Abstract
In this paper, we study the following quasilinear Schrödinger equation:
where \(N\geq 3\), \(1< p< N\), \(-\infty <\alpha <\frac{N-p}{p}\), \(\alpha \leq e\leq \alpha +1\), \(d=1+\alpha -e\), \(p^{*}:=p^{*}(\alpha ,e)=\frac{Np}{N-dp}\) (critical Hardy–Sobolev exponent), V and K are nonnegative potentials, the function a satisfies suitable assumptions, and f is superlinear, which is weaker than the Ambrosetti–Rabinowitz-type condition. By using variational methods we obtain that the quasilinear Schrödinger equation has infinitely many nontrivial solutions.
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1 Introduction
In this paper, we study the following quasilinear Schrödinger equation:
where \(N\geq 3\), \(1< p< N\), \(-\infty <\alpha <\frac{N-p}{p}\), \(\alpha \leq e\leq \alpha +1\), \(d=1+\alpha -e\), \(p^{*}:=p^{*}(\alpha ,e)=\frac{Np}{N-dp}\) is the critical Hardy–Sobolev exponent, V and K are nonnegative potentials, f is of superlinear growth near infinity, and for some positive functions \(h_{1}(x)\in L^{\infty }(\mathbb{R}^{N})\) and \(h_{0}(x)\in L^{p/(p-1)}_{\overline{\alpha }}(\mathbb{R}^{N})\), where \({\overline{\alpha }}=\frac{\alpha p}{p^{*}}\), the function a satisfies \(|a(x,\nabla u)|\leq c_{0}|x|^{-\alpha p}h_{0}(x)+c_{0}(1+|x|^{- \alpha p})h_{1}(x)|\nabla u|^{p-1}\).
Problem (1.1) comes from the quasilinear Schrödinger equation and involves several improvements. Firstly, Duc [17] established the existence of a nontrivial solution to the problem
For different types of \(a(x,\nabla u)\), the quasilinear equation of the form (1.1) has been derived from several physical models. Especially, \(a(x,\nabla u)=|\nabla u|^{p-2}\nabla u\) and \(a(x,\nabla u)=|x|^{-\alpha p}|\nabla u|^{p-2}\nabla u\) were used for the problems of nonlinear diffusion, such as nonlinear optics, plasma physics, condensed matter physics, and so on. We refer the reader to [16, 25] and references therein.
This type of equation has been extensively studied in recent years with a huge variety of hypotheses on the potentials \(V(x)\) and \(K(x)\). For V bounded from below by a positive constant (\(V(x)\geq V_{1}>0\)) and \(K(x)\equiv 1\), we would like to cite [1, 10, 21] and references therein, and in case of \(K(x)\not \equiv 1\), we refer to [18, 23, 25].
If V goes to zero as \(|x|\rightarrow \infty \), that is,
which is called the zero mass case, we can cite [2, 7, 9], which use the same technique as that used in [2]. In the case where K vanishes at infinity, we refer to the papers in [3–5]. The cases of K bounded by a positive constant and unbounded K are considered in [14].
Finally, [11, 12] deal with the comprehensive problems including the potentials V and K. In [3], with more general potentials K and V, the authors obtained an inequality of Hardy type and then the strong convergence in the whole space. As a matter of fact, they have obtained the compact embedding of \(E\subset \mathcal{D}^{1,2}(\mathbb{R}^{N})\) in \(L^{q}_{K}(\mathbb{R}^{N})\) with \(2< q<2^{*}\). Using the same way, the compact embedding of \(E\subset \mathcal{D}^{1,p}_{\alpha }(\mathbb{R}^{N})\) in \(L^{q}_{K,\alpha }(\mathbb{R}^{N})\) is proved in [8] with \(1< p< N\), \(p< q< p^{*}\).
In most of the aforesaid references, the Ambrosetti–Rabinowitz (AR) condition is usually assumed. It is very crucial to ensure the boundedness of the Palais–Smale (PS) sequences of the energy functional. However, there are many functions that do not satisfy the AR condition. So in this paper, to prove that there are infinitely many solutions to quasilinear Schrödinger equation, we develop a superquadratic condition, which is weaker than the condition AR.
There are many difficulties in solving the problem of relationship among nonlinearities, operator, and potentials. To overcome this, we prove the existence of infinitely many solutions to problem (1.1) with compact embedding by using Tang’s methods in [24]. As far as we know, to prove the boundedness of the \((C)_{c}\)-sequence for problem (1.1), we must have compact embedding, so we need to enhance some conditions for potentials \(K(x)\) and \(V(x)\). Before proving our results, we need to make the following assumptions on a, A, V, K, and f.
(1) Functions a and A. We consider continuous functions \(a:\mathbb{R}^{N}\times \mathbb{R}^{N}\mapsto \mathbb{R}^{N}\) and \(A:\mathbb{R}^{N}\times \mathbb{R}^{N}\mapsto \mathbb{R}\) such that \(a(x,\xi )=\frac{\partial A(x,\xi )}{\partial \xi }\). Let \(c_{0}\) and \(c_{1}\) be positive real numbers, and let \(h_{0}(x)\) and \(h_{1}(x)\) be nonnegative measurable real functions in \(\mathbb{R}^{N}\) such that \(h_{0}(x)\in L^{p/(p-1)}_{\overline{\alpha }}(\mathbb{R}^{N})\) with \({\overline{\alpha }}=\frac{\alpha p}{p^{*}}\) and \(h_{1}(x)\in L^{\infty }(\mathbb{R}^{N})\) with \(h_{1}(x)\geq 1\) for a.e. \(x \in \mathbb{R}^{N}\). We introduce the following hypotheses:
- \((A_{1})\):
-
\(|a(x,\xi )|\leq c_{0}|x|^{-\alpha p}h_{0}(x)+c_{0}(1+|x|^{-\alpha p})h_{1}(x)| \xi |^{p-1}\) for a.e. \(x\in \mathbb{R}^{N}\).
- \((A_{2})\):
-
\(c_{1}(1+|x|^{-\alpha p})h_{1}(x)|\xi -\eta |^{p} \leq (a(x,\xi )-a(x, \eta ))(\xi -\eta )\) for all \(\xi , \eta \in \mathbb{R}^{N}\) and a.e. \(x \in \mathbb{R}^{N}\).
- \((A_{3})\):
-
A is subhomogeneous, that is, \(0\leq a(x,\xi )\xi \leq pA(x,\xi )\) for all \(\xi \in \mathbb{R}^{N}\) and a.e. \(x\in \mathbb{R}^{N}\).
- \((A_{4})\):
-
\(a(x,0)=0\), \(A(x,\xi )=A(x,-\xi )\) for a.e. \(x\in \mathbb{R}^{N}\).
Remark 1.1
([20])
The function A can be used in several cases. For example:
-
(i)
\(A(x,\xi )=\frac{1}{p}|\xi |^{p}\).
-
(ii)
\(A(x,\xi )=\frac{h(x)}{p}|\xi |^{p}\) with \(h\in L^{1}_{\mathrm{loc}}(\mathbb{R}^{N})\).
-
(iii)
\(A(x,\xi )=\frac{1}{p}[(1+|\xi |^{2})^{\frac{p}{2}}-1]\) with \(p\geq 2\).
-
(iv)
\(A(x,\xi )=\frac{h(x)}{p}[(1+|\xi |^{2})^{\frac{p}{2}}-1]\) with \(p\geq 2\) and \(h\in L^{\frac{p}{p-1}}(\mathbb{R}^{N})\).
-
(v)
\(A(x,\xi )=\frac{1}{p}|\xi |^{p}+\theta (x) (\sqrt{1+|\xi |^{2}}-1 )\) with a suitable function θ. We get the operator \(\operatorname{div}(|\nabla u|^{p-2}\nabla u)+\operatorname{div} (\theta (x) \frac{\nabla u}{\sqrt{1+|\nabla u|^{2}}} )\), which can be regarded as the sum of the p-Laplacian operator and a degenerate-form mean-curvature operator.
(2) Potentials V and K.
- \((VK1)\):
-
\(V ,K\in \mathcal{C}(\mathbb{R}^{N}, \mathbb{R})\), \(V(x)\geq \min V(x) \geq 1\), \(K(x)\geq \min K(x)\geq 0\), \(K(x)\not \equiv 0\), and \(K(x)\in L^{\infty }(\mathbb{R}^{N})\);
- \((VK2)\):
-
\(\lim_{|x|\to \infty }\frac{K(x)}{V^{\theta }(x)}=0\) for all \(0<\theta <1\).
Example 1.2
([15])
The following functions are typical examples of functions satisfying \((VK1)\) and \((VK2)\):
-
(i)
\(K(x)=2\) and \(V(x)=(|x|+1)^{\frac{1}{\theta }}\) for \(0<\theta <1\).
-
(ii)
\(K(x)=\sin x\) and \(V(x)=[(|x|+1)(|\sin x|+1)]^{ \frac{1}{\theta }}\) for \(0<\theta <1\).
Is easy to check that\(\lim_{|x|\to \infty } \frac{K(x)}{V^{\theta }(x)}=0\), \(K(x)\not \equiv 0\), \(K(x)\in L^{\infty }( \mathbb{R}^{N})\), \(V(x)\geq \min V(x)\geq 1\), and \(K(x)\geq \min K(x)\geq 0\) for all \(0<\theta <1\).
(3) Functions f and F. Let functions \(f:\mathbb{R}^{N+1}\rightarrow \mathbb{R}\) and \(F:\mathbb{R}^{N+1}\rightarrow \mathbb{R}\) such that \(f(x,u)=\frac{\partial F(x,u)}{\partial x}\) for all \(x\in \mathbb{R}\) satisfy the following conditions:
- \((f_{1})\):
-
there exist constants \(c_{1},c_{2}>0\) and \(\beta \in (p,p^{*})\) such that
$$\bigl\vert f(x,u) \bigr\vert \leqslant c_{1} \vert u \vert ^{p-1}+c_{2} \vert u \vert ^{\beta -1}\quad \mbox{for all } (x,u)\in \mathbb{R}^{N} \times \mathbb{R}. $$ - \((f_{2})\):
-
\(\lim_{|u|\to \infty } \frac{|x|^{-\alpha p^{*}}|F(x,u)|}{|u|^{p}}=\infty \) for a.e. \(x \in \mathbb{R}^{N}\), and there exists \(r_{0}\geq 0\) such that
$$F(x,u)\geq 0, \quad \forall (x,u)\in \mathbb{R}^{N} \times \mathbb{R}, \vert u \vert \geq r_{0}. $$ - \((f_{3})\):
-
\(\mathcal{F}(x,u):=\frac{1}{p}uf(x,u)-F(x,u)\geq 0\), and there exist \(c_{0}>0\) and \(\kappa >\frac{N}{dp}\) such that
$$\bigl\vert F(x,u) \bigr\vert ^{\kappa }\leq c_{0} \vert u \vert ^{p\kappa }\mathcal{F}(x,u), \quad \forall (x,u)\in \mathbb{R}^{N} \times \mathbb{R}, \vert u \vert \geq r_{0}. $$ - \((f_{4})\):
-
\(f(x,-u)=-f(x,u)\), \(\forall (x,u)\in \mathbb{R}^{N} \times \mathbb{R}\).
- \((f_{5})\):
-
There exist \(\mu >p\) and \(\varrho >0\) such that
$$\mu F(x,u)\leq uf(x,u)+\varrho \vert u \vert ^{p}, \quad \forall (x,u)\in \mathbb{R}^{N+1}. $$ - \((f_{6})\):
-
There exist \(\mu >p\) and \(r_{1}>0\) such that
$$\mu F(x,u)\leq uf(x,u), \quad \forall (x,u)\in \mathbb{R}^{N+1}, \vert u \vert \geq r_{1}. $$
Example 1.3
([19])
Is easy to check that the following nonlinearities f satisfy \((f_{1})\), \((f_{2})\), \((f_{4})\), and \((f_{6})\):
-
(i)
\(f(x,u)=g(x)|u|^{p-1}u[(p+3)u^{2}-2(p+2)u+p+1]\).
-
(ii)
\(f(x,u)=g(x)|u|^{p-2}u(4|u|^{3}+2u\sin u-4\cos u)\).
-
(iii)
\(f(x,u)=g(x)\sum^{m}_{i=1}b_{i}|u|^{\gamma _{i}}u\), where \(b_{1}>0\), \(b_{i}\in \mathbb{R}\), \(i=2,3,\ldots ,m\), \(\gamma _{1}>\gamma _{2}> \cdots >\gamma _{m}\geq p-2\), \(g\in \mathcal{C}(\mathbb{R}^{N}, \mathbb{R})\), and \(0<\inf_{\mathbb{R}^{N}}g\leq \sup_{\mathbb{R}^{N}}g<\infty \).
Now we are ready to state the main theorems of this paper.
Theorem 1.4
Let \((VK1)\)–\((VK2)\), \((A_{1})\)–\((A_{4})\), and \((f_{1})\)–\((f_{4})\) be satisfied. Then equation (1.1) has infinitely many nontrivial solutions.
Theorem 1.5
Let \((VK1)\)–\((VK2)\), \((A_{1})\)–\((A_{4})\), \((f_{1})\), \((f_{2})\), \((f_{4})\), and \((f_{5})\) be satisfied. The equation (1.1) has infinitely many nontrivial solutions.
It is easy to check that \((f_{1})\) and \((f_{6})\) imply \((f_{5})\). Thus we have the following corollary.
Corollary 1.6
Let \((VK1)\)–\((VK2)\), \((A_{1})\)–\((A_{4})\), \((f_{1})\), \((f_{2})\), \((f_{4})\), and \((f_{6})\) be satisfied. Then equation (1.1) has infinitely many nontrivial solutions.
Remark 1.7
In our theorems, \(F(x,u)\) is allowed to be sign-changing. Even if \(F(x,u)\geq 0\), assumptions \((f_{2})\), \((f_{3})\), \((f_{5})\), and \((f_{6})\) seem to be weaker than the superquadratic conditions obtained in the aforementioned references.
Notations
Considering α and K in equation (1.1), an open set \(B\subset \mathbb{R}\), and a measurable function \(u:B \rightarrow \mathbb{R} \), we use the following notations.
-
\(L^{q}_{\alpha }(B)=\{u: B \rightarrow \mathbb{R} |\int _{B}|x|^{- \alpha p^{*}}|u|^{q} \,\text{d}x<\infty \}\) for \(1\leq q <\infty \).
-
\(L^{q}_{K,\alpha }(B)=\{u: B \rightarrow \mathbb{R} |\int _{B}K(x)|x|^{- \alpha p^{*}}|u|^{q} \,\text{d}x<\infty \}\) for \(1\leq q <\infty \).
-
\(\|u\|_{L^{q}_{K,\alpha }(B)}=(\int _{B}K(x)|x|^{-\alpha p^{*}}|u|^{q} \,\text{d}x)^{\frac{1}{q}}\) for \(1\leq q <\infty \).
-
\(\|u\|_{\mathcal{D}^{1,p}_{\alpha }(B)}=(\int _{B}|x|^{-\alpha p}| \nabla u|^{p} \,\text{d}x)^{\frac{1}{p}}\).
-
\(L^{q}(B)\) is the usual Sobolev space for \(1\leq q <\infty \).
-
We denote by \(o_{n}(1)\) terms that tend to zero as \(n\to \infty \). The weak \((\rightharpoonup )\) and strong \((\rightarrow )\) convergences are always taken as \(n\to \infty \).
-
Hereafter C is a positive constant that can changes its value in a sequence of inequalities.
The remainder of the paper is organized as follows. In Sect. 2, we present variational framework. In Sect. 3, we state and prove the main results of the paper.
2 Variational framework
In this section, we want to use variational methods. So we define a convenient space and functional. We consider the spaces
and
We define
endowed with the norm
with \(k_{0}\) given by the inequality \(A(x,\nabla u)\geq k_{0} h_{1}(x)(1+|x|^{-\alpha p})|\nabla u|^{p}\) for all \(\xi \in \mathbb{R}^{N}\) and a.e. \(x\in \mathbb{R}^{N}\), which will be proved in Lemma 3.2. Evidently, E is continuously embedded into \(\mathcal{D}^{1,p}_{\alpha }(\mathbb{R}^{N})\). By the weighted Caffarelli–Kohn–Nirenberg’s inequality [13]
\(\mathcal{D}^{1,p}_{\alpha }(\mathbb{R}^{N})\) is continuously embedded into \(L^{p^{*}}_{\alpha }(\mathbb{R}^{N})\). Thus we get \(E\hookrightarrow \mathcal{D}^{1,p}_{\alpha }(\mathbb{R}^{N}) \hookrightarrow L^{p^{*}}_{\alpha }(\mathbb{R}^{N})\) for \(N\geq 3\).
In E, we define the following energy functional \(J\in \mathcal{C}^{1}(E,\mathbb{R})\):
Its Gateaux derivative is given by
By condition \((f_{1})\) we have
3 Existence of infinitely many solutions
In this section, we prove the existence of infinitely many solutions for problem (1.1). Next, we give the definition of a \((C)_{c}\)-sequence.
A sequence \(\{u_{n}\}\subset X\) is said to be a \((C)_{c}\)-sequence if \(J(u_{n})\rightarrow c\) and \(\|J^{\prime }(u_{n})\|(1+\|u_{n}\|)\rightarrow 0\), and it is said to satisfy the \((C)_{c}\)-condition if any \((C)_{c}\)-sequence has a convergent subsequence.
To prove our results, we use the following symmetric mountain pass theorem.
Lemma 3.1
Let X be an infinite-dimensional Banach space, \(X=Y\oplus Z\), where Y is finite dimensional. Suppose that \(I\in \mathcal{C}^{1}(X,\mathbb{R})\) satisfies the \((C)_{c}\)-condition for all \(c>0\) and the following conditions:
- \((I_{1})\):
-
\(I(0)=0\), \(I(-u)=I(u)\) for all \(u\in X\);
- \((I_{2})\):
-
there exist constants \(\rho , \theta >0\) such that \(I| _{\partial B_{\rho }\cap Z}\geq \theta \);
- \((I_{3})\):
-
for any finite-dimensional subspace \(\tilde{X}\subset X\), there is \(R=R(\tilde{X})>0\) such that \(I(u)\leq 0\) on \(\tilde{X}\setminus B_{R}\).
Then I possesses an unbounded sequence of critical values.
Lemma 3.2
([20])
The function A satisfies
and there exists \(k_{0}>0\) such that
The following two lemmas discuss the continuous and compact embedding \(E\hookrightarrow L^{q}_{K,\alpha }({\mathbb{R}^{N}})\) for all \(q\in [p,p^{*})\).
Lemma 3.3
Let \((VK1)\)–\((VK2)\) be satisfied. Then E is continuously embedded in \(L^{q}_{K,\alpha }({\mathbb{R}^{N}})\) for all \(q\in [p,p^{*})\), that is, there exists \(\gamma _{q}>0\) such that
Proof
Since \(\frac{K(x)}{V^{\theta }(x)}\rightarrow 0\) as \(|x|\rightarrow \infty \) and \(0<\frac{K(x)}{V(x)}\leq \frac{K(x)}{V^{\theta }(x)}\), we have \(\frac{K(x)}{V(x)}\rightarrow 0\) as \(|x|\rightarrow \infty \). By the continuity of \(V(x)\) and \(K(x)\) there exists \(M>0\) such that \(K(x)\leq MV^{\theta }(x)\leq MV(x)\) for all \(x\in \mathbb{R}^{N}\) and \(0<\theta <1\). If \(q=p\), then the proof is trivial. Fix \(q\in (p,p^{*})\) and choose \(\sigma =\frac{p^{*}-q}{p^{*}-p}\). Then \(q=p\sigma +(1-\sigma )p^{*}\) and \(0<\sigma <1\). From \(\mathcal{D}^{1,p}_{\alpha }(\mathbb{R}^{N})\hookrightarrow L^{p^{*}}_{\alpha }(\mathbb{R}^{N})\), \((VK2)\), (2.1), and Hölder’s inequality we can get the following inequality:
It follows that \(E\hookrightarrow L^{q}_{K,\alpha }({\mathbb{R}^{N}})\) is a continuous embedding. □
Lemma 3.4
Let \((VK1)\)–\((VK2)\) be satisfied. Then E is compactly embedded in \(L^{q}_{K,\alpha }({\mathbb{R}^{N}})\) for all \(q\in [p,p^{*})\).
Proof
From Lemma 3.3 we have \(\frac{K(x)}{V(x)}\rightarrow 0\) as \(|x|\rightarrow \infty \). Hence for any \(\varepsilon >0\), there exists \(R>0\) such that \(K(x)\leq \varepsilon V(x)\) for \(|x|>R\). Let \(\{u_{n}\}\subset E\) be a bounded sequence of E. Going if necessary to a subsequence, we may assume that
Next, we claim that \(u_{n}\rightarrow 0\) in \(L^{p}_{K,\alpha }({\mathbb{R}^{N}})\). Set \(B_{R}(0)=\{x\in \mathbb{R}^{N}: |x|\leq R\}\). Then
Hence, for any \(\varepsilon >0\), we have
from which (3.4) follows. Since \(|s|^{q}/|s|^{p}\rightarrow 0\) as \(s\rightarrow 0\) and \(|s|^{q}/|s|^{p^{*}}\rightarrow 0\) as \(s\rightarrow \infty \), then for any \(\varepsilon >0\), there exists \(C>0\) such that
To prove the lemma for general exponent q, we use an interpolation argument. Let \(u_{n}\rightarrow 0\) in E. We have just proved that \(u_{n}\rightarrow 0\) in \(L^{q}_{K,\alpha }({\mathbb{R}^{N}})\), that is,
Since the embedding \(E\hookrightarrow L^{p^{*}}_{\alpha }(\mathbb{R}^{N})\) is continuous and \(\{u_{n}\}\) is bounded in E, we also have that \(\{u_{n}\}\) is bounded in \(L^{p^{*}}_{\alpha }(\mathbb{R}^{N})\). From (3.5) we have
implying that \(u_{n}\rightarrow 0\) in \(L^{q}_{K,\alpha }({\mathbb{R}^{N}})\). This completes the proof. □
Next, we need the following lemmas to show that J satisfies Lemma 3.1.
Lemma 3.5
Under assumptions \((VK1)\), \((VK2)\), \((f_{1})\), \((f_{2})\), and \((f_{3})\), any sequence \(\{u_{n}\}\subset E\) satisfying
is bounded in E.
Proof
To prove the boundedness of \(\{u_{n}\}\), arguing by contradiction, suppose that \(\|u_{n}\|\rightarrow \infty \) as \(n\to \infty \). Let \(v_{n}=\frac{u_{n}}{\|u_{n}\|}\). Then \(\|v_{n}\|=1\). Observe that for large n,
It follows from (2.2) and (3.2) that
By (3.8) we obtain
For \(0\leq a< b\), let
Passing to a subsequence, we may assume that \(v_{n}\rightharpoonup v\) in E is satisfied. Then by Lemma 3.4, E is compactly embedded in \(L^{q}_{K,\alpha }(\mathbb{R}^{N})\), \(q\in [p,p^{*})\), \(v_{n}\rightarrow v\) in \(L^{q}_{K,\alpha }(\mathbb{R}^{N})\), \(q\in [p,p^{*})\), and \(v_{n}\rightarrow v\) a.e. on \(\mathbb{R}^{N}\).
If \(v=0\), then \(v_{n}\rightarrow 0\) in \(L^{q}_{K,\alpha }(\Omega _{n})\), \(q\in [p,p^{*})\), and \(v_{n}\rightarrow 0\) a.e. on \(\mathbb{R}^{N}\). Hence from (2.4) it follows that
Set \(\kappa ^{\prime }=\frac{\kappa }{\kappa -1}\). Since \(\kappa >\frac{N}{dp}\), we see that \(p\kappa ^{\prime }\in (p,p^{*})\). Hence from \((f_{3})\) and (3.7) we have
Combining (3.11) with (3.12), we have
which contradicts (3.9).
Set \(A:=\{x\in \mathbb{R}^{N}:v(x)\neq 0\}\). If \(v\neq 0\), then \(\operatorname{meas}(A)>0\). For a.e. \(x\in A\), we have \(\lim_{n\to \infty }|u_{n}(x)|=\infty \). Hence \(A\subset \Omega _{n}(r_{0},\infty )\) for large \(n\in \mathbb{N}\), and from Hölder’s inequality, \(h_{0}(x)\in L^{p/(p-1)}_{\overline{\alpha }}(\mathbb{R}^{N})\), \(h_{1}(x) \in L^{\infty }(\mathbb{R}^{N})\), \(h_{1}(x)\geq 1\), (2.2), (2.4), (3.1), \((f_{2})\), and Fatou’s lemma it follows that
which is a contradiction. Thus \(\{u_{n}\}\) is bounded in E. □
Lemma 3.6
Let \(p_{1}, p_{2}>1\), \(r,q\geq 1\), and \(\Omega \subseteq \mathbb{R}^{N}\). Let \(g(x,t)\) be a Carathéodory function on \(\Omega \times \mathbb{R}\) satisfying
where \(a_{1}, a_{2}\geq 0\). If \(u_{n}\rightarrow u\) in \(L^{p_{1}}_{K,\alpha }(\Omega )\cap L^{p_{2}}_{K,\alpha }(\Omega )\) and \(u_{n}\rightarrow u\) for a.e. \(x\in \Omega \). Then for any \(v\in L^{p_{1}q}_{K,\alpha }(\Omega )\cap L^{p_{2}q}_{K,\alpha }( \Omega )\),
Proof
If (3.15) is not true, then there exist a constant \(\varepsilon _{0}>0\) and a subsequence \(\{u_{k_{i}}\}\) such that
Since \(u_{n}\rightarrow u\) in \(L^{p_{1}}_{K,\alpha }(\Omega )\cap L^{p_{2}}_{K,\alpha }(\Omega )\), passing to a subsequence if necessary, we can assume that \(\sum^{\infty }_{i=1}\|u_{k_{i}}-u\|^{p_{1}}_{L^{p_{1}}_{K,\alpha }}<+ \infty \) and \(\sum^{\infty }_{i=1}\|u_{k_{i}}-u\|^{p_{2}}_{L^{p_{2}}_{K,\alpha }}<+ \infty \). Set
Then \(w_{1}\in L^{p_{1}}_{K,\alpha }(\Omega )\) and \(w_{2}\in L^{p_{2}}_{K,\alpha }(\Omega )\). Note that
and
Since \(u_{n}\rightarrow u\) for a.e. \(x\in \Omega \), by (3.17), (3.18), and Lebesgue’s dominated convergence theorem we have
which contradicts (3.16). Hence (3.15) holds. □
Similarly, we can prove the following lemma.
Lemma 3.7
Let \(p_{1}, p_{2}>1\), \(r\geq 1\), and \(\Omega \subseteq \mathbb{R}^{N}\). Let \(g(x,t)\) be a Carathéodory function on \(\Omega \times \mathbb{R}\) satisfying (3.14). If \(u_{n}\rightarrow u\) in \(L^{p_{1}}_{K,\alpha }(\Omega )\cap L^{p_{2}}_{K,\alpha }(\Omega )\) and \(u_{n}\rightarrow u\) for a.e. \(x\in \Omega \), then
Lemma 3.8
Under assumptions \((VK1)\), \((VK2)\), \((f_{1})\), \((f_{2})\), and \((f_{3})\), any sequence \(\{u_{n}\}\subset E\) satisfying (3.6) has a convergent subsequence in E.
Proof
By Lemma 3.5 the sequence \(\{u_{n}\}\) is bounded in E. Going if necessary to a subsequence, we can assume that \(u_{n}\rightharpoonup u\) in E. By Lemma 3.4, \(u_{n}\rightarrow u\) in \(L^{q}_{K,\alpha }({\mathbb{R}^{N}})\) for \(q\in [p,p^{*})\), which, together with Lemma 3.7, yields
Observe that
It is clear that
From (3.20)–(3.22) we have \(\|u_{n}-u\|\rightarrow 0\), \(n\to \infty \). □
Lemma 3.9
Under assumptions \((VK1)\), \((VK2)\), \((f_{1})\), \((f_{2})\), and \((f_{5})\), any sequence \(\{u_{n}\}\subset E\) satisfying (3.6) has a convergent subsequence in E.
Proof
First, we prove that \(\{u_{n}\}\) is bounded in E. To this end, by contradiction set \(\|u_{n}\|\rightarrow \infty \) as \(n\to \infty \). Let \(v_{n}=\frac{u_{n}}{\|u_{n}\|}\). Then \(\|v_{n}\|=1\). By (2.2), (2.3), (3.2), \((A_{3})\), \((f_{5})\), and \(h_{1}(x)\geq 1\), for large \(n\in \mathbb{N}\), we have
which implies
Passing to a subsequence, we may assume that \(v_{n}\rightharpoonup v\) in E. Then by Lemma 3.4, E is compactly embedded in \(L^{q}_{K,\alpha }(\mathbb{R}^{N})\), \(q\in [p,p^{*})\), \(v_{n}\rightarrow v\) in \(L^{q}_{K,\alpha }(\mathbb{R}^{N})\), \(q\in [p,p^{*})\), and \(v_{n}\rightarrow v\) a.e. on \(\mathbb{R}^{N}\). Hence from (3.23) it follows that \(v\neq 0\). By a similar fashion as for (3.13), we can get a contradiction. Thus \(\{u_{n}\}\) is bounded in E. The rest of the proof is the same as that in Lemma 3.8. □
Lemma 3.10
Under assumptions \((VK1)\), \((VK2)\), \((f_{1})\), and \((f_{2})\), for any finite-dimensional subspace \(\tilde{E}\subset E\), we have
Proof
Arguing indirectly, assume that for some sequence \(\{u_{n}\}\subset \tilde{E}\) with \(\|u_{n}\|\rightarrow \infty \), \(n\to \infty \), there exists \(M>0\) such that \(J(u_{n})\geq -M\) for all \(n\in \mathbb{N}\). Set \(v_{n}=\frac{u_{n}}{\|u_{n}\|}\); then \(\|v_{n}\|=1\). Passing to a subsequence, we may assume that \(v_{n}\rightharpoonup v\) in E. Since Ẽ is finite dimensional, \(v_{n}\rightarrow v\in \tilde{E}\) in E, \(v_{n}\rightarrow v\) a.e. on \(\mathbb{R}^{N}\), and thus \(\|v\|=1\). Hence we can get a contradiction by a similar fashion as for (3.13). □
Corollary 3.11
Under assumptions \((VK1)\), \((VK2)\), \((f_{1})\), and \((f_{2})\), for any finite-dimensional subspace \(\tilde{E}\subset E\), there is \(R=R(\tilde{E})>0\) such that
Since E is a reflexive separable Banach space, there exist \(\{e_{n}\}^{\infty }_{n=1}\subset E\) and \(\{e_{n}^{*}\}^{\infty }_{n=1}\subset E^{*}\) such that
For \(k=1,2,\ldots \) , we denote
Lemma 3.12
Let \((VK1)\) and \((VK2)\) be satisfied. Then for any \(p\leq q< p^{*}\), we have
Proof
It is clear that \(0 \leq \eta _{k+1}\leq \eta _{k}\), so that \(\eta _{k}\rightarrow \eta \geq 0\) (\(k\rightarrow \infty \)). For every \(k\in \mathbb{N}\), there exists \(u_{k}\in Z_{k}\) satisfying
Then there exists a subsequence of \(\{u_{k}\}\) (which we still denote by \(u_{k}\)) such that \(u_{k}\rightharpoonup u\) and
which implies that \(u=0\), and so \(u_{k}\rightharpoonup 0\). By the compact embedding \(E\hookrightarrow L^{q}_{K,\alpha }(\mathbb{R}^{N})\) we have \(u_{k}\rightarrow 0\) in \(L^{q}_{K,\alpha }(\mathbb{R}^{N})\). Hence we get \(\eta _{k}\rightarrow 0\). □
By Lemma 3.12 we can choose an integer \(m\geq 1\) such that
Lemma 3.13
Let \((VK1)\), \((VK2)\), and \((f_{1})\) be satisfied. Then there exist constants \(\rho ,\theta >0\) such that \(J| _{\partial B_{\rho }\cap Z_{m}}\geq \theta \).
Proof
From (2.2), (2.4), (3.2), (3.24), and \(h_{1}(x)\geq 1\), for \(u\in Z_{m}\), choosing \(\rho :=\|u\|=\frac{1}{2}\), we get
This completes the proof. □
Proof of Theorem 1.4
Let \(X=E\), \(Y=Y_{m}\), and \(Z=Z_{m}\). By Lemma 3.5, Lemma 3.8, Lemma 3.13, and Corollary 3.11 all conditions of Lemma 3.1 are satisfied. Thus problem (1.1) possesses infinitely many nontrivial solutions. □
Proof of Theorem 1.5
Let \(X=E\), \(Y=Y_{m}\), and \(Z=Z_{m}\). Obviously, the rest of the proof is the same as that of Theorem 1.4 by using Lemma 3.9 instead of Lemmas 3.5 and 3.8. □
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References
Alves, C.O., Soares, S.H.M.: Existence and concentration of positive solutions for a class of gradient systems. Nonlinear Differ. Equ. Appl. 12, 437–457 (2006)
Alves, C.O., Souto, M.A.S.: Existence of solutions for a class of elliptic equations in \(\mathbb{R}^{N}\) with vanishing potentials. J. Differ. Equ. 252, 5555–5568 (2012)
Alves, C.O., Souto, M.A.S.: Existence of solutions for a class of nonlinear Schrödinger equations with potential vanishing at infinity. J. Differ. Equ. 254, 1977–1991 (2013)
Alves, C.O., Souto, M.A.S., Montenegro, M.: Existence of solution for two classes of elliptic problems in \(\mathbb{R}^{N}\) with zero mass. J. Differ. Equ. 252, 5735–5750 (2012)
Ambrosetti, A., Felli, V., Malchiodi, A.: Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity. J. Eur. Math. Soc. 7, 117–144 (2005)
Bartolo, T., Benci, V., Fortunato, D.: Abstract critical point theorems and applications to some nonlinear problems with strong resonance at infinity. J. Nonlinear Anal. 7, 241–273 (1983)
Bastos, W.D., Miyagaki, O.H., Vieira, R.S.: Existence of solutions for a class of degenerate quasilinear equation in \(\mathbb{R}^{N}\) with vanishing potentials. Bound. Value Probl. 2013, 92 (2013)
Bastos, W.D., Miyagaki, O.H., Vieira, R.S.: Positive solution for a class of degenerate quasilinear elliptic equations. Milan J. Math. 82, 213–231 (2014)
Benci, V., Crisant, C.R., Micheletti, A.M.: Existence of solutions of nonlinear Schrödinger equations with \(V(\infty )=0\). Prog. Nonlinear Differ. Equ. Appl. 66, 53–65 (2005)
Berestycki, H., Lions, P.L.: Nonlinear scalar field equations, I existence of a ground state. Arch. Ration. Mech. Anal. 82, 313–345 (1983)
Byeon, J., Wang, Z.Q.: Spherical semiclassical states of a critical frequency for Schrödinger equations with decaying potentials. J. Eur. Math. Soc. 8, 217–228 (2006)
Byeon, J., Wang, Z.Q.: Standing waves for nonlinear Schrödinger equations with singular potentials. Nonlinear Anal. 26, 943–958 (2009)
Caffarelli, L., Kohn, R., Nirenberg, L.: First order interpolations inequalities with weights. Compos. Math. 53, 259–275 (1984)
Catrina, F., Wang, Z.Q.: On the Caffarelli–Kohn–Nirenberg inequalities: sharp constants. Existence (and nonexistence) and symmetry of extremal functions. Commun. Pure Appl. Math. 54, 229–258 (2001)
Chen, J.H., Huang, X.J., Cheng, B.T., Luo, H.X.: Existence and multiplicity of nontrivial solutions for nonlinear Schrödinger equations with unbounded potentials. Filomat 32, 2465–2481 (2018)
Diaz, J.I.: Nonlinear partial differential equations and free boundaries. J. Appl. Math. Mech. (1985)
Duc, D.M., Vu, N.T.: Nonuniformly elliptic equations of p-Laplacian type. Nonlinear Anal. 61, 1483–1495 (2005)
Kryszewski, W., Szulkin, A.: Generalized linking theorem with an application to semilinear Schrödinger equation. Adv. Differ. Equ. 3, 441–472 (1998)
Lin, X.Y., Tang, X.H.: Existence of infinitely many solutions for p-Laplacian equations in \(\mathbb{R}^{N}\). Nonlinear Anal. 92, 72–81 (2013)
Miyagaki, O.H., Santana, C.R., Vieira, R.S.: Ground states of degenerate quasilinear Schrödinger equation with vanishing potentials. J. Nonlinear Anal. 189, 111587 (2019)
Pino, M.D., Felmer, P.: Local mountain passes for semilinear elliptic problems in unbounded domains. Calc. Var. Partial Differ. Equ. 4, 121–137 (1996)
Rabinowitz, P.H.: Minimax Methods in Critical Point Theory with Applications to Differential Equations. Am. Math. Soc., Providence (1986)
Rabinowitz, P.H.: On a class of nonlinear Schrödinger equations. Z. Angew. Math. Phys. 43, 270–291 (1992)
Tang, X.H.: Infinitely many solutions for semilinear Schrödinger equations with sign-changing potential and nonlinearity. J. Math. Anal. Appl. 401, 407–415 (2013)
Wang, Z.Q., Willem, M.: Singular minimization problems. J. Differ. Equ. 2, 307–320 (2000)
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This work was supported by National Natural Science Foundation of China (Grant Nos. 11661053, 11771198, 11961045, and 11901276), the Provincial National Natural Science Foundation of Jiangxi (Grant Nos. 20161BAB201009, 20181BAB201003, 20202BAB201001, and 20202BAB211004).
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Meng, Y., Huang, X. & Chen, J. Infinitely many solutions of degenerate quasilinear Schrödinger equation with general potentials. Bound Value Probl 2021, 45 (2021). https://doi.org/10.1186/s13661-021-01520-x
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DOI: https://doi.org/10.1186/s13661-021-01520-x