On superlinear fractional advection dispersion equation in \(\mathbb{R}^{N}\)

We consider the fractional advection dispersion equation in \(\mathbb {R}^{N}\). The nonlinearity is superlinear but does not satisfy the Ambrosetti-Rabinowitz type condition. We obtain the existence of nontrivial solutions of the equations, improving a recent result of Zhang-Sun-Li (Appl. Math. Model. 38:4062-4075, 2014).

In this paper we are concerned with the following fractional advection dispersion equation:

where \(N>1\), \(\beta\in(0,1)\), \(\alpha=\frac{\beta+1}{2}\), \(M(d\theta)\) is a Borel probability measure on the unit sphere in \(\mathbb{R}^{N}\), \(D_{\theta}^{\beta}\) denotes directional fractional derivative of order β in the direction of the unit vector θ. We recall that the theory of fractional operators has gained a lot of attention in a large scientific community including pure mathematicians, applied scientists, and numerical analysts: see [1–10] and the references therein. Now, we make the following assumptions on the functions M, V, and f.

(H _M ):

There is a constant \(c_{M}>0\) such that \(\int_{|\theta|=1}|\xi \cdot\theta|^{2\alpha}M(d\theta)\geq c_{M}\) for \(\xi\in\mathbb {S}^{N-1}\), where \(\mathbb{S}^{N-1}=\{x\in\mathbb{R}^{N}: |x|=1\}\).

(H _V ):

\(V\in C(\mathbb{R}^{N})\), V is 1-periodic in \(x_{1}, x_{2},\ldots , x_{N}\) and \(0< V_{0}\leq V(x)\leq V_{1}<+\infty\).

(H _f ):

1. (i)

   \(f\in C(\mathbb{R}^{N}\times\mathbb{R})\) is 1-periodic in \(x_{1},x_{2},\ldots, x_{N}\), and

   $$ \lim_{|t|\rightarrow+\infty}\frac{f(x,t)}{|t|^{2_{\alpha}^{*}-1}}=0,\qquad \lim _{|t|\rightarrow+\infty}\frac {F(x,t)}{|t|^{2}}=+\infty $$

   (1)

   uniformly in \(x\in\mathbb{R}^{N}\), where \(F(x,t)=\int_{0}^{t} f(x,s)\,ds\) and \(2_{\alpha}^{*}=\frac{2N}{N-2\alpha}\).

2. (ii)

   \(\lim_{ t \rightarrow0}\frac{f(x,t)}{t}=0\) uniformly with respect to \(x\in\mathbb{R}^{N}\).

3. (iii)

   There exists \(\vartheta\geq1\) such that \(\vartheta G(x,t)\geq G(x,st)\) for \((x,t)\in\mathbb{R}^{N}\times\mathbb{R}\) and \(s\in [0,1]\), where \(G(x,t)=f(x,t)t-2F(x,t)\).

Note that if the nonlinearity is subcritical and superlinear, that is, for some positive constants \(c_{1}>0\) and \(p\in[2,2_{\alpha}^{*}-1)\),

then (1) is satisfied. Our condition (1) is slightly weaker. The condition (1) was first introduced by Liu and Wang [11] and then was used in [12]. The condition (H _f )(iii) is originally due to Jeanjean [13] for semilinear problem in \(\mathbb{R}^{N}\). It is known that (H _f )(iii) is weaker than the condition that

(h₁):

for each \(x\in\mathbb{R}^{N}\), \(\frac{f(x,t)}{|t|}\) is an increasing function of t in \(\mathbb{R}\backslash\{0\}\),

1. (AR)

   There exist \(\mu>2\) such that

   $$0< \mu F(x,t)\leq tf(x,t),\quad \forall x\in\mathbb{R}^{N}, t\neq0. $$

It is well known that the condition (AR) is crucial in verifying the boundedness of the family of the corresponding functionals \((\mathit{PS})_{c}\), \(c\in\mathbb{R}\). This is very crucial in applying the critical point theory. However, there are many functions which are superlinear at infinity, but do not satisfy the condition (AR) for any \(\mu>2\). In fact, the condition (AR) implies that \(F(x,t)\geq C|t|^{\mu}\) for some \(C>0\). Thus, for example the superlinear function \(f(x,t)=t\log(1+|t|)\) does not satisfy the condition (AR). However, it satisfies our condition (H _f ). From the above we can see the (AR) condition is stronger than (H _f ).

We look for solution of (P) in the space

endowed with the norm

The fractional Sobolev space \(J_{M}^{\alpha}(\mathbb{R}^{N})\) is defined as the completion of \(C_{0}^{\infty}(\mathbb{R}^{N})\) with respect to the norm

From Lemma 2.6 in [1], we have \(E\subset J_{M}^{\alpha}(\mathbb {R}^{N})\hookrightarrow L^{p}(\mathbb{R}^{N})\), \(2\leq p\leq2_{\alpha}^{*}\). By conditions (H _f )(i) and (H _f )(ii), there exists a constant \(C> 0\) such that

Hence the energy functional \(\varphi: E\rightarrow\mathbb{R}\) given by

is well defined and of class \(C^{1}\). The derivative of φ is given by

for \(v\in C_{0}^{\infty}(\mathbb{R}^{N})\), where \((\cdot, \cdot)\) denotes the inner product in \(L^{2}(\mathbb{R}^{N})\). Therefore, the critical points of φ are weak solutions of (P). We are now in the position to state our main results.

Theorem 1

Suppose the measure M is symmetric and satisfies (H _M ). If (H _V ) and (H _f ) hold, then problem (P) has at least one nontrivial solution.

Corollary 1

Suppose the measure M is symmetric and satisfies (H _M ). If \(V(x)=V_{0}\), \(f(x,u)=f(u)\) and (H _f ) are satisfied, then problem (P) has at least one nontrivial solution.

This corollary improves a recent result of Zhang-Sun-Li [1], Theorem 3.1. In the some paper [1], in order to get the existence of a nontrivial solution, by substituting the condition (H _M ), Zhang-Sun-Li assumed in addition to (H _f )(ii), (AR), and the well-known subcritical growth condition, that is:

(h₂):

there are positive constants c and \(p\in[2,\frac{N+2\alpha }{N-2\alpha})\) such that

$$\bigl\vert f(x,u)\bigr\vert \leq c\bigl(1+\vert u\vert ^{p} \bigr), \quad x\in\mathbb{R}^{N}, u\in\mathbb{R}. $$

In this paper, we consider problem (P) in the case when the nonlinear term \(f(x, t)\) is superlinear at infinity but does not satisfy the (AR) type condition. Thus, the variational functional φ may possess unbounded \((\mathit{PS})\) sequences. To overcome this difficulty, the Cerami sequences are employed (see Lemma 5).

The rest of this paper is organized as follows. In Section 2, we state some preliminary results that will be used later. We will finish the proof of our main result (Theorem 1) in Section 3.

In this section, we introduce some basic definitions and properties of the fractional calculus which are used further in this paper. For the proofs, which are omitted, we refer the reader to [1] or other texts on basic fractional calculus.

Definition 1

Let \(u:\mathbb{R}^{N}\rightarrow\mathbb{R}\), \(\alpha>0\), θ be a unit vector in \(\mathbb{R}^{N}\). The αth order fractional integral in the direction of θ of u is given by

and the αth order directional derivative in the direction of θ is defined by

where n denotes the smallest integer greater than or equal to α.

Definition 2

Let \(u:\mathbb{R}^{N}\rightarrow\mathbb{R}\), \(\alpha>0\) (or \(\alpha <0\)) be given. Then the αth order fractional integral (derivative) with respect to the measure M is defined as

where \(M(d\theta)\) is a Borel probability measure on the unit sphere in \(\mathbb{R}^{N}\).

Definition 3

Let \(\alpha=\frac{1+\beta}{2}\), a function \(u\in J_{M}^{\alpha}(\mathbb {R}^{N})\) is called of (P) if

for all \(v\in C_{0}^{\infty}(\mathbb{R}^{N})\).

Lemma 1

If \(\alpha\in(0,1)\), then for \(u\in C_{0}^{\infty}(\mathbb{R}^{N})\), we have the following Fourier transform property:

where \(\mathcal{F}(u)(\xi)=\frac{1}{(2\pi)^{\frac{N}{2}}}\int _{\mathbb{R}^{N}}e^{-i\xi\cdot x}u(x)\, dx\).

Lemma 2

Assume that the measure M is symmetric. Let \(\alpha>0\), \(u,v\in J_{M}^{\alpha}(\mathbb{R}^{N})\). Then, for M-a.e. \(\theta\in S^{N-1}\), we have

especially

Throughout this paper, we will use the norm \(\|\cdot\|\) in E. As usual, for \(1\leq\nu<\infty\), we let

For the reader’s convenience, we review the main embedding result for this class of fractional Sobolev spaces.

Lemma 3

([1], Lemma 2.6)

E continuously embedded into \(L^{p}(\mathbb{R}^{N})\) for \(p\in[2,2_{\alpha}^{*}]\), and compactly embedded into \(L_{\mathrm{loc}}^{p}(\mathbb{R}^{N})\) for \(p\in[2,2_{\alpha}^{*})\).

In order to prove Theorem 1, we need the following lemma whose proof is analogous to that of Lemma 1.21 in [17] (see also [18]).

Lemma 4

Let \(r>0\) and \(2\leq p<2_{\alpha}^{*}\). If \(\{u_{n}\}\) is bounded in \(J_{M}^{\alpha}(\mathbb{R}^{N})\) and if

where \(B_{r}(y)=\{x\in\mathbb{R}^{N}: |x-y|< r\}\), then \(u_{n}\rightarrow0\) in \(L^{s}(\mathbb{R}^{N})\) for \(s\in(2,2_{\alpha}^{*})\).

In order to ensure the existence of solutions for the problem (P), our main tool will be the mountain pass theorem [19], p.140, Theorem 6, which will be used in our proof.

Lemma 5

Let X be a Banach space and \(I: X\rightarrow\mathbb{R}\) a continuous, Gâteaux-differentiable function, such that \(I': X\rightarrow X\) is continuous from the norm topology of X to the weak ^∗ topology of \(X^{*}\). Take two points \((z_{0}, z_{1})\) in X and consider the set Γ of all continuous paths from \(z_{0}\) to \(z_{l}\):

Define a number c by \(c=\inf_{\gamma\in\Gamma}\max_{t\in[0,1]}I(\gamma(t))\). Assume there is a closed subset Θ of X such that \(\Theta\cap I^{c}\) separates \(z_{0}\) and \(z_{1}\) with \(I^{c}=\{x\in X: I(x)\geq c\}\). Then there is a sequence \(\{x_{n}\}\) in X such that

In this section, for the notation in Lemma 5, the space \(X=E\), and related functional on E is \(I=\varphi\). Recall that a sequence \(\{u_{n}\}\subset E\) is called a Palais-Smale sequence of φ the level c, a \((\mathit{PS})_{c}\) sequence for short, if \(\varphi (u_{n})\rightarrow c\) and \(\varphi'(u_{n})\rightarrow0\). A sequence \(\{u_{n}\}\subset E\) is called a Cerami sequence ϕ at the level c, a \((C)_{c}\) sequence for short, if

We deduce from (H _f )(iii) that

Let \(t>0\). For \(x\in\mathbb{R}^{N}\), by direct computations we have

Taking into account hypothesis (H _f )(ii) we deduce

So it follows from (5) and (6) that \(F(x,t)\geq0\) for all \(x\in\mathbb{R}^{N}\) and \(t\geq0\). Arguing similarly for the case \(t\leq0\), eventually we obtain

Lemma 6

There exists \(r>0\) and \(\eta_{0}\in E\) such that \(\|\eta_{0}\|>r\) and

Proof

First of all, from (H _f )(i) and (H _f )(ii) it follows that, for all given \(\varepsilon>0\), there exists \(c_{\varepsilon}>0\) such that

Consequently, using Lemma 2, we have

Since \(E\hookrightarrow L^{p}(\mathbb{R}^{N})\), \(2\leq p\leq2_{\alpha}^{*}\), we know there exists a constant \(C>0\) such that \(|u|_{p}\leq C\|u\|\). So

By choosing \(\varepsilon>0\) such that \(-\frac{\cos(\pi\alpha )}{2}-\varepsilon C^{2}\geq-\frac{\cos(\pi\alpha)}{4}\), we obtain

That is, there exist \(r>0\) and \(\rho>0\), such that

Using (1), it is easy to see that for any \(u\neq0\), we have \(\varphi(tu)\rightarrow-\infty\) as \(t\rightarrow+\infty\). Hence, there is a point \(\eta_{0}\in E\backslash\overline{B}_{r}\) such that \(\varphi(\eta_{0})\leq0\). □

By Lemma 6 we see that φ has a mountain pass geometry: that is, setting

we have \(\Gamma\neq\emptyset\). Moreover, it is easy to see that

Take \(F =E\) in Lemma 5. Equation (8) implies that \(\varphi ^{c}\) separates \(\gamma(0)=0\) and \(\gamma(1)=\eta_{0}\), and there exists a \((C)_{c}\) sequence \(\{u_{n}\}\) for φ.

Claim

The sequence \(\{u_{n}\}\) is bounded.

If \(\{u_{n}\}\) is unbounded, up to a subsequence we may assume that

In particular,

Let \(w_{n}=\frac{u_{n}}{\|u_{n}\|}\), then \(\{w_{n}\}\) is bounded in E. We claim that

Otherwise, for some \(\delta>0\), up to a subsequence we have

So we can choose \(\{y_{n}\} \subset\mathbb{R}^{N}\) such that

It is easy to see that the number of points in \(Z^{N}\cap B_{2}(y_{n})\) is less than \(4^{N}\); there exists \(z_{n}\in B_{2}(y_{n})\) such that

Let \(\overline{w}_{n}(x) = w_{n}(x+z_{n})\). Then \(\{\overline{w}_{n}\}\) is also bounded in E. Passing to a subsequence we have

Remark that

and so \(\overline{w}\neq0\). Let \(\overline{u}_{n}=\|u_{n}\|\overline {w}_{n}\). It is easy to see that \(\overline{u}_{n}(x)\rightarrow+\infty\) as \(n\rightarrow+\infty\). Using (1) we have

Recall that \(f(x, u)\) is 1-periodic with respect to x. Using this fact, we obtain

Since the set \(\Omega_{0}= \{x\in\mathbb{R}^{N}:\overline{w}(x)\neq0\} \) has positive Lebesgue measure, using (13) we have

This is impossible. Therefore we have proved (10). Hence, using Lemma 4 and (10), we obtain

Next, we shall derive a contradiction as follows. Given a real number \(R >0\), by (H _f )(i) and (H _f )(ii), for any \(\varepsilon>0\), there exists \(c_{\varepsilon}>0\) such that

Note that \(\|w_{m}\|=1\). So from (14), (15), and the continuity of the embedding \(E\hookrightarrow L^{p}(\mathbb{R}^{N})\) (since \(p\in[2,2_{\alpha}^{*}]\)), we obtain

Now let \(\varepsilon\rightarrow0\), we obtain

Let \(t_{n}\in[0, 1]\) such that \(\varphi(t_{n}u_{n})=\max_{t\in [0,1]}\varphi(tu_{n})\). Given \(m>0\). Since for n large enough we have \((-\frac{4m}{\cos (\pi\alpha)} )^{\frac{1}{2}}\|u_{n}\|^{-1}\in(0,1)\), using (16) with \(R= (-\frac{4m}{\cos(\pi\alpha)} )^{\frac{1}{2}}\), we have

That is, \(\varphi(t_{n}u_{n})\rightarrow+\infty\). But \(\varphi(0)=0\), \(\varphi(u_{n})\rightarrow c\), we see that \(t_{n}\in(0, 1)\), and

Because of hypothesis (H _f )(iii),

This contradicts (9) and consequently we have proved that \(\{ u_{n}\}\) is bounded.

Let

If \(\tau= 0\), using Lemma 4, similarly to (16) we have

Hence using (9) we have \(c= 0\), a contradiction. Therefore \(\tau>0\). Similarly to (12), we can choose a sequence \(\{z_{n}\} \subset Z^{N}\) such that setting \(\overline{u}_{n}(x)=u_{n}(x+z_{n})\), we have

Note that \(\|\overline{u}_{n}\|=\|u_{n}\|\), we see that \(\{\overline {u}_{n}\}\) is bounded. Going if necessary to a subsequence, we obtain

Returning to (18), we see that \(\overline{u}\neq0\). Moreover, by the \(Z^{N}\) invariance of the problem, \(\{\overline{u}_{n}\}\) is also a \((C)_{c}\) sequence of φ. Thus for every \(v\in C_{0}^{\infty}(\mathbb{R}^{N})\), we have

So \(\varphi'(\overline{u})= 0\) and u̅ is a nontrivial solution of (P).

This work is supported by the National Natural Science Foundation of China (No. 11201095), the Youth Scholar Backbone Supporting Plan Project of Harbin Engineering University (No. 307201411008), the Fundamental Research Funds for the Central Universities (No. 2016), Heilong Jiang Postdoctoral Funds for scientific research initiation (No. LBH-Q14044), the Science Research Funds for Overseas Returned Chinese Scholars of Heilongjiang Province (No. LC201502).

Affiliations

1. College of Aerospace and Civil Engineering, Harbin Engineering University, Harbin, 150001, P.R. China
   • Gang-Ling Hou
2. Department of Applied Mathematics, Harbin Engineering University, Harbin, 150001, P.R. China
   • Bin Ge

Corresponding author

Correspondence to Bin Ge.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors contributed equally to this paper. All authors read and approved the final manuscript.

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