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Non-uniformly asymptotically linear fourth-order elliptic problems
Boundary Value Problems volume 2015, Article number: 209 (2015)
Abstract
The existence of multiple solutions for a class of fourth-order elliptic equations with respect to the non-uniformly asymptotically linear conditions is established by using the minimax method and Morse theory.
1 Introduction
Let \(\mathbf{H}=H^{2}(\Omega)\cap H_{0}^{1}(\Omega)\) be a Hilbert space equipped with the inner product
and the deduced norm
Let \(\lambda_{k}\) (\(k=1,2,\ldots\)) denote the eigenvalues and \(\varphi_{k}\) (\(k=1,2,\ldots\)) the corresponding eigenfunctions of the eigenvalue problem
where each eigenvalue \(\lambda_{k}\) is repeated as the multiplicity; recall that \(0<\lambda_{1}<\lambda_{2}\leq \lambda_{3}\leq\cdots \leq\lambda_{k}\rightarrow\infty\) and that \(\varphi_{1}(x)>0\) for \(x\in\Omega\). We can easily observe that \(\Lambda_{k}=\lambda_{k}(\lambda_{k}-c)\), \(k=1,2,\ldots\) , are eigenvalues of the eigenvalue problem
and the corresponding eigenfunctions are still \(\varphi_{k}(x)\).
The set of \(\{\varphi_{k}(x)\} \) is an orthogonal base on space H; thus one may denote an element u of H as \(\sum_{k=1}^{\infty}u_{k}\varphi_{k}\), \(\sum_{k=1}^{\infty}u_{k}^{2}<\infty\).
Assume that \(c<\lambda_{1}\); let us define a norm of \(u\in \mathbf{H}\) as follows:
It is easy to show that the norm \(\|\cdot\|\) is a equivalent norm on H and the following Poincaré inequality holds:
for all \(u\in \mathbf{H}\).
Consider the following Navier boundary value problem:
where \(\Delta ^{2}\) is the biharmonic operator, Ω is a bounded smooth domain in \({\mathbb{R}}^{N}\) (\(N>4\)), and \(c<\lambda_{1}\).
Let f be a continuous function on \(\Omega\times\mathbb{R}\). Suppose that there are measurable real functions \(p(x)\) and \(q(x)\) on Ω such that
If the convergences in (1.2) and (1.3) are uniform in Ω, we say that f is uniformly asymptotically linear at zero and infinity. This case has been studied by many authors under various assumptions on \(p(x)\) and \(q(x)\).
In [1], An and Liu obtained the existence of one non-trivial solution of (1.1), if the following conditions are fulfilled:
-
(AL1)
\(f(x,t) \in C(\overline{\Omega}\times\mathbb{R})\); \(f(x,t)\equiv0\), \(\forall x \in \Omega\), \(t\leq0\), \(f(x,t)\geq0\), \(\forall x \in\Omega\), \(t>0\);
-
(AL2)
\(\frac{f(x,t)}{t}\) is nondecreasing with respect to \(t\geq 0\) for a.e. \(x \in\Omega\);
-
(AL3)
\(\lim_{|t|\rightarrow0} \frac{f(x,t)}{t}=\mu\); \(\lim_{|t|\rightarrow\infty} \frac{f(x,t)}{t}=\nu\) uniformly for a.e. \(x \in\Omega\), where \(\mu<\lambda_{1}(\lambda_{1}-c)<\nu<+\infty\), \(\nu\neq \Lambda_{k}=\lambda_{k}(\lambda_{k}-c)\) are constants.
In [2], under the above similar conditions, Qian and Li established the existence of three non-trivial solutions of (1.1) by use of mountain pass theorem and regularity of critical groups. Similarly, in [3], we also obtained three non-trivial solutions by using mountain pass theorem and Morse theory for problem (1.1) when the nonlinearity f is resonant at infinity or the nonlinearity f is not resonant at infinity. Pu et al. [4] proved the existence and multiplicity of solutions for the fourth Navier boundary value problems with concave term, which is similar to problem (1.1) when nonlinearity f is uniformly asymptotically linear at infinity. In [5], Wei established the existence of multiple solutions for problem (1.1) by means of bifurcation theory. Particularly, Liu and Huang [6] obtained one sign-changing solution for problem (1.1) with uniformly asymptotically linear nonlinearity term.
In the present paper, we study the problem in the case that f may be non-uniformly asymptotically linear. These new aspects with p-Laplacian were first presented by Duc and Huy in [7]. But they discussed asymmetric non-uniformly asymptotically linear situation and their methods are not directly to use the non-uniformly asymptotically linear Navier boundary value problems since \(u\in \mathbf{H}\) does not imply that \(u^{\pm} \in\mathbf{H}\), where \(u^{\pm}=\max\{\pm u,0\}\). Our main results are as follows.
Theorem 1.1
Suppose:
- (H1):
-
\(f(x,0)=0\), \(f(x,t)t\geq0\) for all \(x\in\Omega\), \(t\in\mathbb{R}\).
- (H2):
-
There exists r in the interval \((\frac{N}{4},\infty)\) such that \(q(x)\in L^{r}(\Omega)\) with \(\|q\|_{L^{r}}>0\).
- (H3):
-
There exists a nonnegative measurable function W on Ω such that:
-
(i)
$$\bigl\vert f(x,s)\bigr\vert \leq W(x)|s|,\quad \forall x\in\Omega, \forall s \in \mathbb{R}. $$
-
(ii)
There exists a constant \(K_{W} \) such that
$$\int_{\Omega}W|u|^{2}\,dx \leq K_{W}\|u \|^{2},\quad \forall u\in \mathbf{H}. $$ -
(iii)
For any sequence \(\{u_{m}\}\) converging weakly to u in H, there exists a measurable function g on Ω and a subsequence \(\{u_{m_{k}}\}\) of \(\{u_{m}\}\) having the following properties: \(|u_{m_{k}}|\leq g\) for a.e. \(x \in\Omega\), for any k and
$$\int_{\Omega}Wg^{2}\,dx< \infty. $$
-
(i)
- (H4):
-
\(\gamma(p)>1\) and \(\gamma(q)<1\), where
$$\gamma(p)=\inf \biggl\{ \int_{\Omega}\bigl(|\Delta u|^{2}-c|\nabla u|^{2}\bigr)\,dx, \int_{\Omega}p(x)u^{2}\,dx=1 \biggr\} , $$and the definition of \(\gamma(q)\) is similar.
Then problem (1.1) has at least two non-trivial solutions \(u_{1}\) and \(u_{2}\) such that \(u_{1}>0\), \(u_{2}<0\), \(I(u_{1})>0\), and \(I(u_{2})>0\), where
and
Theorem 1.2
Suppose:
- (H1∗):
-
\(f\in C^{1}(\overline{\Omega}\times{\mathbb{R}},{\mathbb{R}})\), \(f(x,0)=0\), \(f(x,t)t\geq0\) for all \(x\in\Omega\), \(t\in {\mathbb{R}}\).
- (H3∗):
-
There exists a nonnegative measurable function W on Ω with \(W \in L^{\infty}(\Omega)\) such that
$$\bigl\vert f_{s}(x,s)\bigr\vert \leq W(x) $$for all \(x \in \Omega\) and \(s \in \mathbb{R}\).
- (H4∗):
-
\(\gamma(p)>1\).
If \(\Lambda_{k}< q(x)<\Lambda_{k+1}\) for \(k\geq2\), then problem (1.1) has at least three non-trivial solutions.
Here we introduce a non-quadratic condition.
- (H5):
-
\(\lim_{|t|\rightarrow\infty}[tf(x,t)-2F(x,t)]=-\infty\) for every \(x\in\Omega\).
Theorem 1.3
Suppose (H1∗), (H3∗), (H4∗), and (H5) hold. If \(q(x)\equiv\Lambda_{k}\) for \(k\geq2\), then problem (1.1) has at least three non-trivial solutions.
2 Preliminary results
Let u be in H, F, and I be as in (1.4) and (1.5). Put
where
Combining with the knowledge of nonlinear functional analysis, we have the following lemmas.
Lemma 2.1
Under conditions (H1) and (H3), the functionals I and \(I_{+}\) belong to \(C^{1}(\mathbf{H},\mathbb{R})\). Moreover, for every u and v in H,
Proof
We only prove the lemma for I and the case of \(I^{+}\) is similar. From the knowledge of nonlinear functional analysis, we easily imply that the map \(u \mapsto\frac{1}{2}\|u\|^{2}\) is continuously Fréchet differentiable from H to H.
Let
for all \(u\in\mathbf{H}\). We prove that \(G\in C^{1}(\mathbf{H},\mathbb{R})\) by the following steps.
(i) Given \(u, v\in\mathbf{H}\) and \(s\in\mathbb{R}\setminus \{0\}\) with \(|s|\leq1\). Using the mean-value theorem, by (H3)(i), one gets
Using Hölder’s inequality and by (H3)(ii), we have
Since \(W|v|^{2}\) is integrable, combining (2.1), (2.2), by the Lebesgue dominated convergence theorem, we see that G is directional-differentiable on H and
Moreover, by the estimate (2.2), it follows that
Hence, \(DG(u)\) is a continuous linear functional on H and G is Gâteaux-differentiable on H.
(ii) We now prove that DG is continuous on H. Let \(\{u_{n}\}\) converging to u in H. Suppose by contradiction that \(DG(u_{n})\) does not converge to \(D(u)\). Then there exists \(\epsilon>0\), a subsequence of \(\{u_{n}\}\) (it will be also denoted by \(\{u_{n}\}\)) and a sequence \(\{v_{n}\}\in\mathbf{H}\) with \(\|v_{n}\|=1\) such that
By condition (H3)(iii), there exist measurable functions \(g_{1}\), \(g_{2}\), v, and a strictly increasing sequence \(\{n_{k}\}\) of positive integer numbers such that \(Wg_{i}^{2}\) is integrable and
It follows that
and by condition (H3)(i),
Let \(T=W g_{2} g_{1}+W|u|g_{1}\). By (H3)(ii), \(W(x)|u|^{2}\) is integrable on Ω and T is therefore integrable on Ω. Indeed, it follows from Hölder’s inequality that
Using the Lebesgue dominated convergence theorem, we obtain
which contradicts (2.3). □
Lemma 2.2
Under conditions (H1)-(H4), the functional \(I_{+}\) satisfies the (PS) condition.
Proof
Let \(\{u_{n}\}\subset\mathbf{H}\) be a sequence such that \(|I_{+}'(u_{n})|\leq c\), \(\langle I_{+}'(u_{n}),\phi\rangle\rightarrow0\) as \(n\rightarrow\infty\). Note that
for all \(\phi\in\mathbf{H}\).
1. We prove that \(\{\|u_{n}\|\}\) is bounded. Suppose by contradiction that there is a subsequence of \(\{u_{n}\}\) (also denoted by \(\{u_{n}\}\)) such that \(\|u_{n}\|\rightarrow\infty\). Put \(w_{n}=\frac{u_{n}}{\|u_{n}\|}\) for every \(n\in\mathbb{N}\). We have \(\|w_{n}\|=1\) for every n. Without loss of generality, we assume that \(w_{n}\rightharpoonup w\) in H, then \(w_{n}\rightarrow w\) in \(L^{2}(\Omega)\). Hence, \(w_{n} \rightarrow w\) a.e. in Ω. Dividing both sides of (2.4) by \(\|u_{n}\|\), we get
Note that if \(u_{n}(x)=0\) then \(w_{n}(x)=0\) and
Taking \(\phi=w_{n}\) in (2.5), we have
Using (H3)(i), we obtain
By (H3)(iii), using the Lebesgue dominated convergence theorem, we have
and thus
Letting \(n\rightarrow\infty\) in (2.6), we get
which is impossible. Therefore, we conclude that \(w\not\equiv0\).
Set \(D=\{ x: w(x)\neq0\}\). We have \(u_{n}(x)\rightarrow\infty\) for all \(x\in D\). Then condition (1.3) implies that
for all \(x\in D\). Similar to (2.7), we get
Since \(w_{n}\rightharpoonup w\), combining (2.8) and letting \(n\rightarrow\infty\) in (2.5), we have
Then we have
Meanwhile, let \(-\Delta w =u\), by the comparison maximum principle \(w>0\). This contradicts our assumption (H4). So \(\{u_{n}\}\) is bounded in H.
2. We prove that \(\{u_{n}\}\) has a strong convergent subsequence. Since H is a Hilbert space, we only need to prove that \(\|u_{n}\|\rightarrow\|u\|\). We may assume that \(u_{n}\rightharpoonup u\). Using (H3) and the Lebesgue dominated theorem, we obtain
Combining (2.4) and (2.10), we have
□
Remark 2.1
Under conditions (H1∗), (H3∗), and (H4∗), this lemma still holds.
Lemma 2.3
Under conditions (1.2), (H1), (H3), and (H4), there exist positive numbers ρ and η such that \(I_{+}(u)\geq \eta\) for all \(u\in\mathbf{H}\) with \(\|u\|=\rho\).
Proof
We adapt a new method from [7] to prove this conclusion. Suppose by contradiction that for every \(n\in \mathbb{N}\), there exists \(u_{n}\) in H such that \(\|u_{n}\|=n^{-1}\) and
Let \(w_{n}=nu_{n}\); then \(\|w_{n}\|=1\) and we can suppose that \(\{w_{n}\}\) weakly converges to w in H. Dividing both sides of (2.11) by \(\frac{1}{n^{2}}\), one has
Now, we claim that
From (H3)(i), we have
Therefore, by the Lebesgue dominated convergence theorem and (ii), (iii) of (H3),
So, our claim holds. Combining this claim and \(w_{n}\) weakly converging to w, we get
as \(n\rightarrow\infty\).
On the other hand, arguing as in the proof of (2.13), we have
Using (2.14), (2.15) and letting \(n\rightarrow\infty\) in (2.12), we obtain
According to assumption (H4), this leads to a contradiction. □
Remark 2.2
Under conditions (1.2), (H1∗), (H3∗), and (H4∗), this lemma still holds.
Lemma 2.4
Under conditions (H1)-(H4), the functional \(I_{+}\) satisfies
where \(\phi_{1}(q)\) is the first eigenfunction of the eigenvalue problem
Proof
Let \(u=\phi_{1}(q)\). Using principal eigenvalue theorem, we have \(u>0\) a.e. in Ω. By the Lebesgue dominated convergence theorem and (H3), we obtain
Combining (2.17) and (H4), we have
□
Remark 2.3
Under conditions (1.3), (H1∗), (H3∗), and (H4∗), this lemma still holds.
Lemma 2.5
Let \(\mathbf{H}=V\oplus W\), where \(V=E_{\lambda_{1}}\oplus E_{\lambda_{2}}\oplus\cdots\oplus E_{\lambda_{k}}\). If f satisfies (H1∗), (H3∗) and (H4∗) and \(\lambda_{k}\leq q(x)<\lambda_{k+1}\), then:
-
(i)
the functional I is coercive on W, that is,
$$I(u)\rightarrow+\infty\quad \textit{as } \|u\|\rightarrow+\infty, u\in W $$and bounded from below on W,
-
(ii)
the functional I is anti-coercive on V.
Proof
(i) Suppose by contradiction that there exist \(M>0\) and \(\{u_{n}\}\) in H such that \(\|u_{n}\|\rightarrow\infty\) and
Let \(w_{n}=\frac{u_{n}}{\|u_{n}\|}\); then \(\|w_{n}\|=1\). Now, we may assume that \(w_{n}\rightharpoonup w\) in W and \(w_{n}\rightarrow w\) a.e. \(x\in\Omega\). It is obvious that \(w\neq0\).
Dividing both sides of (2.18) by \(\|u_{n}\|^{2}\), we have
By (1.3) and (H3∗), using the Lebesgue dominated theorem and letting \(n\rightarrow\infty\) in (2.19), we get
Since \(w_{n}\rightharpoonup w\), from (2.20)
which leads to a contradiction.
(ii) Case 1. When \(\lambda_{k}< q(x)<\lambda_{k+1}\), similar to the proof of (i), it is easy to verify that the conclusion holds.
Case 2. When \(l=\lambda_{k}\), write \(G(x,t)=F(x,t)-\frac{1}{2}\lambda_{k}t^{2}\), \(g(x,t)=f(x,t)-\lambda_{k}t\). Then for every \(x\in\Omega\), (H5), and (1.3) imply that
and
It follows from (2.21) that for every \(M>0\), there exists a constant \(T>0\) (T depends on x) such that
For \(\tau>0\), we have
Integrating (2.24) over \([t,s]\subset[T,+\infty)\), we deduce that
Letting \(s\rightarrow+\infty\) and using (2.22), we see that \(G(x,t)\geq\frac{M}{2}\), for \(t\in{\mathbb{R}} \), \(t\geq T\). A similar argument shows that \(G(x,t)\geq\frac{M}{2}\), for \(t\in{\mathbb{R}}\), \(t\leq -T\). Hence, for every \(x\in\Omega\), we have
By (2.26), we get
for \(v\in V \) with \(\|v\|\rightarrow+\infty\), where \(v^{-}\in E_{\lambda_{1}}\oplus E_{\lambda_{2}}\oplus \cdots\oplus E_{\lambda_{k-1}}\). □
Lemma 2.6
Under conditions (1.3), (H1∗), and (H3∗), I satisfies the (PS) condition for \(\lambda_{k}< q(x)<\lambda_{k+1}\).
Proof
Let \(\{u_{n}\}\subset\mathbf{H}\) be a sequence such that \(|I'(u_{n})|\leq c\), \(\langle I'(u_{n}),\phi\rangle\rightarrow0\) as \(n\rightarrow\infty\). Note that
for all \(\phi\in\mathbf{H}\).
We prove that \(\{\|u_{n}\|\}\) is bounded. Suppose by contradiction that there is a subsequence of \(\{u_{n}\}\) (also denoted by \(\{u_{n}\}\)) such that \(\|u_{n}\|\rightarrow\infty\). Put \(w_{n}=\frac{u_{n}}{\|u_{n}\|}\) for every \(n\in\mathbb{N}\). We have \(\|w_{n}\|=1\) for every n. Without loss of generality, we assume that \(w_{n}\rightharpoonup w\) in H, then \(w_{n}\rightarrow w\) in \(L^{2}(\Omega)\). Hence, \(w_{n} \rightarrow w\) a.e. in Ω. Dividing both sides of (2.27) by \(\|u_{n}\|\), we get
Note that if \(u_{n}(x)=0\) then \(w_{n}(x)=0\) and
Taking \(\phi=w_{n}\) in (2.28), we have
Using (H3∗), we obtain
From the Lebesgue dominated convergence theorem, we have
and thus
Letting \(n\rightarrow\infty\) in (2.29), we get
which is impossible. Therefore, we conclude that \(w\not\equiv0\).
Set \(D=\{ x: w(x)\neq0\}\). We have \(u_{n}(x)\rightarrow\infty\) for all \(x\in D\). Then condition (1.3) implies that
for all \(x\in D\). Similar to (2.30), we get
Since \(w_{n}\rightharpoonup w\), combining (2.31) and letting \(n\rightarrow\infty\) in (2.28), we have
Then we have
This combined with our assumptions implies that \(w=0\) and it leads to a contradiction. So \(\{u_{n}\}\) is bounded in H. □
Lemma 2.7
Under conditions (1.3), (H1∗), (H3∗), and (H5), the functional I satisfies the (C) condition which is stated in [8] for \(q(x)=\Lambda_{k}\).
Proof
Suppose \({u_{n}}\in\mathbf{H}\) satisfies
According to the proof of Lemma 2.2, it suffices to prove that \(u_{n}\) is bounded in H. Similar to the proof of Lemma 2.6, we have
Therefore \(w\neq0\) is an eigenfunction of \(\lambda_{k}\), then \(|u_{n}(x)|\rightarrow\infty\) for a.e. \(x\in\Omega\). It follows from (H5) that
holds for every \(x\in\Omega\), which implies that
On the other hand, (2.33) implies that
Thus
which contradicts (2.35). Hence \(u_{n}\) is bounded. □
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 reader to [9] for more information on Morse theory.
Let H be a Hilbert space and \(I\in C^{1}(\mathbf{H},{\mathbb{R}})\) be a functional satisfying the (PS) condition or the (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 U be a neighborhood of \(u_{0}\). The group
is said to be the qth critical group of I at \(u_{0}\), where \(I^{c}= \{ u\in\mathbf{H}:I(u)\leq c\}\).
Let \(K:=\{u\in\mathbf{H}:I'(u)=0\}\) 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 [10])
The following result comes from [9, 10] and will be used to prove the result in this article.
Proposition 2.1
[10]
Assume that \(\mathbf{H}=H_{\infty}^{+}\oplus H_{\infty}^{-}\), I is bounded from below on \(H_{\infty}^{+}\) and \(I(u) \rightarrow-\infty\) as \(\|u\|\rightarrow\infty\) with \(u\in H_{\infty}^{-}\). Then
3 Proof of the main result
Proof of Theorem 1.1
By Lemma 2.2, Lemma 2.3, Lemma 2.4, and the mountain pass theorem, 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 (1.1). Similarly, we can obtain another negative critical point \(u_{2}\) of I. □
Proof of Theorem 1.2
By Remarks 2.1 and 2.3 and the mountain pass theorem, 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 (1.1) and satisfies
From conditions (H1∗) and (H3∗), we easily verify that I is \(C^{2}\) on H. Thus, by using the results in [2], we obtain
Similarly, we can obtain another negative critical point \(u_{2}\) of I satisfying
Since \(\gamma(p)>1\), the zero function is a local minimizer of I, then
On the other hand, by Lemma 2.5, Lemma 2.6, and Proposition 2.1, we have
Hence I has a critical point \(u_{3}\) satisfying
Since \(k\geq2\), it follows from (3.2)-(3.6) that \(u_{1}\), \(u_{2}\), and \(u_{3}\) are three different non-trivial solutions of the problem (1.1). □
Proof of Theorem 1.3
By Lemma 2.5, Lemma 2.7, and Proposition 2.1, we can prove the conclusion (3.5). The other proof is similar to that of Theorem 1.2. □
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Acknowledgements
This study was supported by the National NSF (Grant No. 11571268) of China, Natural Science Foundation of Gansu Province China (Grant No. 1506RJZE114) and Planned Projects for Postdoctoral Research Funds of Jiangsu Province (Grant No. 1301038C). The authors are very grateful for reviewers’ valuable comments and suggestions in improving this paper.
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Pei, R., Zhang, J. Non-uniformly asymptotically linear fourth-order elliptic problems. Bound Value Probl 2015, 209 (2015). https://doi.org/10.1186/s13661-015-0473-7
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DOI: https://doi.org/10.1186/s13661-015-0473-7