A multiplicity result for double phase problem in the whole space

1.   Introduction

In recent years, the differential equations and variational problems driven by the so-called double phase operator have been greatly studied. The existence of solutions for double phase problems on bounded domains have been greatly discussed, see for example ^{[1,2,3,4,5,6,7,8,9]}. For unbounded domains, Liu and Dai ^[10], Liu and Winkert ^[11], Robert^[12], Ge and Pucci^[13] and Shen, Wang, Chi and Ge ^[14] investigated the existence and multiplicity of solutions for double phase problem.

In this paper we study the following double phase problem:

where $1 < p < q < N$ and

The first work concerning the ground state solution for problem $(P)$, was that of Liu and Dai ^[10]. More specifically, they studied the existence of at least three nontrivial solutions of $(P)$ under the following assumption on $f$:

$(h_1)$ $f\in C(\mathbb{R}^N\times\mathbb{R}, \mathbb{R})$ and there exists $\gamma\in(q, p^*)$ such that

where $p^* = \frac{Np}{N-p}$, $k(x)\geq 0$, $k\in L^{^\theta}(\mathbb{R}^N)\cap L^{^\infty}(\mathbb{R}^N)$ with $\frac{1}{\theta}+\frac{\gamma}{\tau} = 1$, here $\theta > 1$ and $\tau\in(\gamma, p^*]$.

$(h_2)$ $\lim\limits_{t\rightarrow 0}\frac{f(x, t)}{|t|^{^{p-1}}} = 0$ uniformly in $x$.

$(h_3)$ $\lim\limits_{t\rightarrow +\infty}\frac{F(x, t)}{|t|^{^q}} = +\infty$ uniformly in $x$.

$(h_4)$ $\frac{f(x, t)}{|t|^{^{q-1}}}$ is strictly increasing on $(-\infty, 0)$ and $(0, +\infty)$.

It must be point out that $(h_1)$ is subcritical growth condition, $(h_3)$ means that $f(x, u)$ is superlinear at infinity; $(h_4)$ is a well-known Nehari-type condition. In the present paper, we will further study the existence of two non-trivial weak solutions of $(P)$ under the following sublinear growth condition:

$(h_1)'$ $f\in C(\mathbb{R}^N\times\mathbb{R}, \mathbb{R})$ and there exists $\gamma\in(1, p)$ such that

where $k(x)\geq 0$, $k\in L^{^\theta}(\mathbb{R}^N)\cap L^{^\infty}(\mathbb{R}^N)$ with $\frac{1}{\theta}+\frac{\gamma}{p^*} = 1$.

$(h_5)$ There exists a $C > 1$ large enough, $c_{_0} > 0$, $x_{_0}\in\mathbb{R}^N$, $0 < r < 1$ such that $f(x, t) = 0$, for any $x\in\mathbb{R}^N$, $0 < |t|\leq \delta$ and

where $0 < \delta < \min\Big\{ \frac{1}{2} \Big(\frac{c_{_0}pr^{^q}}{\gamma 2^{^{\gamma+1}}(C^{^q}+r^{^q})m_\mu }\Big)^{^{\frac{1}{p-\gamma}}}, \frac{1}{2} \Big\}$ and $m_\mu = \max\big\{1, \sup\limits_{x\in B_{_{2r}}(x_{_0})}\mu(x)\big\}$.

Remark 1.1. There are many functions $f(x, t)$ satisfying $(h_1)'$ and $(h_5)$. For example,

where $k_1(x)\geq 0$, $k\in C(\mathbb{R}^N)\cap L^{^\theta}(\mathbb{R}^N)\cap L^{^\infty}(\mathbb{R}^N)$ with $\frac{1}{\theta}+\frac{\gamma}{p^*} = 1$ and $\inf\limits_{x\in B_r(x_0)}k(x)\geq c_0 > 0$. Indeed,

Hence, we have $|f(x, t)|\leq k(x)|t|^{\gamma-1}$ with $k(x) = k_1(x)(1+\frac{1}{\delta^{\gamma-1}})$ and $f(x, t) = k_1(x)(t-\delta)^{\gamma-1}\geq (t-\delta)^{\gamma-1}\inf\limits_{x\in B_{_r}(x_{_0})}k_1(x) = c_0(t-\delta)^{\gamma-1}$ for all $x\in B_{_r}(x_{_0})$ and $\delta < t\leq1$.

The main result of this paper establishes the following Theorem 1.2.

Theorem 1.2. Assume that hypotheses (1.1), $(h_1)'$ and $(h_5)$ hold. Then the problem $(P)$ has at least two distinct nontrivial weaksolutions $u_{_0}, \widetilde{u}_{_0}$ in $W^{^{1, H}}(\mathbb{R}^N)$ and $\widetilde{u}_{_0}(x)\leq u_{_0}(x)$ for a.e. $x\in \mathbb{R}^N$.

Sketch of the proof. We introduce the following functions

for all $(x, t)\in \mathbb{R}^N\times[0, +\infty)$. Now, let us consider the Musielak-Orlicz space

endowed with the norm

and the usual Musielak-Orlicz Sobolev space

equipped with the Luxemburg norm given by

Under Assumption 1.1, we have the following facts:

(see [10,Theorem 2.7 (ii)]) and the following continuous embedding hold

(see [10,Theorem 2.7 (iii)]); and from [10,Proposition 2.6] we directly obtain that

where $\rho(u): = \int_{\mathbb{R}^N}\big[H(x, |\nabla u|)+H(x, |u|)\big]dx.$

We introduce the following two functionals in $W^{^{1, H}}(\mathbb{R}^N):$

where $F(x, t) = \int_{_0}^{^t} f(x, s)ds$. Consider the $C^{^1}$-functional $\varphi: W^{^{1, H}}(\mathbb{R}^N)\rightarrow \mathbb{R}$ defined by

We split the proof into several steps.

Step 1. The functional $\varphi$ is weakly lower semi-continuous in $W^{^{1, H}}(\mathbb{R}^N).$

First, by Proposition 3.1 (ii) in ^[10], we known that $K$ is weakly continuous in $W^{^{1, H}}(\mathbb{R}^N).$ Thus, it is enough to show that functional $J$ is weakly lower semi-continuous in $W^{^{1, H}}(\mathbb{R}^N).$ Let $u_n\rightharpoonup u$ weakly in $W^{^{1, H}}(\mathbb{R}^N).$ Since $J$ is convex, we deduced that the following inequality holds:

Then we get that

which implies that

Step 2. The functional $\varphi$ is coercive.

Set $M = \max\Big\{1, \Big(\frac{2p|k|_{_\infty} }{\gamma}\Big)^{^{\frac{1}{p-\gamma}}}\Big\}$. Then for any $u\in W^{^{1, H}}(\mathbb{R}^N)$, we have

where $\Omega_1 = \{x\in\mathbb{R}^N: |u(x)|\geq M\}$ and $\Omega_2 = \mathbb{R}^N\backslash \Omega_1$.

On the one hand, it is easy to compute directly that

On the other hand, by using Young's inequality, for $\varepsilon\in(0, 1)$ we estimate

Then we deduce that

Let $0 < \varepsilon < \min\Big\{1, \big(\frac{p^{*}M^{^{p-p^*}}}{2p}\big)^{^{\frac{\gamma}{p^{*}}}}\Big\}$. Then

Consequently, using (1.6) and (1.7) in (1.5) finally yields we obtain that

so that by (1.4) it follows that $\varphi(u)\rightarrow +\infty$ as $\|u\|\rightarrow +\infty$.

Therefore, using Steps 1 and 2, and applying the Weierstrass Theorem, we deduce that there exists a global minimizer $u_{_0}\in W^{^{1, H}}(\mathbb{R}^N)$ of $\varphi$. The following Step 3 to show that $u_{_0}\neq 0$.

Step 3. We have $\varphi(u_{_0}) = \inf\limits_{u\in W^{^{1, H}}(\mathbb{R}^N)}\varphi(u) < 0$.

Let $\xi\in C_{_0}^{^\infty}(B_{_{2r}}(x_{_0}))$ such that $\xi(x)\equiv 1$, $x\in B_{_r}(x_{_0})$; $0\leq\xi(x)\leq 1$, $|\nabla\xi(x)|\leq \frac{C}{r}$, $x\in\mathbb{R}^N$. Denote $t = 2\delta$, then by assumption $(h_5)$, we obtain

and so

It follows from Step 3 that $u_{_0}\in W^{^{1, H}}(\mathbb{R}^N)$ is a non-trivial weak solution of problem $(P)$. It remains to show that there exists another non-trivial weak solution of problem $(P)$.

Step 4. There exists a critical point $\widetilde{u}_0 \in W^{^{1, H}}(\mathbb{R}^N)$ of $\varphi$.

Let

and $\widetilde{F}(x, t) = \int_{_0}^{^t}\widetilde{f}(x, s)ds$. Then it follows from $f\in C(\mathbb{R}^N\times\mathbb{R}, \mathbb{R})$ that $\widetilde{f}(x, t): \mathbb{R}^N\times\mathbb{R}\rightarrow \mathbb{R}$ is a Carathéodory function and

Similarly to Proposition 3.1 (i) in ^[10], we get that the functional

is of class $C^{^1}(W^{^{1, H}}(\mathbb{R}^N), \mathbb{R})$, and

for all $u, v\in W^{^{1, H}}(\mathbb{R}^N).$ Next, we define the functional $\widetilde{\varphi}: W^{^{1, H}}(\mathbb{R}^N)\rightarrow \mathbb{R}$ by

The same arguments as those used for functional $\varphi$ imply that $\widetilde{\varphi}\in C^{^1}(W^{1, H}(\mathbb{R}^N), \mathbb{R})$ and $\widetilde{\varphi}$ is coercive. And by the definition of $\widetilde{\varphi}$, we get

In the following, we determine a critical point $\widetilde{u}_0 \in W^{^{1, H}}(\mathbb{R}^N)$ of $\widetilde{\varphi}$, such that $\widetilde{\varphi}(\widetilde{u}_{_0}) > 0$ via the Mountain Pass Theorem.

First, we will show that there exists $0 < r_{_0} < \min\{1, \|u_{_0}\|\}$ such that

Using $(h_1)'$ and $(h_5)$, for any $u\in W^{^{1, H}}(\mathbb{R}^N)$ with $0 < \|u\| < \min\{1, \|u_{_0}\|\}$ we have

where $\Omega_3 = \{x\in\mathbb{R}^N: |u(x)|\leq |u_{_0}(x)|\}\cap\{x\in\mathbb{R}^N:|u(x)| > \delta\}$, $\Omega_4 = \{x\in\mathbb{R}^N: |u(x)| > |u_{_0}(x)|\}\cap\{x\in\mathbb{R}^N:|u(x)| > \delta\}$. Since $q < p^*$, then there exists $q < \tau < p^*$ such that $W^{^{1, H}}(\mathbb{R}^N)$ is continuously embedded in $L^{^\tau}(\mathbb{R}^N)$. Thus, there exists a positive constant $C_{_\tau}$ such that

Using Hölder's inequality and the above estimate, we obtain

where $\frac{1}{\tau'}+\frac{q}{\tau} = 1.$

By inequalities (1.9) and (1.10), we infer that it is enough to show that

in order to prove (1.8). Indeed, taking into account the fact that $k\in L^{^{\infty}}(\mathbb{R}^N)$, yields

which implies that

In view of Mountain Pass Theorem (see Ambrosetti-Rabinowitz^[15] with the variant given by Theorem 1.15 in Willem^[16]), there exists a sequence $\{u_n\} \subset W^{^{1, H}}(\mathbb{R}^N)$, such that

where $c = \inf\limits_{\lambda\in\Gamma}\max\limits_{t\in[0, 1]} \widetilde{\varphi}(\lambda(t))$, and

Since the functional $\widetilde{\varphi}$ is coercive, we obtain that $\{u_n\}$ is bounded in $W^{^{1, H}}(\mathbb{R}^N)$, and passing to a subsequence, still denoted by $\{u_n\}$, we may assume that there exists a $\widetilde{u}_{_0}\in W^{^{1, H}}(\mathbb{R}^N)$, such that $u_n\rightharpoonup \widetilde{u}_{_0}$ weakly in $W^{^{1, H}}(\mathbb{R}^N)$. By (1.3), we deduce that

Thus, there is a positive constant $M > 0$ such that

We first will prove that the $u_n \rightarrow \widetilde{u}_{_0}$ in $W^{^{1, H}}(\mathbb{R}^N)$. Recall that

Then it is enough to show that

Denote $\Omega_{_j} = \{x\in\mathbb{R}^N: |x|\leq j\}$ and $\Omega^{^c}_{_j} = \mathbb{R}^N\backslash \Omega_{_j}$, $j\in\mathbb{ N}$. Then by the fact that $k\in L^{^\theta}(\mathbb{R}^N)$, we deduce that

and so for given $\varepsilon \in (0, 1)$, there exists $j_{_0} > 0$ big enough such that

We also known that $u_n\rightarrow \widetilde{u}_{_0}$ in $L^{^\gamma}(\Omega_{_{j_{_0}}})$ because the embedding $W^{^{1, H}}(\Omega_{_{j_{_0}}})\hookrightarrow L^{^\gamma}(\Omega_{_{j_{_0}}})$ is compact. It follows that there exists $n_{_0} > 0$, such that

By a straightforward computation we deduce that

Applying Hölder's inequality and condition $(h_1)'$, we have

and

Consequently, we obtain that

when $n\geq n_{_0}$. By the arbitrariness of $\varepsilon$, we get

Noting that

Then we obtain

Due to Proposition 1.2 (ii) in ^[10], we have that $u_n\rightarrow \widetilde{u}_{_0}$ in $W^{^{1, H}}(\mathbb{R}^N)$. Since $\widetilde{\varphi}\in C^{^1}(W^{^{1, H}}(\mathbb{R}^N), \mathbb{R}^N)$, we observe that $\widetilde{u}_{_0}$ is a non-trivial critical point of $\widetilde{\varphi}$ because $\widetilde{\varphi}(\widetilde{u}_{_0}) = c > 0$ and $\widetilde{\varphi}'(\widetilde{u}_{_0}) = 0$.

Finally, we will show that $\widetilde{u}_{_0}(x)\leq u_{_0}(x)$ for a.e. $x\in\mathbb{R}^N$. Indeed, it is easy to check that

where $(\widetilde{u}_{_0}-u_{_0})^+ = \max\{0, \widetilde{u}_{_0}-u_{_0}\}$ and $[\widetilde{u}_{_0}\geq u_{_0}] = \{x\in\mathbb{R}^N: \widetilde{u}_{_0}(x)\geq u_{_0}(x)\}$. Obviously, the each term on the right hand side of above equality is non-negative, then we conclude that

which implies that $\widetilde{u}_{_0}(x) = u_{_0}(x)$ for a.e. $x\in\{x\in\mathbb{R}^N: \widetilde{u}_{_0}(x)\geq u_{_0}(x)\}$. Consequently, $\widetilde{u}_{_0}(x)\leq u_{_0}(x)$, for a.e. $x \in\mathbb{R}^N$. This immediately yields

Then we obtain

which yields that $\widetilde{u}_{_0}$ is a critical point of $\varphi$, and so a weak solution of problem $(P)$. Recall that $\varphi(\widetilde{u}_{_0}) = c > 0 > \varphi(u_{_0})$. Thus we see that $\widetilde{u}_{_0}$ is non-trivial. Therefore, $\widetilde{u}_{_0}\neq u_{_0}$ and this completes the proof of Theorem 1.2.

2.   Conclusions

In this paper, we have discussed a class of sublinear double phase problem in $\mathbb{R}^{^N}$. Some new criteria to guarantee that the existence of two non-trivial weak solutions for the considered problem $(P)$ is established by using the Weierstrass Theorem and Mountain Pass Theorem. Our results are obtained to improve and supplement some corresponding results.

Conflict of interest

All authors declare no conflicts of interest in this paper.