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Singular potential biharmonic problem with fixed energy
Boundary Value Problems volume 2016, Article number: 54 (2016)
Abstract
We investigate multiple solutions for the perturbation of a singular potential biharmonic problem with fixed energy. We get a theorem that shows the existence of at least one nontrivial weak solution under some conditions and fixed energy on which the corresponding functional of the equation satisfies the Palais-Smale condition. We obtain this result by variational method and critical point theory.
1 Introduction and statement of main result
Let Ω be a simply connected bounded domain of \(R^{n}\) with smooth boundary ∂Ω, \(n\ge3\). Let C be a closed interval containing 0 in R, and \(D=R^{+}\backslash C\) be the complement of C in \(R^{+}\). Let χ be any curve in Ω, \(x:S^{1}\to\Omega\subset R^{n}\) be a \(C^{4}\) curve such that \(x(t)\in \chi\subset\Omega\), and \(u\circ x:S^{1}\to R\) be the composition function of u and x such that \((u\circ x)(t)=u(x(t))\in D=R^{+}\backslash C\) for all \(t\in S^{1}\). Then \(u\circ x\) is a \(C^{4}\) function. Let \(c\in R\), Δ be the elliptic operator, and \(\Delta ^{2}\) be the biharmonic operator. Let us introduce the following subset of \(L^{q}(S^{1},R)\):
Then H is the loop space on D. Let us endow H with the norm
Then H is a Hilbert space. In this paper, we investigate the existence and multiplicity of weak solutions \(u\circ x\in H\) for the perturbation of the biharmonic equation with singular potential
where Λ, p, and q are real constants such that \(2< q< p\) and \(q<\frac{2n}{n-2}\). Throughout this paper, we deal with (1.1) with fixed energy
where h is a positive constant.
We assume that:
-
(A1)
(fixed energy) there exists a positive constant \(h>0\) such that
$$\begin{aligned}& \Lambda u\bigl(x(t)\bigr)+\frac{1}{q}\bigl\vert u\bigl(x(t)\bigr)\bigr\vert ^{q}-\frac{1}{p}\frac {1}{|u(x(t))|^{p}} \\& \quad =h, \quad u\bigl(x(t) \bigr)\in D=R^{+}\backslash C, x(t)\in\chi\subset\Omega, \forall t\in S^{1}; \end{aligned}$$ -
(A2)
there exists a neighborhood Z of C in R such that, for some constant \(A>0\),
$$-\frac{1}{2}\Lambda u\bigl(x(t)\bigr)^{2}-\frac{1}{q}\bigl\vert u\bigl(x(t)\bigr)\bigr\vert ^{q}+\frac {1}{p} \frac{1}{|u(x(t))|^{p}}\ge\frac{A}{d^{2}(u(x(t)),C)} $$for \(u(x(t))\in Z\), \(x(t)\in\chi\subset\Omega\), \(\forall t\in S^{1}\).
Our problems are characterized as singular biharmonic problems with singularity at \(\{u=0\}\). We recommend the book [1] for the singular elliptic problems. Many authors considered the biharmonic boundary value problem or the fourth-order elliptic boundary value problems. In particular, Choi and Jung [2] showed that the problem
has at least two nontrivial solutions when \(c<\lambda_{1}\), \(\lambda_{1}(\lambda_{1}-c)< b<\lambda_{2}(\lambda_{2}-c)\), and \(s<0\) or when \(\lambda_{1}< c<\lambda_{2}\), \(b<\lambda_{1}(\lambda_{1}-c)\), and \(s>0\). We obtained these results by using the variational reduction method. In [3], by using degree theory we also proved that when \(c<\lambda_{1}\), \(\lambda_{1}(\lambda_{1}-c)< b<\lambda_{2}(\lambda_{2}-c)\), and \(s<0\), problem (1.2) has at least three nontrivial solutions. Tarantello [4] also studied the problem
She showed that if \(c<\lambda_{1}\) and \(b\ge\lambda_{1}(\lambda _{1}-c)\), then problem (1.3) has a negative solution. She obtained this result by degree theory. Micheletti and Pistoia [5] also proved that if \(c<\lambda_{1}\) and \(b\ge \lambda_{2}(\lambda_{2}-c)\), then problem (1.3) has at least three solutions by variational linking theorem and Leray-Schauder degree theory. In this paper, we essentially work with variational techniques: We first prove that the associated functional of (1.1) satisfies the Palais-Smale condition, and then we use critical point theory.
Let \(\lambda_{1}<\lambda_{2}\le\cdots\le\lambda_{k}\le\cdots\) be the eigenvalues of the eigenvalue problem \(-\Delta u=\lambda u\) in Ω, \(u=0\) on ∂Ω, and let \(\phi_{k}\) be eigenfunctions corresponding to the eigenvalues \(\lambda_{k}\), \(k\ge 1\), suitably normalized with respect to the \(L^{2}(\Omega)\) inner product, where each eigenvalue \(\lambda_{k}\) is repeated with its multiplicity. We note that \(\phi_{1}(x)>0\) for \(x\in\Omega\). Then the eigenvalue problem
has infinitely many eigenvalues
and eigenfunctions \(\phi_{k}\) corresponding to the eigenvalues \(\nu _{k}=\lambda_{k}(\lambda_{k}-c)\), \(k\ge1\), suitably normalized with respect to the \(L^{2}(\Omega)\) inner product. We note that there exists a constant \(D>0\) such that \(\|u\| _{L^{q}(S^{1},R)}\le D\|u\|_{H}\) for \(q\ge1\) because \(\lambda _{i}(\lambda_{i}-c)\to\infty\) as \(i\to\infty\). In this paper we are trying to find weak solutions of equation (1.1) in H. The weak solutions of (1.1) in H satisfy
for all \(v\circ x\in H\). We shall show in Section 2 that there exists a one-to-one correspondence between weak solutions of (1.1) and critical points of the continuous and Frechét-differentiable functional
where
The Euler equation for J is (1.1).
Our main result is as follows.
Theorem 1.1
(Fixed energy problem)
Assume that \(\lambda_{k}< c<\lambda _{k+1}\), \(2< q< p\), \(q<\frac{2n}{n-2}\), \(\lambda_{k+m}(\lambda _{k+m}-c)<\Lambda<\lambda_{k+m+1}(\lambda_{k+m+1}-c)\), \(k\ge1\), \(m\ge1\), and conditions (A1) and (A2) hold. Then (1.1) has at least one nontrivial weak solution \(u(x)\) such that
For the proof of Theorem 1.1, we apply the variational technique. Under the assumptions of Theorem 1.1, we show that the functional \(J(u\circ x)\) satisfies the Palais-Smale condition, so that we can use the variational linking method in critical point theory. The outline of the proof of Theorem 1.1 is as follows. In Section 2, we introduce the eigenvalues and eigenfunctions of the eigenvalue problem \(\Delta^{2} u+c\Delta u-\Lambda u=\Lambda_{k} u\) in Ω, \(u=0\), \(\Delta u=0\) on ∂Ω, introduce the eigenspaces spanned by the eigenfunctions of \(\Lambda_{k}=\lambda_{k}(\lambda_{k}-c)-\Lambda\), investigate the properties of eigenspaces and prove that the functional \(J(u\circ x)\) satisfies the Palais-Smale condition. In Section 3, we divide the whole space H into two subspaces \(H^{+}(S,R)\) and \(H^{-}(S,R)\), find some inequalities of \(J(u\circ x)\) on two linked sublevel sets, and prove Theorem 1.1.
2 Eigenspace and Palais-Smale condition
Let us consider the eigenvalue problem
Let \(\Lambda_{i}\), \(i\ge1\), be eigenvalues of the eigenvalue problem (2.1), that is,
If \(\lambda_{k+m}(\lambda_{k+m}-c)<\Lambda<\lambda_{k+m+1}(\lambda _{k+m+1}-c)\), then
and
Let \(c_{\lambda_{i}(\lambda_{i}-c)}\) be eigenvectors of \(\lambda (\lambda-c)-\Lambda\) corresponding to the eigenvalues \(\Lambda_{i}\). Let us set
Then
Lemma 2.1
Assume that \(\lambda _{k}< c<\lambda_{k+1}\), \(2< q< p\), \(q<\frac{2n}{n-2}\), \(\lambda_{k+m}(\lambda_{k+m}-c)<\Lambda< \lambda _{k+m+1}(\lambda_{k+m+1}-c)\), \(k\ge1\), \(m\ge1\), and that conditions (A1) and (A2) hold. Let \(u\circ x\in L^{q}(S^{1},R)\) and \(\Lambda (u\circ x)+|u\circ x|^{q-1}+\frac{1}{|u\circ x|^{p+1}}\in L^{q}(S^{1},R)\). Then all the solutions of
belong to H.
Proof
Equation (1.1) can be rewritten as
Then there exists a constant \(D_{1}>0\) such that
Thus,
and the lemma is proved. □
Lemma 2.2
Assume that \(\lambda _{k}< c<\lambda_{k+1}\), \(2< q< p\), \(q<\frac{2n}{n-2}\), \(\lambda_{k+m}(\lambda_{k+m}-c)<\Lambda< \lambda _{k+m+1}(\lambda_{k+m+1}-c)\), \(k\ge1\), \(m\ge1\), and that conditions (A1) and (A2) hold. Then the functional \(J(u\circ x)\) is continuous and Fréchet differentiable with Fréchet derivative in H,
Moreover \(DJ\in C\), that is, \(J\in C^{1}\).
Proof
First, we shall prove that \(J(u\circ x)\) is continuous. Let \(u\circ x, v\circ x\in H\). Then since \(u(x(t)), v(x(t))\in D=R^{+}\backslash C\), \(\forall t\in S^{1}\), it follows that \(u(x(t))>0\), \(v(x(t))>0\), and \(u(x(t))+v(x(t))>0\). Thus, \(\frac {1}{|u(x(t))|^{p}}\) and \(\frac{1}{|u(x(t))+v(x(t))|^{p}}\) are well defined, continuous, and \(C^{1}\). Thus, we have
Thus, we have
Then there exist constants \(D_{1}>0\), \(D_{2}>0\), and \(D_{3}>0\) such that
Since \(u(x(t)), v(x(t))\in D\), we have that \(u(x(t))>0\), \(v(x(t))>0\) and \(u(x(t))+v(x(t))>0\). Thus, \(\frac{1}{|u(x(t))|^{p}}\) and \(\frac{1}{|u(x(t))+v(x(t))|^{p}}\) are well defined, continuous, and \(C^{1}\). By the mean value theorem we have
Thus, there exist constants \(D_{4}>0\), \(D_{5}>0\), \(D_{6}>0\), and \(D_{7}>0\) such that
Thus, we have
Next, we shall prove that \(J(u\circ x)\) is Fréchet differentiable. Let \(u\circ x, v\circ x\in H\). Then since \(u(x(t)), v(x(t))\in D\), it follows that \(u(x(t))>0\), \(v(x(t))>0\), and \(u(x(t))+v(x(t))>0\). Thus, \(\frac{1}{|u(x(t))|^{p}}\) and \(\frac{1}{|u(x(t))+v(x(t))|^{p}}\) are well defined, continuous, and \(C^{1}\). Thus, we have
By (2.3) and the same arguments as in the proof of the continuity of \(J(u\circ x)\) we have
Thus, \(J\in C^{1}\). □
Lemma 2.3
Assume that \(\lambda _{k}< c<\lambda_{k+1}\), \(2< q< p\), \(q<\frac{2n}{n-2}\), \(\lambda_{k+m}(\lambda_{k+m}-c)<\Lambda< \lambda _{k+m+1}(\lambda_{k+m+1}-c)\), \(k\ge1\), \(m\ge1\), and that conditions (A1) and (A2) hold. Then for any sequence \((u_{n}\circ x)_{n}\subset H\) such that \(u_{n}\circ x\rightharpoonup u\circ x\) weakly in H with \(u\circ x\in\partial H\), we have \(J(u_{n}\circ x)\to\infty\).
Proof
We claim that
By (A2) there exists a neighborhood Z of C in R such that for any \(u(x(t))\in Z\), \(\{-\frac{1}{2}\Lambda u_{n}(x(t))^{2}-\frac {1}{q}|u_{n}(x(t))|^{q}+\frac{1}{p}\frac{1}{|u_{n}(x(t))|^{p}}\}\) is bounded from below. Thus, it suffices to prove that there exists an interval \([t_{1},t_{2}]\subset[0,2\pi]\) such that
Since \(u\circ x\in\partial H\), there exists \(t^{*}\in[0,2\pi]\) such that \(u(x(t^{*}))\in\partial D\). By (A2) there exist constants \(A>0\) and \(B>0\) such that
Thus, we have
for all \(\delta>0\). On the other hand, we have
It follows from (2.4) that
Thus,
as \(\delta\to0\). Since the embedding \(H(S^{1},R)\hookrightarrow C(S^{1},R)\) is compact, we have
By Fatou’s lemma we have
as \(\delta\to0\). Thus,
so that \(J(u\circ x)\longrightarrow+\infty\), and the lemma is proved. □
Lemma 2.4
Assume that \(\lambda _{k}< c<\lambda_{k+1}\), \(2< q< p\), \(q<\frac{2n}{n-2}\), \(\lambda_{k+m}(\lambda_{k+m}-c)<\Lambda< \lambda _{k+m+1}(\lambda_{k+m+1}-c)\), \(k\ge1\), \(m\ge1\), and that conditions (A1) and (A2) hold. Then if \(\|u_{n}\circ x\|_{H}\to\infty\) and \((u_{n}\circ x)_{n}\) is a sequence in H such that
then there exist \((u_{h_{n}}\circ x)_{ n}\) and \(z\circ x\) in H such that
Proof
Since
the sequence \((\frac{\int^{2\pi}_{0}[ (|u_{n}(x(t))|^{q-1}+\frac {1}{|u_{n}(x(t))|^{p+1}} ) \cdot u_{n}(x(t))- (\frac {2}{q}|u_{n}(x(t))|^{q}-\frac{2}{p}\frac{1}{|u_{n}(x(t))|^{p}} )]x^{\prime}(t)\,dt}{\|u_{n}\circ x\|_{H}})_{n}\) is bounded, and there exists a constant \(C_{1}>0\) such that
Then we have
We note that
Then there exist constants \(C_{2}>0\) and \(C_{3}>0\) such that
where \(l=-1+\frac{q-1}{q}=-\frac{1}{q}<0\). Note that since \((\frac{\|u_{n}\circ x\|^{q}_{L^{q}(S^{1},R)}}{\| u_{n}\circ x\|_{H}} )^{\frac{q-1}{q}}\) is bounded, it follows from \(l<0\) that the right-hand side of (2.6) is bounded from above and
Thus, by (2.5),
and
Thus, the sequence \((\frac{\int^{2\pi}_{0}[ (|u_{n}(x(t))|^{q-1}+\frac{1}{|u_{n}(x(t))|^{p+1}} ) \cdot u_{n}(x(t))]x^{\prime}(t)\,dt}{\|u_{n}\circ x\|_{H}})_{n}\) is bounded. When \(2< q<\frac{2n}{n-2}\), the embedding \(H\hookrightarrow L^{q}(\Omega)\) is compact, Thus there exists a subsequence \((u_{h_{n}}\circ x)_{n}\) such that
We note that \(0<|u_{h_{n}}(x(t))|^{q-1}+\frac {1}{|u_{h_{n}}(x(t))|^{p+1}}\le\|u_{h_{n}}(x(t))\|^{q-1}_{L^{q}(S^{1},R)} +\|\frac{1}{|u_{h_{n}}(x(t))|^{p+1}}\|_{L^{q}(S^{1},R)}<\infty\). It follows from (2.7) that there exists \(z\circ x\) in H such that
Thus, the lemma is proved. □
Let us set
Then
We note that if \(\lambda_{k+m}(\lambda_{k+m}-c)<\Lambda<\lambda _{k+m+1}(\lambda_{k+m+1}-c)\), \(k\ge1\), \(m\ge1\), then
\(\operatorname{dim}H^{-}(S^{1},R)<\infty\), \(H^{0}(S^{1},R)=\emptyset\), and
Now, we shall prove that \(J(u\circ x)\) satisfies \((\mathit{P.S.})_{c}\) condition for \(c\in R\).
Lemma 2.5
(Palais-Smale condition)
Assume that \(\lambda_{k}< c<\lambda_{k+1}\), \(2< q< p\), \(q<\frac {2n}{n-2}\), \(\lambda_{k+m}(\lambda_{k+m}-c)<\Lambda<\lambda _{k+m+1}(\lambda_{k+m+1}-c)\), \(k\ge1\), \(m\ge1\), and that conditions (A1) and (A2) hold. Then \(J(u\circ x)\) satisfies \((\mathit{P.S.})_{c}\) condition for any \(c\in R\): if \((u_{n}\circ x)_{n}\in H\) is any sequence such that \(J(u_{n}\circ x)\to c\) and \(DJ(u_{n}\circ x)\to0\), then \((u_{n}\circ x)_{ n}\) has a convergent subsequence \((u_{n_{i}}\circ x)\) such that
Proof
Let \(c\in R\), and let \((u_{n}\circ x)_{n}\subset H\) be a sequence such that \(J(u_{n}\circ x)\to c\) and
or, equivalently,
where \((\Delta^{2}+c\Delta)^{-1}\) is a compact operator. We shall show that \((u_{n}\circ x)_{n}\) has a convergent subsequence. We claim that \(\{u_{n}\circ x\}\) is bounded in H. By contradiction we suppose that \(\|u_{n}\circ x\|_{H}\to\infty\) and set \(w_{n}\circ x=\frac{u_{n}\circ x}{\|u_{n}\circ x\|_{H}}\). Since \((w_{n}\circ x)_{n}\) is bounded, up to a subsequence, \((w_{n}\circ x)_{n}\) converges weakly to some \(w_{0}\circ x\) in H. Since \(J(u_{n}\circ x)\to c\) and \(DJ(u_{n}\circ x)\to0\), we have
Thus, we have
By Lemma 2.4 and (2.8) there exist \((u_{h_{n}}\circ x)_{ n}\) and \(z\circ x\) in H such that
Thus, we have \(w_{0}\circ x=0\), which is absurd because \(\|w_{0}\circ x\|_{H}=1\). Thus, \(\{u_{n}\circ x\}\) is bounded in H. Thus, \((u_{n}\circ x)_{n}\) has a convergent subsequence converging weakly to some \(u\circ x\) in H. We claim that this subsequence of \((u_{n}\circ x)_{n}\) converges strongly to \(u\circ x\). Since \(DJ(u_{n}\circ x)\to0\), we have
We claim that the mapping \(u_{n}\circ x\to\mapsto(|u_{n}\circ x|^{q-1}+\frac{1}{|u_{n}\circ x|^{p+1}})_{n}\) is compact. Since the embedding \(H\hookrightarrow L^{q}(S^{1},R)\) is compact for \(2< q<\frac{2n}{n-2}\), the mapping \(H\to L^{q}(S^{1},R):u_{n}\circ x\mapsto\int^{2\pi}_{0}(|u_{n}(x(t))|^{q-1}+\frac {1}{|u_{n}(x(t))|^{p+1}})u_{n}(x(t))x^{\prime}(t)\,dt\) is compact. Thus, the sequence \((\int^{2\pi}_{0}(|u_{n}(x(t))|^{q-1}+\frac {1}{|u_{n}(x(t))|^{p+1}})u_{n}(x(t))x^{\prime}(t)\,dt)_{n}\) has a convergent subsequence that converges to \(\int^{2\pi }_{0}(|u(x(t))|^{q-1}+\frac{1}{|u(x(t))|^{p+1}})u(x(t))x^{\prime}(t)\,dt\). Because \(\{u_{n}\circ x\}\) is bounded and the subsequence of \((u_{n}\circ x)_{n}\) converges weakly to some \(u\circ x\) in H, \((|u_{n}\circ x|^{q-1}+\frac{1}{|u_{n}\circ x|^{p+1}})_{n}\) has a convergent subsequence. Since \((|u_{n}\circ x|^{q-1}+\frac {1}{|u_{n}\circ x|^{p+1}})_{n}\) has a convergent subsequence, the subsequence of \((\Delta^{2}+c\Delta-\Lambda)(u_{n}\circ x)\) converges. Since \((\Delta^{2}+c\Delta-\Lambda)^{-1}\) is compact, the sequence \((u_{n}\circ x)_{n}\) has a subsequence converging strongly to \(u\circ x\) in H. □
3 Proof of Theorem 1.1
Let us set again
Then
We note that if \(\lambda_{k+m}(\lambda_{k+m}-c)<\Lambda<\lambda _{k+m+1}(\lambda_{k+m+1}-c)\), \(k\ge1\), \(m\ge1\), then
\(\operatorname{dim}H^{-}(S^{1},R)<\infty\), \(H^{0}(S^{1},R)=\emptyset\), and
Let us set
Let us define
Lemma 3.1
Under the assumptions of Theorem 1.1, there exists a large number \(R>0\) such that if \(e\circ x\in\partial B_{1}\cap H_{\lambda _{k+m+1}(\lambda_{k+m+1}-c)}(S^{1},R)\subset\partial B_{1}\cap H^{+}(S^{1},R)\) and \(u\circ x\in\partial Q=\partial(\bar{B_{R}}\cap H^{-}(S^{1},R)\oplus\{r(e\circ x)\mid 0< r< R\})\), then
Proof
Let us choose elements \(e\circ x\in\partial B_{1}\cap H_{\lambda _{k+m+1}(\lambda_{k+m+1}-c)}(S^{1},R)\subset\partial B_{1}\cap H^{+}(S^{1},R)\) and \(u\circ x\in H^{-}(S^{1},R)\oplus\{r(e\circ x)\mid r>0\}\). Then we have
If \(u\circ x\in\partial Q\), then since \(2< p\), there exists a constant CÌ„ such that \(\int^{2\pi}_{0}\frac{1}{p}\frac {1}{|u(x(t))|^{p}}x^{\prime}(t)\,dt<\bar{C}\). Thus, we have
Since \(2< q\), there exists a large number \(R>0\) such that if \(u\circ x\in\partial Q\), then \(J(u\circ x)<0\). Thus, we have \(\sup_{u\circ x\in\partial Q}J(u\circ x)<0\). Moreover, if \(u\circ x\in Q\), then \(J(u\circ x)\le\frac{1}{2}\Lambda_{k+m+1}\|u\circ x\| ^{2}_{L^{q}(S^{1},R)}+\bar{C}<\infty\). □
Lemma 3.2
Under the assumptions of Theorem 1.1, there exists a small number \(r>0\) such that
Proof
Let \(u\circ x\in\partial B_{r}\cap H^{+}(S^{1},R)\). Then we have
Since \(2< q\), there exists a small number \(r>0\) such that if \(u\circ x\in\partial B_{r}\cap H^{+}(S^{1},R)\), then \(J(u\circ x)>0\). Thus, \(\inf_{u\circ x\in\partial B_{r}\cap H^{+}(S^{1},R)}J(u\circ x)>0\). Moreover, if \(u\circ x\in B_{r}\cap H^{+}(S^{1},R)\), then \(J(u\circ x)\ge-\frac{1}{q}\|u\circ x\|^{q}_{L^{q}(S^{1},R)}>-\infty\). Thus, \(\inf_{u\circ x\in B_{r}\cap H^{+}(S^{1},R)}J(u\circ x)>-\infty\). So the lemma is proved. □
Let us define
Lemma 3.3
Under the assumptions of Theorem 1.1, we have
Proof
By Lemma 3.1 we have
By Lemma 3.2 we have
Thus, the lemma is proved. □
Proof of Theorem 1.1
Assume that \(\lambda_{k}< c<\lambda_{k+1}\), \(2< q< p\), \(q<\frac {2n}{n-2}\), \(\lambda_{k+m}(\lambda_{k+m}-c)<\Lambda<\lambda _{k+m+1}(\lambda_{k+m+1}-c)\), \(k\ge1\), \(m\ge1\), and that conditions (A1) and (A2) hold. Note that \(J(u\circ x)\) is continuous and Fréchet differentiable in H and \(DJ\in C\). By Lemma 2.5, \(J(u\circ x)\) satisfies \((\mathit{P.S.})_{c}\) condition for \(c\in R\). We claim that \(c=\inf_{h\in\Gamma}\sup_{u\circ x\in Q}J(h(u\circ x))>0\) is a critical value of \(J(u\circ x)\), that is, \(J(u\circ x)\) has a critical point \(u_{0}\circ x\) such that
In fact, by contradiction we suppose that \(c>0\) is not a critical value of \(J(u\circ x)\). Then by Theorem A.4 in [6], for any \(\bar{\epsilon }\in(0,c)>0\), there exist a constant \(\epsilon\in(0,\bar{\epsilon })\) and a deformation \(\eta\in C([0,1]\times H,H)\) such that:
-
(i)
\(\eta(0,u\circ x)=u\circ x\) for all \(u\circ x\in H\),
-
(ii)
\(\eta(s,u\circ x)=u\circ x\) for all \(s\in[0,1]\) if \(J(u\circ x)\notin[c-\bar{\epsilon},c+\bar{\epsilon}]\),
-
(iii)
\(J(\eta(1,u\circ x))\le c-\epsilon\) if \(J(u\circ x)\le c+\epsilon\).
We can choose \(h\in\Gamma\) such that
and
This leads to \(J(h(u\circ x))\notin[c-\bar{\epsilon},c+\bar {\epsilon}]\). Thus, by (ii),
Hence, \(\eta(1,h(u\circ x))\in\Gamma\). By (iii) and the definition of c,
which is a contradiction. Thus, c is a critical value of \(J(u\circ x)\). So \(J(u\circ x)\) has a critical point \(u_{0}\circ x\) with a critical value
such that
By Lemma 2.3,
Thus, (1.1) has at least one nontrivial solution \(u_{0}\) such that \(u_{0}(x(t))\neq0 \), and Theorem 1.1 is proved. □
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Acknowledgements
This work was supported by Inha University Research Grant.
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Jung, T., Choi, QH. Singular potential biharmonic problem with fixed energy. Bound Value Probl 2016, 54 (2016). https://doi.org/10.1186/s13661-016-0564-0
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DOI: https://doi.org/10.1186/s13661-016-0564-0