Existence, nonexistence, and multiplicity of solutions for the fractional $p\&q$-Laplacian equation in $\mathbb{R}^{N}$

In this paper, we study the existence, nonexistence, and multiplicity of solutions to the following fractional $p\&q$-Laplacian equation:

where λ is a real parameter, $(-\Delta)_{p}^{s} $ and $(-\Delta )_{q}^{s} $ are the fractional $p\&q$-Laplacian operators with $0< s<1<q<p, r>1$ and $sp< N$, and the functions $a(x), b(x),\mu(x)$, and $h(x)$ are nonnegative in ${\mathbb {R}}^{N}$. Three cases on $p,q,r,m$ are considered: $p< m< r< p_{s}^{*}$, $\max\{p,r\}< m< p_{s}^{*}$, and $1< m< q< r< p_{s}^{*}$. Using variational methods, we prove the existence, nonexistence, and multiplicity of solutions to Eq. (0.1) depending on $\lambda, p,q,r,m$ and the integrability properties of the ratio $h^{r-p}/\mu^{m-p}$. Our results extend the previous work in Bartolo et al. (J. Math. Anal. Appl. 438:29-41, 2016) and Chaves et al. (Nonlinear Anal. 114:133-141, 2015) to the fractional $p\&q$-Laplacian equation (0.1).

In this paper, we study the existence, nonexistence, and multiplicity of solutions to the following fractional $p\&q$-Laplacian equation:

where $(-\Delta)_{p}^{s} $ and $(-\Delta)_{q}^{s} $ are the fractional $p\&q$-Laplacian operators with $0< s<1<q<p, r>1$ and $sp< N$. The nonlinearity $f(x,u)=\lambda h(x)\vert u\vert ^{m-2}u-\mu(x)\vert u\vert ^{r-2}u$ can be seen as a competitive interplay of two nonlinearities. The coefficients $a(x), b(x),\mu(x), h(x)$ are assumed to be positive in ${\mathbb {R}}^{N}$, and other exact assumptions will be given further.

The fractional t-Laplacian operator $(-\Delta)_{t}^{s}$ with $0< s<1<t$ and $st< N$ is defined along a function $\varphi\in C_{0}^{\infty}({\mathbb {R}}^{N})$ as

where $B_{\varepsilon}(x):=\{y\in {\mathbb {R}}^{N}:\vert x-y\vert <\varepsilon\}$; see [3–6] and the references therein.

When $p=q$, Eq. (1.1) is reduced to the fractional p-Laplacian equation

and when $s=1$, Eq. (1.1) is the $p\&q$-Laplacian equation

Equation (1.4) comes from a general reaction-diffusion system

where $D(u)=\vert \nabla u\vert ^{p-2}+\vert \nabla u\vert ^{q-2}$. This system has a wide range of applications in physics and related sciences such as biophysics, plasma physics, and chemical reaction design. In such applications, the function u describes a concentration, and the first term on the right-hand side of (1.5) corresponds to the diffusion with a diffusion coefficient $D(u)$, whereas the second one is the reaction and relates to sources and loss processes. Typically, in chemical and biological applications, the reaction term $f(x, u)$ is a polynomial of u with variable coefficients [7, 8].

The solution of (1.4) has been studied by many authors; for example, see [1, 2, 7, 9–13] and the references therein. In the literature cited, the authors always assume that the potentials $a(x), b(x)$ satisfy one of the following conditions:

(A₁):

$a(x),b(x)\in C(\mathbb{R}^{N})$ and $a(x),b(x)\ge c_{0}$ in ${\mathbb {R}}^{N}$ for some constant $c_{0}>0$. Furthermore, for each $d>0$, $\operatorname{meas}(\{x\in\mathbb{ R}^{N}: a(x), b(x)\le d\})<\infty$.

(A₂):

$\lim _{\vert x\vert \to\infty}a(x)=+\infty, \lim _{\vert x\vert \to\infty}b(x)=+\infty$.

(A₃):

$a(x),b(x)\ge c_{0}>0$ in ${\mathbb {R}}^{N}$, and $a(x)^{-1}, b(x)^{-1}\in L^{1}(\mathbb{R}^{N})$.

It is well known that one of assumptions (A₁), (A₂), and (A₃) guarantees that the embedding $W^{1,t}({\mathbb {R}}^{N})\hookrightarrow L^{r}({\mathbb {R}}^{N})$ is compact for each $t\le r< t^{*}=\frac{tN}{N-t}$ with $1< t< N$. As far as we know, there are few papers that deal with a general bounded potential case for problem (1.4).

Now let us recall some advances of our problem. Pucci and Rădulescu [14] first studied the nonnegative solutions of the equation

where $h(x)>0$ satisfies

and $\lambda>0$, and $2\le p< m<\min\{r,p^{*}\}$ with $p^{*}=pN/(N-p)$ if $N>p$ and $p*=\infty$ if $N\le p$. They showed the nonexistence of nontrivial solutions to (1.6) if λ is small enough and the existence of at least two nontrivial solutions for (1.6) if λ is large enough.

Autuori and Pucci [15] generalized (1.6) to the quasilinear elliptic equation

where $A(x,\nabla u)$ acts like the p-Laplacian, $\max\{2,p\}< m<\min\{r,p^{*}\}$, and the coefficients ω and h are related by the integrability condition

By imposing a strong convexity condition of the p-Laplacian type on the potential of A, the authors extend completely the result of [14]. Moreover, Autuori and Pucci [15] proposed two open questions: the deletion of the restriction $\max\{2,p\}< m$ and the replacement of (1.9) by the assumption that $\omega(\omega/h)^{(m-p)/(r-m)}$ is in $L^{N/p}({\mathbb {R}}^{N})$.

Later, Autuori and Pucci [16] studied the existence and multiplicity of solution to the following elliptic equation involving the fractional Laplacian:

where $\lambda>0$, $0< s<1$, $2s< N$, $2< m<\min\{r,2_{s}^{*}\}$, $2_{s}^{*}=2N/(N-2s)$, and $(-\Delta)^{s}$ is the fractional Laplacian operator. The coefficients ω and h are related by condition (1.9). The authors proved the existence of entire solutions of (1.10) by using a direct variational method and the mountain pass theorem.

More recently, Xiang et al. [6] investigated the fractional p-Laplacian equation

where $\lambda>0$, $p< m<\min\{r,p_{s}^{*}\}$, $p_{s}^{*}=pN/(N-ps)$, and $a(x)$ and $b(x)$ are related by the condition $a(a/b)^{(r-p)/(m-r)}\in L^{N/ps}({\mathbb {R}}^{N})$.

Up to now, it is worth noting that there is much attention on equations like (1.6), (1.8), and (1.11) with $1< m< r$. From the papers mentioned, it is natural to ask whether the existence, nonexistence, and multiplicity of solutions to Eq. (1.1) is admitted if $1< r< m< p_{s}^{*}$ and $1< m< r< p_{s}^{*}$ ? Clearly, equations like (1.6), (1.8), and (1.11) are contained in (1.1).

In this paper, motivated by [5, 6], we will answer this interesting question, extend the $p\&q$-Laplacian (1.4), which has been studied deeply in [1, 2], to the fractional $p\&q$-Laplacian equation (1.1), and investigate the existence, nonexistence, and multiplicity of solutions depending on λ and according to the integrability properties of the ratio $h^{r-p}/\mu^{m-p}$.

For this purpose, we apply a version of symmetric mountain pass lemma in [17]. Also, we adapt some ideas developed by Pucci et al. [18] and Xiang et al. [6]. Note that although the idea was earlier used for other problems, the adaptation to the procedure to our problem is not trivial at all since the parameters $r,m$ satisfy $1< r< m$ and we must consider our problem for a suitable space, and so we need more delicate estimates and new technique. Our results, which are new even in the canonical case $p=q=2$, generalize the main results of [1, 2] in several directions. Furthermore, we weaken the conditions in those papers, and assumptions (A₁)-(A₃) are not necessary for our results.

In order to state our main theorems, we recall some fractional Sobolev spaces and norms. The fractional Sobolev space $W^{s,t}({\mathbb {R}}^{N})$ ($0< s<1<t$) with $st< N$ is defined by

This space is endowed with the natural norm

whereas $[u]_{s,t}$ denotes the Gagliardo seminorm given by

The spaces $X_{p}$ and $X_{q}$ denote the completion of $C_{0}^{\infty}({\mathbb {R}}^{N})$ with respect to the norms

respectively, in which the functions $a(x), b(x)$ satisfy

(H₀):

$a(x),b(x)\in C({\mathbb {R}}^{N})$ and $a(x),b(x)\ge c_{0}>0$ in ${\mathbb {R}}^{N}$ for some constant $c_{0}$.

In general, let $\Vert u\Vert _{t,\rho}=(\int_{{\mathbb {R}}^{N}}\rho \vert u\vert ^{t}\,dx)^{1/t}$ with $t\ge1$ and $\rho=\rho(x)\ge0, \neq0$ a.e. in ${\mathbb {R}}^{N}$. In particular, denote $\Vert u\Vert _{t}=(\int_{{\mathbb {R}}^{N}}\vert u\vert ^{t}\,dx)^{1/t}$ or $\Vert u\Vert _{L^{t}(\Omega)}=(\int_{\Omega} \vert u\vert ^{t}\,dx)^{1/t}$ with the domain $\Omega\subset {\mathbb {R}}^{N}$. Let $E=X_{p}\cap X_{q}$ with $1< q< p< N$. The norm of $u\in E$ is defined by

Lemma 1.1

[3, 19]

Let $0< s<1<t$ with $st< N$. In addition, assume (H₀). Then, $Y_{t}\equiv W^{s,t}({\mathbb {R}}^{N})$ is a uniformly convex Banach space, and there exists a positive constant $S_{0}=S_{0}(N,t,s)$ such that

and

where $t=p$ or q, $t_{s}^{*}=\frac{tN}{N-ts}$ is the fractional critical exponent, and $S_{r}$ is a constant depending on $s,r,t,N$. For convenience, we denote $S_{t_{s}^{*}}$ by $S_{0}$. Consequently, the space $Y_{t}$ is continuously embedded in $L^{r}({\mathbb {R}}^{N})$ for any $r\in [t,t_{s}^{*}]$. Moreover, the embedding $Y_{t}\hookrightarrow L^{r}({\mathbb {R}}^{N})$ is locally compact whenever $1< r< t_{s}^{*}$.

Clearly, from definitions (1.13) and (1.15) and assumption (H₀), we see that

Let $J(u):E\rightarrow\mathbb{R} $ be the energy functional associated to Eq. (1.1) defined by

where the norms $\Vert \cdot \Vert _{X_{p}}$ and $\Vert \cdot \Vert _{X_{p}}$ are defined by (1.15).

From the embedding inequalities (1.18) and assumptions (H₀)-(H₄) below, we see that the functional J is well defined and $J\in C^{1}(E,\mathbb{R})$ with

A function $u\in E$ is said to be a (weak) solution of Eq. (1.1) if $J'(u)\varphi=0$ for any $\varphi\in E$.

Throughout this paper, we let $0< s<1<q<p$ with $sp< N$. Our main results are as follows.

Theorem 1.2

Assume (H₀) and

(H₁):

$p< m< p_{s}^{*}$; $h(x)$ is a positive weight satisfying $h(x)\in L^{\gamma}({\mathbb {R}}^{N})$ with $\gamma=\frac{p_{s}^{*}}{p_{s}^{*}-m}$.

(H₂):

$p< m< r< p_{s}^{*}$; the functions $\mu(x)$ and $h(x)$ are positive and $\mu(x), h(x)\in L_{\mathrm{loc}}^{1}({\mathbb {R}}^{N})$. Furthermore, $h(x)$ and $\mu(x)$ are related by the condition

$$\begin{aligned} \int_{{\mathbb {R}}^{N}} \biggl[\frac{h(x)^{(r-p)/(r-m)}}{\mu (x)^{(m-p)/(r-m)}} \biggr]^{N/ps}\,dx=H_{1} \in {\mathbb {R}}^{+}. \end{aligned}$$

(1.22)

1. (i)

   only the trivial weak solution if $\lambda<\lambda_{1}$;

2. (ii)

   at least two nontrivial weak solutions if $\lambda\ge \lambda_{2}$.

Theorem 1.3

Let $\max\{p,r\}< m< p_{s}^{*}$. Assume that (H₀) and $(H_{1})$ hold. In addition, suppose that $\mu(x)$ are nonnegative and $\mu(x)\in L_{\mathrm{loc}}^{1}({\mathbb {R}}^{N})$. Then Eq. (1.1) admits

1. (i)

   only the trivial solution if $\lambda\le0$;

2. (ii)

   infinitely many weak solutions $u_{n}\in E$ such that $J(u_{n})\to\infty$ as $n\to\infty$ if $\lambda>0$.

Theorem 1.4

Let $0< s<1<m<q\le p<p_{s}^{*}$ and $q\le r< p_{s}^{*}$. Assume (H₀) and

(H₃):

$\mu(x)\ge0$ in ${\mathbb {R}}^{N}$ and $\mu(x)\in L^{\sigma }_{\mathrm{loc}}({\mathbb {R}}^{N})$ with $\sigma=\frac{p_{s}^{*}}{p_{s}^{*}-r}$;

(H₄):

$h(x)\in L^{\delta}(\mathbb{R}^{N})$ with $\delta =\frac {q}{q-m}$, and there exist $d_{0}>0$ and $x_{0}=(x^{0}_{1},x^{0}_{2},\ldots,x^{0}_{N})\in \mathbb{R}^{N}$ such that $h(x)>0$ in $B_{d_{0}}(x_{0})$, where $B_{d_{0}}(x_{0})=\{x\in\mathbb{R}^{N}: \vert x-x_{0}\vert < d_{0}\}$.

Then Eq. (1.1) with $\lambda>0$ admits infinitely many solutions $u_{n}\in E$ with $u_{n}\to0$ in E.

Remark 1.5

From Theorem 1.2, we know that it still remains an open problem to verify whether $\lambda_{1}=\lambda_{2}$. In addition, the nonlinear function $f(x,u)=\lambda h(x)\vert u\vert ^{m-2}u-\mu(x)\vert u\vert ^{r-2}u$ with $p< m< r$ fails to satisfy the Ambrosetti-Rabinowitz condition. Furthermore, for $s=1$ in (1.1), our results and context are more general than those in [1, 2].

The paper is organized as follows: In Section 2, we give some preliminaries, will set up the variational framework for problem (1.1), and prove that the functional associated to (1.1) satisfies the $(PS)_{c}$ condition. The proofs of Theorems 1.2 and 1.3 are given in Section 2. Finally, Section 3 is devoted to the proof of Theorem 1.4.

To prove our main results, we need to establish some lemmas.

Lemma 2.1

Let (H₀) and one of assumptions (H₁) and (H₄) hold. Then, if $\{u_{n}\}$ is a bounded sequence in E, then there exists $u\in E\cap L^{m}({\mathbb {R}}^{N},h)$ such that, up to a subsequence, $u_{n}\to u$ strongly in $L^{m}({\mathbb {R}}^{N},h)$ as $n\to\infty$.

Proof

We first choose a constant $\beta>0$ such that $\Vert u_{n}\Vert _{E}\le\beta$ for all $n\ge1$. If (H₁) is satisfied, then for any $\varepsilon>0$, there exists $R_{0}>0$ such that

Then, it follows from the Hölder inequality and Lemma 1.1 that, for $R\ge R_{0}$,

By Lemma 1.1, up to a subsequence, we obtain $u_{n}\to u$ strongly in $L^{m}(B_{R_{0}})$ and $u_{n}(x)\to u(x)$ a.e. in $B_{R_{0}}$ as $n\to\infty$. Thus $h(x)\vert u_{n}(x)-u(x)\vert ^{m}\to0$ a.e. in $B_{R_{0}}$ as $n\to\infty$. Similarly, for each measurable subset $\Omega\subset B_{R_{0}}$, we have

Since $h(x)\in L^{\gamma}({\mathbb {R}}^{N})$, we obtain that the sequence $\{h(x)\vert u_{n}(x)-u(x)\vert ^{m}\}$ is uniformly integrable and bounded in $L^{1}(B_{R_{0}})$. Furthermore, an application of the Vitali convergence theorem gives

Then the conclusion that $u_{n}\to u$ strongly in $L^{m}({\mathbb {R}}^{N},h)$ follows from (2.2) and (2.4).

If (H₄) is satisfied, then for any $\varepsilon>0$, there exists $R_{0}>0$ such that

and

Similarly, we can derive (2.4). Then combining (2.4) with (2.6), we have $u_{n}\to u$ in $L^{m}({\mathbb {R}}^{N},h)$. □

Lemma 2.2

Let (H₀) and one of assumptions (H₁) and (H₄) hold. If $\{u_{n}\}$ is a bounded $(PS)_{c}$ sequence of the functional J defined by (1.20), then the functional J satisfies $(PS)_{c}$ condition.

Proof

Let $\{u_{n}\}$ be a $(PS)_{c}$ sequence, that is,

Since the sequence $\{u_{n}\}$ is bounded in E, there exists a subsequence, still denoted by $\{u_{n}\}$, such that

where $t=p$ or q. We now prove that $u_{n}\to u$ in E. Let $\varphi \in E$ be fixed and denote by $T_{\varphi}$ the linear functional on E defined by

where $A_{\varphi}(v)$ and $B_{\varphi}(v)$ are the linear functionals defined by

respectively. Clearly, by the Hölder inequality, $T_{\varphi}$ is also continuous, and

Furthermore, the fact that $u_{n}\rightharpoonup u$ weakly in E implies that $\lim _{n\to\infty}A_{u}(u_{n}-u)= \lim _{n\to\infty }B_{u}(u_{n}-u)=0$, and so

On the other hand, as $n\to\infty$, we have

where

From (2.13) and $Z_{n}\ge0$, we obtain, for large n,

Note that, by Lemma 2.1, $P_{n} \to0$ as $n\to\infty$.

Let us now recall the well-known vector inequalities: for all $\xi,\eta\in {\mathbb {R}}^{N}$,

where $c_{p}$ and $C_{p}$ are positive constants depending only on p. Assume first that $p>q\ge2$. Then by (2.16) we have $\Vert u_{n}-u\Vert _{p,a}^{p}\le c_{p}\Phi_{n}$ and

Similarly, we have $\Vert u_{n}-u\Vert _{q,b}^{q}\le c_{q} \Psi_{n}$ and

Let $C_{0}=\min\{c_{p}^{-1},c_{q}^{-1}\}$. By (2.17) and (2.18) we see that

Then the application of (2.15) yields

In conclusion, $u_{n}\to u$ in E as $n\to\infty$.

Finally, it remains to consider the case $1< p<2$. By (2.8) there exists $\beta>0$ such that $\Vert u_{n}\Vert _{E}\le\beta$ for all $n\ge 1$. Now from (2.16) and the Hölder inequality it follows that

and

where we have applied the inequality

and $D_{p}=2C_{p}\beta^{p(2-p)/2}$. Similarly, for $1< q<2$, we have

with $D_{q}=2C_{q}\beta^{q(2-q)/2}$. Then, by (2.21), (2.22), and (2.24) we get

with some $C_{1}>0$. Then (2.15) and (2.25) imply that $u_{n}\to u$ in E as $n\to\infty$. Therefore, J satisfies the $(PS)_{c}$ condition, and we complete the proof of Lemma 2.2. □

Lemma 2.3

Under the assumptions of Theorem  1.2, suppose that $u\in E$ is a nontrivial weak solution of (1.1). Then there exists $\lambda_{1}>0$ such that $\lambda\ge\lambda_{1}$.

Proof

Since $u\in E$ is a nontrivial weak solution of (1.1), we have $J'(u)\varphi=0$ for all $\varphi\in E$. In particular, choosing $\varphi=u$, we have

By the Young inequality with $\epsilon>0$ we see that

Taking $0<\alpha<\beta, c=k_{1}>0, d=t^{\alpha}, \tau=\frac{\beta}{\alpha}$, $\epsilon=(k_{2}\beta/\alpha)^{-\alpha/(\beta-\alpha)}$, $k_{2}>0$, it follows from (2.27) that

with $k_{0}=(1-\alpha/\beta)(\beta/\alpha)^{-\alpha/(\beta-\alpha)}<1$. Furthermore, let $k_{1}=\lambda h(x), k_{2}=\frac{1}{2}\mu(x)$, $\alpha=m-p$, and $\beta=r-p$. Then from (2.28) we obtain

where $c_{1}=2^{(m-p)/(r-m)}$ and $g(x)=[h(x)^{r-p}/\mu(x)^{m-p}]^{\frac{1}{r-m}}$. By (H₂) we know $g(x)\in L^{\frac{N}{sp}}({\mathbb {R}}^{N})$. So, the application of (1.17) and (2.29) yields

with $G=\Vert g\Vert _{L^{\frac{N}{sp}}({\mathbb {R}}^{N})}$. Then, from (2.26) and (2.30) we see that

This implies that $\lambda\ge\lambda_{1}\equiv (c_{1}^{-1}S_{0}^{-p}G^{-1})^{(r-m)/(r-p)}$ and completes the proof of Lemma 2.3. □

Lemma 2.4

Under the assumptions of Theorem  1.2, the functional J is coercive in E.

Proof

Letting $k_{1}=\frac{\lambda}{m}h(x), k_{2}=\frac{1}{2r}\mu(x)$, $\alpha=m-p, \beta=r-p$, and $t=\vert u(x)\vert $ in (2.28), we conclude that

where $c_{2}=(2r)^{\frac{m-p}{r-m}}m^{\frac{p-r}{r-m}}\lambda^{\frac{r-p}{r-m}}$ and $g(x)=[h(x)^{r-p}/\mu(x)^{m-p}]^{\frac{1}{r-m}}$. Since $g(x)\in L^{\frac{N}{ps}}({\mathbb {R}}^{N})$, for any small $\varepsilon>0$, there exists $R_{1}>0$ such that

and

where $S_{0}$ is the embedding constant in (1.17). So, it follows from (2.32)-(2.34) that

For fixed $R_{1}>0$ and for any $\tau>0$ and $\omega>0$, we decompose $B_{R_{1}}=X\cup Y\cup Z$ as follows:

Obviously, the sets $X, Y$, and Z are Lebesgue measurable. Note that the assumption $h(x),\mu(x)\in L_{\mathrm{loc}}^{1}({\mathbb {R}}^{N})$ implies that $\operatorname{meas}(Y)\to0$ as $\tau\to0$ and $\operatorname{meas}(Z)\to0$ as $\omega\to\infty$.

On the other hand, letting $k_{1}= \frac{\lambda}{m}h(x), k_{2}=\frac{1}{2r}\mu(x), t=\vert u(x)\vert , \alpha=m$, and $\beta=r$ in (2.28), we derive

with $c_{3}=(2r)^{m/r}(\lambda/m)^{1+m/r}, g_{1}(x)=[h(x)/\mu(x)]^{m/r}$. Then,

where $C_{1}=C_{1}(\omega,\tau,R)>0$ is a constant. Furthermore, it follows from (2.32) and (2.34) that

For any $\varepsilon>0$, we can choose large $\omega>0$ and small $\tau>0$ such that $\operatorname{meas}(Y\cup Z)$ is so small that

From (1.13) and (2.38)-(2.40) we obtain

Thus, combining (2.34) and (2.35) with (2.41) yields

where $0<2\varepsilon S_{0}^{p}\le1/2p$. Hence, J is coercive in E. □

Lemma 2.5

Under the assumptions of Theorem  1.2, there exists $u\in E$ such that $d=J(u)=\inf_{v\in E}J(v)$ and u is a weak solution of (1.1).

Proof

By Lemma 2.4 we see that $d>-\infty$. Let $\{u_{n}\}$ be a minimizing sequence for d in E, which is bounded in E by Lemma 2.4. Without loss of generality, we may assume that $\{u_{n}\}$ is nonnegative, converges to weakly to some u in E, and $u_{n}(x)\to u(x)$ a.e. in ${\mathbb {R}}^{N}$. Moreover, by the weak lower semicontinuity of the norms we have

Then from Lemma 2.1 and (2.43) it follows

On the other hand, since $u\in E$, we have that $J(u)\ge d$, which shows that $J(u)=d$. Therefore, u is a global minimum for J, and hence it is a critical point, namely a weak solution of (1.1). □

Lemma 2.6

Under the assumptions of Theorem  1.2, there exists $\lambda_{2}>0$ such that for all $\lambda>\lambda_{2}$, Eq. (1.1) admits a global nontrivial minimum $u_{0}\in E$ of J with $J(u_{0})<0$.

Proof

Clearly, $J(0)=0$. Consider the constrained minimization problem

Let $u_{n}$ be a minimizing sequence of (2.45), which is clearly bounded in E, so that we can assume, without loss of generality, that it converges weakly to some $u_{0}\in E$ with $\Vert u_{0}\Vert _{m,h}^{m}=m$ and

Thus, $J(u_{0})=\lambda_{2}-\lambda<0$ for any $\lambda>\lambda_{2}$, and

This completes the proof. □

Next, we show that if $\lambda>\lambda_{2}$, then problem (1.1) admits a second nontrivial weak solution $e\neq u_{0}$ by the mountain pass theorem.

Lemma 2.7

Suppose that assumptions (H₀)-(H₁) are satisfied. Then, for all $e\in E$ and $\lambda>0$, there exist $\alpha>0$ and $\rho\in (0,\Vert e\Vert _{E})$ such that $J(u)\ge\alpha$ for all $u\in E$ with $\Vert u\Vert _{E}=\rho$.

Proof

Let $u\in E$. From (H₁), (1.18), and (1.19) with $t=p$ we obtain

Then,

where $H=\Vert h\Vert _{\gamma}, \Vert u\Vert _{E}=\rho$, and

so that

Thus, we finish the proof of Lemma 2.7. □

Lemma 2.8

Under the assumptions of Theorem  1.2 and $\lambda>\lambda_{2}$, Eq. (1.1) admits a nontrivial weak solution $u\in E$ such that $J(u)>0$.

Proof

By Lemma 2.6, for all $\lambda>\lambda_{2}$, there exists a nontrivial weak solution $u_{0}\in E$ with $J(u_{0})<0$. Taking $e=u_{0}$ in Lemma 2.7, we get that J satisfies the geometrical structure of Theorem A.3 of [15]. Thus, for all $\lambda>\lambda_{2}$ there exists a sequence $\{u_{n}\}\subset E$ such that

where

Since J is coercive in E, the sequence $\{u_{n}\}$ is bounded in E. By Lemma 2.2 there exists a subsequence, still denoted by $\{u_{n}\}$, such that $u_{n}\to u$ in E as $n\to\infty$. Therefore, $J(u)=\lim_{n\to\infty}J(u_{n})=c>0$, and $J'(u)\varphi=\lim_{n\to\infty}J'(u_{n})\varphi=0$ for all $\varphi \in E$. So, u is a weak solution of (1.1) with $J(u)>0$. □

Proof of Theorem 1.2

The application of Lemma 2.2 shows that problem (1.1) has only a trivial solution if $\lambda<\lambda_{1}$. By Lemmas 2.6 and 2.8 it follows that, for all $\lambda>\lambda_{2}$, problem (1.1) admits at least two nontrivial weak solutions in E, one with negative energy and the other with positive energy. This completes the proof of Theorem 1.2. □

Proof of Theorem 1.3

We first prove, under the assumptions in Theorem 1.3, that any $(PS)_{c}$ sequence $\{u_{n}\}$ is bounded in E. Let the sequence $\{u_{n}\}$ satisfy (2.7). Then, for large n, we have

Since $m>\max\{p,r\}$, it follows from (1.19) that $\{\Vert u_{n}\Vert _{E}\}$ is bounded. Furthermore, by Lemma 2.2 there exists a subsequence of $\{u_{n}\}$, still denoted by $\{u_{n}\}$, and $u\in E$ such that $u_{n}\to u$ in E and J satisfies the $(PS)_{c}$ condition.

From (2.26) it follows that if $u\in E$ is a nontrivial solution, then $\lambda>0$. This proves part (i). In the following, we prove part (ii). We now verify the conditions in Theorem 6.5 in [17]. Clearly, the functional J defined by (1.20) is even, and $J(0)=0$. By Lemma 2.7 there exist $\alpha,\rho>0$ such that $J(u)\ge\alpha$ for all $u\in E$ with $\Vert u\Vert _{E}=\rho$.

On the other hand, for any finite-dimensional subspace $E_{0}\subset E$, it is well known that any norms in $E_{0}$ are equivalent. So, there exist $d_{1},d_{2}>0$ such that

Then, from (1.20) we have

Since $\lambda>0$ and $m>\max\{p,r\}$, there exists $R=R(E_{0})>\rho$ such that $J(u)<0$ for $u\in E_{0}$ and $\Vert u\Vert _{E}\ge R$. Therefore, all conditions are verified. Then an application of Theorem 6.5 in [17] shows that Eq. (1.1) admits infinitely many solutions $u_{n}\in E$ with $J(u_{n})\to\infty$ as $n\to\infty$. This completes the proof of Theorem 1.3. □

In this section, we give a proof of Theorem 1.4. The main tool for this purpose is the following symmetric mountain pass lemma. First, we introduce the concept of genus.

Definition 3.1

[17]

Let E be a Banach space, and A a subset of E. The set A is said to be symmetric if $u\in E$ implies $-u\in E$. For a closed symmetric set A that does not contain the origin, we define the genus $\gamma(A)$ of A as the smallest integer k such that there exists an odd continuous mapping from A to $\mathbb{R}^{k}\setminus\{0\}$. If such k does not exist, then we define $\gamma(A)=\infty$. We set $\gamma(\emptyset)=0$. Let $\Gamma_{k}$ denote the family of closed symmetric subsets A of E such that $0\notin A$ and $\gamma(A)\ge k$.

Lemma 3.1

[20] (Symmetric mountain pass lemma)

Let E be an infinite-dimensional Banach space and $J\in C^{1}(E,\mathbb{R})$ such that:

1. (I)

   J is even and bounded from below, $J(0)=0$, and J verifies the $(PS)_{c}$ condition.

2. (II)

   for each $k\in\mathbb{N}$, there exists $A_{k}\in\Gamma_{k}$ such that $\sup _{u\in A_{k}}J(u)<0$.

Then one of the following two results holds:

1. (1)

   there exists a sequence $\{u_{k}\}$ such that $J'(u_{k})=0, J(u_{k})<0$, and $\{u_{k}\}$ converges to zero.

2. (2)

   there exist two sequences $\{u_{k}\}$ and $\{v_{k}\}$ such that $J'(u_{k})=0, J(u_{k})=0, u_{k}\neq0, \lim_{k\to\infty}u_{k}=0, J'(v_{k})=0, J(v_{k})<0, \lim_{k\to\infty}J(v_{k})=0$, and $\{v_{k}\}$ converges to a nonzero limit.

We now establish the following:

Lemma 3.2

Let the assumptions in Theorem  1.4 be satisfied. Then, for each $k\in\mathbb{N}$, there exists $A_{k}\in\Gamma_{k}$ such that

Proof

We use the following geometric construction introduced by Kajikiya [20]. Let $d_{0}$ and $x_{0}=(x_{1}^{0},x_{2}^{0},\ldots,x_{N}^{0})$ be fixed by assumption (H₄) and consider the cube

We choose small $d>0$ such that the cube $D(d)\subset\Omega_{0}:= B_{d_{0}}(x_{0})$. Note that $h(x)>0$ in $D(d)$. Fix $k\in\mathbb{N}$ arbitrarily. Let $n\in\mathbb{N}$ be the smallest integer such that $n^{N}\ge k$. We divide $D(d)$ equally into $n^{N}$ small cubes, denoted $D_{i}$, $1\le i\le n^{N}$, by planes parallel to each face of $D(d)$. The edge of $D_{i}$ has the length of $z=\frac{2d}{n}$. We construct new cubes $E_{i}$ in $D_{i}$ such that $E_{i}$ has the same center as that of $D_{i}$. The faces of $E_{i}$ and $D_{i}$ are parallel, and the edge of $E_{i}$ has the length $\frac{d}{n}$. Then, let the functions $\psi_{i}(x)\in C^{1}(\mathbb{R}^{N}), 1\le i\le k$, be such that

Denote

and

Clearly, $V_{k}$ is the surface of k-dimensional, cube and $W_{k}$ is a closed symmetric set in E such that $0\notin W_{k}$. It is easy to see that $V_{k}$ is homeomorphic to the sphere $S^{k-1}$ by an odd mapping (take, e.g., the radial projection $V_{k}\to S^{k-1}$). Hence, $\gamma(V_{k})=k$. Moreover, $\gamma(W_{k})=\gamma(V_{k})=k$ because the mapping $(t_{1},t_{2},\ldots,t_{k})\longmapsto\sum_{i=1}^{k}t_{i}\psi_{i}(x)$ is homeomorphic and odd. On the other hand, since $W_{k}$ is bounded in E, there is a constant $\alpha_{k}>0$ such that

Let $z>0$ and $u=\sum_{i=1}^{k}t_{i}\psi_{i}(x)\in W_{k}$. Then,

where $S_{r}$ is the embedding constant in (1.18), and $\sigma =p_{s}^{*}/(p_{s}^{*}-r)$. By (3.7) there exists an integer $j\in[1,k]$ such that $\vert t_{j}\vert =1$ and $\vert t_{i}\vert \le1$ for $i\neq j$. Hence,

Since $\psi_{j}(x)=1$ for $x\in E_{j}$ and $\vert t_{j}\vert =1$, we have

On the over hand, since $D(d)\subset\Omega_{0}$, by $(H_{4})$ we obtain

Then, it follows from (3.7)-(3.10) that

Since $h(x)>0$ in $E_{i}$ and $m\in(1,q)$, we have

We fix $z>0$ small such that

which completes the proof of (3.1) and thus of Lemma 3.2. □

Proof of Theorem 1.4

Evidently, $J(0)=0$, and J is an even functional. Then, by Lemma 2.2, J satisfies the $(PS)_{c}$ condition. Furthermore, by Lemma 3.2 conditions (I) and (II) in Lemma 3.1 are satisfied. Thus, by Lemma 3.1 problem (1.1) admits infinitely many solutions $u_{n}\in E$ with $u_{n}\to0$ in E. Thus, the proof of Theorem 1.4 is finished. □

This work is supported by the Fundamental Research Funds for the Central Universities of China (2015B31014).

Affiliations

1. College of Science, Hohai University, Nanjing, 210098, P.R. China

   • Caisheng Chen
   •  & Jinfeng Bao

2. College of Science, Nanjing Forestry University, Nanjing, 210037, P.R. China

   • Jinfeng Bao

Corresponding author

Correspondence to Caisheng Chen.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The main idea of this paper was proposed by CSC. JFB prepared the manuscript in part. All steps of the proofs in this research are performed by CSC. All authors read and approved the final manuscript.

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