## Boundary Value Problems

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# Infinitely many solutions for class of Neumann quasilinear elliptic systems

Boundary Value Problems20122012:54

DOI: 10.1186/1687-2770-2012-54

Accepted: 6 May 2012

Published: 6 May 2012

## Abstract

We investigate the existence of infinitely many weak solutions for a class of Neumann quasilinear elliptic systems driven by a (p1, ..., p n )-Laplacian operator. The technical approach is fully based on a recent three critical points theorem.

AMS subject classification: 35J65; 34A15.

### Keywords

infinitely many solutions Neumann system critical point theory variational methods

## 1 Introduction

$\left\{\begin{array}{cc}-{\Delta }_{{p}_{i}}{u}_{i}+{a}_{i}\left(x\right){\left|{u}_{i}\right|}^{{p}_{i}-2}u=\lambda {F}_{{u}_{i}}\left(x,{u}_{1},\dots ,{u}_{n}\right)\hfill & \mathsf{\text{in}}\phantom{\rule{2.77695pt}{0ex}}\Omega ,\hfill \\ \frac{\partial {u}_{i}}{\partial \nu }=0\hfill & \mathsf{\text{on}}\phantom{\rule{0.3em}{0ex}}\partial \Omega \hfill \end{array}\right\$
(1)

for i = 1, ..., n, where Ω N (N ≥ 1) is a non-empty bounded open set with a smooth boundary ∂Ω, p i > N for i = 1, ..., n, ${\Delta }_{{p}_{i}}{u}_{i}=\mathsf{\text{div}}\left({\left|\nabla {u}_{i}\right|}^{{p}_{i}-2}\nabla {u}_{i}\right)$ is the p i -Laplacian operator, a i L (Ω) with ess infΩ a i > 0 for i = 1, ..., n, λ > 0, and F: Ω × n is a function such that the mapping (t1, t2,..., t n ) → F (x, t1, t2,..., t n ) is in C1 in n for all $x\in \Omega ,{F}_{{t}_{i}}$ is continuous in Ω × n for i = 1,..., n, and F (x, 0,..., 0) = 0 for all x Ω and ν is the outward unit normal to ∂Ω. Here, ${F}_{{t}_{i}}$ denotes the partial derivative of F with respect to t i .

Precisely, under appropriate hypotheses on the behavior of the nonlinear term F at infinity, the existence of an interval Λ such that, for each λ Λ, the system (1) admits a sequence of pairwise distinct weak solutions is proved; (see Theorem 3.1). We use a variational argument due to Ricceri which provides certain alternatives in order to find sequences of distinct critical points of parameter-depending functionals. We emphasize that no symmetry assumption is required on the nonlinear term F (thus, the symmetry version of the Mountain Pass theorem cannot be applied). Instead of such a symmetry, we assume a suitable oscillatory behavior at infinity on the function F.

We recall that a weak solution of the system (1) is any $u=\left({u}_{1},...,{u}_{n}\right)\in {W}^{1,{p}_{1}}\left(\Omega \right)×...×{W}^{1,{p}_{n}}\left(\Omega \right),$ such that
$\begin{array}{c}\underset{\Omega }{\int }\sum _{i=1}^{n}\left({\left|\nabla {u}_{i}\left(x\right)\right|}^{{p}_{i}-2}\nabla {u}_{i}\left(x\right)\nabla {v}_{i}\left(x\right)+{a}_{i}\left(x\right){\left|{u}_{i}\left(x\right)\right|}^{{p}_{i}-2}{u}_{i}\left(x\right){v}_{i}\left(x\right)\right)dx\hfill \\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}-\lambda \underset{\Omega }{\int }\sum _{i=1}^{n}{F}_{{u}_{i}}\left(x,{u}_{1}\left(x\right),...{u}_{n}\left(x\right)\right){v}_{i}\left(x\right)dx=0\hfill \end{array}$

for all $v=\left({v}_{1},...,{v}_{n}\right)\in {W}^{1,{p}_{1}}\left(\Omega \right)×...×{W}^{1,{p}_{n}}\left(\Omega \right).$

For a discussion about the existence of infinitely many solutions for differential equations, using Ricceri's variational principle [1]and its variants [2, 3] we refer the reader to the articles [416].

For other basic definitions and notations we refer the reader to the articles [1722]. Here, our motivation comes from the recent article [8]. We point out that strategy of the proof of the main result and Example 3.1 are strictly related to the results and example contained in [8].

## 2 Preliminaries

Our main tool to ensure the existence of infinitely many classical solutions for Dirichlet quasilinear two-point boundary value systems is the celebrated Ricceri's variational principle [[1], Theorem 2.5] that we now recall as follows:

Theorem 2.1. Let X be a reflexive real Banach space, let Φ, Ψ: X be two Gâteaux differentiable functionals such that Φ is sequentially weakly lower semicontinuous, strongly continuous, and coercive and Ψ is sequentially weakly upper semicontinuous. For every r > inf X Φ, let us put
$\phi \left(r\right):=\underset{u\in {\Phi }^{-1}\left(\left]-\infty ,r\right[\right)}{inf}\frac{{sup}_{v\in {\Phi }^{-1}\left(\left]-\infty ,r\right[\right)}\Psi \left(v\right)-\Psi \left(u\right)}{r-\Phi \left(u\right)}$
and
$\gamma :=\underset{r\to +\infty }{lim inf}\phi \left(r\right),\phantom{\rule{1em}{0ex}}\delta :=\underset{r\to {\left({inf}_{X}\Phi \right)}^{+}}{lim inf}\phi \left(r\right).$

Then, one has

(a) for every r > inf X Φ and every $\lambda \in \left]0,\frac{1}{\phi \left(r\right)}\right[$, the restriction of the functional I λ = Φ - λ Ψ to Φ-1(] - ∞, r[) admits a global minimum, which is a critical point (local minimum) of I λ in X.

(b) If γ < +∞ then, for each $\lambda \in \left]0,\frac{1}{\gamma }\right[$, the following alternative holds:

either

(b1) I λ possesses a global minimum,

or

(b2) there is a sequence {u n } of critical points (local minima) of I λ such that
$\underset{n\to +\infty }{lim}\Phi \left({u}_{n}\right)=+\infty .$

(c) If δ < +∞ then, for each $\lambda \in \left]0,\frac{1}{\delta }\right[$, the following alternative holds:

either

(c1) there is a global minimum of Φ which is a local minimum of I λ ,

or

(c2) there is a sequence {u n } of pairwise distinct critical points (local minima) of I λ that converges weakly to a global minimum of Φ.

We let X be the Cartesian product of n Sobolev spaces ${W}^{1,{p}_{1}}\left(\Omega \right)$, ${W}^{1,{p}_{2}}\left(\Omega \right)$,... and ${W}^{1,{p}_{n}}\left(\Omega \right)$, i.e., $X={\prod }_{i=1}^{n}{W}^{1,{p}_{i}}\left(\Omega \right)$, equipped with the norm
$∥\left({u}_{1},{u}_{2},\dots ,{u}_{n}\right)∥=\sum _{i=1}^{n}{∥{u}_{i}∥}_{{p}_{i}},$
where
$\begin{array}{c}{∥{u}_{i}∥}_{{p}_{i}}={\left(\underset{\Omega }{\int }{\left|\nabla {u}_{i}\left(x\right)\right|}^{{p}_{i}}+{a}_{i}\left(x\right){\left|{u}_{i}\left(x\right)\right|}^{{p}_{i}}dx\right)}^{\frac{1}{{p}_{i}}},\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}i=1,\dots ,n.\hfill \\ \phantom{\rule{1em}{0ex}}C=max\left\{\underset{{u}_{i}\in {W}^{1,{p}_{i}}\left(\Omega \right)\\left\{0\right\}}{sup}\frac{{sup}_{x\in \Omega }{\left|u\left(x\right)\right|}^{{p}_{i}}}{{∥{u}_{i}∥}_{{p}_{i}}^{{p}_{i}}};\phantom{\rule{2.77695pt}{0ex}}i=1,\dots ,n\right\}.\hfill \end{array}$
(2)
Since p i > N for 1 ≤ i ≤ n, one has C < +∞. In addition, if Ω is convex, it is known [23] that
$\underset{{u}_{i}\in {W}^{1,{p}_{i}}\left(\Omega \right)\\left\{0\right\}}{sup}\frac{{sup}_{x\in \Omega }\left|{u}_{i}\left(x\right)\right|}{{∥{u}_{i}∥}_{{p}_{i}}}\le {2}^{\frac{{p}_{i}-1}{{p}_{i}}}max\left\{{\left(\frac{1}{{∥{a}_{i}∥}_{1}}\right)}^{\frac{1}{{p}_{i}}};\frac{\mathsf{\text{diam}}\left(\Omega \right)}{{N}^{\frac{1}{{p}_{i}}}}{\left(\frac{{p}_{i}-1}{{p}_{i}-N}m\left(\Omega \right)\right)}^{\frac{{p}_{i}-1}{{p}_{i}}}\frac{{∥{a}_{i}∥}_{\infty }}{{∥{a}_{i}∥}_{1}}\right\}$

for 1 ≤ i ≤ n, where ||·||1 = ∫Ω|·(x)| dx, ||·|| = supxΩ|·(x)| and m(Ω) is the Lebesgue measure of the set Ω, and equality occurs when Ω is a ball.

In the sequel, let $\underset{¯}{p}=min\left\{{p}_{i};1\le i\le n\right\}$.

For all γ > 0 we define
$K\left(\gamma \right)=\left\{\left({t}_{1},\dots ,{t}_{n}\right)\in {ℝ}^{n}:\sum _{i=1}^{n}\left|{t}_{i}\right|\le \gamma \right\}.$
(3)

## 3 Main results

We state our main result as follows:

Theorem 3.1. Assume that

(A1)
$\begin{array}{c}\phantom{\rule{1em}{0ex}}\underset{\xi \to +\infty }{lim inf}\frac{{\int }_{\Omega }{{sup}_{\left({t}_{1},...,{t}_{n}\right)}}_{\in K\left(\xi \right)}F\left(x,{t}_{1},\dots ,{t}_{n}\right)dx}{{\xi }^{\underset{-}{p}}}\hfill \\ <{\left(\sum _{i=1}^{n}{\left({p}_{i}C\right)}^{\frac{1}{{p}_{i}}}\right)}^{\underset{-}{p}}\underset{\begin{array}{c}\left({t}_{1},\dots ,{t}_{n}\right)\to \infty \hfill \\ \left({t}_{1},\dots ,{t}_{n}\right)\in {ℝ}_{+}^{n}\hfill \end{array}}{lim sup}\frac{{\int }_{\Omega }F\left(x,{t}_{1},\dots ,{t}_{n}\right)dx}{{\sum }_{i=1}^{n}\frac{{∥{a}_{i}∥}_{1}{\left|{t}_{i}\right|}^{{p}_{i}}}{{p}_{i}}}\hfill \end{array}$

where $K\left(\xi \right)=\left\{\left({t}_{1},\dots ,{t}_{n}\right)|{\sum }_{i=1}^{n}\left|{t}_{i}\right|\le \xi \right\}$ (see (3)).

Then, for each
$\begin{array}{c}\hfill \lambda \in \Lambda :=\hfill \\ \hfill \left]\frac{1}{lim{sup}_{\begin{array}{c}\left({t}_{1},\dots ,{t}_{n}\right)\to \infty \\ \left({t}_{1},\dots ,{t}_{n}\right)\in {ℝ}_{+}^{n}\end{array}}\frac{{\int }_{\Omega }F\left(x,{t}_{1},\dots ,{t}_{n}\right)dx}{\sum _{i=1}^{n}\frac{||{a}_{i}|{|}_{1}|{t}_{i}|}{{p}_{i}}}},\frac{{\left({\sum }_{i=1}^{n}{\left({p}_{i}C\right)}^{\frac{1}{{p}_{{}_{i}}}}\right)}^{\underset{-}{p}}}{lim\phantom{\rule{2.77695pt}{0ex}}{inf}_{\xi \to +\infty }\frac{{\int }_{\Omega }{sup}_{\left({t}_{1},...,{t}_{n}\right)\in K\left(\xi \right)}F\left(x,{t}_{1},\dots ,{t}_{n}\right)dx}{{\xi }^{\underset{-}{p}}}}\right[\hfill \end{array}$

the system (1) has an unbounded sequence of weak solutions in X.

Proof. Define the functionals Φ, Ψ: X for each u = (u1, ..., u n ) X, as follows
$\Phi \left(u\right)=\sum _{i=1}^{n}\frac{{∥{u}_{i}∥}_{{p}_{i}}^{{p}_{i}}}{{p}_{i}}$
and
$\Psi \left(u\right)=\underset{\Omega }{\int }F\left(x,{u}_{1}\left(x\right),\dots ,{u}_{n}\left(x\right)\right)dx.$
It is well known that Ψ is a Gâteaux differentiable functional and sequentially weakly lower semicontinuous whose Gâteaux derivative at the point u X is the functional Ψ'(u) X*, given by
${\Psi }^{\prime }\left(u\right)\left(v\right)=\underset{\Omega }{\int }\sum _{i=1}^{n}{F}_{{u}_{i}}\left(x,{u}_{1}\left(x\right),\dots ,{u}_{n}\left(x\right)\right){v}_{i}\left(x\right)dx$
for every v = (v1, ..., v n ) X, and Ψ': XX* is a compact operator. Moreover, Φ is a sequentially weakly lower semicontinuous and Gâteaux differentiable functional whose Gâteaux derivative at the point u X is the functional Φ' (u) X*, given by
${\Phi }^{\prime }\left({u}_{1},\dots ,{u}_{n}\right)\left({v}_{1},\dots ,{v}_{n}\right)\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\underset{\Omega }{\int }\sum _{i=1}^{n}\left({\left|\nabla {u}_{i}\left(x\right)\right|}^{{p}_{i}-2}\nabla {u}_{i}\left(x\right)\nabla {v}_{i}\left(x\right)+{a}_{i}\left(x\right){\left|{u}_{i}\left(x\right)\right|}^{{p}_{i}-2}{u}_{i}\left(x\right){v}_{i}\left(x\right)\right)\phantom{\rule{0.3em}{0ex}}dx$

for every v = (v1, ..., v n ) X. Furthermore, (Φ')-1: X*X exists and is continuous.

Put I λ : = Φ - λ Ψ. Clearly, the weak solutions of the system (1) are exactly the solutions of the equation ${I}_{\lambda }^{\prime }\left({u}_{1},\dots ,{u}_{n}\right)=0$. Now, we want to show that
$\gamma <+\infty .$
Let {ξ m } be a real sequence such that ξ m → +∞ as m → ∞ and
$\begin{array}{c}\phantom{\rule{1em}{0ex}}\underset{m\to \infty }{lim}\frac{{\int }_{\Omega }{sup}_{\left({t}_{1},\dots ,{t}_{n}\right)\in K\left({\xi }_{m}\right)}F\left(x,{t}_{1},\dots ,{t}_{n}\right)dx}{{\xi }_{m}^{\underset{-}{p}}}\hfill \\ =\underset{\xi \to +\infty }{lim inf}\frac{{\int }_{\Omega }{sup}_{\left({t}_{1},\dots ,{t}_{n}\right)\in K\left(\xi \right)}F\left(x,{t}_{1},\dots ,{t}_{n}\right)dx}{{\xi }^{\underset{-}{p}}}.\hfill \end{array}$
Put ${r}_{m}=\frac{{\xi }_{m}^{\underset{-}{p}}}{{\left({\sum }_{i=1}^{n}{\left({p}_{i}C\right)}^{\frac{1}{{p}_{i}}}\right)}^{\underset{-}{p}}}$ for all m . Since
$\underset{x\in \Omega }{sup}{\left|{u}_{i}\left(x\right)\right|}^{{p}_{i}}\le C{∥{u}_{i}∥}_{{p}_{i}}^{{p}_{i}}$
for each ${u}_{i}\in {W}^{1,{p}_{i}}\left(\Omega \right)$ for 1 ≤ in, we have
$\underset{x\in \Omega }{sup}\sum _{i=1}^{n}\frac{{\left|{u}_{i}\left(x\right)\right|}^{{p}_{i}}}{{p}_{i}}\le C\sum _{i=1}^{n}\frac{{∥{u}_{i}∥}_{{p}_{i}}^{{p}_{i}}}{{p}_{i}}.$
(4)
for each u = (u1, u2, ..., u n ) X. This, for each r > 0, together with (4), ensures that
Hence, an easy computation shows that ${\sum }_{i=1}^{n}\left|{u}_{i}\right|\le {\xi }_{m}$ whenever u = (u1, ..., u n ) Φ-1(] - ∞, r m ]). Hence, one has
$\begin{array}{cc}\phi \left({r}_{m}\right)\hfill & =\underset{u\in {\Phi }^{-1}\left(\left]-\infty ,{r}_{m}\right[\right)}{inf}\frac{\left({sup}_{v\in {\Phi }^{-1}\left(\left]-\infty ,{r}_{m}\right[\right)}\Psi \left(v\right)\right)-\Phi \left(u\right)}{{r}_{m}-\Phi \left(u\right)}\hfill \\ \le \frac{{sup}_{v\in {\Phi }^{-1}\left(\left]-\infty ,{r}_{m}\right[\right)}\Psi \left(v\right)}{{r}_{m}}\hfill \\ \le \frac{{\int }_{\Omega }{sup}_{\left({t}_{1},\dots ,{t}_{n}\right)\in K\left({\xi }_{m}\right)}F\left(x,{t}_{1},\dots ,{t}_{n}\right)dx}{\frac{{\xi }_{m}^{\underset{-}{p}}}{{\left({\sum }_{i=1}^{n}{\left({p}_{i}C\right)}^{\frac{1}{{p}_{i}}}\right)}^{\underset{-}{p}}}}.\hfill \end{array}$
Therefore, since from Assumption (A1) one has
$\underset{\xi \to +\infty }{lim inf}\frac{{\int }_{\Omega }{sup}_{\left({t}_{1},\dots ,{t}_{n}\right)\in K\left(\xi \right)}F\left(x,{t}_{1},\dots ,{t}_{n}\right)dx}{{\xi }^{\underset{-}{p}}}<\infty ,$
we deduce
$\begin{array}{c}\hfill \phantom{\rule{1em}{0ex}}\gamma \le \underset{m\to +\infty }{lim inf}\phi \left({r}_{m}\right)\hfill \\ \hfill \le {\left(\sum _{i=1}^{n}{\left({p}_{i}C\right)}^{\frac{1}{{p}_{i}}}\right)}^{\underset{-}{p}}\underset{\xi \to +\infty }{lim inf}\frac{{\int }_{\Omega }{sup}_{\left({t}_{1},\dots ,{t}_{n}\right)\in K\left(\xi \right)}F\left(x,{t}_{1},\dots ,{t}_{n}\right)dx}{{\xi }^{\underset{-}{p}}}<+\infty .\hfill \end{array}$
(5)
Assumption (A1) along with (5), implies
$\Lambda \subseteq \left]0,\frac{1}{\gamma }\right[.$

Fix λ Λ. The inequality (5) concludes that the condition (b) of Theorem 2.1 can be applied and either I λ has a global minimum or there exists a sequence {u m } where u m = (u1m, ..., u nm ) of weak solutions of the system (1) such that limm→∞||(u1m, ..., u nm )|| = +.

Now fix λ Λ and let us verify that the functional I λ is unbounded from below. Arguing as in [8], consider n positive real sequences ${\left\{{d}_{i,m}\right\}}_{i=1}^{n}$ such that $\sqrt{{\sum }_{i=1}^{n}{d}_{i,m}^{2}}\to +\infty$ as m

and
$\underset{m\to +\infty }{lim}\frac{{\int }_{\Omega }F\left(x,{d}_{1,m},\dots ,{d}_{n,m}\right)dx}{{\sum }_{i=1}^{n}\frac{{d}_{i,m}^{{p}_{i}}}{{p}_{i}}}=\underset{\begin{array}{c}\left({t}_{1},\dots ,{t}_{n}\right)\to \infty \\ \left({t}_{1},\dots ,{t}_{n}\right)\in {ℝ}_{+}^{n}\end{array}}{lim sup}\frac{{\int }_{\Omega }F\left(x,{t}_{1},\dots ,{t}_{n}\right)dx}{{\sum }_{i=1}^{n}\frac{{∥{a}_{i}∥}_{1}{\left|{t}_{i}\right|}^{{p}_{i}}}{{p}_{i}}}.$
(6)
For all m define w m (x) = (d1, m, ..., dn, m). For any fixed m , w m X and, in particular, one has
$\Phi \left({w}_{m}\right)=\sum _{i=1}^{n}\frac{{d}_{i,m}^{{p}_{i}}{∥{a}_{i}∥}_{1}}{{p}_{i}}.$
Then, for all m ,
${I}_{\lambda }\left({w}_{m}\right)=\Phi \left({w}_{m}\right)-\lambda \Psi \left({w}_{m}\right)=\sum _{i=1}^{n}\frac{{d}_{i,m}^{{p}_{i}}{∥{a}_{i}∥}_{1}}{{p}_{i}}-\lambda \underset{\Omega }{\int }F\left(x,{d}_{1,m},\dots ,{d}_{n,m}\right)dx.$
Now, if
$\underset{\begin{array}{c}\left({t}_{1},\dots ,{t}_{n}\right)\to \infty \\ \left({t}_{1},\dots ,{t}_{n}\right)\in {ℝ}_{+}^{n}\end{array}}{lim sup}\frac{{\int }_{\Omega }F\left(x,{t}_{1},\dots ,{t}_{n}\right)dx}{{\sum }_{i=1}^{n}\frac{{∥{a}_{i}∥}_{1}|{t}_{i}{|}^{{p}_{i}}}{{p}_{i}}}<\infty ,$
we fix $\mathit{\epsilon }\in \left]\frac{1}{\lambda lim{sup}_{\begin{array}{c}\left({t}_{1},\dots ,{t}_{n}\right)\to \infty \\ \left({t}_{1},\dots ,{t}_{n}\right)\in {ℝ}_{+}^{n}\end{array}}\frac{{\int }_{\Omega }F\left(x,{t}_{1},\dots ,{t}_{n}\right)dx}{\sum _{i=1}^{n}\frac{{∥{a}_{i}∥}_{1}{\left|{t}_{i}\right|}^{{p}_{i}}}{{p}_{i}}}},1\right[$. From (6) there exists τ ε such that
$\begin{array}{c}\hfill \underset{\Omega }{\int }F\left(x,{d}_{1,m},\dots ,{d}_{n,m}\right)dx\hfill \\ \hfill >\mathit{\epsilon }\underset{\begin{array}{c}\left({t}_{1},\dots ,{t}_{n}\right)\to \infty \\ \left({t}_{1},\dots ,{t}_{n}\right)\in {ℝ}_{+}^{n}\end{array}}{lim sup}\frac{{\int }_{\Omega }F\left(x,{t}_{1},\dots ,{t}_{n}\right)dx}{{\sum }_{i=1}^{n}\frac{{∥{a}_{i}∥}_{1}{\left|{t}_{i}\right|}^{{p}_{i}}}{{p}_{i}}}\left(\sum _{i=1}^{n}\frac{{d}_{i,m}^{{p}_{i}}{∥{a}_{i}∥}_{1}}{{p}_{i}}\right)\phantom{\rule{1em}{0ex}}\forall m>{\tau }_{\mathit{\epsilon }},\hfill \end{array}$
therefore
${I}_{\lambda }\left({w}_{m}\right)\le \left(1-\lambda \mathit{\epsilon }\underset{\begin{array}{c}\left({t}_{1},\dots ,{t}_{n}\right)\to \infty \\ \left({t}_{1},\dots ,{t}_{n}\right)\in {ℝ}_{+}^{n}\end{array}}{lim sup}\frac{{\int }_{\Omega }F\left(x,{t}_{1},\dots ,{t}_{n}\right)dx}{{\sum }_{i=1}^{n}\frac{{∥{a}_{i}∥}_{1}{\left|{t}_{i}\right|}^{{p}_{i}}}{{p}_{i}}}\right)\sum _{i=1}^{n}\frac{{d}_{i,m}^{{p}_{i}}{∥{a}_{i}∥}_{1}}{{p}_{i}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\forall m>{\tau }_{\mathit{\epsilon }},$
and by the choice of ε, one has
$\underset{m\to +\infty }{lim}\left[\Phi \left({w}_{m}\right)-\lambda \Psi \left({w}_{m}\right)\right]=-\infty .$
If
$\underset{\xi \to +\infty }{lim sup}\frac{{\int }_{\Omega }F\left(x,{t}_{1},\dots ,{t}_{n}\right)dx}{{\sum }_{i=1}^{n}\frac{{∥{a}_{i}∥}_{1}{\left|{t}_{i}\right|}^{{p}_{i}}}{{p}_{i}}}=\infty ,$
let us consider $K>\frac{1}{\lambda }$. From (6) there exists τ K such that
$\underset{\mathrm{\Omega }}{\int }F\left(x,{d}_{1,m},\dots ,{d}_{n,m}\right)dx>K\sum _{i=1}^{n}\frac{{d}_{i,m}^{{p}_{i}}{∥{a}_{i}∥}_{1}}{{p}_{i}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\forall m>{\tau }_{K},$
therefore
${I}_{\lambda }\left({w}_{m}\right)\le \left(1-\lambda K\right)\sum _{i=1}^{n}\frac{{d}_{i,m}^{{p}_{i}}{∥{a}_{i}∥}_{1}}{{p}_{i}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\forall m>{\tau }_{K},$
and by the choice of K, one has
$\underset{m\to +\infty }{lim}\left[\Phi \left({w}_{m}\right)-\lambda \Psi \left({w}_{m}\right)\right]=-\infty .$
Hence, our claim is proved. Since all assumptions of Theorem 2.1 are satisfied, the functional I λ admits a sequence {u m = (u1m, ..., u nm )} X of critical points such that
$\underset{m\to \infty }{lim}∥\left({u}_{1m},\dots ,{u}_{nm}\right)∥=+\infty ,$

and we have the conclusion.   □

Here, we give a consequence of Theorem 3.1.

Corollary 3.2. Assume that

(A2) ${lim inf}_{\xi \to +\infty }\frac{{\int }_{\Omega }{sup}_{\left({t}_{{}_{1}},\dots ,{t}_{n}\right)\in K\left(\xi \right)}F\left(x,{t}_{1},\dots ,{t}_{n}\right)dx}{{\xi }^{\underset{-}{p}}}<{\left({\sum }_{i=1}^{n}{\left({p}_{i}C\right)}^{\frac{1}{{p}_{i}}}\right)}^{\underset{-}{p}};$

(A3) ${lim sup}_{\begin{array}{c}\left({t}_{1},\dots ,{t}_{n}\right)\to \infty \\ \left({t}_{1},\dots ,{t}_{n}\right)\in {ℝ}_{+}^{n}\end{array}}\frac{{\int }_{\Omega }F\left(x,{t}_{1},\dots ,{t}_{n}\right)dx}{{\sum }_{i=1}^{n}\frac{{∥{a}_{i}∥}_{1}{\left|{t}_{i}\right|}^{{p}_{i}}}{{p}_{i}}}>1.$

Then, the system
$\left\{\begin{array}{cc}-{\Delta }_{{p}_{i}}{u}_{i}+{a}_{i}\left(x\right){\left|{u}_{i}\right|}^{{p}_{i}-2}u={F}_{{u}_{i}}\left(x,{u}_{1},\dots ,{u}_{n}\right)\hfill & in\phantom{\rule{2.77695pt}{0ex}}\Omega ,\hfill \\ \frac{\partial {u}_{i}}{\partial \nu }=0\hfill & on\partial \Omega \hfill \end{array}\right\$

for 1 ≤ i ≤ n, has an unbounded sequence of classical solutions in X.

Now, we want to present the analogous version of the main result (Theorem 3.1) in the autonomous case.

Theorem 3.3. Assume that

(A4)
$\begin{array}{c}\phantom{\rule{1em}{0ex}}\underset{\xi \to +\infty }{lim inf}\frac{{sup}_{\left({t}_{1},\dots ,{t}_{n}\right)\in K\left(\xi \right)}F\left({t}_{1},\dots ,{t}_{n}\right)}{{\xi }^{\underset{-}{p}}}\hfill \\ <{\left(\sum _{i=1}^{n}{\left({p}_{i}C\right)}^{\frac{1}{{p}_{i}}}\right)}^{\underset{-}{p}}\underset{\begin{array}{c}\left({t}_{1},\dots ,{t}_{n}\right)\to \infty \hfill \\ \left({t}_{1},\dots ,{t}_{n}\right)\in {ℝ}_{+}^{n}\hfill \end{array}}{lim sup}\frac{F\left({t}_{1},\dots ,{t}_{n}\right)}{{\sum }_{i=1}^{n}\frac{{∥{a}_{i}∥}_{1}{\left|{t}_{i}\right|}^{{p}_{i}}}{{p}_{i}}}\hfill \end{array}$

where $K\left(\xi \right)=\left\{\left({t}_{1},\dots ,{t}_{n}\right)|{\sum }_{i=1}^{n}\left|{t}_{i}\right|\le \xi \right\}$ (see (3)).

Then, for each
$\begin{array}{c}\lambda \in \mathrm{\Lambda }:=\\ \left]\frac{1}{\frac{F\left({t}_{1},\dots ,{t}_{n}\right)}{lim{sup}_{\begin{array}{c}\left({t}_{{}_{1}},\dots ,{t}_{n}\right)\to \infty \\ \left({t}_{{}_{1}},\dots ,{t}_{n}\right)\in {ℝ}_{+}^{n}\end{array}}\sum _{i=1}^{n}\frac{{∥{a}_{i}∥}_{1}\left|{t}_{i}\right|}{{p}_{i}}}},\frac{{\left({\sum }_{i=1}^{n}{\left({p}_{i}C\right)}^{\frac{1}{{p}_{i}}}\right)}^{\underset{-}{p}}}{lim\phantom{\rule{2.77695pt}{0ex}}{inf}_{\xi \to +\infty }\frac{{sup}_{\left({t}_{{}_{1}},\dots ,{t}_{{}_{n}}\right)\in K\left(\xi \right)}F\left({t}_{{}_{1}},\dots ,{t}_{n}\right)}{{\xi }^{\underset{-}{p}}}}\right[\end{array}$
the system
$\left\{\begin{array}{cc}-{\Delta }_{{p}_{i}}{u}_{i}+{a}_{i}\left(x\right){\left|{u}_{i}\right|}^{{p}_{i}-2}u=\lambda {F}_{{u}_{i}}\left({u}_{1},\dots ,{u}_{n}\right)\hfill & in\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\Omega ,\hfill \\ \frac{\partial {u}_{i}}{\partial \nu }=0\hfill & on\partial \Omega \hfill \end{array}\right\$

has an unbounded sequence of weak solutions in X.

Proof. Set F (x, u1, ..., u n ) = F (u1, ..., u n ) for all x Ω and (u1, ..., u n ) n . The conclusion follows from Theorem 3.1. □

Remark 3.1. We observe in Theorem 3.1 we can replace ξ → +∞ and (t1, ..., t n ) → (+, ..., +∞) with ξ → 0+ (t1, ..., t n ) → (0+, ..., 0+), respectively, that by the same way as in the proof of Theorem 3.1 but using conclusion (c) of Theorem 2.1 instead of (b), the system (1) has a sequence of weak solutions, which strongly converges to 0 in X.

Finally, we give an example to illustrate the result.

Example 3.1. Let Ω 2 be a non-empty bounded open set with a smooth boundary ϑΩ and consider the increasing sequence of positive real numbers given by
${a}_{n}:=2,\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}{a}_{n+1}:=n!{\left({a}_{n}\right)}^{\frac{5}{4}}+2$
for every n ≥ 1. Define the function
$F\left({t}_{1},{t}_{2}\right)=\left\{\begin{array}{cc}{\left({a}_{n+1}\right)}^{5}{e}^{-\frac{1}{1-\left[{\left({t}_{1}-{a}_{n+1}\right)}^{2}+{\left({t}_{2}-{a}_{n+1}\right)}^{2}\right]}}\hfill & \left({t}_{1},{t}_{2}\right)\in {\bigcup }_{n\ge 1}B\left(\left({a}_{n+1},{a}_{n+1}\right),1\right),\hfill \\ 0\hfill & \mathsf{\text{otherwise}}\hfill \end{array}\right\$
(7)

where B((an+1, an+1), 1)) be the open unit ball of center (an+1, an+1). We observe that the function F is non-negative, F (0, 0) = 0, and F C1(2). We will denote by f and g, respectively, the partial derivative of F respect to t1 and t2. For every n , the restriction F on B((an+1, an+1), 1) attains its maximum in (an+1, an+1) and F (an+1, an+1) = (an+1)5,

then
$\underset{n\to +\infty }{lim sup}\frac{F\left({a}_{n+1},{a}_{n+1}\right)}{\frac{{a}_{n+1}^{3}}{3}+\frac{{a}_{n+1}^{4}}{4}}=+\infty$
So
$\underset{\left({t}_{1},{t}_{2}\right)\to \left(+\infty ,+\infty \right)}{lim sup}\frac{F\left({t}_{1},{t}_{2}\right)}{\frac{{\left|{t}_{1}\right|}^{3}}{3}+\frac{{\left|{t}_{2}\right|}^{4}}{4}}=+\infty$
On the other by setting y n = an+1- 1 for every n , one has
$\underset{\left({t}_{1},{t}_{2}\right)\in K\left({y}_{n}\right)}{sup}F\left({t}_{1},{t}_{2}\right)={a}_{n}^{5}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\forall n\in ℕ$
Then
$\underset{n\to \infty }{lim}\frac{{sup}_{\left({t}_{1},{t}_{2}\right)\in K\left({y}_{n}\right)}F\left({t}_{1},{t}_{2}\right)}{{\left({a}_{n+1}-1\right)}^{3}}=0,$
and hence
$\underset{\xi \to \infty }{lim inf}\frac{{sup}_{\left({t}_{1},{t}_{2}\right)\in K\left(\xi \right)}F\left({t}_{1},{t}_{2}\right)}{{\xi }^{3}}=0.$
Finally
$\begin{array}{c}\hfill 0=\underset{\xi \to +\infty }{lim inf}\frac{{sup}_{\left({t}_{1},{t}_{2}\right)\in K\left(\xi \right)}F\left({t}_{1},{t}_{2}\right)}{{\xi }^{3}}\hfill \\ \hfill <{\left({\left(3C\right)}^{\frac{1}{3}}+{\left(4C\right)}^{\frac{1}{4}}\right)}^{3}\underset{\left({t}_{1},{t}_{2}\right)\to {\left(+\infty ,+\infty \right)}_{\left({t}_{{}_{1}},{t}_{{}_{2}}\right)\in {ℝ}_{+}^{n}}}{lim sup}\frac{F\left({t}_{1},{t}_{2}\right)}{\frac{{\left|{t}_{1}\right|}^{3}}{3}+\frac{{\left|{t}_{2}\right|}^{4}}{4}}=+\infty .\hfill \end{array}$
So, since all assumptions of Theorem 3.3 is applicable to the system
$\left\{\begin{array}{cc}-{\Delta }_{3}u+\left|u\right|u=\lambda f\left(u,v\right)\hfill & \mathsf{\text{in}}\phantom{\rule{2.77695pt}{0ex}}\Omega ,\hfill \\ -{\Delta }_{4}v+{\left|v\right|}^{2}g=\lambda g\left(u,v\right)\hfill & \mathsf{\text{in}}\phantom{\rule{2.77695pt}{0ex}}\Omega ,\hfill \\ \frac{\partial u}{\partial \nu }=\frac{\partial v}{\partial \nu }=0\hfill & \mathsf{\text{on}}\partial \Omega \hfill \end{array}\right\$

for every λ [0, +[.

## Authors’ Affiliations

(1)
Department of Mathematics, Science and Research Branch, Islamic Azad University (IAU)
(2)
Department of Mathematics, Faculty of Basic Sciences, University of Mazandaran

## References

1. Ricceri B: A general variational principle and some of its applications. J Comput Appl Math 2000, 113: 401-410. 10.1016/S0377-0427(99)00269-1
2. Bonanno G, Molica Bisci G: Infinitely many solutions for a boundary value problem with discontinuous nonlinearities. Bound Value Probl 2009, 2009: 1-20.
3. Marano S, Motreanu D: Infinitely many critical points of non-differentiable functions and applications to the Neumann-type problem involving the p -Laplacian. J Diff Equ 2002, 182: 108-120. 10.1006/jdeq.2001.4092
4. Bonanno G, D'Aguì G: On the Neumann problem for elliptic equations involving the p -Laplacian. J Math Anal Appl 2009, 358: 223-228. 10.1016/j.jmaa.2009.04.055
5. Bonanno G, Di Bella B: Infinitely many solutions for a fourth-order elastic beam equation. Nonlinear Diff Equ Appl NoDEA 2011, 18: 357-368. 10.1007/s00030-011-0099-0
6. Bonanno G, Molica Bisci G: A remark on perturbed elliptic Neumann problems. Studia Univ "Babeş-Bolyai", Mathematica 2010., LV(4):Google Scholar
7. Bonanno G, Molica Bisci G: Infinitely many solutions for a Dirichlet problem involving the p -Laplacian. Proc Royal Soc Edinburgh 2010, 140A: 737-752.
8. Bonanno G, Molica Bisci G, O'Regan D: Infinitely many weak solutions for a class of quasilinear elliptic systems. Math Comput Model 2010, 52: 152-160. 10.1016/j.mcm.2010.02.004
9. Bonanno G, Molica Bisci G, Rădulescu V: Infinitely many solutions for a class of nonlinear eigenvalue problems in Orlicz-Sobolev spaces. C R Acad Sci Paris, Ser I 2011, 349: 263-268. 10.1016/j.crma.2011.02.009
10. Candito P: Infinitely many solutions to the Neumann problem for elliptic equations involving the p -Laplacian and with discontinuous nonlinearities. Proc Edin Math Soc 2002, 45: 397-409.
11. Candito P, Livrea R: Infinitely many solutions for a nonlinear Navier boundary value problem involving the p -biharmonic. Studia Univ "Babeş-Bolyai", Mathematica 2010., LV(4):Google Scholar
12. Dai G: Infinitely many solutions for a Neumann-type differential inclusion problem involving the p ( x )-Laplacian. Nonlinear Anal 2009, 70: 2297-2305. 10.1016/j.na.2008.03.009
13. Fan X, Ji C: Existence of infinitely many solutions for a Neumann problem involving the p(x)-Laplacian. J Math Anal Appl 2007, 334: 248-260. 10.1016/j.jmaa.2006.12.055
14. Kristály A: Infinitely many solutions for a differential inclusion problem in N . J Diff Equ 2006, 220: 511-530. 10.1016/j.jde.2005.02.007
15. Li C: The existence of infinitely many solutions of a class of nonlinear elliptic equations with a Neumann boundary conditions for both resonance and oscillation problems. Nonlinear Anal 2003, 54: 431-443. 10.1016/S0362-546X(03)00100-7
16. Ricceri B: Infinitely many solutions of the Neumann problem for elliptic equations involving the p -Laplacian. Bull Lond Math Soc 2001, 33(3):331-340. 10.1017/S0024609301008001
17. Afrouzi GA, Heidarkhani S: Existence of three solutions for a class of Dirichlet quasi-linear elliptic systems involving the ( p1, ..., p n )-Laplacian. Nonlinear Anal 2009, 70: 135-143. 10.1016/j.na.2007.11.038
18. Afrouzi GA, Heidarkhani S, O'Regan D: Three solutions to a class of Neumann doubly eigenvalue elliptic systems driven by a ( p1, ..., p n )-Laplacian. Bull Korean Math Soc 2010, 47(6):1235-1250. 10.4134/BKMS.2010.47.6.1235
19. Bonanno G, Heidarkhani S, O'Regan D: Multiple solutions for a class of Dirichlet quasilinear elliptic systems driven by a (p, q)-Laplacian operator. Dyn Syst Appl 2011, 20: 89-100.
20. Heidarkhani S, Tian Y: Multiplicity results for a class of gradient systems depending on two parameters. Nonlinear Anal 2010, 73: 547-554. 10.1016/j.na.2010.03.051
21. Heidarkhani S, Tian Y: Three solutions for a class of gradient Kirchhoff-type systems depending on two parameters. Dyn Syst Appl 2011, 20: 551-562.
22. Zeidler E: Nonlinear Functional Analysis and its Applications. Volume II. Springer, New York; 1985.
23. Bonanno G, Candito P: Three solutions to a Neumann problem for elliptic equations involving the p-Laplacian. Arch Math (Basel) 2003, 80: 424-429.