Skip to main content

Multiple solutions for a class of nonlocal quasilinear elliptic systems in Orlicz–Sobolev spaces

Abstract

In this paper, we study some results on the existence and multiplicity of solutions for a class of nonlocal quasilinear elliptic systems. In fact, we prove the existence of precise intervals of positive parameters such that the problem admits multiple solutions. Our approach is based on variational methods.

Introduction

In this article, we are interested in establishing the existence of multiple solutions to the following Kirchhof-type systems in Orlicz–Sobolev spaces

$$ \textstyle\begin{cases} -M_{i} {(}\int _{\Omega }\Phi _{i}( \vert \nabla u_{i} \vert )\,dx {)} {(} \operatorname{div}(\alpha _{i}( \vert \nabla u_{i} \vert )\nabla u_{i}) {)}=\lambda F_{u_{i}}(x,u_{1}, \dots , u_{n}) \quad \text{in } \Omega , \\ u_{i}=0 \quad \text{on } \partial \Omega , \end{cases} $$
(1.1)

for \(1\leq i\leq n\), where Ω is a bounded domain in \(\mathbb{R}^{N}\) (\(N\geq 3\)), with smooth boundary Ω and λ is a positive parameter, \(F:\Omega \times \mathbb{R}^{n}\rightarrow \mathbb{R}\) is a measurable function with respect to \(x\in \Omega \) for every \((t_{1}, \dots , t_{n})\in \mathbb{R}^{n}\) and is \(C^{1}\) with respect to \((t_{1}, t_{2}, \dots , t_{n})\in \mathbb{R}^{n}\) for a.e. \(x\in \Omega \); \(F_{t_{i}}\) denotes the partial derivative of F with respect to \(t_{i}\). Also \(M_{i}: \mathbb{R}\rightarrow \mathbb{R}\) (\(i=1, 2, \dots , n\)), are continuous and increasing functions satisfying the following boundedness condition:

(M):

There exist positive numbers \(m_{i}^{0}\), \(M_{i}^{0}\) such that

$$ m_{i}^{0}\leq M_{i}(t)\leq M_{i}^{0}, \quad \text{for all } t\geq 0\ (i=1, 2, \dots , n). $$
Throughout this article we assume that for \(i=1, \dots , n\), the functions \(\alpha _{i}: (0, +\infty )\rightarrow \mathbb{R}\) are such that the mappings \(\varphi _{i}:\mathbb{R}\rightarrow \mathbb{R}\) defined by

$$ \varphi _{i}(t)= \textstyle\begin{cases} \alpha _{i}( \vert t \vert )t& \text{for } t\neq 0, \\ 0& \text{for } t=0, \end{cases} $$

are odd, strictly increasing homeomorphisms from \(\mathbb{R}\) onto \(\mathbb{R}\). For the functions \(\varphi _{i}\) above, let us define \(\Phi _{i}(t)=\int _{0}^{t}\varphi _{i}(s)\,ds\) for all \(t\in \mathbb{R}\).

Notice that if \(i=1\), then problem (1.1) becomes

$$ \textstyle\begin{cases} -M {(}\int _{\Omega }\Phi ( \vert \nabla u \vert )\,dx {)} {(}\operatorname{div}( \alpha ( \vert \nabla u \vert )\nabla u) {)}=\lambda f(x,u) \quad \text{in } \Omega , \\ u=0 \quad \text{on } \partial \Omega . \end{cases} $$
(1.2)

It should be mentioned that if \(\varphi (t)=p|t|^{p-2}t\) for all \(t\in \mathbb{R}\), \(p>1\) then problem (1.2) becomes the well-known p-Kirchhoff-type equation

$$ \textstyle\begin{cases} -M(\int _{\Omega } \vert \nabla u \vert ^{p}\,dx)\Delta _{p}u=\lambda f(x, u)\quad \text{in } \Omega , \\ u=0\quad \text{on } \partial \Omega . \end{cases} $$
(1.3)

Problem (1.3) is related to the stationary problem

$$ \rho \frac{\partial ^{2}u}{\partial t^{2}}-\biggl(\frac{\rho _{0}}{h}+ \frac{E}{2L} \int ^{L}_{0} \biggl\vert \frac{\partial u}{\partial x} \biggr\vert ^{2}\,dx\biggr) \frac{\partial ^{2}u}{\partial x^{2}}=0, $$

where ρ, \(\rho _{0}\), h, E, L are constants, for \(0< x< L\), \(t \geq 0\), and where \(u=u(x,t)\) is the lateral displacement at the space coordinate x and time t, E the Young modulus, ρ the mass density, h the cross-section area, L the length, and \(\rho _{0}\) the initial axial tension, proposed by Kirchhoff [17] as an extension of the classical D’Alembert’s wave equation for free vibrations of elastic strings. This is an example of a nonlinear problem. One can refer to [35, 9, 13, 14, 2023, 2628, 3032] for more relevant problems and techniques.

Now, we recall some basic facts about Orlicz and Orlicz–Sobolev spaces (see [2, 29] and the references therein). Let \(\varphi _{i}\) and \(\Phi _{i}\) be as introduced at the beginning of the paper. Set

$$ \Phi ^{*}_{i}(t)= \int _{0}^{t}\varphi ^{-1}_{i}(s) \,ds, \quad \text{for all } t\in \mathbb{R}. $$

We see that \(\Phi _{i}\), for \(1\leq i\leq n\), are Young functions, that is, \(\Phi _{i}(0)=0\), \(\Phi _{i}\) are convex, and \(\lim_{t\rightarrow \infty }\Phi _{i}(t)=+\infty \).

Also, since \(\Phi _{i}(t)=0\) if and only if \(t=0\),

$$ \lim_{t\rightarrow 0}\frac{\Phi _{i}(t)}{t}=0\quad \text{and}\quad \lim _{t\rightarrow \infty }\frac{\Phi _{i}(t)}{t}=+\infty , $$

then \(\Phi _{i}\) are called N-functions. The functions \(\Phi _{i}^{*}\), for \(1\leq i\leq n\) are called the complementary functions of \(\Phi _{i}\) and they satisfy

$$ \Phi _{i}^{*}(t)=\sup \bigl\{ st-\Phi _{i}(s); s\geq 0\bigr\} , \quad \text{for all } t \geq 0. $$

We observe that \(\Phi _{i}^{*}\) are also N-functions and the following Young’s inequality holds:

$$ st\leq \Phi _{i}(s)+\Phi _{i}^{*}(t),\quad \text{for all } s,t\geq 0. $$

We define the numbers

$$ (p_{i})_{0}:=\inf_{t>0} \frac{t\varphi (t)}{\Phi (t)}, \quad \text{and}\quad (p_{i})^{0}:=\sup _{t>0}\frac{t\varphi (t)}{\Phi (t)}. $$

Throughout this paper, we assume the following condition:

$$ N< (p_{i})_{0}\leq \frac{t\varphi _{i}(t)}{\Phi _{i}(t)} \leq (p_{i})^{0}< \infty , \quad \text{for all } t>0. $$
(1.4)

The Orlicz spaces \(L_{\Phi _{i}}(\Omega )\), for \(1\leq i\leq n\), defined by the N-functions \(\Phi _{i}\) are the spaces of measurable functions \(u:\Omega \rightarrow \mathbb{R}\) such that

$$ \Vert u \Vert _{L_{\Phi _{i}}}:=\sup \biggl\{ \biggl\vert \int _{\Omega } u(x)v(x)\,dx \biggr\vert : \int _{ \Omega } \Phi _{i}^{*}\bigl( \bigl\vert v(x) \bigr\vert \bigr)\,dx\leq 1\biggr\} < \infty . $$

Then \((L_{\Phi _{i}}(\Omega ),\|\cdot \|_{L_{\Phi _{i}}})\) are Banach spaces whose norms are equivalent to the Luxemburg norm

$$ \Vert u \Vert _{\Phi _{i}}:=\inf \biggl\{ k>0; \int _{\Omega }\Phi _{i}\biggl( \frac{u(x)}{k} \biggr)\,dx \leq 1 \biggr\} . $$

For Orlicz spaces, the Hölder’s inequality takes the form

$$ \int _{\Omega }uv \,dx \leq 2 \Vert u \Vert _{L_{\Phi _{i}}} \Vert v \Vert _{L_{\Phi _{i}^{*}}} \quad \text{for all } u\in L_{\Phi _{i}}( \Omega ) \text{ and } v\in L_{ \Phi _{i}^{*}}(\Omega ), 1\leq i\leq n. $$

The Orlicz–Sobolev spaces \(W^{1, \Phi _{i}}(\Omega )\), \(1\leq i\leq n\) are the spaces defined by

$$ W^{1,{\Phi _{i}}}(\Omega )=\biggl\{ u\in L_{\Phi _{i}}(\Omega ), \frac{\partial u}{\partial x_{j}}\in L_{\Phi _{i}}(\Omega ), j=1, \dots , N\biggr\} . $$

These are Banach spaces with respect to the norms:

$$ \Vert u \Vert _{1, \Phi _{i}}:= \Vert u \Vert _{\Phi _{i}}+\bigl\| | \nabla u|\bigr\| _{\Phi _{i}}, \quad 1\leq i\leq n. $$

Now, we introduce the Orlicz–Sobolev spaces \(W^{1,\Phi _{i}}_{0}(\Omega )\), for \(1\leq i\leq n\), as the closure of \(C^{\infty }_{0}(\Omega )\) in \(W^{1,\Phi _{i}}(\Omega )\) which can be renormed by equivalent norms:

$$ \Vert u \Vert _{{i}}:=\bigl\| |\nabla u|\bigr\| _{\Phi _{i}}. $$

The relation (1.4) implies that \(\Phi _{i}\) and \(\Phi _{i}^{*}\), for \(1\leq i\leq n\), both satisfy the \(\Delta _{2}\)-condition [1, 12], i.e.,

$$ \Phi _{i}(2t)\leq k\Phi _{i}(t) \quad \text{for all } t\geq 0, $$

where k is a positive constant. Furthermore, we assume that \(\Phi _{i}\) satisfy in the following conditions:

$$ \text{For each $x\in \bar{\Omega}$, the functions $t\rightarrow \Phi _{i}(x, \sqrt{t})$ are convex for all $t\in [0, \infty )$}. $$
(1.5)

Condition \(\Delta _{2}\) for \(\Phi _{i}\) assures that for each \(i\in \{1, \dots , n\}\) the Orlicz spaces \(L_{\Phi _{i}}(\Omega )\) are separable. Also the \(\Delta _{2}\) condition and (1.5) assure that \(L_{\Phi _{i}}(\Omega )\) are uniformly convex spaces, and thus reflexive Banach spaces (see [25, Proposition 2.2]), implying that Orlicz–Sobolev spaces \(W_{0}^{1,\Phi _{i}}(\Omega )\), \(i\in \{1, \dots , n\}\) are reflexive Banach spaces also [16].

We define the space \(X:=\prod_{i=1}^{n}W^{1,\Phi _{i}}_{0}(\Omega )\) for problem (1.1) which is a reflexive Banach space with respect to the norm

$$ \Vert u \Vert =\sum_{i=1}^{n} \Vert u_{i} \Vert _{i},\quad u=(u_{1}, \dots , u_{n})\in X. $$

Remark 1.1

In [12] we see that the Orlicz–Sobolev spaces \(W_{0}^{1,\Phi _{i}}(\Omega )\), \(i=1, \dots , n\), are continuously embedded in \(W_{0}^{1, (p_{i})_{0}}(\Omega )\). On the other hand, since we assume that \((p_{i})_{0}>N\), we conclude that \(W_{0}^{1, (p_{i})_{0}}(\Omega )\) are compactly embedded in \(C^{0}(\bar{\Omega })\), see [19]. Thus, we have that \(W_{0}^{1,\Phi _{i}}(\Omega )\) are compactly embedded in \(C^{0}(\bar{\Omega })\).

So, \(X\hookrightarrow C^{0}(\bar{\Omega })\times \cdots \times C^{0}( \bar{\Omega })\) is compact. We set a constant \(C>0\) such that

$$ C:=\max \biggl\{ \sup_{u_{i}\in W_{0}^{1,\Phi _{i}}\setminus \{0\}} \frac{\max_{x\in \bar{\Omega }} \vert u_{i}(x) \vert ^{(p_{i})^{0}}}{ \Vert u_{i} \Vert _{i}^{(p_{i})^{0}}} : \text{for } 1\leq i\leq n \biggr\} < +\infty . $$
(1.6)

Proposition 1.1

([24, Lemma 1])

Let \(u\in W_{0}^{1, \Phi _{i}}(\Omega )\), then the following relations hold:

  1. (I)

    \(\|u\|_{i}^{(p_{i})_{0}}\leq \int _{\Omega }\Phi _{i}(|\nabla u(x)|)\,dx \leq \|u\|_{i}^{(p_{i})^{0}}\) if \(\|u\|_{i}>1\), \(i=1, \dots , n\),

  2. (II)

    \(\|u\|_{i}^{(p_{i})^{0}}\leq \int _{\Omega }\Phi _{i}(|\nabla u(x)|)\,dx \leq \|u\|_{i}^{(p_{i})_{0}}\) if \(\|u\|_{i}<1\), \(i=1, \dots , n\).

Proposition 1.2

([21, Lemma 2.1])

Let \(u\in W_{0}^{1,\Phi _{i}}(\Omega ) \) and

$$ \int _{\Omega }\Phi _{i}\bigl( \bigl\vert \nabla u(x) \bigr\vert \bigr)\,dx \leq r $$

for some \(0< r<1\). Then one has \(\|u\|_{i}<1\).

Proposition 1.3

([7, Remark 2.1])

Let \(u\in W_{0}^{1,\Phi _{i}}(\Omega ) \) be such that \(\|u\|_{i}=1\). Then

$$ \int _{\Omega }\Phi _{i}\bigl( \bigl\vert \nabla u(x) \bigr\vert \bigr)\,dx=1. $$

Our aim is to prove the existence and multiplicity solutions for problem (1.1); so we study problem (1.1) by using the results as follows.

First, we recall the following three critical points theorem, obtained by G. Bonanno and S.A. Marano in [8].

Theorem 1.1

Let X be a reflexive real Banach space, \(J:X\rightarrow \mathbb{R}\) be a sequentially weakly lower semicontinuous and continuously Gâteaux differentiable functional that is bounded on bounded subsets of X and whose Gâteaux derivative admits a continuous inverse on \(X^{*}\), and let \(I:X\rightarrow \mathbb{R}\) be a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact and satisfies \(J(0)=I(0)=0\). Assume that there exist \(r>0\) and \(\bar{v}\in X\), with \(r< J(\bar{v})\) such that:

  1. (a1)

    \(\frac{\sup_{J^{-1}(-\infty , r]}I(u)}{r}< \frac{I(\bar{v})}{J(\bar{v})}\);

  2. (a2)

    for each \(\lambda \in \Lambda _{r}:=\,]\frac{J(\bar{v})}{I(\bar{v})}, \frac{r}{\sup_{J^{-1}(-\infty , r]}I(u)}[\) the functional \(J-\lambda I\) is coercive.

Then, for each compact interval \([\alpha , \beta ]\subseteq \Lambda _{r}\), there exists \(\rho >0\) with the following property: for every \(\lambda \in [\alpha , \beta ]\), the equation

$$ J^{\prime }(u)-\lambda I^{\prime }(u)=0 $$

has at least three solutions in X whose norms are less than ρ.

Here, we recall a multiple critical points theorem of Bonanno et al. [6].

Theorem 1.2

Let X be a reflexive real Banach space, let \(J, I:X \rightarrow \mathbb{R}\) be two Gâteaux differentiable functionals such that J is strongly continuous, sequentially weakly lower semicontinuous and coercive, and I is sequentially weakly upper semicontinuous. For every \(r>\inf_{X} J\), let

$$\begin{aligned}& {\varphi }(r):=\inf_{u \in J^{-1}(-\infty ,r)} \frac{\sup_{v \in J^{-1}(-\infty ,r)} I(v)- I(u)}{r-J(u)}, \\& \gamma :=\liminf_{r\rightarrow +\infty }{\varphi }(r),\qquad \delta := \liminf _{r \rightarrow (\inf _{X} J)^{+}} {\varphi }(r). \end{aligned}$$

Then the following properties hold:

  1. (a)

    If \(\gamma <+\infty \), then for each \(\lambda \in\, ]0, \frac{1}{\gamma }[\), either

    1. (a1)

      \(h_{\lambda }:=J-\lambda I\) possesses a global minimum, or

    2. (a2)

      there is a sequence \(\{u_{n}\}\) of critical points (local minima) of \(h_{\lambda }\) such that

      $$ \lim_{n \rightarrow +\infty }J(u_{n})=+\infty ; $$
  2. (b)

    If \(\delta <+\infty \), then for each \(\lambda \in\, ]0, \frac{1}{\delta }[\), either

    1. (b1)

      there is a global minimum of J that is a local minimum of \(h_{\lambda }\), or

    2. (b2)

      there is a sequence \(\{u_{n}\}\) of pairwise distinct critical points (local minima) of \(h_{\lambda }\) that weakly converges to a global minimum of J with

      $$ \lim_{n \rightarrow +\infty }J(u_{n})=\inf_{u\in X }J(u). $$

Main results

Definition 2.1

We say that \(u=(u_{1}, u_{2}, \dots , u_{n})\) is a weak solution to the system (1.1) if \(u=(u_{1}, u_{2}, \dots , u_{n})\in X\) and

$$ \begin{aligned} &\sum_{i=1}^{n}M_{i} {\biggl(} \int _{\Omega }\Phi _{i}\bigl( \bigl\vert \nabla u_{i}(x) \bigr\vert \bigr)\,dx {\biggr)} \int _{\Omega }\alpha _{i}\bigl( \bigl\vert \nabla u_{i}(x) \bigr\vert \bigr) \nabla u_{i}(x)\nabla v_{i}(x) \,dx \\ &\quad {}-\lambda \int _{\Omega }\sum_{i=1}^{n}F_{u_{i}} \bigl(x, u_{1}(x), \dots , u_{n}(x)\bigr) v_{i}(x) \,dx=0, \end{aligned} $$

for every \(v=(v_{1}, v_{2}, \dots , v_{n}) \in X\).

Set \(\overline{p}:=\max \{(p_{i})^{0} : i=1, \dots , n\}\), \(m_{0}:=\min \{m_{i}^{0} : i=1, \dots , n\}\) and \(m_{1}:=\max \{M_{i}^{0} : i=1, \dots , n\}\). For all \(\sigma >0\), we define the set

$$ Q(\sigma ):=\Biggl\{ (t_{1}, \dots , t_{n})\in \mathbb{R}^{n}: \sum_{i=1}^{n} \vert t_{i} \vert \leq \sigma \Biggr\} . $$

We need the following proposition in the proof of the main results.

Proposition 2.1

Let \(T:X \rightarrow X^{*}\) be the operator defined by

$$ \begin{aligned} T(u_{1}, \dots , u_{n}) (v_{1}, \dots , v_{n})={}&\sum _{i=1}^{n}M_{i} {\biggl(} \int _{\Omega }\Phi _{i}\bigl( \bigl\vert \nabla u_{i}(x) \bigr\vert \bigr)\,dx {\biggr)} \\ &{}\times \int _{\Omega }\alpha _{i}\bigl( \bigl\vert \nabla u_{i}(x) \bigr\vert \bigr)\nabla u_{i}(x) \nabla v_{i}(x) \,dx, \end{aligned} $$

for every \(u=(u_{1}, \dots , u_{n})\), \(v=(v_{1}, \dots , v_{n})\in X\). Then T admits a continuous inverse on \(X^{*}\), where \(X^{*}\) denotes the dual of X.

Proof

By applying the Minty–Browder theorem [33, Theorem 26.A(d)], it is sufficient to verify that T is coercive, hemicontinuous, and uniformly monotone. Since

$$ (p_{i})_{0}\leq \frac{t\varphi _{i}(t)}{\Phi _{i}(t)}, \quad \text{for all } t>0, $$

by Proposition 1.1, for each \(u\in X\) with \(\|u_{i}\|_{i}>1\), we have

$$ \begin{aligned} &T(u_{1}, \dots , u_{n}) (u_{1}, \dots , u_{n}) \\ &\quad =\sum_{i=1}^{n}M_{i} { \biggl(} \int _{\Omega }\Phi _{i}\bigl( \bigl\vert \nabla u_{i}(x) \bigr\vert \bigr)\,dx {\biggr)} \int _{\Omega }\alpha _{i}\bigl( \bigl\vert \nabla u_{i}(x) \bigr\vert \bigr) \bigl\vert \nabla u_{i}(x) \bigr\vert ^{2} \,dx \\ &\quad \geq \sum_{i=1}^{n}M_{i} {\biggl(} \int _{\Omega }\Phi _{i}\bigl( \bigl\vert \nabla u_{i}(x) \bigr\vert \bigr)\,dx {\biggr)} \int _{\Omega }\Phi _{i}\bigl( \bigl\vert \nabla u_{i}(x) \bigr\vert \bigr) \,dx \\ &\quad \geq m_{0}\sum_{i=1}^{n} \Vert u_{i} \Vert _{i}^{2(p_{i})_{0}}, \end{aligned} $$

so if \((p_{i})_{0}>N\) then T is coercive. The fact that T is hemicontinuous can be verified using standard arguments. Similar to proof given in [18, Lemma 3.2], T is strictly monotone. Therefore, in view of Minty–Browder theorem, there exists \(T^{-1}:X^{*}\rightarrow X\), and, by a similar method as that given in [10], one has that \(T^{-1}\) is continuous. □

Now, we define the energy functional of problem (1.1) by \(h_{\lambda }:X\rightarrow \mathbb{R}\):

$$ h_{\lambda }(u)=J(u)-\lambda I(u), $$

for all \(u=(u_{1}, \dots , u_{n})\in X\), where

$$\begin{aligned}& J(u)=\sum_{i=1}^{n} \hat{M_{i}}\biggl( \int _{\Omega }\Phi _{i}\bigl( \bigl\vert \nabla u_{i}(x) \bigr\vert \bigr)\,dx\biggr), \qquad \hat{M_{i}}(t)= \int _{0}^{t}{M_{i}}(s) \,ds, \quad i=1, 2, \dots , n, \\& I(u)= \int _{\Omega }F\bigl(x, u_{1}(x), \dots , u_{n}(x)\bigr) \,dx. \end{aligned}$$

Note that the weak solutions of (1.1) are exactly the critical points of \(h_{\lambda }\). Similar arguments as in [25, Lemma 4.2] imply that J and I are continuously Gâteaux differentiable functionals and whose Gâteaux differentials at the point \(u=(u_{1}, \dots , u_{n})\in X\) are the functionals \(J^{\prime }(u)\) and \(I^{\prime }(u)\) given by

$$\begin{aligned}& J^{\prime }(u) (v)=\sum_{i=1}^{n}M_{i} {\biggl(} \int _{\Omega }\Phi _{i}\bigl( \bigl\vert \nabla u_{i}(x) \bigr\vert \bigr)\,dx {\biggr)} \int _{\Omega }\alpha _{i}\bigl( \bigl\vert \nabla u_{i}(x) \bigr\vert \bigr) \nabla u_{i}(x)\nabla v_{i}(x) \,dx, \\& I^{\prime }(u) (v)= \int _{\Omega }\sum_{i=1}^{n}F_{u_{i}} \bigl(x, u_{1}(x), \dots , u_{n}\bigr) v_{i}(x) \,dx. \end{aligned}$$

Moreover, \(I^{\prime }: X\rightarrow X^{*}\) is a compact derivative. For this purpose, it is enough to show that \(I^{\prime }\) is strongly continuous on X, so for a fixed \((u_{1}, u_{2}, \dots , u_{n})\in X\), let \((u_{1k}, u_{2k}, \dots , u_{nk})\rightharpoonup (u_{1}, u_{2}, \dots , u_{n})\) weakly in X as \(k\rightarrow +\infty \). Since X is compactly embedded in \(C^{0}(\bar{\Omega })\times \cdots \times C^{0}(\bar{\Omega })\), we have that \((u_{1k}, u_{2k}, \dots , u_{nk})\) converges uniformly to \((u_{1}, u_{2}, \dots , u_{n})\) on Ω as \(k\rightarrow +\infty \). Since \(F(x, \cdot , \dots , \cdot )\) is \(C^{1}\) in \(\mathbb{R}^{n}\) for every \(x\in \Omega \), and the partial derivatives of F are continuous in \(\mathbb{R}^{n}\) for every \(x\in \Omega \), \(F_{u_{i}}(x, u_{1k}, \dots , u_{nk})\rightarrow F_{u_{i}}(x, u_{1}, \dots , u_{n})\) strongly as \(k\rightarrow +\infty \), thus \(I^{\prime }(u_{1k}, \dots , u_{nk})\rightarrow I^{\prime }(u_{1}, \dots , u_{n})\) strongly as \(k\rightarrow +\infty \). So \(I^{\prime }\) is strongly continuous on X, which implies that \(I^{\prime }\) is a compact operator [33].

Lemma 2.1

J is coercive and sequentially weakly lower semicontinuous.

Proof

For all \(t\geq 0\), we have

$$ J(u)\geq \sum_{i=1}^{n}m^{0}_{i} \biggl( \int _{\Omega }\Phi _{i}\bigl( \bigl\vert \nabla u_{i}(x) \bigr\vert \bigr)\,dx\biggr), \quad i=1, 2, \dots , n, $$

and, by Proposition 1.1, for all \(u\in X\) with \(\|u_{i}\|_{i}>1\), we have

$$ J(u)\geq \sum_{i=1}^{n}m_{0} \Vert u_{i} \Vert _{i}^{(p_{i})_{0}}, $$

from which it follows that J is coercive. Moreover, since \(\Phi _{i}\) for \(1\leq i\leq n\) are convex, J is a convex functional, and thus it is sequentially weakly lower semicontinuous. □

Three weak solutions

Theorem 2.1

Assume that condition (M) holds and

  1. (h1)

    \(F(x, 0, \dots , 0)=0\), for a.e. \(x\in \Omega \).

  2. (h2)

    There exist \(\alpha (x)\in L^{1}(\Omega )\) and n positive constants \(\beta _{i}\), with \(\beta _{i}<(p_{i})_{0}\) for \(1\leq i\leq n\), such that

    $$ 0\leq F(x, t_{1}, \dots , t_{n})\leq \alpha (x) { \Biggl(}1+\sum_{i=1}^{n} \vert t_{i} \vert ^{ \beta _{i}} {\Biggr)}, $$

    for a.e. \(x\in \Omega \), \((t_{1}, \dots , t_{n})\in \mathbb{R}^{n}\).

  3. (h3)

    There exist \(x_{0}\in \Omega \), \(D>0\), \(\delta >0\), \(0< b_{i}<(\frac{C}{m_{0}})^{\frac{1}{(p_{i})^{0}}}\), and

    $$ m_{0}\frac{\pi ^{\frac{N}{2}}}{\Gamma (1+\frac{N}{2})}\biggl(\frac{D}{2} \biggr)^{N}\bigl(2^{N}-1\bigr) \sum _{i=1}^{n}\Phi _{i}\biggl( \frac{2\delta }{D}\biggr)>1 $$

    such that

    $$ \begin{aligned} \int _{\Omega }\sup_{|t_{1}|< b_{1}, \dots , |t_{n}|< b_{n}} F(x, t_{1}, \dots , t_{n})\,dx< {}&\frac{\min \{\frac{m_{0}}{C}(b_{i})^{(p_{i})^{0}}: 1\leq i\leq n\}}{m_{1}\frac{\pi ^{\frac{N}{2}}}{\Gamma (1+\frac{N}{2})}(\frac{D}{2})^{N}(2^{N}-1)\sum_{i=1}^{n}\Phi _{i}(\frac{2\delta }{D})} \\ &{}\times \int _{B(x_{0}, \frac{D}{2})} F(x, \delta , \dots , \delta ) \,dx. \end{aligned} $$

    Furthermore, set

    $$ \begin{aligned} &\underline{\lambda }:= \frac{m_{1}\frac{\pi ^{\frac{N}{2}}}{\Gamma (1+\frac{N}{2})}(\frac{D}{2})^{N}(2^{N}-1)\sum_{i=1}^{n}\Phi _{i}(\frac{2\delta }{D})}{\int _{B(x_{0}, \frac{D}{2})} F(x, \delta , \dots , \delta ) \,dx}, \\ &\overline{\lambda }:= \frac{\min \{\frac{m_{0}}{C}(b_{i})^{(p_{i})^{0}}: 1\leq i\leq n\}}{\int _{\Omega }\sup_{|t_{1}|< b_{1}, \dots , |t_{n}|< b_{n}} F(x, t_{1}, \dots , t_{n})\,dx}, \end{aligned} $$

then, for each \(\lambda \in \Lambda :=(\underline{\lambda }, \overline{\lambda })\), problem (1.1) possesses at least three distinct weak solutions in X.

Proof

Our aim is to apply Theorem 1.1 to our problem, so we check that the functionals J, I satisfy the conditions of Theorem 1.1. We set \(u_{0}=(0, \dots , 0)\). Then by the definitions of I, J and from (h1), we have \(J(u_{0})=I(u_{0})=0\). Let \(x_{0}\in \Omega \), \(D>0\), and take

$$ w(x)= \textstyle\begin{cases} 0 &x\in \Omega \setminus B(x_{0}, D), \\ \delta , &x\in B(x_{0}, \frac{D}{2}), \\ \frac{2\delta }{D}(D- \vert x-x_{0} \vert ) &x\in B(x_{0}, D)\setminus B(x_{0}, \frac{D}{2}). \end{cases} $$

Let \(\bar{u}=(w(x), \dots , w(x))\) and \(r=\min \{\frac{m_{0}}{C}(b_{i})^{(p_{i})^{0}}: 1\leq i\leq n\}\). Clearly, \(\bar{u}\in X\) and from (h3) we have

$$ \begin{aligned} J(\bar{u})&=\sum_{i=1}^{n} \hat{M_{i}}\biggl( \int _{\Omega } \Phi _{i}\bigl( \bigl\vert \nabla w(x) \bigr\vert \bigr)\,dx\biggr) \\ &\geq \sum_{i=1}^{n} m_{i}^{0} \int _{\Omega }\Phi _{i}\bigl( \bigl\vert \nabla w(x) \bigr\vert \bigr)\,dx \\ &\geq \sum_{i=1}^{n}m_{0} \Phi _{i}\biggl(\frac{2\delta }{D}\biggr) \frac{\pi ^{\frac{N}{2}}}{\Gamma (1+\frac{N}{2})} \biggl(\frac{D}{2}\biggr)^{N}\bigl(2^{N}-1\bigr) \\ & > r. \end{aligned} $$

On the other way, when \(J(u)\leq r\) for \(u=(u_{1}, \dots , u_{n})\in X\),

$$ \sum_{i=1}^{n}\hat{M_{i}} {\biggl(} \int _{\Omega }\Phi \bigl( \bigl\vert \nabla u_{i}(x) \bigr\vert \bigr)\,dx {\biggr)}\leq r. $$

Hence, since \(0< b_{i}<(\frac{C}{m_{0}})^{\frac{1}{(p_{i})^{0}}}\), using Propositions 1.1 and 1.2, we have

$$ m_{0} \Vert u_{i} \Vert ^{(p_{i})^{0}}_{i}< r, $$

and from (1.6) we obtain

$$ \bigl\vert u_{i}(x) \bigr\vert < \biggl(\frac{Cr}{m_{0}} \biggr)^{\frac{1}{(p_{i})^{0}}}=b_{i}, \quad \text{for } 1\leq i\leq n. $$

Therefore, for every \(u=(u_{1}, \dots , u_{n})\in X\),

$$ \begin{aligned} \sup_{u\in J^{-1}(-\infty , r)}I(u)&=\sup _{u\in J^{-1}(- \infty , r)} \int _{\Omega } F\bigl(x, u_{1}(x), \dots , u_{n}(x)\bigr) \,dx \\ &\leq \int _{\Omega }\sup_{|t_{1}|\leq b_{1}, \dots , |t_{n}|\leq b_{n}} F(x, t_{1}, \dots , t_{n}) \,dx. \end{aligned} $$

On the other hand, we have

$$ \begin{aligned} J(\bar{u})&=\sum_{i=1}^{n} \hat{M_{i}}\biggl( \int _{\Omega } \Phi _{i}\bigl( \bigl\vert \nabla w(x) \bigr\vert \bigr)\,dx\biggr) \\ &\leq m_{1}\frac{\pi ^{\frac{N}{2}}}{\Gamma (1+\frac{N}{2})}\biggl( \frac{D}{2} \biggr)^{N}\bigl(2^{N}-1\bigr)\sum _{i=1}^{n}\Phi _{i}\biggl( \frac{2\delta }{D}\biggr) \end{aligned} $$

and

$$ I(\bar{u})> \int _{B(x_{0}, \frac{D}{2})} F(x, \delta , \dots , \delta ) \,dx. $$

So, from (h3), we have

$$ \frac{\sup_{u\in J^{-1}(-\infty , r)}I(u)}{r}\leq \frac{\int _{\Omega }\sup_{|t_{1}|\leq b_{1}, \dots , |t_{n}|\leq b_{n}} F(x, t_{1}, \dots , t_{n}) \,dx}{\min \{\frac{m_{0}}{C}(b_{i})^{(p_{i})^{0}} :1\leq i\leq n\}}< \frac{I(\bar{u})}{J(\bar{u})}. $$

Hence, we observe that the condition (a1) of Theorem 1.1 is satisfied.

From (h2), it follows that the function \(J-\lambda I\) is coercive for every positive parameter λ, in particular for every

$$ \lambda \in \Lambda \subseteq \biggl(\frac{J(\bar{u})}{I(\bar{u})}, \frac{r}{\sup_{J(u)\leq r} I(u)}\biggr), $$

so the condition (a2) of Theorem 1.1 holds. Then all the assumptions of Theorem 1.1 are fulfilled. By Theorem 1.1, we know that there exist an open interval \(\Lambda \subseteq [0, \infty )\) and a positive constant ρ such that, for any \(\lambda \in \Lambda \), problem (1.1) has at least three weak solutions whose norms are less than ρ. □

Theorem 2.2

Assume that conditions (M) and (h1) hold and consider the following:

  1. (h4)

    \(F(x, t_{1}, \dots , t_{n})\geq 0\) for every \((x, t_{1}, \dots , t_{n})\in \Omega \times \mathbb{R}_{+}^{n}\).

  2. (h5)

    There exist \(x_{0}\in \Omega \) and values \(D, \varrho >0\) such that \(\overline{B(x_{0}, D)}\subseteq \Omega \),

    $$ \lim_{t\rightarrow 0^{+}}\frac{\Phi _{i}(t)}{t^{(p_{i})^{0}}}< \varrho , $$
    (2.1)

    and for

    $$ \begin{aligned} &A:=\liminf_{\sigma \rightarrow 0^{+}} \frac{\int _{\Omega }\sup_{(t_{1}, \dots , t_{n})\in Q(\sigma )}F(x, t_{1}, \dots , t_{n}) \,dx}{\sigma ^{\overline{p}}}, \\ &B:=\limsup_{(t_{1}, \dots , t_{n})\rightarrow (0^{+}, \dots , 0^{+})} \frac{\int _{B(x_{0}, \frac{D}{2})}F(x, t_{1}, \dots , t_{n}) \,dx}{\sum_{i=1}^{n}{t_{i}}^{(p_{i})^{0}}}, \end{aligned} $$

    one has

    $$ A < L B, $$

    where \(L=\min \{L_{(p_{i})^{0}}, i=1, 2, \dots , n\}\),

    $$ L_{(p_{i})^{0}}= \frac{\Gamma (1+\frac{N}{2})}{ {(}\sum_{i=1}^{n}(\frac{C}{m_{0}})^{\frac{1}{(p_{i})^{0}}} {)}^{\overline{p}}m_{1}\varrho \pi ^{\frac{N}{2}}(\frac{2}{D})^{(p_{i})^{0}-N}(2^{N}-1)}. $$
    (2.2)

Then for every

$$ \lambda \in \Lambda := \frac{1}{ {(}\sum_{i=1}^{n}(\frac{C}{m_{0}})^{\frac{1}{(p_{i})^{0}}} {)}^{\overline{p}}}\biggl( \frac{1}{LB}, \frac{1}{A}\biggr), $$

problem (1.1) admits a sequence of pairwise distinct weak solutions which strongly converges to zero in X.

Proof

We apply the part (b) of Theorem 1.2 and show that \(\delta <\infty \). Let \(\{\sigma _{k}\}\) be a sequence of positive numbers such that \(\lim_{k\rightarrow +\infty }\sigma _{k}=0\) then

$$ \begin{aligned} &\lim_{k\rightarrow +\infty } \frac{\int _{\Omega }\sup_{(t_{1}, \dots , t_{n})\in Q(\sigma _{k})} F(x, t_{1}, \dots , t_{n}) \,dx}{\sigma _{k}^{\overline{p}}} \\ &\quad =\liminf_{\sigma \rightarrow 0^{+}} \frac{\int _{\Omega }\sup_{(t_{1}, \dots , t_{n})\in Q(\sigma )} F(x, t_{1}, \dots , t_{n}) \,dx}{\sigma ^{\overline{p}}} \\ &\quad =A< +\infty . \end{aligned} $$
(2.3)

Putting

$$\begin{aligned}& r_{k}= \frac{\sigma _{k}^{\overline{p}}}{ {(}\sum_{i=1}^{n}(\frac{C}{m_{0}}) ^{\frac{1}{(p_{i})^{0}}} {)}^{\overline{p}}}\quad \text{for all } k\in \mathbb{N}, \\& \begin{aligned} J^{-1}\bigl(]-\infty , r_{k}[ \bigr)&:=\bigl\{ u=(u_{1}, u_{2}, \dots , u_{n}) \in X : J(u)< r_{k}\bigr\} \\ &\subseteq \Biggl\{ u\in X: \sum_{i=1}^{n} \hat{M_{i}}\biggl( \int _{\Omega }\Phi _{i}\bigl( \vert \nabla u_{i} \vert \bigr)\,dx\biggr)\leq {r_{k}}\Biggr\} , \end{aligned} \end{aligned}$$

by Propositions 1.1 and 1.2, for k large enough (\(0< r_{k}<1\)),

$$ m_{0} \Vert u_{i} \Vert _{i}^{(p_{i})^{0}}< r_{k}, $$

and from (1.6) we have \(\max_{x\in \bar{\Omega }}|u_{i}(x)|^{(p_{i})^{0}}\leq C\|u_{i}\|_{i}^{(p_{i})^{0}}\). Then we obtain for all \(x\in \Omega \),

$$ \bigl\vert u_{i}(x) \bigr\vert \leq \biggl( \frac{Cr_{k}}{m_{0}}\biggr)^{\frac{1}{(p_{i})^{0}}}. $$

Thus

$$ \sum_{i=1}^{n} \bigl\vert u_{i}(x) \bigr\vert \leq \sum_{i=1}^{n} \biggl(\frac{Cr_{k}}{m_{0}}\biggr)^{ \frac{1}{(p_{i})^{0}}}\leq r_{k}^{\frac{1}{\overline{p}}} \sum_{i=1}^{n}\biggl( \frac{C}{m_{0}}\biggr)^{\frac{1}{(p_{i})^{0}}}\leq \sigma _{k}. $$

Then we have

$$ J^{-1}(-\infty , r_{k})\subseteq \Biggl\{ u\in X: \sum _{i=1}^{n} \bigl\vert u_{i}(x) \bigr\vert \leq \sigma _{k}\Biggr\} . $$

From condition (h1), we have \(\min_{X}J=J(0, \dots , 0)=I(0, \dots , 0)=0\).

$$ \begin{aligned} \varphi (r_{k})&=\inf _{u\in J^{-1}(]-\infty , r_{k}[)} \frac{\sup_{v\in J^{-1}(]-\infty , r_{k}[)}I(v)- I(u) }{r_{k}- J(u)} \\ &\leq \frac{\sup_{v\in J^{-1}(]-\infty , r_{k}[)}I(v)}{r_{k}} \\ &\leq {\Biggl(}\sum_{i=1}^{n}\biggl( \frac{C}{m_{0}}\biggr)^{\frac{1}{(p_{i})^{0}}} {\Biggr)}^{\overline{p}} \frac{\int _{\Omega }\sup_{(t_{1}, t_{2}, \dots , t_{n})\in Q(\sigma _{k})}F(x, t_{1}, \dots , t_{n}) \,dx}{\sigma _{k}^{\overline{p}}}. \end{aligned} $$
(2.4)

Let \(\delta :=\liminf_{r\rightarrow 0^{+}}\varphi (r)\). It follows from (2.3) and (2.4) that

$$ \begin{aligned} \delta &\leq \liminf_{k\rightarrow +\infty } \varphi (r_{k}) \\ &\leq {\Biggl(}\sum_{i=1}^{n}\biggl( \frac{C}{m_{0}}\biggr)^{\frac{1}{(p_{i})^{0}}} {\Biggr)}^{\overline{p}}\lim _{k\rightarrow +\infty } \frac{\int _{\Omega }\sup_{ (t_{1}, t_{2}, \dots , t_{n})\in Q(\sigma _{k})}F(x, t_{1}, \dots , t_{n}) \,dx}{\sigma _{k}^{\overline{p}}} \\ &\leq {\Biggl(}\sum_{i=1}^{n}\biggl( \frac{C}{m_{0}}\biggr)^{\frac{1}{(p_{i})^{0}}} {\Biggr)}^{\overline{p}}A< +\infty . \end{aligned} $$

So \(\Lambda \subseteq\, ]0, \frac{1}{\delta }[\). For a fixed \(\lambda \in \Lambda \), we claim that the functional \(h_{\lambda }\) is unbounded from below. Indeed, since

$$ \frac{1}{\lambda }< {\Biggl(}\sum_{i=1}^{n} \biggl(\frac{C}{m_{0}}\biggr)^{ \frac{1}{(p_{i})^{0}}} {\Biggr)}^{\overline{p}}L B, $$

we can consider n positive real sequences \(\{d_{i, k}\}_{i=1}^{n}\) and \(\eta >0\) such that \(\sqrt{\sum_{i=1}^{n}d_{i, k}^{2}}\rightarrow 0\) as \(k\rightarrow +\infty \) and

$$ \frac{1}{\lambda }< \eta < L {\Biggl(}\sum _{i=1}^{n}\biggl(\frac{C}{m_{0}} \biggr)^{ \frac{1}{(p_{i})^{0}}} {\Biggr)}^{\overline{p}} \frac{\int _{B(x_{0}, \frac{D}{2})}F(x, d_{1, k}, \dots , d_{n, k}) \,dx}{\sum_{i=1}^{n}d_{i, k}^{(p_{i})^{0}}}. $$
(2.5)

Let \(\{u_{k}(x)=(u_{1k}, u_{2k}, \dots , u_{nk})\}\subseteq X\) be a sequence defined by

$$ u_{ik}(x)= \textstyle\begin{cases} 0,& x\in \bar{\Omega }\setminus B(x_{0}, D), \\ \frac{2d_{i, k}}{D} {(}D- \vert x-x_{0} \vert {)},& x\in B(x_{0}, D) \setminus B(x_{0}, \frac{D}{2}), \\ d_{i,k},& x\in B(x_{0}, \frac{D}{2}), \end{cases} $$

for \(1\leq i\leq n\). Then

$$ \begin{aligned} J(u_{k})&=\sum _{i=1}^{n}\hat{M_{i}}\biggl( \int _{\Omega }\Phi _{i}\bigl( \vert \nabla u_{ik} \vert \bigr)\,dx\biggr) \\ &< m_{1} \int _{B(x_{0}, D)\setminus B(x_{0}, \frac{D}{2})}\Phi _{i}\biggl( \frac{2d_{i, k}}{D} \biggr)\,dx. \end{aligned} $$

Moreover, from (2.1) and since \(\lim_{k\rightarrow \infty } \frac{2d_{i, k}}{D}=0\), there exist \(\zeta >0\) and \(n_{i}\in \mathbb{N}\) \(i=1, \dots , n\) such that \(\frac{2d_{i, k}}{D}\in (0, \zeta )\), and

$$\begin{aligned}& \Phi _{i}\biggl(\frac{2d_{i, k}}{D} \biggr)< \varrho \biggl(\frac{2}{D}\biggr) ^{(p_{i})^{0}} d_{i, k}^{(p_{i})^{0}} \quad \text{for all } n\geq n_{i}\ (i=1, \dots , n), \\& \int _{B(x_{0}, D)\setminus B(x_{0}, \frac{D}{2})}\Phi _{i}\biggl( \frac{2d_{i, k}}{D} \biggr)\,dx< \frac{\pi ^{\frac{N}{2}}}{\Gamma (1+{\frac{N}{2}})}\varrho \biggl( \frac{2}{D}\biggr) ^{(p_{i})^{0}-N} d_{i, k}^{(p_{i})^{0}}\bigl(2^{N}-1 \bigr). \end{aligned}$$

From (2.2), for all \(n\geq \max \{n_{1}, \dots , n_{2}\}\), we have

$$ \begin{aligned} J(u_{k})\leq \frac{1}{ {(}\sum_{i=1}^{n}(\frac{C}{m_{0}})^{\frac{1}{(p_{i})^{0}}} {)}^{\overline{p}}} \sum_{i=1}^{n} \frac{d_{i, k}^{(p_{i})^{0}}}{L_{{(p_{i})^{0}}}}. \end{aligned} $$
(2.6)

By (h4), we have

$$ I(u_{k})= \int _{\Omega } F(x, u_{1k}, \dots , u_{nk}) \,dx\geq \int _{B(x_{0}, \frac{D}{2})} F(x, d_{1, k}, \dots , d_{n, k}) \,dx. $$
(2.7)

By (2.5), (2.6), and (2.7), we have

$$\begin{aligned} h_{\lambda }(u_{k})&=J(u_{k})- \lambda I(u_{k}) \\ &\leq \frac{1}{ {(}\sum_{i=1}^{n}(\frac{C}{m_{0}})^{\frac{1}{(p_{i})^{0}}} {)}^{\overline{p}}} \sum_{i=1}^{n} \frac{d_{i, k}^{(p_{i})^{0}}}{L_{{(p_{i})^{0}}}}- \lambda \int _{B(x_{0}, \frac{D}{2})} F(x, d_{1, k}, \dots , d_{i, k}) \,dx \\ &< \frac{1-\lambda \eta }{L {(}\sum_{i=1}^{n}(\frac{C}{m_{0}})^{\frac{1}{(p_{i})^{0}}} {)}^{\overline{p}}} \sum_{i=1}^{n}{d_{i, k}^{(p_{i})^{0}}} \\ &< 0=h_{\lambda }(0, \dots , 0), \end{aligned}$$

for every \(n\in \mathbb{N}\) large enough. Then \((0, \dots , 0)\) is not a local minimum of \(h_{\lambda }\). Thus, owing to the fact that \((0, \dots , 0)\) is the unique global minimum of J, there exists a sequence \(\{u_{k}=(u_{1k}, \dots , u_{nk})\}\) of pairwise distinct critical points of \(h_{\lambda }\) such that \(\lim_{k\rightarrow +\infty }\|u_{k}\|=0\), and this completes the proof. □

We illustrate this abstract existence result with the following example.

Example 2.1

Let \(\Omega \subset \mathbb{R}^{3}\) be a bounded domain with \(|\Omega |=1\) and assume \(i=2\). Similar to [15, Remark 3.6], we have

$$ \varphi _{1}(t)= \textstyle\begin{cases} \frac{ \vert t \vert ^{4}t}{\log (1+ \vert t \vert )}&\text{if } t\neq 0, \\ 0&\text{if } t=0. \end{cases} $$

By [11, Example 3], one has \((p_{1})_{0}=5<(p_{1})^{0}=6\). Thus the condition (1.4) is satisfied. Moreover, owing to

$$ \lim_{t\rightarrow 0^{+}}\frac{1}{t^{5}} \int _{0}^{t} \frac{ \vert s \vert ^{4}s}{\log (1+|s|)}\,ds= \frac{1}{ 5}, $$

the condition (2.1) is also fulfilled (for example, take \(\varrho =\frac{1}{N}=\frac{1}{3}\)). Now let

$$ \varphi _{2}(t)=\log \bigl(1+ \vert t \vert ^{2} \bigr) \vert t \vert ^{2}t, \quad t\in \mathbb{R}. $$

Then by [11, Example 2], one has \((p_{2})_{0}=4<(p_{2})^{0}=6\). So the condition (1.4) is satisfied. Moreover, owing to

$$ \lim_{t\rightarrow 0^{+}}\frac{1}{t^{4}} \int _{0}^{t}\log \bigl(1+ \vert s \vert ^{2}\bigr) \vert s \vert ^{2}s\,ds=0, $$

the condition (2.1) is also fulfilled (here we take \(\varrho =\frac{1}{N}=\frac{1}{3}\), again). So we see that with the above choices, \(\varphi _{1}\) and \(\varphi _{2}\) satisfy the assumptions of Theorem 2.2. Let \(F:\mathbb{R}^{2}\rightarrow [0, \infty )\) be a continuous function defined by

$$\begin{aligned}& F(s, t)= \textstyle\begin{cases} s^{6}(1+\sin (\ln (1+ \vert t \vert ))), &(s, t)\neq (0, 0), \\ 0, &(s, t)=(0, 0), \end{cases}\displaystyle \\& A=\liminf_{\sigma \rightarrow 0^{+}} \frac{\int _{\Omega }\max_{ \vert s \vert + \vert t \vert \leq \sigma }F(s, t)\,dx }{\sigma ^{6}}= \vert \Omega \vert \liminf_{\sigma \rightarrow 0^{+}} \frac{\max_{ \vert s \vert + \vert t \vert \leq \sigma }F(s, t) }{\sigma ^{6}}=2, \\& B=\limsup_{s, t\rightarrow 0^{+}} \frac{\int _{B(x_{0}, \frac{D}{2})}F(s, t) \,dx}{s^{6}+t^{6}}= \biggl\vert B \biggl(x_{0}, \frac{D}{2}\biggr) \biggr\vert \limsup _{s, t\rightarrow 0^{+}} \frac{F(s, t)}{s^{6}+t^{6}}= \biggl\vert B \biggl(x_{0}, \frac{D}{2}\biggr) \biggr\vert . \end{aligned}$$

Then

$$ \lambda _{1}=\frac{7m_{1}}{3}\biggl(\frac{2}{D} \biggr)^{6}>0 \quad \text{and} \quad \lambda _{2}= \frac{m_{0}}{2^{7}C}>0, $$

with this condition \(2^{13}<\frac{3m_{0}D^{6}}{7m_{1}C}\). Then for \(\lambda \in\, ]\lambda _{1}, \lambda _{2}[\), the following system:

$$ \textstyle\begin{cases} -M_{1} {(}\int _{\Omega }\Phi _{1}( \vert \nabla u \vert )\,dx {)}\operatorname{div}( \frac{ \vert \nabla u \vert ^{4}}{\log (1+ \vert \nabla u \vert )}\nabla u)=\lambda F_{u}(x,u,v) \quad \text{in } \Omega , \\ -M_{2} {(}\int _{\Omega }\Phi _{2}( \vert \nabla v \vert )\,dx {)}\operatorname{div}( \log (1+ \vert \nabla v \vert ^{2}) \vert \nabla v \vert ^{2}\nabla v)=\lambda F_{v}(x,u,v) \quad \text{in } \Omega , \\ u=v=0\quad \text{on } \partial \Omega , \end{cases} $$

admits a sequence of pairwise distinct weak solutions which strongly converges to zero in \(W_{0}^{1, \Phi _{1}}(\Omega )\times W_{0}^{1, \Phi _{2}}(\Omega )\).

Availability of data and materials

Not applicable.

References

  1. 1.

    Adams, R.A.: Sobolev Spaces. Academic Press, New York (1975)

    Google Scholar 

  2. 2.

    Adams, R.A., Fournier, J.: Sobolev Spaces. Academic Press, London (2003)

    Google Scholar 

  3. 3.

    Allegue, O., Bezzarga, M.: Three solutions for a class of quasilinear elliptic systems in Orlicz–Sobolev spaces. Complex Var. Elliptic Equ. 58(9), 1215–1227 (2013)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Alves, C.O., Santos, J.A.: Multivalued elliptic equation with exponential critical growth in \(\mathbb{R}^{2}\). J. Differ. Equ. 261(9), 4758–4788 (2016)

    Article  Google Scholar 

  5. 5.

    Behboudi, F., Razani, A., Oveisiha, M.: Existence of a mountain pass solution for a nonlocal fractional \((p, q)\)-Laplacians problem. Bound. Value Probl. 2020, 149 (2020)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Bonanno, G., Bisci, G.M.: Infinitely many solutions for a boundary value problem with discontinuous nonlinearities. Bound. Value Probl. 2009, 670675 (2009)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Bonanno, G., Bisci, G.M., Rădulescu, V.D.: Quasilinear elliptic non-homogeneous Dirichlet problems through Orlicz–Sobolev spaces. Nonlinear Anal. 75(12), 4441–4456 (2012)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Bonanno, G., Marano, S.A.: On the structure of the critical set of non-differentiable functions with a weak compactness condition. Appl. Anal. 89, 1–10 (2010)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Cheng, B., Wu, X., Liu, J.: Multiplicity of solutions for nonlocal elliptic system of \((p, q)\)-Kirchhoff type. Abstr. Appl. Anal. 2011, 526026 (2011)

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Chung, N.T.: Three solutions for a class of nonlocal problems in Orlicz–Sobolev spaces. J. Korean Math. Soc. 250(6), 1257–1269 (2013)

    MathSciNet  MATH  Google Scholar 

  11. 11.

    Clément, P., de Pagter, B., Sweers, G., de Thélin, F.: Existence of solutions to a semilinear elliptic system through Orlicz–Sobolev spaces. Mediterr. J. Math. 1(3), 241–267 (2004)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Clément, P., García Huidobro, M., Manásevich, R., Schmitt, K.: Mountain pass type solutions for quasilinear elliptic equations. Calc. Var. Partial Differ. Equ. 11, 33–62 (2000)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Cowan, C., Razani, A.: Singular solutions of a p-Laplace equation involving the gradient. J. Differ. Equ. 269, 3914–3942 (2020)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Figueiredo, G.M., Santos, J.A.: Existence of least energy nodal solution with two nodal domains for a generalized Kirchhoff problem in an Orlicz–Sobolev space. Math. Nachr. 290(4), 583–603 (2017)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Heidarkhani, S., Caristi, G., Ferrara, M.: Perturbed Kirchhoff-type Neumann problems in Orlicz–Sobolev spaces. Comput. Math. Appl. 71(10), 2008–2019 (2016)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Hudzik, H.: The problem of separability, duality, reflexivity and of comparison for generalized Orlicz–Sobolev spaces \((W_{M}^{k} (\Omega ))\). Comment. Math. 21(2), 315–324 (1979)

    Google Scholar 

  17. 17.

    Kirchhoff, G.: Mechanik. Teubner, Leipzig (1883)

    Google Scholar 

  18. 18.

    Kristály, A., Mihăilescu, M., Rădulescu, V.: Two non-trivial solutions for a non-homogeneous Neumann problem: an Orlicz–Sobolev space setting. Proc. R. Soc. Edinb., Sect. A 139(2), 367–379 (2009)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Kurdila, A.J., Zabarankin, M.: Convex Functional Analysis, Systems Control: Foundations Applications. Birkhäuser, Basel (2005)

    Google Scholar 

  20. 20.

    Li, Q., Yang, Z.: Existence of positive solutions for a quasilinear elliptic systems of p-Kirchhoff type. Differ. Equ. Appl. 6(1), 73–80 (2014)

    MathSciNet  MATH  Google Scholar 

  21. 21.

    Makvand Chaharlang, M., Razani, A.: Existence of infinitely many solutions for a class of nonlocal problems with Dirichlet boundary condition. Commun. Korean Math. Soc. 34(1), 155–167 (2019)

    MathSciNet  MATH  Google Scholar 

  22. 22.

    Makvand Chaharlang, M., Razani, A.: A fourth order singular elliptic problem involving p-biharmonic operator. Taiwan. J. Math. 23(3), 589–599 (2019)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Makvand Chaharlang, M., Razani, A.: Two weak solutions for some Kirchhoff-type problem with Neumann boundary condition. Georgian Math. J. (2020). https://doi.org/10.1515/gmj-2019-2077

    Article  MATH  Google Scholar 

  24. 24.

    Mihăilescu, M., Rădulescu, V.: Eigenvalue problems associated to nonhomogeneous differential operators in Orlicz–Sobolev spaces. Anal. Appl. 6(1), 1–16 (2008)

    MathSciNet  Article  Google Scholar 

  25. 25.

    Mihăilescu, M., Rădulescu, V.: Neumann problems associated to nonhomogeneous differential operators in Orlicz–Sobolev spaces. Ann. Inst. Fourier 58(6), 2087–2111 (2008)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Mihăilescu, M., Repovs, D.: Multiple solutions for a nonlinear and non-homogeneous problem in Orlicz–Sobolev spaces. Appl. Math. Comput. 217, 6624–6632 (2011)

    MathSciNet  MATH  Google Scholar 

  27. 27.

    Ragusa, M.A.: Elliptic boundary value problem in vanishing mean oscillation hypothesis. Comment. Math. Univ. Carol. 40(4), 651–663 (1999)

    MathSciNet  MATH  Google Scholar 

  28. 28.

    Ragusa, M.A., Tachikawa, A.: Boundary regularity of minimizers of \(p(x)\)-energy functionals. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 33(2), 451–476 (2016)

    MathSciNet  Article  Google Scholar 

  29. 29.

    Rao, M.M., Ren, Z.D.: Applications of Orlicz Spaces. Monographs and Textbooks in Pure and Applied Mathematics, vol. 250. Dekker, New York (2002)

    Google Scholar 

  30. 30.

    Rasouli, S.H.: Existence of solutions for singular \((p, q)\)-Kirchhoff type systems with multiple parameters. Electron. J. Differ. Equ. 2016, 69 (2016)

    MathSciNet  Article  Google Scholar 

  31. 31.

    Razani, A.: Subsonic detonation waves in porous media. Phys. Scr. 94, 085209 (2019)

    Article  Google Scholar 

  32. 32.

    Safari, F., Razani, A.: Existence of radially positive solutions for Neumann problem on the Heisenberg group. Bound. Value Probl. 2020, 88 (2020)

    Article  Google Scholar 

  33. 33.

    Zeidler, E.: Nonlinear Functional Analysis and Applications. Nonlinear Monotone Operators, vol. II/B. Springer, New York (1990)

    Google Scholar 

Download references

Acknowledgements

Not applicable.

Funding

Not available.

Author information

Affiliations

Authors

Contributions

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

Corresponding author

Correspondence to A. Razani.

Ethics declarations

Competing interests

The authors declare that they have no competing interests.

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Heidari, S., Razani, A. Multiple solutions for a class of nonlocal quasilinear elliptic systems in Orlicz–Sobolev spaces. Bound Value Probl 2021, 22 (2021). https://doi.org/10.1186/s13661-021-01496-8

Download citation

MSC

  • 35J35
  • 35D30
  • 35J92
  • 34B16

Keywords

  • Orlicz–Sobolev spaces
  • Kirchhoff-type problems
  • Variational methods
  • Infinitely many solutions