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GRADIENT NONLINEAR ELLIPTIC SYSTEMS DRIVEN BY A (p, q)-LAPLACIAN OPERATOR.

1. INTRODUCTION

Let [OMEGA] [subset] [R.sup.N] (N [greater than or equal to] 3) be a non-empty bounded open set with a smooth boundary [partial derivative][OMEGA]. In this paper, we study the following gradient nonlinear elliptic system with Dirichlet conditions

(1.1) [mathematical expression not reproducible]

where p, q > 1, [lambda] is a positive real parameter, by [[DELTA].sub.s] we denote the s-Laplacian operator defined [[DELTA].sub.s]u = div([[absolute value of ([nabla]u)].sup.s- 2][nabla]u) for all u [member of] [W.sup.1,s.sub.0]([OMEGA]) (s = p, q). In the statement of system (1.1) the reaction term F : [bar.[OMEGA]] x [R.sup.2] [right arrow] R is a [C.sup.1]-function such that F(x, 0, 0) = 0 for every x [member of] [OMEGA], [F.sub.u], [F.sub.v] denote the partial derivatives of F with respect to u and v respectively. We suppose, moreover that a, b [member of] [L.sup.[infinity]]([OMEGA]) and

(1.2) [mathematical expression not reproducible].

In recent years the existence and structure of solutions for problem driven by p-Laplacian have found many interest and different approaches have been developed. The variational methods are used to obtain weak solutions as critical points of a suitable energy function. This approach is employed to deal with systems of gradient type, i.e. the nonlinearities are the gradient of a [C.sup.1] functional. We refer the reader to [15] for a complete overview on this subject and [1], [2], [3], [4], [5], [9] [10], [12], [17], [18] and the references therein for more developments. In [5], the authors study the existence of critical points of the energy functional for which these points are the solutions of a quasilinear elliptic system involving (p, q)-Laplacian with 1 < p, q < N. They consider subcritical growth conditions, and under suitable conditions on the nonlinearity, prove the existence of non-trivial solutions according to various cases: sublinear, superlinear and resonant case. In [10] and [17] the authors get the existence of three solutions for a class of quasilinear elliptic systems involving (p, q)- Laplacian with p, q > N. In [1], the authors generalize the results obtained in [17] to systems involving ([p.sub.1], [p.sub.2], ..., [p.sub.n])-Laplacian. Similar studies, but under different boundary conditions, can be found in [3], [4] (mixed boundary conditions). It is worth noticing that in [17] precise values of parameter [lambda] are not established. In the study of nonlinear elliptic systems in order to obtain non-zero solutions, non-variational approaches have also been used under a different set of assumptions (as for instance the monotony of the nonlinearity) and we refer to [14, 16] and the references therein for further details.

The aim of paper is to determine the existence of multiple solutions as the parameter [lambda] > 0 varies in an appropriate interval. In this work, without losing generality, we suppose that 1 < q [less than or equal to] p < N.

The paper is arranged as follows. First, we obtain the existence of one non-zero weak solution to system (1.1) without assuming any asymptotic condition neither at zero nor at infinity (see Theorem 3.1). Next we prove the existence of at least two non zero weak solutions by using the Ambrosetti-Rabinowitz condition (see Theorem 3.2). Finally, we present an existence result three solutions under an appropriate condition on the nonlinear term F (see Theorem 3.3). Moreover the case in which F is autonomuos is presented and some examples are given.

2. PRELIMINARIES

In this section, we recall definitions and theorems used in the paper.

Let (X, [parallel] * [parallel]) be a real Banach space and [PHI], [PSI] : X [right arrow] R be two Gateaux differentiable functionals and r [member of]] - [infinity], +[infinity]]. We say that functional I = [PHI] - [PSI] satisfies the Palais-Smale condition cut off upper at r (in short [(PS).sup.[r]]-condition) if any sequence {[u.sub.n]} in X such that

[mathematical expression not reproducible]

has a convergent subsequence.

When r = +[infinity] the previous definition coincides with the classical (PS)- condition, while if r < [infinity] such condition is more general than the classical one. We refer to [6] for more details.

We say that the functional I satisfies the weak Palais-Smale condition ((WPS)- condition) if any bounded sequence {[u.sub.n]} in X such that ([[alpha].sub.1]) and ([[alpha].sub.2]) hold, admits a convergent subsequence.

Our main tool is the local minimum theorem obtained in [6]. We recall here its version presented in [7].

Theorem 2.1 ([7, Theorem 2.3]). Let X be a real Banach space, and let [PHI], [PSI] : X [right arrow] R be two continuously Gateaux differentiable functionals such that [inf.sub.X] [PHI] = [PHI](0) = [PSI](0) = 0. Assume that there exist [gamma] [member of] R and [bar.u] [member of] X, with 0 < [PHI]([bar.u]) < [gamma], such that

(2.1) [mathematical expression not reproducible]

and, for each [mathematical expression not reproducible] the functional [I.sub.[lambda]] = [PHI] - [lambda][PSI] satisfied [(PS).sup.[r]]-condition.

Then, for each [mathematical expression not reproducible].

Now, we also recall a recent result obtained in [9] that insures the existence of at least two non-zero critical points for differentiable functionals.

Theorem 2.2 (8, Theorem 2.1]). Let X be a real Banach space and let [PHI], [PSI] : X [right arrow] R be two continuously Gateaux differentiable functionals such that [inf.sub.X] [PHI] = [PHI](0) = [PSI](0) = 0. Assume that there exist [gamma] [member of] R and [bar.u] [member of] X, with 0 < [PHI]([bar.u]) < [gamma], such that

[mathematical expression not reproducible]

and for each [mathematical expression not reproducible], the functional [I.sub.[lambda]] = [PHI] - [lambda][psi] satisfies (PS)-condition and it is unbounded from below.

Then, for each [mathematical expression not reproducible], the functional [I.sub.[lambda]] admits two non-zero critical points [u.sub.[lambda],1], [u.sub.[lambda],2] such that [I.sub.[lambda]]([u.sub.[lambda],1]) < 0 < [I.sub.[lambda]]([u.sub.[lambda],2]).

Finally, we point out an other result which insures the existence of at least three critical points. Theorem 2.3. has been obtained in [6], it is a more precise version of Theorem 3.2 of [8] and Theorem 3.6 of [11].

Theorem 2.3 ([9, Theorem 2.1]). Let X be a real Banach space, [PHI], [PSI] : X [right arrow] R be two continuously Gateaux differentiable functionals with $ bounded from below and [PHI](0) = [PSI](0) = 0.

Assume that there exist [gamma] [member of] R and [bar.u] [member of] X, with 0 < [gamma] < [PHI]([bar.u]), such that

(i) [mathematical expression not reproducible]

(ii) for each [mathematical expression not reproducible] the functional [I.sub.[lambda]] = [PHI] - [lambda][PSI] is bounded from below and satisfies (PS)-condition.

Then, for each [lambda] [member of] A, the functional [I.sub.[lambda]] = [PHI] - [lambda][PSI] has at least three distinct critical points in X.

Throughout in the paper, we suppose that the following condition holds (H) there exist two non negative constants [a.sub.1], [a.sub.2] and two constants s [member of] [1, pN/N-p[ and r [member of] [1, qN/N-q[ such that

[mathematical expression not reproducible]

for every (x, t) [member of] [OMEGA] x [R.sup.2].

Clearly, from (H) follows

(2.2) [mathematical expression not reproducible] for every (x, t) [member of] [OMEGA] x [R.sup.2].

In fact, there exists 0 < [theta] < 1 such that

[absolute value of (F(x, t))] = [absolute value of (F(x, t) - F(x, 0))] = [absolute value of ([nabla]F(x, [theta]t) x t)]

by using (H), we have

[mathematical expression not reproducible].

We consider the Sobolev space X = [W.sup.1,p.sub.0] ([OMEGA]) x [W.sup.1,q.sub.0]([OMEGA]) endowed with the norm

[mathematical expression not reproducible]

for all (u, v) [member of] X, where

[mathematical expression not reproducible],

that are, taking into account (1.2), equivalent to the usual one.

A function (u,v) [member of] X is said a weak solution to system (1.1) if

[mathematical expression not reproducible]

for every ([w.sub.1], [w.sub.2]) [member of] X.

Now, consider 1 < h < N and put [h.sup.*] = hN/N-h. Denote by [GAMMA] the Gamma function defined by

[GAMMA](s) = [[integral].sup.+[infinity].sub.0] [z.sup.s-1][e.sup.z]dz, [for all]s > 0.

From the Sobolev embedding theorem, for every u [member of] [W.sup.1,h.sub.0]([OMEGA]) there exists a constant c(N, h) [member of] [R.sub.+] such that

(2.3) [mathematical expression not reproducible]

the best constant that appears in (2.3) is

[mathematical expression not reproducible]

(see [19]).

Fixing s [member of] [1, [h.sup.*][ in virtue of Sobolev embedding theorem, for every u [member of] [W.sup.1,h.sub.0]([OMEGA]), there exists a positive constant [mathematical expression not reproducible] such that

(2.4) [mathematical expression not reproducible]

and, in virtue of Rellich theorem the embedding is compact.

By using Holder's inequality, we have

(2.5) [mathematical expression not reproducible]

where [mu]([OMEGA]) denotes the Lebesgue measure of the set [OMEGA]. Now, we put

(2.6) [mathematical expression not reproducible]

where the constants s and r are given by (H).

Moreover, let

(2.7) [mathematical expression not reproducible].

Simple calculations show that there is [x.sub.0] [member of] [OMEGA] such that B([x.sub.0], D) [subset or equal to] [OMEGA].

Finally, we set

(2.8) k = [[pi].sup.N/2][D.sup.N]/[THETA](1 + N/2),

(2.9) [sigma] = [2.sup.p]/q max{1/[D.sup.p](1 - 1/[2.sup.N]) + [[parallel]a[parallel].sub.[infinity]], 1/[D.sub.q](1 - 1/[2.sup.N]) + [[parallel]b[parallel].sub.[infinity]]}.

(2.10) [tau] = [2.sup.q]/p min {1/[D.sup.p](1 - 1/[2.sup.N]) + [a.sub.0], 1/[D.sub.q](1 - 1/[2.sup.N]) + [b.sub.0]}.

In order to study problem (1.1), we will use the functionals [PHI], [PSI] : X [right arrow] R defined by putting

(2.11) [mathematical expression not reproducible]

for every (u, v) [member of] X and put [I.sub.[lambda]] = [PHI] - [lambda][PSI] for [lambda] > 0.

Clearly, [PHI] is a coercive, weakly sequentially lower semicontinuous, continuously Gateaux differentiable and its derivative at point (u,v) [member of] X is defined by

[mathematical expression not reproducible]

for every ([w.sub.1], [w.sub.2]) [member of] X.

Moreover, [PSI] is well defined, weakly sequentially upper semicontinuous continuously Gateaux differentiable with compact derivative and its derivative at point (u, v) [member of] X is defined by

[PSI](u, v)([w.sub.1], [w.sub.2]) = [[integral].sub.[OMEGA]] [[F.sub.u](x, u(x), v(x))[w.sub.1](x) + [F.sub.v](x, u(x), v(x))[w.sub.2](x)]dx,

for every ([w.sub.1], [w.sub.2]) [member of] X.

A critical point for the functional [I.sub.[lambda]] := [PHI] - [lambda][PSI] is any (u, v) [member of] X such that

[PHI]'(u, v)([w.sub.1], [w.sub.2]) - [lambda][PSI]'(u, v)([w.sub.1], [w.sub.2]) = 0 [for all]([w.sub.1], [w.sub.2]) [member of] X.

Hence, the critical points for functional [I.sub.[lambda]] := [PHI] - [lambda][PSI] are exactly the weak solutions to system (1.1).

We have the following result

Lemma 2.4. Fix [lambda] > 0 the functional [I.sub.[lambda]] = [PHI] - [lambda][PSI] satisfies the (WPS)-condition.

Proof. Fixed [lambda] > 0, we claim that the functional [I.sub.[lambda]] = [PHI] - [lambda][PSI] satisfies the (WPS)- condition. For this end, let {([u.sub.n], [v.sub.n])} be a bounded sequence in X such that [I.sub.[lambda]]([u.sub.n], [v.sub.n]) is bounded and [mathematical expression not reproducible] for all ([[omega].sub.1], [[omega].sub.2]) [member of] X and where [[epsilon].sub.n] [right arrow] [0.sup.+]. Hence, taking a subsequence if necessary, we have

([u.sub.n], [v.sub.n]) [??] (u, v) in X,

[u.sub.n] [right arrow] u in [L.sup.[alpha]]([OMEGA]) for all [alpha] [member of] [1, p*[

[v.sub.n] [right arrow] v in [L.sup.[beta]]([OMEGA]) for all [beta] [member of] [1, q*[

From the previous relation, written with u1 = u and u2 = v we infer

(2.12) [mathematical expression not reproducible].

We observe that

[mathematical expression not reproducible]

and, bearing in mind that for all a, b [member of] R and p > 1,

[[absolute value of (a)].sup.p-1][absolute value of (b)] [less than or equal to] p - 1/p [[absolute value of (a)].sup.p] + 1/p [[absolute value of ([beta])].sup.p]

one has

(2.13) [mathematical expression not reproducible].

Moreover, by using (H) we have

[mathematical expression not reproducible]

where [alpha] = [p.sup.*]/[p.sup.*] - s + 1 and [beta] = [q.sup.*]/[q.sup.*] - r + 1, hence observing that [alpha] < [p.sup.*] and [beta] < [q.sup.*], we obtain

(2.14) [mathematical expression not reproducible].

From (2.12) and (2.13) we obtain

[mathematical expression not reproducible],

from this, taking into account (2.14) we have

[mathematical expression not reproducible]

thus, since X is uniformly convex, Proposition III.30 of [13] ensures that {([u.sub.n], [v.sub.n])} converges to (u, v) in X. Hence our claim is proved.

3. MAIN RESULTS

By using the notation of Section 2 we have our main results

Theorem 3.1. We suppose that (H) holds and assume that

([i.sub.1]) F(x, t) [greater than or equal to] 0 for every (x, t) [member of] [OMEGA] x [R.sup.2.sub.+] where [R.sup.2.sub.+] = {t = ([t.sub.1], [t.sub.2]) [member of] [R.sup.2] : [t.sub.i] [greater than or equal to] 0 i = 1, 2};

([i.sub.2]) there exist two positive constants [gamma] and [delta] with

[[delta].sup.p] + [[delta].sup.q] < q[gamma]/[kappa][sigma]

such that

[mathematical expression not reproducible]

where [a.sub.1], [a.sub.2], s and r are given by (H) and k, [sigma] are given by (2.8) and (2.9).

Then, for each [mathematical expression not reproducible], the system (1.1) has at least one non-zero weak solution.

Proof. Our goal is to apply Theorem 2.1. Consider the Sobolev space X and the operators defined in (2.11).

Taking into account (2.2), it follows that

(3.1) [mathematical expression not reproducible].

Let [gamma] [member of]]0, +[infinity][, then for every (u, v) [member of] X such that [PHI](u, v) < [gamma], by using (2.4) and (2.6) we get

(3.2) [PSI](u, v) [less than or equal to] [a.sub.1][c.sub.1,1]([(p[gamma]).sup.1/p] + [(q[gamma]).sup.1/q]) + [a.sub.2][c.sub.r,s] [((p[gamma]).sup.s/p]/s + [(q[gamma]).sup.r/q]/r).

Hence, from (3.2), we have

(3-3) [mathematical expression not reproducible],

for every [gamma] > 0.

Now, we choose the function ([bar.u], [bar.u]) defined by putting

(3.4) [mathematical expression not reproducible]

Clearly ([bar.u], [bar.u]) [member of] X and by using (2.8), (2.9) and (2.10) we have

(3.5) [[delta].sup.p] + [[delta].sup.q]/p [kappa][tau] < [PHI]([bar.u], [bar.u]) < [[delta].sup.p] + [[delta].sup.q]/q [kappa][sigma].

In virtue of (3.5) and bearing in mind that [[delta].sup.p] + [[delta].sup.q] < q[gamma]/[kappa][sigma] we obtain

0 < [PHI]([bar.u], [bar.u]) < [gamma]

and by using ([i.sub.1]) we have

(3.6) [mathematical expression not reproducible].

Hence, by (3.5) and (3.6), one has

(3.7) [PSI]([bar.u], [bar.u])/[PHI]([bar.u], [bar.u]) [greater than or equal to] q/[2.sup.N][sigma] [inf.sub.x[member of][OMEGA]] F(x, [delta], [delta])/[[delta].sup.p] + [[delta].sup.q]

By using (3.3), (3.7) and taking into account ([i.sub.2]), we get

[mathematical expression not reproducible]

Moreover, let be [r.sub.2] > 0 and {([u.sub.n], [v.sub.n])} a sequence in X such that ([[alpha].sub.3]) holds, since $ is coercive we have that {([u.sub.n], [v.sub.n])} is bounded. Then by using Lemma 2.1. we obtain that (WPS)-condition implies [mathematical expression not reproducible].

Therefore, all the assumptions of Theorem 2.1 are satisfied. So, for each

[mathematical expression not reproducible]

the functional [I.sub.[lambda]] has at least one non-zero critical point that is weak solution of system (1.1).

The following result, in which Ambrosetti-Rabinowitz condition is also used, ensures the existence at least two non-zero weak solutions.

Theorem 3.2. We suppose that (H) holds. Assume that

([j.sub.1]) F(x, t) [greater than or equal to] 0 for every (x, t) [member of] [OMEGA] x [R.sup.2.sub.+] where [R.sup.2.sub.+] = {t = ([t.sub.1], [t.sub.2]) [member of] [R.sup.2] : [t.sub.i] [greater than or equal to] 0 i = 1, 2};

([j.sub.2]) there are two positive constants [gamma] and [delta] with

[[delta].sup.p] + [[delta].sup.q] < q[gamma]/[kappa][sigma],

such that

[mathematical expression not reproducible]

where [a.sub.1], [a.sub.2], s and r are given by (H) and [kappa], [sigma] are given by (2.8) and (2.9), and that there are two positive constants [mu] > p and R such that

(AR) 0 < [mu]F(x, t) [less than or equal to] t x [[nabla].sub.t]F(x, t)

for all x [member of] [OMEGA] and [absolute value of (t)] > R.

Then for each [mathematical expression not reproducible], the system (1.1) has at least two non-zero weak solutions.

Proof. Our goal is to apply Theorem 2.2. Consider the Sobolev space X and the operators defined in (2.11) taking into account that the regolarity assumptions on [PHI] and [PSI] are satisfied. Arguing as in the proof of Theorem 3.1, put ([bar.u], [bar.u]) as in (3.4), by using ([i.sub.1]), ([j.sub.2]), (3.5) and bearing in mind that [[delta].sup.p] + [[delta].sup.q] > q[gamma]/[kappa][sigma], we obtain

0 < [PHI]([bar.u], [bar.u]) < [gamma]

and

[mathematical expression not reproducible], from (AR.), by standard computations, there is a positive constant C such that

(3.8) F(x, t) [greater than or equal to] C[[absolute value of (t)].sup.[mu]]

[for all]x [member of] [OMEGA], [absolute value of (t)] > R.

From (3.8) it follows that [I.sub.[lambda]] is unbounded from below.

Now, by using Lemma 2.4 to verify (PS)-condition it is enough to prove that any sequence of Palais-Smale is bounded. To this end, taking into account (AR) one has

(3.9) [mathematical expression not reproducible]

where C is a constant.

If {([u.sub.n], [v.sub.n])} is not bounded from (3.9) we have a contradiction.

Therefore, all conditions of Theorem 2.2 are satisfied, then the system (1.1), for each [mathematical expression not reproducible], admits at least two non-zero weak solutions.

Now, we point out the following result on the existence of at least three weak solutions.

Theorem 3.3. We suppose that (H) holds and assume that

([j.sub.1]) F(x, t) [greater than or equal to] 0 for every (x, t) [member of] [OMEGA] x [R.sup.2.sub.+] where [R.sup.2.sub.+] = {t = ([t.sub.1], [t.sub.2]) [member of] [R.sup.2] : [t.sub.i] [greater than or equal to] 0 i = 1, 2};

([h.sub.2]) there exist three positive constants [alpha], [beta] and b with [alpha] < p and [beta] < q such that

F(x, [t.sub.1], [t.sub.2]) [less than or equal to] b(1 + [[absolute value of ([t.sub.1])].sup.[alpha]] + [[absolute value of ([t.sub.2])].sup.[beta]])

for almost every x [member of] [OMEGA] and for every ([t.sub.1], [t.sub.2]) [member of] [R.sup.2.sub.+];

([h.sub.3]) there exist two positive constants [gamma] and [delta] with

[[delta].sup.p] + [[delta].sup.q] > p[gamma]/[kappa][tau],

such that

[mathematical expression not reproducible]

where [a.sub.1], [a.sub.2], s and r are given by (H) and [kappa], [sigma] are given by (2.8) and (2.9).

Then, for each [mathematical expression not reproducible], the system (1.1) has at least three weak solutions.

Proof. Our goal is to apply Theorem 2.3. Consider the Sobolev space X and the operators defined in (2.11) taking into account that the regolarity assumptions on [PHI] and [PSI] are satisfied, our aim is to verify (i) and (ii). Arguing as in the proof of Theorem 3.1, put (U,U) as in (3.4), by using (3.5) and bearing in mind that [[delta].sup.P] + [[delta].sup.q] > p[gamma]/[kappa][tau], we obtain

[PHI]([bar.u], [bar.u]) > [gamma] > 0.

Therefore, the assumption (i) of Theorem 2.3 is satisfied.

We prove that the functional [I.sub.[lambda]] = [PHI] - [gamma][PSI] is coercive for all positive parameter, in fact by using condition (h2) we have

[mathematical expression not reproducible].

We observe that the functional [I.sub.[lambda]] = [PHI] - [lambda][PSI] is bounded from below because it is coercive and weakly sequentially lower semicontinuous.

Now, by using Lemma 2.4 to verify (PS)-condition it is enough to observe that since the functional [I.sub.[lambda]] = [PHI] - [lambda][PSI] is coercive any sequence of Palais-Smale is bounded. Then also condition (ii) holds. Hence all the assumptions of Theorem 2.3 are satisfied.

So, for each [mathematical expression not reproducible], the functional [I.sub.[lambda]] has at least three distinct critical points that are weak solutions of system (1.1).

Now, we point out the case when F does not depend on x G Q, we consider the following system

(3.10) [mathematical expression not reproducible],

we have the following result.

Corollary 3.4. Let F : [R.sup.2] [right arrow] R be a nonnegative and [C.sup.1]- function satisfying (H) and assume that

[mathematical expression not reproducible].

Then, there is [lambda]* > 0 such that, for each [lambda] [member of]]0, [lambda]*[, the problem (3.10) admits at least one non-zero weak solution.

Proof. Fix

[mathematical expression not reproducible]

where the constants [a.sub.1], [a.sub.2], [c.sub.1,1] and [c.sub.r,s] are given by condition (H) and (2.6).

By using (2.8) and (2.9) and taking into account that

[mathematical expression not reproducible]

we obtain that for each [lambda] [member of]]0, [lambda]*[ there exists [bar.h] > 0 such that F(t, t)/[t.sup.q] > [2.sup.N][sigma]/[t.sup.q] for each [absolute value of (t)] < [bar.h].

Now, consider 0 < [delta] < min{[bar.h], [(q/2[kappa][sigma].sup.1/q] we have

F([delta], [delta])/[[delta].sup.p] + [[delta].sup.q] > [2.sup.N][sigma]/q[lambda] > [2.sup.N][sigma]/q[lambda]*

[[delta].sup.p] + [[delta].sup.q] < q/[kappa][sigma].

Then, by choosing [gamma] = 1 all assumptions of Theorem 3.1 are satisfied and the proof is complete.

Corollary 3.5. Let F : [R.sup.2] [right arrow] R be a nonnegative and [C.sup.1]- function satisfying (H), (AR) and assume that

[mathematical expression not reproducible].

Then, there is [lambda]* > 0 such that, for each [lambda] [member of]]0, [lambda]*[, the problem (3.10) admits at least two non-zero weak solutions.

Proof. Fix

[mathematical expression not reproducible]

where the constants [a.sub.1], [a.sub.2], [c.sub.1,1] and [c.sub.r,s] are given by condition (H) and (2.6).

The conclusion follows arguing as in the proof of Corollary 3.4 taking into account Theorem 3.2.

Now, we present some examples that illustrate our results.

Example 3.6. Let [OMEGA] be an open ball of radius one in [R.sup.6].

Consider the function F : [R.sup.2] [right arrow] R defined by

F ([t.sub.1], [t.sub.2]) = log(1 + [t.sup.2.sub.1] + [t.sup.2.sub.2]).

We observe that

[mathematical expression not reproducible]

then, choosing q = 3, p = 4, s = r = 2 [a.sub.1] = 0 and [a.sub.2] = 2, the condition (H) holds.

We observe

[mathematical expression not reproducible].

Then by using Corollary 3.4, put

[lambda]* = 0, 46

[for all][lambda] [member of]]0, [lambda]*[ the following system

[mathematical expression not reproducible]

admits at least one non-zero weak solution in X = [W.sup.1,4.sub.0]([OMEGA]) x [W.sup.1,3.sub.0]([OMEGA]).

Example 3.7. Let [OMEGA] be an open ball of radius one in [R.sup.6].

Consider the function F : [R.sup.2] [right arrow] R defined by

[mathematical expression not reproducible].

We observe that

[mathematical expression not reproducible]

then, choosing p = q = 3, r = s = 4, [a.sub.1] = 3 and [a.sub.2] = 6, the condition (H) holds. Moreover, choose [mu] = 4 and R = 1 we have

[mathematical expression not reproducible]

for every ([t.sub.1], [t.sub.2]) [member of] [R.sup.2] with [absolute value of (([t.sub.1], [t.sub.2]))] > 1. We observe

[mathematical expression not reproducible].

Then by using Corollary 3.5, put [lambda]* = 0.061, [for all][lambda] [member of]]0, [lambda]*[ the following system

[mathematical expression not reproducible]

admits at least two non-zero weak solutions in X = [W.sup.1,3.sub.0]([OMEGA]) x [W.sup.1,3.sub.0]([OMEGA]).

Acknowledgments. The authors are members of the "Gruppo Nazionale per l'Analisi Matematica, la Probabilita e le loro Applicazioni (GNAMPA)" of the "Istituto Nazionale di Alta Matematica (INdAM)".

Received September 13, 2017

REFERENCES

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DIEGO AVERNA (a), GABRIELE BONANNO (b), AND ELISABETTA TORNATORE (c)

(a) Department of Mathematics and Computer Science, University of Palermo, Via Archirafi, 90123--Palermo, Italy diego.averna@unipa.it

(b) Department of Engineering, University of Messina, c.da Di Dio Sant'Agata, 98166--Messina, Italy bonanno@unime.it

(c) Department of Mathematics and Computer Science, University of Palermo, Via Archirafi, 90123--Palermo, Italy elisa.tornatore@unipa.it

Dedicated with great esteem to Professor R. P. Agarwal
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Author:Averna, Diego; Bonanno, Gabriele; Tornatore, And Elisabetta
Publication:Dynamic Systems and Applications
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Date:Mar 1, 2017
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