Multiple solutions for nonhomogeneous Schrödinger–Kirchhoff type equations involving the fractional p-Laplacian in ${\mathbb {R}}^N$

Abstract

In this paper we investigate the existence of multiple solutions for the nonhomogeneous fractional p-Laplacian equations of Schrödinger–Kirchhoff type

$$\begin{aligned} M\left( \iint _{R^{2N}}\frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}dxdy\right) (-\varDelta )^s_pu+V(x)|u|^{p-2}u=f(x,u)+g(x) \end{aligned}$$

in ${\mathbb {R}}^N$, where $(-\varDelta )^s_p$ is the fractional p-Laplacian operator, with $0<s<1<p<\infty$ and $ps<N$, the nonlinearity $f:{\mathbb {R}}^N\times {\mathbb {R}}\rightarrow {\mathbb {R}}$ is a Carathéodory function and satisfies the Ambrosetti–Rabinowitz condition, $V:{\mathbb {R}}^N\rightarrow {\mathbb {R}}^+$ is a potential function and $g:{\mathbb {R}}^N\rightarrow {\mathbb {R}}$ is a perturbation term. We first establish Batsch–Wang type compact embedding theorem for the fractional Sobolev spaces. Then multiplicity results are obtained by using the Ekeland variational principle and the Mountain Pass theorem.

Appendix

In this section we show that the Banach space W defined in the Introduction is a uniformly convex Banach space. We prefer to give full details for completeness, even if it could be readily seen.

Lemma 10

$W=(W,\Vert \cdot \Vert _{W})$ is a uniformly convex Banach space.

Proof

Clearly W is complete with respect to the norm $\Vert \cdot \Vert _{W}$. Indeed, let $\{u_n\}_n$ be a Cauchy sequence in W. Thus, for any $\varepsilon >0$ there exists $\mu _\varepsilon >0$ such that if $n,m\ge \mu _\varepsilon$, then

$$\begin{aligned} V_0\Vert u_n-u_m\Vert _{L^p({\mathbb {R}}^N)}^p\le \Vert u_n-u_m\Vert ^p_{W}<\varepsilon . \end{aligned}$$

(5.1)

By the completeness of $L^p({\mathbb {R}}^N)$, there exists $u\in L^p({\mathbb {R}}^N)$ such that $u_n\rightarrow u$ strongly in $L^p({\mathbb {R}}^N)$ as $n\rightarrow \infty$. So, there exists a subsequence $\{u_{n_k}\}$ in W such that $u_{n_k}\rightarrow u$ a.e. in ${\mathbb {R}}^N$ as $k\rightarrow \infty$ (see [9, Theorem 4.9]). Therefore, by the Fatou Lemma and the second inequality in (5.1) with $\varepsilon =1$, we have

$$\begin{aligned} \iint _{{\mathbb {R}}^{2N}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+ps}}dxdy&\le \liminf _{k\rightarrow \infty }\iint _{{\mathbb {R}}^{2N}}\frac{|u_{n_k}(x)-u_{n_k}(y)|^{p}}{|x-y|^{N+ps}}dxdy\\&\le \liminf _{k\rightarrow \infty }\left( \Vert u_{n_k}-u_{\mu _1}\Vert _{W}+\Vert u_{\mu _1}\Vert _{W}\right) ^p\\&\le (1+\Vert u_{\mu _1}\Vert _{W})^p<\infty . \end{aligned}$$

Thus, $u\in W$. Let $n\ge \mu _\varepsilon$, by the second inequality in (5.1) and the Fatou Lemma, we get

$$\begin{aligned} \Vert u_n-u\Vert _{W}^p\le \liminf _{k\rightarrow \infty }\Vert u_n-u_{n_k}\Vert _{W}^p\le \varepsilon , \end{aligned}$$

that is, $u_n\rightarrow u$ strongly in W as $n\rightarrow \infty$.

Next, we prove that $(W, \Vert \cdot \Vert _{W})$ is uniformly convex. To this aim fix $\varepsilon \in (0,2)$ and let $u, v\in W$, with $\Vert u\Vert _{W}=\Vert v\Vert _{W}=1$ and $\Vert u-v\Vert _{W}\ge \varepsilon$.

Case $p\ge 2$. By the following inequality (see [1])

$$\begin{aligned} \left| \frac{a+b}{2}\right| ^{p}+\left| \frac{a-b}{2}\right| ^{p}\le \frac{1}{2}\left( |a|^{p}+|b|^{p}\right) \quad \forall \ a,\ b\in {\mathbb {R}}, \end{aligned}$$

it is easy to verify that

$$\begin{aligned}&\left\| \frac{u+v}{2}\right\| ^p_{W}+\left\| \frac{u-v}{2}\right\| ^p_{W}\nonumber \\&\quad =\left[ \frac{u+v}{2}\right] ^p_{s,p}+\left[ \frac{u-v}{2}\right] ^p_{s,p} +\left\| \frac{u+v}{2}\right\| ^p_{p,V}+\left\| \frac{u-v}{2}\right\| ^p_{p,V}\nonumber \\&\quad \le \frac{1}{2}\left( \left[ u\right] ^p_{s,p}+\left[ v\right] ^p_{s,p}+\left\| u\right\| ^p_{p,V}+\left\| v\right\| ^p_{p,V}\right) \nonumber \\&\quad =\frac{1}{2}\Vert u\Vert ^p_{W}+\frac{1}{2}\Vert v\Vert ^p_{W}=1. \end{aligned}$$

(5.2)

It follows from (5.2) that $\left\| (u+v)/2\right\| ^p_{W}\le 1-(\varepsilon /2)^p.$ Taking $\delta = \delta (\varepsilon )$ such that $1-(\varepsilon /2)^p = (1-\delta )^p$, we obtain that $\left\| (u+v)/2\right\| _{W}\le 1-\delta .$

Case $1<p<2$. First, we notice that

$$\begin{aligned} {[}u]^{p^\prime }_{s,p}=\left[ \iint _{{\mathbb {R}}^{2N}}\left( \left( \left| u(x)-u(y)\right| \cdot \left| x-y\right| ^{\frac{-N-ps}{p}}\right) ^{p^\prime }\right) ^{p-1}dxdy\right] ^{\frac{1}{p-1}}, \end{aligned}$$

here $p^\prime =p/(p-1)$. Using the reverse Minkowski inequality (see [1, Theorem 2.13]) and $(V_{1})$, we get

$$\begin{aligned}&\left\| \frac{u+v}{2}\right\| _{W}^{p^\prime }+\left\| \frac{u-v}{2}\right\| _{W}^{p^\prime }\nonumber \\&\quad =\left\| \left( \left| \frac{u(x)+v(x)}{2}-\frac{u(y)+v(y)}{2}\right| |x-y|^{\frac{-N-ps}{p}}\right) ^{p^\prime }\right\| _{L^{p-1}({\mathbb {R}}^{2N})}\nonumber \\&\qquad +\left\| \left( V(x)^{\frac{1}{p}} \left| \frac{u(x)+v(x)}{2}\right| \right) ^{p^\prime } \right\| _{L^{p-1}({\mathbb {R}}^{2N})}\nonumber \\&\qquad +\left\| \left( \left| \frac{u(x)+v(x)}{2}-\frac{u(y)+v(y)}{2}\right| |x-y|^{\frac{-N-ps}{p}}\right) ^{p^\prime }\right\| _{L^{p-1}({\mathbb {R}}^{2N})}\nonumber \\&\qquad +\left\| \left( V(x)^{\frac{1}{p}} \left| \frac{u(x)-v(x)}{2}\right| \right) ^{p^\prime } \right\| _{L^{p-1}({\mathbb {R}}^{2N})}\nonumber \\&\quad \le \left\{ \iint _{{\mathbb {R}}^{2N}}\left[ \left( \left| \frac{u(x)-u(y)}{2}+\frac{v(x)-v(y)}{2}\right| |x-y|^{\frac{-N-ps}{p}}\right) ^{p^\prime }\right. \right. \nonumber \\&\qquad +\left( V(x)^{\frac{1}{p}} \left| \frac{u(x)+v(x)}{2}\right| \right) ^{p^\prime }\nonumber \\&\qquad +\left( \left| \frac{u(x)-u(y)}{2}-\frac{v(x)-v(y)}{2}\right| |x-y|^{\frac{-N-ps}{p}}\right) ^{p^\prime }\nonumber \\&\qquad \left. \left. +\left( V(x)^{\frac{1}{p}} \left| \frac{u(x)-v(x)}{2}\right| \right) ^{p^\prime }\right] ^{p-1}dxdy\right\} ^{\frac{1}{p-1}}. \end{aligned}$$

(5.3)

By the following inequality (see [1])

$$\begin{aligned} \left| \frac{a+b}{2}\right| ^{p^\prime }+\left| \frac{a-b}{2}\right| ^{p^\prime }\le \left( \frac{1}{2}\left( \left| a\right| ^{p}+\left| b\right| ^{p}\right) \right) ^{\frac{1}{p-1}}\quad \hbox {for all }a,\ b\in {\mathbb {R}}, \end{aligned}$$

we obtain in addition to (5.3) that

$$\begin{aligned} \left\| \frac{u+v}{2}\right\| _{W}^{p^\prime }+\left\| \frac{u-v}{2}\right\| _{W}^{p^\prime } \le \left( \frac{1}{2}\Vert u\Vert ^p_{W}+\frac{1}{2}\Vert v\Vert ^p_{W}\right) ^{p^\prime -1}=1. \end{aligned}$$

(5.4)

Here we apply the following inequality:

$$\begin{aligned} \left( a^{\frac{1}{p-1}}+b^{\frac{1}{p-1}}\right) ^{p-1}\le a+b\quad \hbox {for all } a,\ b\ge 0, \end{aligned}$$

thanks to $1<p<2$. From (5.4) we have

$$\begin{aligned} \left\| \frac{u+v}{2}\right\| ^{p^\prime }_{W}\le 1-\left( \frac{\varepsilon }{2}\right) ^{p^\prime }. \end{aligned}$$

Taking $\delta =\delta (\varepsilon )$ such that $1-(\varepsilon /2)^{p^\prime }=(1-\delta )^{p^\prime }$, we get the desired claim. $\square$

Remark 2

By Theorem 1.21 of [1], the space W is a reflexive Banach space. With the same arguments as Lemma 10, it easily follows that $W^{s,p}({\mathbb {R}}^N)$ is also a uniformly convex Banach space, and hence $W^{s,p}({\mathbb {R}}^N)$ is a reflexive Banach space.