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])
Let F satisfy ({F}_{1})({F}_{4}), then for any fixed \lambda \in \mathbb{R},
{C}_{q}(\mathrm{\Phi},\mathrm{\infty})\cong 0,\phantom{\rule{1em}{0ex}}\mathit{\text{for all}}\phantom{\rule{0.1em}{0ex}}q\in \mathbb{Z}.
(5.1)
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 \u2a7d1\}, {S}_{1}=\partial {B}_{1}. Modifying the arguments in [5], we get the following facts:
The following arguments are from [10]. As \mathrm{\Phi}(0)=0, it follows from (5.2) and (5.3) that for each z\in {S}_{1}, there is a unique \tau (z)>0 such that
\mathrm{\Phi}(\tau (z)z)=a.
(5.4)
By (5.4) and the implicit function theorem, we have that \tau \in C({S}_{1},\mathbb{R}). Define
\pi (z)=\{\begin{array}{cc}1,\hfill & \text{if}\mathrm{\Phi}(z)\u2a7da,\hfill \\ {\parallel z\parallel}^{1}\tau ({\parallel z\parallel}^{1}z),\hfill & \text{if}\mathrm{\Phi}(z)a,z\ne 0.\hfill \end{array}
Then \pi \in C(E\setminus \{0\},\mathbb{R}). Define a map \varrho :[0,1]\times E\setminus \{0\}\to E\setminus \{0\} by
\varrho (t,z)=(1t)z+t\pi (z)z.
(5.5)
Clearly, ϱ is continuous, and for all z\in E\setminus \{0\} with \mathrm{\Phi}(z)>a, by (5.4),
\mathrm{\Phi}(\varrho (1,z))=\mathrm{\Phi}\left(\pi \left({\parallel z\parallel}^{1}z\right){\parallel z\parallel}^{1}z\right)=a.
Therefore,
\varrho (1,z)\in {\mathrm{\Phi}}^{a}\phantom{\rule{1em}{0ex}}\text{for all}z\in E\setminus \{0\},\phantom{\rule{2em}{0ex}}\varrho (t,z)=z\phantom{\rule{1em}{0ex}}\text{for all}t\in [0,1],z\in {\mathrm{\Phi}}^{a},
and so {\mathrm{\Phi}}^{a} is a strong deformation retract of E\setminus \{0\}. Hence,
{C}_{q}(\mathrm{\Phi},\mathrm{\infty}):={H}_{q}(E,{\mathrm{\Phi}}^{a})\cong {H}_{q}(E,E\setminus \{0\})\cong {H}_{q}({B}_{1},{S}_{1})\cong 0,\phantom{\rule{1em}{0ex}}q\in \mathbb{Z}
since {S}_{1} is contractible, which follows from the fact that dimE=\mathrm{\infty}. □
Lemma 5.2 Let F satisfy ({F}_{1})({F}_{3}).

(1)
For \lambda \in ({\lambda}_{k}^{A},{\lambda}_{k+1}^{A}), {C}_{q}(\mathrm{\Phi},0)\cong {\delta}_{q,{\ell}_{k}}\mathbb{F}.

(2)
For \lambda \in ({\lambda}_{k+1}^{A},{\lambda}_{k+2}^{A}), {C}_{q}(\mathrm{\Phi},0)\cong {\delta}_{q,{\ell}_{k+1}}\mathbb{F}.

(3)
For \lambda ={\lambda}_{k+1}^{A}, if F(x,z)\u2a7d0 for z small, then {C}_{q}(\mathrm{\Phi},0)\cong {\delta}_{q,{\ell}_{k}}\mathbb{F}.

(4)
For \lambda ={\lambda}_{k+1}^{A}, if F(x,z)\u2a7e0 for z small, then {C}_{q}(\mathrm{\Phi},0)\cong {\delta}_{q,{\ell}_{k+1}}\mathbb{F}.
Proof By ({F}_{2}), we have
\u3008{\mathrm{\Phi}}^{\u2033}(0)y,y\u3009={Q}_{\lambda}(y)={\int}_{\mathrm{\Omega}}{\mathrm{\nabla}y}^{2}\lambda (A(x)y,y)\phantom{\rule{0.2em}{0ex}}dx,\phantom{\rule{1em}{0ex}}y\in E.

(1)
When \lambda \in ({\lambda}_{k}^{A},{\lambda}_{k+1}^{A}), z=0 is a nondegenerate critical point of Φ with the Morse index {m}_{0}={\ell}_{k}, thus {C}_{q}(\mathrm{\Phi},0)\cong {\delta}_{q,{\ell}_{k}}\mathbb{F}.

(2)
When \lambda \in ({\lambda}_{k+1}^{A},{\lambda}_{k+2}^{A}), z=0 is a nondegenerate critical point of Φ with the Morse index {m}_{0}={\ell}_{k+1}, thus {C}_{q}(\mathrm{\Phi},0)\cong {\delta}_{q,{\ell}_{k+1}}\mathbb{F}.

(3)
When \lambda ={\lambda}_{k+1}^{A}, z=0 is a degenerate critical point of Φ with the Morse index {m}_{0}={\ell}_{k} and the nullity {n}_{0}=dimE({\lambda}_{k+1}^{A}), {m}_{0}+{n}_{0}={\ell}_{k+1}.
Assume that F(x,z)\u2a7d0 for z\u2a7d\sigma with \sigma >0 small. We will show that Φ has a local linking structure at z=0 with respect to E={E}_{k}\oplus {E}_{k}^{\perp}. If this has been done, then by Proposition 2.2, we have {C}_{q}(\mathrm{\Phi},0)\cong {\delta}_{q,{\ell}_{k}}\mathbb{F}.
Now, Φ can be written as
\mathrm{\Phi}(z)=\frac{1}{2}{Q}_{{\lambda}_{k+1}^{A}}(z){\int}_{\mathrm{\Omega}}F(x,z)\phantom{\rule{0.2em}{0ex}}dx.
By ({F}_{2}) and ({F}_{3}), for \u03f5>0, there is {C}_{\u03f5}>0 such that
F(x,z)\u2a7d\frac{1}{2}\u03f5{z}^{2}+{C}_{\u03f5}{z}^{p},\phantom{\rule{1em}{0ex}}z\in {\mathbb{R}}^{2},x\in \mathrm{\Omega}.
Hence, for z\in {E}_{k}, we have that
\mathrm{\Phi}(z)\u2a7d\frac{{\lambda}_{k}^{A}{\lambda}_{k+1}^{A}}{2{\lambda}_{k}^{A}}{\parallel z\parallel}^{2}+\frac{\u03f5}{2}{z}_{2}^{2}+{C}_{\u03f5}{z}_{p}^{p}.
Since {E}_{k} is finite dimensional, all norms on {E}_{k} are equivalent, hence for r>0 small,
\mathrm{\Phi}(z)\u2a7d0,\phantom{\rule{1em}{0ex}}\text{for}z\in {E}_{k},\parallel z\parallel \u2a7dr.
By ({F}_{3}), we have that for some C>0,
F(x,z)\u2a7dC{z}^{p},\phantom{\rule{1em}{0ex}}z\u2a7e\sigma ,x\in \mathrm{\Omega}.
(5.6)
For z\in {E}_{k}^{\perp}, we write z=y+w where y\in E({\lambda}_{k+1}^{A}) and w\in {E}_{k+1}^{\perp}. Then
\mathrm{\Phi}(z)\u2a7e\frac{{\lambda}_{k+2}^{A}{\lambda}_{k+1}^{A}}{2{\lambda}_{k+2}^{A}}{\parallel w\parallel}^{2}{\int}_{\mathrm{\Omega}}F(x,z)\phantom{\rule{0.2em}{0ex}}dx.
(5.7)
For x\in \mathrm{\Omega} with z(x)\u2a7e\sigma, we have w(x)\u2a7e\frac{2}{3}z(x). Hence, by (5.6) and the Poincaré inequality, we have for various constants C>0,
For x\in \mathrm{\Omega} with z(x)\u2a7d\sigma, F(x,z(x))\u2a7d0. Therefore,
\begin{array}{rcl}\mathrm{\Phi}(z)& \u2a7e& \frac{{\lambda}_{k+2}^{A}{\lambda}_{k+1}^{A}}{2{\lambda}_{k+2}^{A}}{\parallel w\parallel}^{2}{\int}_{\{x:z(x)\u2a7d\sigma \}}F(x,z(x))\phantom{\rule{0.2em}{0ex}}dxC{\parallel w\parallel}^{p}\\ \u2a7e& \frac{{\lambda}_{k+2}^{A}{\lambda}_{k+1}^{A}}{2{\lambda}_{k+2}^{A}}{\parallel w\parallel}^{2}C{\parallel w\parallel}^{p}.\end{array}
(5.9)
Since p>2, for r>0 small,
\mathrm{\Phi}(z)>0,\phantom{\rule{1em}{0ex}}\text{for}z=y+w\text{with}w\ne 0,\parallel z\parallel \u2a7dr.
(5.10)
For z=y\in E({\lambda}_{k+1}^{A}), it must hold that
\mathrm{\Phi}(y)={\int}_{\mathrm{\Omega}}F(x,y(x))\phantom{\rule{0.2em}{0ex}}dx>0\phantom{\rule{1em}{0ex}}\text{if}0\parallel y\parallel \u2a7dr.
(5.11)
Here we use a potential convention that (GS)_{
λ
} has finitely many solutions and then 0 is isolated. Otherwise, one would have that as r>0 small, \parallel y\parallel \u2a7dr implies y(x)\u2a7d\delta for all x\in \mathrm{\Omega}, \mathrm{\nabla}F(x,y)\equiv 0 for all x\in \mathrm{\Omega}. Thus, 0 would not be an isolated critical point of Φ and (GS)_{
λ
} would have infinitely many nontrivial solutions. By (5.10) and (5.11), we verify that
\mathrm{\Phi}(z)>0,\phantom{\rule{1em}{0ex}}\text{for}z\in {E}_{k}^{\perp},0\parallel z\parallel \u2a7dr.
Applying Proposition 2.2, we obtain
{C}_{q}(\mathrm{\Phi},0)={\delta}_{q,{\ell}_{k}}\mathbb{F}.
(4) When F(x,z)\u2a7e0 for z small, a similar argument shows that Φ has a local linking structure at z=0 with respect to E={E}_{k+1}\oplus {E}_{k+1}^{\perp}. By Proposition 2.2, it follows that {C}_{q}(\mathrm{\Phi},0)\cong {\delta}_{q,{\ell}_{k+1}}\mathbb{F}. □
Finally, we prove the theorems.
Proof of Theorem 1.1 It follows from ({F}_{5}) that F(x,z)\u2a7e0 for z>0 small. By Theorem 3.4 for the part \lambda \in ({\lambda}_{k+1}^{A}\delta ,{\lambda}_{k+1}^{A}), (GS)_{
λ
} has a nontrivial solution {z}^{\lambda} satisfying
{C}_{{\ell}_{k+1}+1}(\mathrm{\Phi},{z}^{\lambda})\ncong 0.
(5.12)
By Theorem 4.2(1), (GS)_{
λ
} has two nontrivial solutions {z}_{\lambda}^{i} (i=1,2) with their Morse indices satisfying
{\ell}_{k}\u2a7dm\left({z}_{\lambda}^{i}\right)\u2a7dm\left({z}_{\lambda}^{i}\right)+n\left({z}_{\lambda}^{i}\right)\u2a7d{\ell}_{k+1},\phantom{\rule{1em}{0ex}}i=1,2.
From Proposition 2.1(2), we have that
{C}_{q}(\mathrm{\Phi},{z}_{\lambda}^{i})\cong 0,\phantom{\rule{1em}{0ex}}q\notin [{\ell}_{k},{\ell}_{k+1}],i=1,2.
(5.13)
From (5.12) and (5.13), we see that {z}^{\lambda}\ne {z}_{\lambda}^{i} (i=1,2). The proof is complete. □
Proof of Theorem 1.2 With the same argument as above, it follows from Theorem 4.2(2) and Theorem 3.4 for the part \lambda \in ({\lambda}_{k+1}^{A},{\lambda}_{k+1}^{A}+\delta ). We omit the details. □
Proof of Theorem 1.3 By Theorem 3.4 for the part \lambda \in ({\lambda}_{k+1}^{A}\delta ,{\lambda}_{k+1}^{A}], (GS)_{
λ
} has a solution {z}^{\lambda} with its energy \mathrm{\Phi}({z}^{\lambda})\u2a7e\alpha >0 and
{C}_{{\ell}_{k+1}+1}(\mathrm{\Phi},{z}^{\lambda})\ncong 0.
(5.14)
By Lemma 5.1 and Lemma 5.2(3), we have that
Assume that (GS)_{
λ
} has only two solutions 0 and {z}^{\lambda}. Choose a,b\in \mathbb{R} such that a<0<b<\mathrm{\Phi}({z}^{\lambda}). Then by the deformation and excision properties of singular homology (see [12]), we have
\{\begin{array}{c}{C}_{q}(\mathrm{\Phi},\mathrm{\infty})\cong {H}_{q}(E,{\mathrm{\Phi}}^{a});\hfill \\ {C}_{q}(\mathrm{\Phi},0)\cong {H}_{q}({\mathrm{\Phi}}^{b},{\mathrm{\Phi}}^{a});\hfill \\ {C}_{q}(\mathrm{\Phi},{z}^{\lambda})\cong {H}_{q}(E,{\mathrm{\Phi}}^{b}).\hfill \end{array}
(5.17)
By (5.17), the long exact sequences for the topological triple (E,{\mathrm{\Phi}}^{b},{\mathrm{\Phi}}^{a}) read as
\cdots \to {C}_{q+1}(\mathrm{\Phi},\mathrm{\infty})\to {C}_{q+1}(\mathrm{\Phi},{z}^{\lambda})\to {C}_{q}(\mathrm{\Phi},0)\to {C}_{q}(\mathrm{\Phi},\mathrm{\infty})\to \cdots .
(5.18)
We deduce by (5.15) and (5.18) that
{C}_{q+1}(\mathrm{\Phi},{z}^{\lambda})\cong {C}_{q}(\mathrm{\Phi},0),\phantom{\rule{1em}{0ex}}\text{for}q\in \mathbb{Z}.
(5.19)
Take q={\ell}_{k+1} in (5.19), then
{C}_{{\ell}_{k+1}+1}(\mathrm{\Phi},{z}^{\lambda})\cong {C}_{{\ell}_{k+1}}(\mathrm{\Phi},0)\cong 0
which contradicts (5.14). The proof is complete. □
We finally remark that Theorem 1.1 is valid for \lambda \in ({\lambda}_{1}^{A}\delta ,{\lambda}_{1}^{A}), from which one sees that z=0 is a local minimizer of Φ.