Superlinear gradient system with a parameter

In this paper, we study the existence of multiple solutions to the gradient system

with the functions $a,b,c\in C(\overline{\mathrm{\Omega }},\mathbb{R})$ satisfying the conditions that $b(x)⩾0$ for all $x\in \overline{\mathrm{\Omega }}$, which means A is cooperative and that ${max}_{x\in \overline{\mathrm{\Omega }}}\{a,c\}>0$.

We impose the following assumptions on the function F:

($F_1$) $F\in C^2(\mathrm{\Omega }\times {\mathbb{R}}^2,\mathbb{R})$.

($F_2$) $F(x,0)=0$, $\mathrm{\nabla }F(x,0)=0$, $F_z^{″}(x,0)=0$ for $x\in \mathrm{\Omega }$.

for all $x\in \mathrm{\Omega }$, $|z|⩾M$.

($F_5$) $F_z^{″}(x,z)>0$ for $|z|>0$ small and $x\in \mathrm{\Omega }$.

($F_6$) $F_z^{″}(x,z)<0$ for $|z|>0$ small and $x\in \mathrm{\Omega }$.

Here and in the sequel, 0 is used to denote the origin in various spaces, $|\cdot |$ and $(\cdot ,\cdot )$ denote the norm and the inner product in ${\mathbb{R}}^2$, Bz denotes the matrix product in ${\mathbb{R}}^2$ for a $2\times 2$ matrix B and $z=(u,v)\in {\mathbb{R}}^2$. For two symmetric matrices B and C in ${\mathbb{R}}^2$, $B>C$ means that $B-C$ is positive definite.

is well defined and is of class $C^2$ (see [1]) with derivatives

for $z=(u,v)$, $w=(\varphi ,\psi )$, $z_1=(u_1,v_1)$, $z_2=(u_2,v_2)\in E$. Therefore, the solutions to (GS) _λ are exactly critical points of Φ in E.

By ($F_2$) the system (GS) _λ admits a trivial solution $z=0$ for any fixed parameter $\lambda \in \mathbb{R}$. We are interested in finding nontrivial solutions to (GS) _λ . The existence of nontrivial solutions of (GS) _λ depends on the behaviors of F near zero and infinity. The purpose of this paper is to find multiple nontrivial solutions to (GS) _λ with superlinear term when the trivial solution $z=0$ acts as a local saddle point of the energy functional Φ in the sense that the parameter λ is close to a higher eigenvalue of the linear gradient system with the given weight matrix A

such that ${\lambda }_k^A\to \mathrm{\infty }$ as $k\to \mathrm{\infty }$.

Denote by $F^{-}$ the negative part of F, i.e., $F^{-}(x,z)=max\{-F(x,z),0\}$.

We will prove the following theorems.

Theorem 1.1 Assume ($F_1$)-($F_4$), ($F_5$) and let $k⩾1$ be fixed. Then there is $\delta >0$ such that when ${sup}_{(x,z)\in \mathrm{\Omega }\times {\mathbb{R}}^2}{F}^{-}(x,z)⩽\delta$, for all $\lambda \in ({\lambda }_{k+1}^A-\delta ,{\lambda }_{k+1}^A)$, (GS) _λ has at least three nontrivial solutions in E.

Theorem 1.2 Assume ($F_1$)-($F_4$), ($F_6$) and let $k⩾1$ be fixed. Then there is $\delta >0$ such that when ${sup}_{(x,z)\in \mathrm{\Omega }\times {\mathbb{R}}^2}{F}^{-}(x,z)⩽\delta$, for all $\lambda \in ({\lambda }_{k+1}^A,{\lambda }_{k+1}^A+\delta )$, (GS) _λ has at least three nontrivial solutions in E.

Theorem 1.3 Assume ($F_1$)-($F_4$) and $F⩽0$ for $x\in \mathrm{\Omega }$, $z\in {\mathbb{R}}^2$ with $|z|$ small. Then there is $\delta >0$ such that when ${sup}_{(x,z)\in \mathrm{\Omega }\times {\mathbb{R}}^2}{F}^{-}(x,z)⩽\delta$, for all $\lambda \in ({\lambda }_{k+1}^A-\delta ,{\lambda }_{k+1}^A]$, (GS) _λ has at least two nontrivial solutions in E.

We give some comments and comparisons. The superlinear problems have been studied extensively via variational methods since the pioneering work of Ambrosetti and Rabinowitz [4]. Most known results on elliptic superlinear problems are contributed to a single equation with Dirichlet boundary data. Let us mention some historical progress on a single equation. When the trivial solution 0 acted as a local minimizer of the energy functional, one positive solution and one negative solution were obtained by using the mountain-pass theorem in [4] and the cut-off techniques; and a third solution was constructed in a famous paper of Wang [5] by using a two dimensional linking method and a Morse theoretic approach. When the trivial solution 0 acted as a local saddle point of the energy functional, the existence of one nontrivial solution was obtained by applying a critical point theorem, which is now well known as the generalized mountain-pass theorem, built by Rabinowitz in [6] under a global sign condition (see [1]). Some extensions were done in [7, 8]via local linking. More recently, in the work of Rabinowitz, Su and Wang [9], multiple solutions have been obtained by combining bifurcation methods, Morse theory and homological linking when 0 is a saddle point in the sense that the parameter λ is very close to a higher eigenvalue of the related linear operator.

In the current paper, we build multiplicity results for superlinear gradient systems by applying the ideas constructed in [9]. These results are new since, to the best of our knowledge, no multiplicity results for gradient systems have appeared in the literature for the case that $z=0$ is a saddle point of Φ.

We give some explanations regarding the conditions and conclusions. The assumptions ($F_1$)-($F_4$) are standard in the study of superlinear problems. ($F_5$) and ($F_6$) are used for bifurcation analysis. It sees that ($F_5$) implies that F is positive near zero, while ($F_6$) implies that F must be negative near zero. The local properties of F near zero are necessary for constructing homological linking. When $F⩾0$, for any parameter λ in a bounded interval, say in $[{\lambda }_k^A,{\lambda }_{k+1}^A)$, one can use the same arguments as in [1] to construct linking starting from ${\lambda }_{k+1}^A$. In our theorems, we do not require the global sign condition $F⩾0$. When the parameter λ is close to the eigenvalue ${\lambda }_{k+1}^A$, the homological linking will be constructed starting from ${\lambda }_{k+2}^A$ and this linking is different from the one in [1]. This reveals the fact that when λ is close to ${\lambda }_{k+1}^A$ from the right-hand side, the linking starting from ${\lambda }_{k+2}^A$ can still be constructed even if F is negative somewhere. The conditions similar to ($F_5$) and ($F_6$) were first introduced in [10] where multiple periodic solutions for the second-order Hamiltonian systems were studied via the ideas in [9]. Since we treat a different problem in the current paper, we need to present the detailed discussions although some arguments may be similar to those in [9, 10].

The paper is organized as follows. In Section 2, we collect some basic abstract tools. In Section 3, we get solutions by linking arguments and give partial estimates of homological information. In Section 4, we get solutions by bifurcation theorem and give the estimates of the Morse index. The final proofs of Theorems 1.1-1.3 are given in Section 5.

In this section, we give some preliminaries that will be used to prove the main results of the paper. We first collect some basic results on the Morse theory for a $C^2$ functional defined on a Hilbert space.

Let E be a Hilbert space and $\mathrm{\Phi }\in C^2(E,\mathbb{R})$. Denote $\mathcal{K}=\{z\in E|{\mathrm{\Phi }}^{\prime }(z)=0\}$, ${\mathrm{\Phi }}^c=\{z\in E|\mathrm{\Phi }(z)⩽c\}$, ${\mathcal{K}}_c=\{z\in \mathcal{K}|\mathrm{\Phi }(z)=c\}$ for $c\in \mathbb{R}$. We say that Φ satisfies the (PS) _c condition at the level $c\in \mathbb{R}$ if any sequence $\{z_n\}\subset E$ satisfying $\mathrm{\Phi }(z_n)\to c$, ${\mathrm{\Phi }}^{\prime }(z_n)\to 0$ as $n\to \mathrm{\infty }$, has a convergent subsequence. Φ satisfies (PS) if Φ satisfies (PS) _c at any $c\in \mathbb{R}$.

is called the q th critical group of Φ at $z_0$, where $H_{\ast }(A,B)$ denotes a singular relative homology group of the pair $(A,B)$ with coefficients field $\mathbb{F}$ (see [11, 12]).

is called the q th critical group of Φ at infinity (see [13]).

We call $M_q:={\sum }_{z\in \mathcal{K}}dim{C}_q(\mathrm{\Phi },z)$ the q th Morse-type numbers of the pair $(E,{\mathrm{\Phi }}^a)$ and ${\beta }_q:=dim{C}_q(\mathrm{\Phi },\mathrm{\infty })$ the Betti numbers of the pair $(E,{\mathrm{\Phi }}^a)$. The core of the Morse theory [11, 12] is the following relations between $M_q$ and ${\beta }_q$:

If $\mathcal{K}=\mathrm{\varnothing }$, then ${\beta }_q=0$ for all q. Since $M_q⩾{\beta }_q$ for each $q\in \mathbb{Z}$, it follows that if ${\beta }_{q_{\ast }}\ne 0$ for some $q_{\ast }\in \mathbb{Z}$, then Φ must have a critical point $z_{\ast }$ with $C_{q_{\ast }}(\mathrm{\Phi },z_{\ast })\ncong 0$. If $\mathcal{K}=\{z_{\ast }\}$, then $C_q(\mathrm{\Phi },\mathrm{\infty })\cong C_q(\mathrm{\Phi },z_{\ast })$ for all q. Thus, if $C_q(\mathrm{\Phi },\mathrm{\infty })\ncong C_q(\mathrm{\Phi },z_{\ast })$ for some q, then Φ must have a new critical point. One can use critical groups to distinguish critical points obtained by other methods and use the Morse equality to find new critical points.

For the critical groups of Φ at an isolated critical point, we have the following basic facts (see [11, 12]).

Proposition 2.2 ([15, 16])

Then $C_q(\mathrm{\Phi },0)\cong {\delta }_{q,\kappa }\mathbb{Z}$ for either $\kappa =m_0$ or $\kappa =m_0+n_0$.

The concept of local linking was introduced in [7]. In [15] a partial result was given for a $C^1$ functional. The above result was obtained in [16].

Now, we recall an abstract linking theorem which is from [1, 12, 15].

Proposition 2.3 ([1, 12, 15])

Let E be a real Banach space with $E=X\oplus Y$ and $\ell =dimX$ be finite. Suppose that $\mathrm{\Phi }\in C^1(E,\mathbb{R})$ satisfies (PS) and

where $B_{\rho }=\{z\in E|\parallel z\parallel ⩽\rho \}$,

We note here that under the framework of Proposition 2.3, $S_{\rho }$ and ∂Q homotopically link with respect to the direct sum decomposition $E=X\oplus Y$. $S_{\rho }$ and ∂Q are also homologically linked. The conclusion (2.3) follows from Theorems 1.1′ and 1.5 of Chapter II in [12]. (See also [15].)

Then the following variational inequalities hold:

We refer to [2, 3] for more properties related to the eigenvalue problem (${\mathrm{L}}_A$) and the operator $T_A$.

In this section, we give the existence a nontrivial solution of (GS) _λ by applying homological linking arguments and then give some estimate of its Morse index. The following lemmas are needed.

Lemma 3.1 Assume that F satisfies ($F_1$)-($F_4$), then for any fixed $\lambda \in \mathbb{R}$, the functional Φ satisfies the (PS) condition.

Therefore,

Therefore, $\{z_n\}$ is bounded in E. The proof is complete. □

The constants α and ρ are independent of $\lambda ⩽{\lambda }_{\ast }$. The proof is complete. □

The proof is complete. □

Now, we apply Proposition 2.3 to get the following existence result with partial homological information.

Since $S_{k+1}$ and ∂Q homotopically link with respect to the decomposition $E=E_{k+1}\oplus E_{k+1}^{\perp }$, and $dim{V}_{k+1}={\ell }_{k+1}+1$, it follows from Proposition 2.3 that Φ has a critical point $z^{\ast }\in E$ with positive energy $\mathrm{\Phi }(z^{\ast })⩾\alpha >0$ and its critical group satisfying (3.13). The proof is complete. □

Therefore, when a global sign condition $F⩾0$ is present, as λ is close to ${\lambda }_{k+1}^A$ from the left-hand side, two linkings can be constructed and two nontrivial solutions can be obtained. The question is how to distinguish $z^{\ast }$ from $z_{\ast }$. Theorem 3.4 includes the case that for λ close to ${\lambda }_{k+1}^A$ from the right-hand side, the linking with respect to $E_{k+1}\oplus E_{k+1}^{\perp }$ is constructed provided the negative values of F are small. This phenomenon was first observed in [9].

In this section, we get two solutions for (GS) _λ via bifurcation arguments [1]. We first cite the bifurcation theorem in [1].

Proposition 4.1 (Theorem 11.35 in [1])

We apply Proposition 4.1 to get two nontrivial solutions of (GS) _λ for λ close to an eigenvalue of (${\mathrm{L}}_A$) and then give the estimates of the Morse index.

Proof Under the assumptions ($F_1$)-($F_2$), for each eigenvalue ${\lambda }_j^A$ of (${\mathrm{L}}_A$), $({\lambda }_j^A,0)$ is a bifurcation point of (GS) _λ (see [1]).

We denote the distinct eigenvalues of (4.6) by ${\mu }_1(z)<{\mu }_2(z)<\cdots <{\mu }_i(z)<\cdots$ as $z\ne 0$. By ($F_2$), if we take $z=0$, then for each $i\in \mathbb{N}$, there is $j\in \mathbb{N}$ such that ${\mu }_i(0)={\lambda }_j^A$. By (4.5), the standard variational characterization of the eigenvalues of (4.6) shows that ${\mu }_i(z)$ is less than the corresponding j th ordered eigenvalue ${\lambda }_j^A$ of (${\mathrm{L}}_A$). Furthermore, ${\mu }_i(z)\to {\lambda }_j^A$ as $z\to 0$ in E. By (4.3) and (4.4), we see that z is a solution of (4.6) with eigenvalue λ. It must be that $\lambda <{\lambda }_{k+1}^A$ since λ is close to ${\lambda }_{k+1}^A$. Therefore, the case (ii) of Proposition 4.1 occurs under the given conditions. This proves the case (1). The existence for the case (2) is proved in the same way.

By ($F_2$) and (4.7), there exists $\delta >0$ such that when $0<|\lambda -{\lambda }_{k+1}^A|<\delta$,

Therefore, the Morse index $m(z_{\lambda })$ and the nullity $n(z_{\lambda })$ of $z_{\lambda }$ satisfy (4.2). The proof is complete. □

In this section, we give the proof of main theorems in this paper. We first compute the critical groups of Φ at both infinity and zero.

Lemma 5.1 (see [5])

Proof The idea of the proof comes from the famous paper [5]. We include a sketched proof in an abstract version. Given $\lambda \in \mathbb{R}$, denote $B_1=\{z\in E:\parallel z\parallel ⩽1\}$, $S_1=\partial B_1$. Modifying the arguments in [5], we get the following facts:

(5.2)

(5.3)

since $S_1$ is contractible, which follows from the fact that $dimE=\mathrm{\infty }$. □