Nonautonomous fractional Hamiltonian system with critical exponential growth

Abstract

In this paper, we study the following nonlocal nonautonomous Hamiltonian system on whole ${\mathbb {R}}$

$$\begin{aligned} \left\{ \begin{array}{ll} (-\Delta )^\frac{1}{2}~ u +u=Q(x) g(v)&{}\quad \text{ in } {\mathbb {R}},\\ (-\Delta )^\frac{1}{2}~ v+v = P(x)f(u)&{}\quad \text{ in } {\mathbb {R}}, \end{array}\right. \end{aligned}$$

where $(-\Delta )^\frac{1}{2}$ is the square root Laplacian operator. We assume that the nonlinearities f, g have critical growth at $+\,\infty$ in the sense of Trudinger–Moser inequality and the nonnegative weights P(x) and Q(x) vanish at $+\infty$. Using suitable variational method combined with the generalized linking theorem, we obtain the existence of at least one positive solution for the above system.

Appendix: Validation of Remark 1.2

In this appendix, we show the claim that under the assumptions (H2) and (H5), f (respectively g) belongs to the class ${\mathbb {C}}_{\mathrm P}$ (respectively ${\mathbb {C}}_{\mathrm Q}$) and that if $f(t)=O(t^2)$ then $u_n\rightharpoonup 0$ and $\{\int _{\mathbb {R}} P(x)f(u_n)u_n \,\mathrm {d}x\}$ bounded imply that $\int _{\mathbb {R}} P(x)f(u_n)\,\mathrm {d}x\rightarrow 0$.

Proof

Let $\{u_n\} \subset H^{1/2, 2}({\mathbb {R}})$ be a sequence such that

$$\begin{aligned} u_n\rightharpoonup 0,\;\; \text {and}\;\; \int _{\mathbb {R}} P(x)f(u_n)u_n \,\mathrm {d}x<C. \end{aligned}$$

(5.20)

For any given $\epsilon >0$, using (H2) and (H5), there exists $c_o=c_o(\epsilon )>0$ sufficiently small and $M>0$ sufficiently large such that

$$\begin{aligned} F(t)\le \epsilon |t|^2 \quad \text {for}\quad |t|<c_o \quad \text { and } \quad F(t)\le \epsilon f(t)t \quad \text { for } t>M. \end{aligned}$$

Hence from (5.20), we have

$$\begin{aligned} \begin{aligned} \int _{{\mathbb {R}}} P(x)F(u_n)\, \mathrm {d}x&\le \epsilon \int _{\{u_n\le c_o\}}P(x)|u_n|^2\, \mathrm {d}x+\int _{\{c_o\le u_n\le M\}}P(x)F(u_n)\, \mathrm {d}x\\&\quad +\epsilon \int _{\{u_n\ge M\}}P(x)f(u_n)u_n\, \mathrm {d}x\\&\le \epsilon C+\int _{\{c_o\le u_n\le M\}}P(x)F(u_n)\, \mathrm {d}x. \end{aligned} \end{aligned}$$

(5.21)

Further for $L=L(\epsilon )>0$ large enough such that $P(x)<\epsilon$ for $x\in B^c_L(0)$, we have

$$\begin{aligned} \int _{\{c_o\le u_n\le M\}\cap B^c_L(0)}P(x)F(u_n)\, \mathrm {d}x\le \epsilon C, \end{aligned}$$

(5.22)

where C is independent of n. Indeed,

$$\begin{aligned} c_o^2| \{c_o\le u_n\le M\}|\le \int _{\{c_o\le u_n\le M\}}|u_n|^2 \,\mathrm {d}x \le \int _{{\mathbb {R}}}|u_n|^2\,\mathrm {d}x \le C. \end{aligned}$$

Now by Lebesgue theorem, for a such fixed $L>0$, we have also

$$\begin{aligned} \int _{\{c_o\le u_n\le M\}\cap [-L, L]}P(x)F(u_n)\, \mathrm {d}x\rightarrow 0. \end{aligned}$$

(5.23)

Gathering (5.21)–(5.23), we get

$$\begin{aligned} \int _{{\mathbb {R}}} P(x)F(u_n) \, \mathrm {d}x \rightarrow 0. \end{aligned}$$

It finishes the proof of the first part of the claim. Next, we show the second statement of the claim. From $f(t)=O(t^2)$ and for $c_o$, $M>0$ respectively small and large enough, we have

$$\begin{aligned} \int _{\mathbb {R}} P(x) f(u_n)\,\mathrm {d}x&=\int _{\{u_n< c_o\}}P(x) f(u_n) \,\mathrm {d}x+\int _{\{c_o\le u_n\le M\}}P(x) f(u_n) \,\mathrm {d}x\\&\quad + 1/M\int _{\{u_n> M\}} P(x)f(u_n)u_n \,\mathrm {d}x\\&\le C \int _{\mathbb {R}} P(x) u_n^2 \,\mathrm {d}x +\int _{\{c_o\le u_n\le M\}}P(x) f(u_n) \,\mathrm {d}x+C/M. \end{aligned}$$

Using Lemma 2.1 and estimating the second integral in the above inequality in a similar way as in (5.22) and (5.23), we get the required result. $\square$

Remark 5.1

Let the function P be defined as $P(x)=\frac{1}{(|x|+1)^\epsilon }$, for $\epsilon >0$ sufficiently small. Consider the sequence $\{u_n\}\subset H^{1/2,2}({\mathbb {R}})$ such that

$$\begin{aligned} u_n(x)=\displaystyle {\left\{ \begin{array}{ll} \frac{1}{n^\alpha },&{} \text{ for } x\in [n,2n],\\ \frac{(x-(n-1))}{n^\alpha },&{} \text{ for } x\in [n-1,n],\\ \frac{((2n+1)-x)}{n^\alpha },&{} \text{ for } x\in [2n,2n+1],\\ 0&{} \text{ elsewhere } \end{array}\right. } \end{aligned}$$

with $\alpha \in [1/2,1)$. Then, by straighforward calculations, we can prove $\{u_n\}$ is bounded in $H^{1/2,2}({\mathbb {R}})$. Furthermore, as $n\rightarrow \infty$

$$\begin{aligned} \int _{{\mathbb {R}}}P(x)u_n^q\,\mathrm {d}x\rightarrow 0 \end{aligned}$$

if and only if q satisfies $\alpha q+\epsilon >1$. Therefore if f is of $O(t^{1+\ell })$ near 0 with $0<\ell \le \frac{1-\alpha -\epsilon }{\alpha }$, we easily get that $u_n\rightharpoonup 0$ weakly in $H^{1/2,2}({\mathbb {R}})$ and

$$\begin{aligned} \int _{{\mathbb {R}}}P(x)f(u_n)u_n\,\mathrm {d}x\rightarrow 0 \end{aligned}$$

as $n\rightarrow \infty$. However, $\int _{{\mathbb {R}}}P(x)f(u_n)\,\mathrm {d}x\rightarrow 0$ is not verified.