As in [29], we introduce the variational framework associated with problem (1.1) under (V) which holds also for the case that \(V\in L^{N/2}(\Omega )\).
Denote by \(\mathcal{A}\) the self-adjoint extension of the operator \(-\Delta +V\) with domain \(\mathfrak{D}(\mathcal{A})\) \((C_{0}^{\infty }(\Omega )\subset \mathfrak{D}(\mathcal{A}) \subset L^{2}( \Omega ))\). Let \(\{\mathcal{E}(\lambda ): -\infty \le \lambda \le +\infty \}\) and \(|\mathcal{A}|\) be the spectral family and the absolute value of \(\mathcal{A}\), respectively, and \(|\mathcal{A}|^{1/2}\) be the square root of \(|\mathcal{A}|\). Set \(\mathcal{U}=id-\mathcal{E}(0) -\mathcal{E}(0-)\). Then \(\mathcal{U}\) commutes with \(\mathcal{A}\), \(|\mathcal{A}|\) and \(|\mathcal{A}|^{1/2}\), and \(\mathcal{A} = \mathcal{U}|\mathcal{A}|\) is the polar decomposition of \(\mathcal{A}\) (see [8, Theorem IV 3.3]). Let \(E= \mathfrak{D}(|\mathcal{A}|^{1/2})\) and
$$ E^{-}=\mathcal{E}(0-)E,\qquad E^{0}= \bigl[ \mathcal{E}(0)-\mathcal{E}(0-) \bigr]E,\qquad E^{+}= \bigl[\mathcal{E}(+ \infty )-\mathcal{E}(0) \bigr]E. $$
(2.1)
For any \(u\in E\), it is easy to see that \(u={u}^{-}+{u}^{0}+{u}^{+}\), where
$$ \begin{aligned} &u^{-}:=\mathcal{E}(0-)u\in E^{-},\qquad u^{0}:=\bigl[\mathcal{E}(0)- \mathcal{E}(0-)\bigr]u\in E^{0}, \\ &u^{+}:=\bigl[\mathcal{E}(+\infty )- \mathcal{E}(0)\bigr]u\in E^{+}. \end{aligned} $$
(2.2)
Define an inner product
$$ (u,v)=\bigl( \vert \mathcal{A} \vert ^{1/2}u, \vert \mathcal{A} \vert ^{1/2}v\bigr)_{L^{2}}+ \bigl(u^{0},v^{0}\bigr)_{L^{2}},\quad \forall u,v\in E, $$
(2.3)
and the corresponding norm
$$ \Vert u,v \Vert = \bigl\Vert \vert \mathcal{A} \vert ^{1/2}u \bigr\Vert _{2}+ \bigl\Vert u^{0} \bigr\Vert _{2}, \quad \forall u\in E, $$
(2.4)
where \((\cdot ,\cdot )_{L^{2}}\) denotes the inner product of \(L^{2}(\Omega )\), \(\|\cdot \|_{s}\) stands for the usual \(L^{s}(\Omega )\) norm. Then \(E\subset H_{0}^{1}(\Omega )\) is a Hilbert space. Clearly, \(C_{0}^{\infty }(\Omega )\) is dense in E.
The following lemma was established in [29, Lemmas 2.4, 2.5, Remark 2.8].
Lemma 2.1
Let (V) be satisfied. Then, for the inner products \((\cdot ,\cdot )\) and \((\cdot ,\cdot )_{L^{2}}\) on E, we have
$$ E^{-}\perp E^{0},\qquad E^{-} \perp E^{+},\qquad E^{0}\perp E^{+},\qquad \dim \bigl( \mathcal{E}(M)E \bigr)< +\infty ,\quad \forall M \ge 0, $$
(2.5)
and
$$ E^{0}=\operatorname{Ker} (\mathcal{A}),\qquad \mathcal{A}u^{-}=- \vert \mathcal{A} \vert u^{-},\qquad \mathcal{A}u^{+}= \vert \mathcal{A} \vert u^{+},\quad \forall u\in E \cap \mathfrak{D}(\mathcal{A}). $$
(2.6)
Moreover, E is compactly embedded in \(L^{s}(\Omega )\) for \(1\le s<2^{*}\), where \(2^{*}:=2N/(N-2)\) if \(N\geq 3\) and \(2^{*}:=+\infty \) if \(N=1\) or 2.
For the case that \(V\in L^{N/2}(\Omega )\), spectrum of \(\mathcal{A}\) consists of only eigenvalues numbered in \(-\infty <\mu _{1}\leq \mu _{2}\leq \cdots \leq \mu _{n}\leq 0<\mu _{n+1} \leq \cdots \rightarrow +\infty \) (counted with multiplicity) with the corresponding system of eigenfunctions \({\{e_{n}\}}\) forming an orthogonal basis in \(L^{2}(\Omega )\); see [8, Theorem VI.1.4] or [31]. In particular,
$$ E^{-}=\operatorname{span}\{e_{1},\ldots ,e_{n^{-}}\},\qquad E^{0}=\operatorname{span}\{e_{n^{-}+1}, \ldots ,e_{\bar{n}}\},\qquad E^{+}=\overline{\operatorname{span} \{e_{\bar{n}+1},\ldots \}}, $$
(2.7)
where
$$ n^{-}:=\sharp \{i: \lambda _{i}< 0\},\qquad n^{0}:=\sharp \{i: \lambda _{i}=0\},\qquad \bar{n}:=n^{-}+n^{0}. $$
(2.8)
Moreover, similar to [31, Lemmas 2.1, 2.3], we have the following lemmas.
Lemma 2.2
Under assumption \(V\in L^{N/2}(\Omega )\) with \(N\geq 3\), the norm \(\|\cdot \|\) in \(E=H_{0}^{1}(\Omega )\) is equivalent to the usual Sobolev norm \(\|\cdot \|_{1,2}\) in \(H_{0}^{1}(\Omega )\), and E is compactly embedded in \(L^{s}(\Omega )\) for \(1\le s<2^{*}\).
Lemma 2.3
Suppose that \(V\in L^{N/2}(\Omega )\) or (V) holds. Then E is compactly embedded in \(L^{s}(\Omega )\) for \(1\le s<2^{*}\), and there exists \(\tau _{s}>0\) such that
$$ \Vert u \Vert _{s}\le \tau _{s} \Vert u \Vert , \quad \forall u\in E. $$
(2.9)
By (S1) we have
$$ \bigl\vert F(x,u) \bigr\vert \leq c \bigl( \vert u \vert + \vert u \vert ^{p} \bigr),\quad \forall (x,u)\in \Omega \times \mathbb {R}. $$
(2.10)
Under (S1) and assumptions of Lemma 2.3, the functional Φ defined by (1.2) is of class \(C^{1}(E,\mathbb {R})\). Moreover, by virtue of (2.3) and (2.6), one has
$$ \Phi (u)=\frac{1}{2} \bigl( \bigl\Vert u^{+} \bigr\Vert ^{2}- \bigl\Vert u^{-} \bigr\Vert ^{2} \bigr)- \int _{ \Omega }F(x, u)\,\mathrm{d}x, \quad \forall u\in E, $$
(2.11)
and
$$ \bigl\langle \Phi '(u), v \bigr\rangle = \bigl(u^{+}, v^{+}\bigr)-\bigl(u^{-}, v^{-}\bigr)- \int _{ \Omega }f(x, u) v\,\mathrm{d}x, \quad \forall u, v\in E. $$
(2.12)
Before presenting the critical point theorem used in this paper, we give some notions.
Let X be a Banach space and \(I\in C^{1}(X, \mathbb {R})\) a functional. A sequence \(\{u_{n}\}\subset X\) is called a (PS) sequence (or (PS)c sequence) if
$$ \bigl\{ I(u_{n})\bigr\} \text{ is bounded}\qquad \bigl(\text{or } I(u_{n})\to c\bigr),\qquad I'(u_{n}) \rightarrow 0. $$
(2.13)
The functional I is said to satisfy (PS) condition (or (PS)c condition) if each (PS) sequence (or (PS)c sequence) has a convergent subsequence.
A subset \(A\subset X\) is said to be symmetric if \(u\in A\) implies that \(-u\in A\). For a closed symmetric set A which does not contain the origin, we define a genus \(\gamma (A)\) of A by the smallest integer k such that there exists an odd continuous mapping from A to \(\mathbb {R}^{k}\setminus \{0\}\). If there does not exist such a k, we define \(\gamma (A)=\infty \). Moreover, we set \(\gamma (\emptyset )=0\). Let \(\Gamma _{k}\) denote the family of closed symmetric subset A of X such that \(0\notin A\) and \(\gamma (A)\ge k\).
Theorem 2.4
([12, Theorem 1], [16, Theorem 1.1])
Let X be an infinite dimensional Banach space, and \(I\in C^{1}(X, \mathbb {R})\) satisfies (H1) and (H2) below.
-
(H1)
I is even, bounded from below, \(I(0)=0\) and I satisfies the (PS) condition;
-
(H2)
for each \(k\in \mathbb {N}\), there exists an \(A_{k}\in \Gamma _{k}\) such that \(\sup_{u\in A_{k}}I(u)<0\).
Then there exists a sequence of critical points \(\{u_{k}\}\) such that \(I(u_{k})\le 0\), \(u_{k}\neq 0\) and \(\lim_{k\to \infty }u_{k}=0\).
Remark 2.5
As we see from the proof of [16, Theorem 1.1], the above theorem holds also if we use the (PS)c condition with \(c\le 0\) instead of the (PS) condition in (H1).
Under \(V\in L^{N/2}(\Omega )\) or (V), it follows from (2.5) or (2.7) that \(\operatorname{dim} (E^{-}\oplus E^{0} )< \infty \). We choose an orthonormal basis \(\{\xi _{j}\}^{k_{0}}_{j=1}\) for \(E^{-}\), an orthonormal basis \(\{\xi _{j}\}^{l_{0}}_{j=k_{0}+1}\) for \(E^{0}\) and an orthonormal basis \(\{\xi _{j}\}^{\infty }_{j=l_{0}+1}\) for \(E^{+}\), where \(k_{0}, l_{0}\in \mathbb {N}\) and \(1\le k_{0}< l_{0}<\infty \). Then \(\{\xi _{j}\}^{\infty }_{j=1}\) is an orthonormal basis of E. Define
$$ X_{j}:=\mathbb {R}\xi _{j},\qquad Y_{k}:=\bigoplus _{j=1}^{k}X_{j},\qquad Z_{k}:=\bigoplus _{j=k+1}^{\infty }X_{j},\quad k\in \mathbb {Z}. $$
(2.14)
Proof of Theorem 1.2
Consider the truncated functional
$$ I(u)=\frac{1}{2} \Vert u \Vert ^{2}-\frac{1}{2} \biggl( \bigl\Vert u^{-} \bigr\Vert ^{2}+ \bigl\Vert u^{0} \bigr\Vert _{2}^{2} +2 \int _{{\Omega }}F(x,u)\,\mathrm{d}x \biggr)\phi \bigl( \Vert u \Vert ^{2} \bigr),\quad u\in E, $$
(2.15)
where \(\phi :\mathbb {R}^{+}\to [0,1]\) is a smooth functional such that \(\phi (t)=1\) for \(t\in [0,1]\), \(\phi (t)=0\) for \(t\ge 2\), \(\phi '(t)\le 0\) and \(|\phi '(t)|\le 2\), \(\forall t\ge 0\). Obviously, \(I\in C^{1}(E,\mathbb {R})\) and \(I(0)=0\) by (S2). If we can prove that I satisfies (H1) and (H2), using Theorem 2.4, I admits a sequence of critical points \(\{u_{k}\}\) such that \(I(u_{k})\le 0\), \(u_{k}\neq 0\) and \(u_{k}\to 0\) as \(k\to \infty \). So does Φ by the fact \(\Phi (u)=I(u)\) for \(\|u\|\le 1\). Obviously, I is even by (S3)
Note that \(I(u)=\frac{1}{2}\|u\|^{2}\) if \(\|u\|\ge \sqrt{2}\). Then the functional I is coercive, i.e.
$$ I(u)\to +\infty , \quad \text{as } \Vert u \Vert \to \infty , $$
(2.16)
which implies that I is bounded from below and satisfies (PS)c condition with \(c\le 0\). Indeed, any sequence \(\{u_{n}\}\subset E\) satisfying (2.13) is bounded by (2.16). Passing to a subsequence, we may assume that \(u_{n}\rightharpoonup u\) in E. By Lemma 2.3, \(u_{n}\to u\) in \(L^{s}(\Omega )\) for \(s\in [1,2^{*})\), and \(u^{-}_{n}\to u^{-}\), \(u^{0}_{n}\to u^{0}\) in E since \(\operatorname{dim} (E^{-}\oplus E^{0} )<\infty \). It follows from (S1) and (2.9) that
$$\begin{aligned}& \int _{\Omega } \bigl( \bigl\vert f(x,u_{n}) \bigr\vert + \bigl\vert f(x,u) \bigr\vert \bigr) \vert u_{n}-u \vert \,\mathrm{d}x \\& \quad \le c \int _{\mathbb {R}^{N}} \bigl( \vert u_{n} \vert + \vert u \vert + \vert u_{n} \vert ^{p-1}+ \vert u \vert ^{p-1} \bigr) \vert u_{n}-u \vert \,\mathrm{d}x \\& \quad \le c \bigl( \Vert u_{n} \Vert _{2}+ \Vert u \Vert _{2} \bigr) \Vert u_{n}-u \Vert _{2} +c \bigl( \Vert u_{n} \Vert ^{p-1}_{p}+ \Vert u \Vert ^{p-1}_{p} \bigr) \Vert u_{n}-u \Vert _{p} \\& \quad \le c\tau ^{2}_{2} \bigl( \Vert u_{n} \Vert + \Vert u \Vert \bigr) \Vert u_{n}-u \Vert _{2} +c \tau ^{p-1}_{p} \bigl( \Vert u_{n} \Vert ^{p-1}+ \Vert u \Vert ^{p-1} \bigr) \Vert u_{n}-u \Vert _{p} \\& \quad = o(1). \end{aligned}$$
(2.17)
By (2.15), direct computation shows that
$$\begin{aligned} \bigl\langle I'(u),v \bigr\rangle = & \biggl[ 1- \biggl( \bigl\Vert u^{-} \bigr\Vert ^{2}+ \bigl\Vert u^{0} \bigr\Vert _{2}^{2} +2 \int _{{\Omega }}F(x,u)\,\mathrm{d}x \biggr) \phi ' \bigl( \Vert u \Vert ^{2}\bigr) \biggr]( u, v) \\ &{} - \biggl[\bigl( u^{-},v^{-}\bigr) + \bigl(u^{0},v\bigr)_{L^{2}}+ \int _{\Omega }f(x,u)v \,\mathrm{d}x \biggr]\phi \bigl( \Vert u \Vert ^{2}\bigr), \quad u, v \in E. \end{aligned}$$
(2.18)
It follows from (2.13) and (2.15) that
$$ - \biggl( \bigl\Vert u_{n}^{-} \bigr\Vert ^{2}+ \bigl\Vert u_{n}^{0} \bigr\Vert _{2}^{2} +2 \int _{{\Omega }}F(x,u_{n}) \,\mathrm{d}x \biggr)\phi \bigl( \Vert u_{n} \Vert ^{2} \bigr) = 2 I(u_{n})- \Vert u_{n} \Vert ^{2}\le 0, $$
which implies that
$$ \bigl\Vert u_{n}^{-} \bigr\Vert ^{2}+ \bigl\Vert u_{n}^{0} \bigr\Vert _{2}^{2} +2 \int _{{\Omega }}F(x,u_{n}) \,\mathrm{d}x \ge 0. $$
(2.19)
Note that \(\phi '(t)\le 0\) and \(|\phi '(t)|/2, \phi (t)\le 1\) for any \(t\ge 0\), then, by (2.9), (2.17), (2.18) and (2.19), one has
$$\begin{aligned}& \Vert u_{n}-u \Vert ^{2} \\& \quad = \bigl\langle I'(u_{n})-I'(u),u_{n}-u \bigr\rangle + \biggl( \bigl\Vert u_{n}^{-} \bigr\Vert ^{2}+ \bigl\Vert u_{n}^{0} \bigr\Vert _{2}^{2} +2 \int _{{\Omega }}F(x,u_{n})\,\mathrm{d}x \biggr) \\& \qquad {} \times \phi '\bigl( \Vert u_{n} \Vert ^{2}\bigr) ( u_{n}, u_{n}-u) - \biggl( \bigl\Vert u^{-} \bigr\Vert ^{2}+ \bigl\Vert u^{0} \bigr\Vert _{2}^{2} +2 \int _{{\Omega }}F(x,u)\,\mathrm{d}x \biggr) \phi ' \bigl( \Vert u \Vert ^{2}\bigr) ( u, u_{n}-u) \\& \qquad {} +\phi \bigl( \Vert u_{n} \Vert \bigr) \biggl[ \bigl\Vert u_{n}^{-}-u^{-} \bigr\Vert ^{2}+ \bigl\Vert u_{n}^{0}-u^{0} \bigr\Vert ^{2}_{L^{2}}+ \int _{\Omega } \bigl(f(x,u_{n})-f(x,u) \bigr) (u_{n}-u)\,\mathrm{d}x \biggr] \\& \qquad {} + \bigl[\phi \bigl( \Vert u_{n} \Vert \bigr)-\phi \bigl( \Vert u \Vert \bigr) \bigr] \biggl[\bigl(u^{-},u^{-}_{n}-u^{-} \bigr)+\bigl(u^{0},u_{n}^{0}-u^{0} \bigr)_{L^{2}} + \int _{\Omega }f(x,u) (u_{n}-u)\,\mathrm{d}x \biggr] \\& \quad \le o(1) + \biggl( \bigl\Vert u_{n}^{-} \bigr\Vert ^{2}+ \bigl\Vert u_{n}^{0} \bigr\Vert _{2}^{2} +2 \int _{{ \Omega }}F(x,u_{n})\,\mathrm{d}x \biggr) \phi '\bigl( \Vert u_{n} \Vert ^{2}\bigr) ( u_{n}, u_{n}-u) \\& \qquad {} + \int _{\Omega } \bigl( \bigl\vert f(x,u_{n}) \bigr\vert + \bigl\vert f(x,u) \bigr\vert \bigr) \vert u_{n}-u \vert \,\mathrm{d}x +2 \int _{\Omega } \bigl\vert f(x,u) \bigr\vert \vert u_{n}-u \vert \,\mathrm{d}x \\& \quad = o(1) + \biggl( \bigl\Vert u_{n}^{-} \bigr\Vert ^{2}+ \bigl\Vert u_{n}^{0} \bigr\Vert _{2}^{2} +2 \int _{{ \Omega }}F(x,u_{n})\,\mathrm{d}x \biggr) \phi '\bigl( \Vert u_{n} \Vert ^{2}\bigr) \Vert u_{n}-u \Vert ^{2} \\& \qquad {} + \biggl( \bigl\Vert u_{n}^{-} \bigr\Vert ^{2}+ \bigl\Vert u_{n}^{0} \bigr\Vert _{2}^{2} +2 \int _{{ \Omega }}F(x,u_{n})\,\mathrm{d}x \biggr) \phi '\bigl( \Vert u_{n} \Vert ^{2}\bigr) ( u, u_{n}-u) \\& \quad = o(1) + \biggl( \bigl\Vert u_{n}^{-} \bigr\Vert ^{2}+ \bigl\Vert u_{n}^{0} \bigr\Vert _{2}^{2} +2 \int _{{ \Omega }}F(x,u_{n})\,\mathrm{d}x \biggr) \phi '\bigl( \Vert u_{n} \Vert ^{2}\bigr) \Vert u_{n}-u \Vert ^{2} \\& \quad \le o(1). \end{aligned}$$
(2.20)
Thus I satisfies the (PS)c condition with \(c\le 0\).
For any fixed \(l_{0}+1\le k\in \mathbb {N}\), where \(l_{0}=\operatorname{dim} (E^{-}\oplus E^{0} )<\infty \). By (2.14) and the equivalence of the norms on finite dimensional spaces, there exist constants \(c_{k}, d_{k}>0\) such that
$$ \Vert u \Vert _{2}\ge c_{k} \Vert u \Vert ,\qquad \operatorname{ess} \sup_{x\in \Omega } \bigl\vert u(x) \bigr\vert := \Vert u \Vert _{\infty }\le {d_{k}} { \Vert u \Vert } ,\quad \forall u\in Y_{k}, $$
(2.21)
(S2) implies the existence of a constant \(r\in (0,1)\) such that \(F(x,u)\ge c^{-2}_{k}|u|^{2}\) for all \(|u|\le r\) and a.e. \(x\in \Omega \). Then, for any \(u\in Y_{k}\) with \(\|u\|=l_{k}:={2}^{-1}\min \{ 1,r d_{k}^{-1} \} \), one has
$$\begin{aligned} I(u) = & \frac{1}{2} \bigl\Vert u^{+} \bigr\Vert ^{2}-\frac{1}{2} \bigl\Vert u^{-} \bigr\Vert ^{2}- \int _{ \Omega }F(x,u)\,\mathrm{d}x \\ \le & \frac{1}{2} \bigl\Vert u^{+} \bigr\Vert ^{2}- \int _{\Omega }F(x,u)\,\mathrm{d}x \le \frac{1}{2} \Vert u \Vert ^{2}-\frac{1}{c_{k}^{2}} \Vert u \Vert _{2}^{2} \\ \le & \frac{1}{2} \Vert u \Vert ^{2}- \frac{c_{k}^{2}}{c_{k}^{2}} \Vert u \Vert ^{2} = - \frac{1}{2} \Vert u \Vert ^{2} \\ = & -\frac{1}{2}l^{2}_{k}, \end{aligned}$$
(2.22)
which implies that
$$ \bigl\{ u\in Y_{k}: \Vert u \Vert =l_{k}\bigr\} \subset \biggl\{ u\in E: I(u)\le - \frac{1}{2}l^{2}_{k} \biggr\} . $$
(2.23)
Let \(A_{k}:= \{ u\in E : I(u)\le -{2}^{-1}l^{2}_{k} \} \). Then
$$ \gamma (A_{k})\ge \gamma \bigl(\bigl\{ u \in Y_{k}: \Vert u \Vert =l_{k}\bigr\} \bigr) \ge k. $$
(2.24)
Clearly \(A_{k}\in \Gamma _{k}\) and \(\sup_{u\in A_{k} }I(u)\le -{2}^{-1}l^{2}_{k}<0\). Then we get the result from Theorem 2.4. □