F}_{+}(x,t)={\int}_{0}^{t}{f}_{+}(x,s)\phantom{\rule{0.2em}{0ex}}ds, then {I}_{+}\in {C}^{1}(E,\mathbb{R}).
Lemma 2.1
LetN>4and{\phi}_{1}>0be a{\mu}_{1}eigenfunction with\parallel {\phi}_{1}\parallel =1and 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)\ge \alpha for all u\in E with \parallel u\parallel =\rho.

(ii)
{I}_{+}(t{\phi}_{1})\to \mathrm{\infty} as t\to +\mathrm{\infty}.
Proof
By (SCPI), (H_{2}) and (H_{3}), for any \epsilon >0, there exist {A}_{1}={A}_{1}(\epsilon ), {B}_{1}={B}_{1}(\epsilon ) and l>2{\mu}_{1} such that for all (x,s)\in \mathrm{\Omega}\times \mathbb{R},
Choose \epsilon >0 such that ({f}_{0}+\epsilon )<{\mu}_{1}. By (4), the Poincaré inequality and the Sobolev inequality {u}_{{p}^{\ast}}^{{p}^{\ast}}\le K{\parallel u\parallel}^{{p}^{\ast}}, we get
So, part (i) is proved if we choose \parallel u\parallel =\rho >0 small enough.
On the other hand, from (5) we have
Thus part (ii) is proved. □
Lemma 2.2
(see [[12]])
Let\mathrm{\Omega}\subset {\mathbb{R}}^{4}be a bounded domain. Then there exists a constantC>0such that
and this inequality is sharp.
Lemma 2.3
LetN=4and{\phi}_{1}>0be a{\mu}_{1}eigenfunction with\parallel {\phi}_{1}\parallel =1and 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)\ge \alpha for all u\in E with \parallel u\parallel =\rho.

(ii)
{I}_{+}(t{\phi}_{1})\to \mathrm{\infty} as t\to +\mathrm{\infty}.
Proof
By (SCE), (H_{2}) and (H_{3}), for any \epsilon >0, there exist {A}_{1}={A}_{1}(\epsilon ), {B}_{1}={B}_{1}(\epsilon ), \kappa >0, q>2 and l>2{\mu}_{1} such that for all (x,s)\in \mathrm{\Omega}\times \mathbb{R},
Choose \epsilon >0 such that ({f}_{0}+\epsilon )<{\mu}_{1}. By (6), the Holder inequality and Lemma 2.2, we get
where r>1 is sufficiently close to 1, \parallel u\parallel \le \sigma and \kappa r{\sigma}^{2}<32{\pi}^{2}. So, part (i) is proved if we choose \parallel u\parallel =\rho >0 small enough.
On the other hand, from (7) we have
Thus part (ii) is proved. □
Lemma 2.4
For the functional I defined by (3), if condition (H_{4}) holds, and for any\{{u}_{n}\}\in Ewith
then there is a subsequence, still denoted by\{{u}_{n}\}, such that
Proof
This lemma is essentially due to [[14]]. We omit it here. □
Proofs of the main results
Proof of Theorem 1.1
By Lemma 2.1 and Proposition 2.1, there exists a sequence \{{u}_{n}\}\subset E such that
Clearly, (9) implies that
To complete our proof, we first need to verify that \{{u}_{n}\} is bounded in E. Assume \parallel {u}_{n}\parallel \to +\mathrm{\infty} as n\to \mathrm{\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}(\mathrm{\Omega}).
We claim that if \parallel {u}_{n}\parallel \to +\mathrm{\infty} as n\to +\mathrm{\infty}, then {w}^{+}(x)\equiv 0. In fact, we set {\mathrm{\Omega}}_{1}=\{x\in \mathrm{\Omega}:{w}^{+}=0\}, {\mathrm{\Omega}}_{2}=\{x\in \mathrm{\Omega}:{w}^{+}>0\}. Obviously, by (11), {u}_{n}^{+}\to +\mathrm{\infty} a.e. in {\mathrm{\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 {\mathrm{\Omega}}_{2} and K>0 can be chosen large enough, so {\mathrm{\Omega}}_{2}=0 and {w}^{+}\equiv 0 in Ω. However, if {w}^{+}\equiv 0, then {lim}_{n\to +\mathrm{\infty}}{\int}_{\mathrm{\Omega}}F(x,{w}_{n}^{+})\phantom{\rule{0.2em}{0ex}}dx=0 and consequently
By \parallel {u}_{n}\parallel \to +\mathrm{\infty} as n\to +\mathrm{\infty} and in view of (11), we observe that {s}_{n}\to 0, 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 convergence subsequence. In fact, we can suppose that
Now, since f has the improved subcritical growth on Ω, for every \epsilon >0, we can find a constant C(\epsilon )>0 such that
then
Similarly, since {u}_{n}\rightharpoonup u in E, {\int}_{\mathrm{\Omega}}{u}_{n}u\phantom{\rule{0.2em}{0ex}}dx\to 0. Since \epsilon >0 is arbitrary, we can conclude that
By (10), we have
So we have {u}_{n}\to u in E which means that {I}_{+} satisfies {(C)}_{{c}^{\ast}}. Thus, from the strong maximum principle, we obtain that the functional {I}_{+} has a positive critical point {u}_{1}, i.e., {u}_{1} is a positive solution of problem (1). Similarly, we also obtain a negative solution {u}_{2} for problem (1). □
Proof of Theorem 1.2
It follows from the assumptions that I is even. Obviously, I\in {C}^{1}(E,\mathbb{R}) and I(0)=0. By the proof of Theorem 1.1, we easily prove that I(u) satisfies condition {(C)}_{{c}^{\ast}} ({c}^{\ast}>0). Now, we can prove the theorem by using the symmetric mountain pass theorem in [[15]–[17]].
Step 1. We claim that condition (i) holds in Theorem 9.12 (see [[16]]). Let {V}_{1}={E}_{{\mu}_{1}}\oplus {E}_{{\mu}_{2}}\oplus \cdots \oplus {E}_{{\mu}_{k}}, {V}_{2}=E\setminus {V}_{1}. For all u\in {V}_{2}, by (SCPI), we have
where a\in (0,1) is defined by
Choose \rho =\rho (k)=\parallel u\parallel so that the coefficient of {\rho}^{2} in the above formula is \frac{1}{4}. Therefore
for u\in \partial {B}_{\rho}\cap {V}_{2}. Since {\lambda}_{k}\to \mathrm{\infty} as k\to \mathrm{\infty}, \rho (k)\to \mathrm{\infty} as k\to \mathrm{\infty}. Choose k so that \frac{1}{4}{\rho}^{2}>2{c}_{6}. Consequently
Hence, our claim holds.
Step 2. We claim that condition (ii) holds in Theorem 9.12 (see [[16]]). By (H_{3}), there exists large enough M such that
So, for any u\in E\setminus \{0\}, we have
Hence, for every finite dimension subspace \tilde{E}\subset E, there exists R=R(\tilde{E}) such that
and our claim holds. □
Proof of Theorem 1.3
By Lemma 2.3, the geometry conditions of the mountain pass theorem (see Proposition 2.1) for the functional {I}_{+} hold. So, we only need to verify condition {(C)}_{{c}^{\ast}}. Similar to the previous part of the proof of Theorem 1.1, we easily know that {(C)}_{{c}^{\ast}} sequence \{{u}_{n}\} is bounded in E. Next, we prove that \{{u}_{n}\} has a convergence subsequence. Without loss of generality, suppose that
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 proof of Theorem 1.1, we have {u}_{n}\to u in E, which means that {I}_{+} satisfies {(C)}_{{c}^{\ast}}. Thus, from the strong maximum principle, we obtain that the functional {I}_{+} has a positive critical point {u}_{1}, i.e., {u}_{1} is a positive solution of problem (1). Similarly, we also obtain a negative solution {u}_{2} for problem (1). □
Proof of Theorem 1.4
Combining the proof of Theorem 1.2 and Theorem 1.3, we easily prove it. □
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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).
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Pei, R., Zhang, J. Biharmonic equations with improved subcritical polynomial growth and subcritical exponential growth. Bound Value Probl 2014, 162 (2014). https://doi.org/10.1186/s136610140162y
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DOI: https://doi.org/10.1186/s136610140162y