Dirichlet and Neumann Problems Related to Nonlinear Elliptic Systems: Solvability, Multiple Solutions, Solutions with Positive Components
Luisa Toscano1and Speranza Toscano1
Academic Editor: D. O'Regan
Received01 Feb 2012
Accepted02 Apr 2012
Published15 Aug 2012
Abstract
We study the solvability of Dirichlet and Neumann problems for different classes of nonlinear elliptic systems depending on parameters and with nonmonotone operators, using existence theorems related to a general system of variational equations in a reflexive Banach space. We also point out some regularity properties and the sign of the found solutions components. We often prove the existence of at least two different solutions with positive components.
1. Introduction
In this paper, we present some significant applications of the results got in [1] to Dirichlet problems (Section 2) of the type:
and to Neumann problems (Section 3) of the type:
where are real parameters, is a bounded connected open set of with regular boundary , and is the outward orthogonal unitary vector to .
The study deals with the solvability of the problems, the existence of multiple solutions with all the components not identically equal to zero and, in the homogeneous case, the existence of solutions with positive components, bounded and locally HΓΆlderian with their first derivatives. It is suitable to recall the problem studied in [1] with some notations and hypotheses.
Letwith and; let us consider the following problem.
Problem (). Find such that
Obviously Problem (P) means to find the critical points of the Euler functional:
where .
Let us set
About Problem (P), using Lagrange multipliers and the βfibering method,β different existence theorems have been proved in [1]. They base on one of the following hypotheses: (;
(;
( is not empty and bounded in W.
Remark 1.1. In this paper, we use some existence theorems ([1], Theorems 2.1, 2.2, 3.1, and 3.2), in which as , in relation to a set , we suppose for each with , there exist and the real functions such that and , and as , for all , for all , for some . The condition () assures that for the solutions of Problem (P), found with the method used in the recalled theorems, we have if .
Let be an open, bounded, connected and set with . Let the Lebesgue measure on otherwise.
Let us assume
Moreover we consider the functionals (as in ()) such that
Let us use the notation instead of .
As let , respectively, the first eigenvalue and the first eigenfunction of the problem:
Let us remember that [4] with in ;; is simple, that is, each eigenfunction of (2.3) related to is of the type with ; is isolate, that is, there exists such that is the only eigenvalue of (2.3) belonging to .
Remark 2.1. About the results related to problem (2.3), it is sufficient to suppose and as . This holds also for the results of this section if we limit to consider only the parameters nonnegative.
Let us start by presenting some sufficient conditions such that , and hold.
Using the variational characterization of it is easy to verify the following proposition.
Proposition 2.2. If , then () holds. Consequently, () holds when .
When ββfor some , it is possible to fulfil () with an additional condition on . Let . For any let
and let us suppose There exists .
Proposition 2.3. Let () holds with . Let . If we fix the parameters set with , then there exists such that () also holds for any
Proof. Arguing by contradiction, for any there exist and such that
Set , we have
moreover, since , there exists such that (within a subsequence)
Taking into account that is weakly continuous in , from (2.6) as we get
Since
from (2.9), we deduce that
Let us add that for some , since if we have the contradiction . Then , and consequently from (). This last inequality contradicts (2.8).
In the same way the following propositions can be proved.
Proposition 2.4. Let () holds with . Let . Then, there exists such that (i14) also holds for any .
Let us pass to () and suppose() there exist and such thatand for any .
Proposition 2.5. If (i22) holds with, then
Moreover, if we fix the parameters set with , then there exists such that
Proof. Let us prove (2.12). Let with if , then . Let , we have
Let us prove (2.13). Arguing by contradiction, for any there exist with and , where if , such that
Relation (2.15) implies that there exists strictly increasing such that
Let , we have
Then, as we get
From (2.18), we get that . Then since () inequality holds, which contradicts (2.19).
Proposition 2.6. If () holds with , then
The proof as in Proposition 2.5.
Remark 2.7. The applications we now show, except the first one, deal with systems with equations. We consider the functionals with , and we suppose .
Application 2.8. Let . Let us consider the problem
where
Evidently
Let us advance the conditions:
Let us note that (Propositions 2.2, 2.4, and 2.6)
Proposition 2.9 (see [1], Theorems 2.1, 2.2, 4.1, and 4.2; Remarks 2.1, 2.3, 4.1, and 4.4; Proposition A.3; [5, 6]). Under assumptions (2.22) we have:(i)When (2.24) holds, with [resp. (2.24) and (2.25) hold, with ] problem (2.21) has at least two weak solutions and), and it results in ;(ii)When (2.25) holds, with problem (2.21) has at least two weak solutions ), and it results in . Consequently, when (2.24) and (2.25) hold, with problem (2.21) has at least four different weak solutions.
Remark 2.10. Our results include the ones of DrΓ‘bek and Pohozaev [2] when .
Application 2.11. Let us consider the system:
where
System (2.27) is included among Problem (P) with:
Let us advance the conditions (compatible):
there exist and a constant such that and
Then (Propositions 2.2, 2.3, and 2.5)
and set
Taking into account that and , from ([1], Theorem 2.1, Remark 2.1, and Theorem 4.1) we get the following proposition.
Proposition 2.12. Under assumptions (2.28) we have:(i)When (2.31) holds, ((2.30) and (2.31) hold resp.), choosing as in (2.32) (resp. (2.33)) system (2.27) has at least two weak solutions and with as ; (ii)When (2.30) holds, choosing as in (2.34) system (2.27) has at least two weak solutions and . Consequently, when (2.30) and (2.31) hold, with and system (2.27) has at least four different weak solutions.
The following proposition is obvious.
Proposition 2.13. The following relations hold:
Proposition 2.14. If , then as :
Proof. It is easy to prove that
where . Then (Proposition A.3) and consequently [5] . Let us note that is a weak supersolution to the equation:
Then, since (2.35), it must be [6] .
Let us continue the analysis of system (2.27) under the condition:
then
Hence (Proposition 2.5) if and :
Proposition 2.15. Under assumptions (2.28) and (2.39), choosing as in (2.41) system (2.27) has at least two weak solutions and with as .
Proof. Thanks to ([1], Theorem 4.1), there exists such that
where . Reasoning by contradiction, let, for example, . Since and from (2.39) , setting we have
then . This implies that ([1], see the proof of Theorem 4.1) is a weak solution of system (2.27). Then from which too as . Condition (2.39) holds in particular when
Proposition 2.16. Replacing in Proposition 2.15 (2.39) with (2.44), it is right to say that and as . Consequently, if
Proof. Set , as in Proposition 2.15 is a weak solution to system (2.27). Let us add that since (2.44), there exists (Proposition 2.6) such that
Then the existence of is assured also choosing as in (2.46), and the conclusions of Proposition 2.16 hold.
Application 2.17. Let us set
where
Let us consider the system:
We advance the conditions
Therefore,
Then ([1], Theorems 2.1 and 4.1, and Remarks 2.1 and 4.1).
Proposition 2.18. Under assumption (2.48), we have:(i)When (2.50) holds, ((2.50) and (2.51) hold resp.), if system (2.49) has at least two weak solutions and with as ;(ii)When (2.51) holdssystem (2.49) has at least two weak solutions and with as . Consequently, when (2.50) and (2.51) hold, with system (2.49) has at least four different weak solutions.
In order to establish some properties of and it is useful to recall that ([1], Theorems 2.1 and 4.1)
Proposition 2.19. When , we have
besides
Proof. The relation comes from Proposition A.3. Then [5] . About (2.56), it is sufficiently (Remark 1.1) to prove that
Let with . Since
let (Proposition A.1) with such that
where is the characteristic function of . Set such that
with it results in
Proposition 2.20. When , we have
Proof. We can get (2.62) from Proposition A.3 and [5]. About (2.63), it is sufficiently [6] to prove that as . Reasoning by contradiction, let, for example, . We note that
Let us suppose and set . Then
Set such that and taking into account (2.54), we get the contradiction:
Proposition 2.21. When , we allow that as :
Proof. The assumption implies that
Let, for example, and . Set and as, with , we have
If , set as , with, it results in
This method let us to find . Then, if for some , with as in (2.68) we have from (2.53) (resp. (2.54)) . Consequently ([1], see the proof of Theorem 2.1 (resp. Theorem 4.1)) is a weak solution of system (2.49). Therefore [6] as .
Application 2.22. Let us assume , and as in Application 2.17,
where
Let us consider the system:
under almost one of the conditions:
By using some results ([1], Theorems 2.2 and 4.2, and Remarks 2.3 and 4.4), we can advance a proposition similar to Proposition 2.18 replacing in particular with and with. Thanks to Proposition A.3 and a result of [5], for the solutions and to system (2.73), we have
We continue to analyze the properties of and . To this aim we recall that ([1], Theorems 2.2 and 4.2), set for each , we have:
Besides with , it results in
Proposition 2.23. When as , then
Proof. It is sufficiently (Remark 1.1) to prove that
Let with . As in Proposition 2.19, it is possible to find such that with , it results in
Proposition 2.24. When as , then
Proof. It is sufficiently [6] to prove that . Reasoning by contradiction, let, for example, and such that
Since
there exist an open ball of with centre included in and a unique functional belongs to such that
Then, the functional
belongs to , and we have
Then, for (2.78)
Now, let us remark that with , it results in
Then, since
we have. Consequently,
from which we get the contradiction:
Proposition 2.25. When , we allow that
Proof. We reason as in Proposition 2.21, taking into account (2.77) and (2.78) ([1], see proofs of Theorems 2.2 and 4.2).
Application 2.26. Let for each :
where
Let us consider the system:
Let us introduce the conditions:
Then (Propositions 2.2, 2.3 and 2.5)
Since ([1], Theorems 2.2 and 4.2; Remarks 2.3 and 4.4), we get the following proposition.
Proposition 2.27. Under assumption (2.95) we have:(i)When (2.97) holds ((2.97) and (2.98) hold, resp.), set as in (2.99) (resp. (2.100)) system (2.96) has at least two weak solutions and with as ((ii)When (2.98) holds, set as in (2.101) system (2.96) has at least two weak solutions and with as . Consequently, when (2.97) and (2.98) hold, with and system (2.96) has at least four different weak solutions.
We remark that (Proposition A.3, [5]) as :
Moreover, since is a weak supersolution of the equation:
where , we have [6]
Proposition 2.28. It results in
Proof. Since (2.104), we must show that
About (2.106), it is sufficient (Remark 1.1) to prove that
Let with . Let be a compact set such that
From Proposition A.1, there exists , with , such that
Then, with , we have
Let us prove (2.107). We recall that ([1], Theorem 4.2):
where as in Application 2.22. Reasoning by contradiction, let . Then, for some and consequently from (2.104) . Let , with , such that . Let us consider the function:
Since
we have
We note that . In fact, if and, being , is bounded (else (within a subsequence) . Then (within a subsequence) with , from which . We add that belongs to , and its derivative has the form:
Hence, set , it results in
As in Proposition 2.24, we introduce the open ball with centre included in and the functionals and belonging to . Chosen such that , we have
and consequently . Then, taking into account (2.112), with such that , we get the contradiction:
Proposition 2.29. If , then
Proof. In fact,
Application 2.30. Let for each :
where
Let as . Let . Set as and , let us consider the system:
under at least one of the following conditions
Evidently, about the validity of () we choose as in Application 2.26.
Proposition 2.31 (see [1], Theorem 3.2). Under assumptions (2.123), (2.125) ((2.125) and (2.126), resp.), if and is sufficiently small, for as in (2.99) (resp. (2.100)) system (2.124) has at least one weak solution , .
Let us note that
Application 2.32. Let , and for each :
under one of the following assumptions:
Set as in Application 2.30. Let us consider the system:
Let us verify that
Let with, for example, . Let and such that . Let us suppose and set . Then,
Proposition 2.33. Under assumption (2.129) (resp. (2.130)), system (2.131) with has at least two weak solutions and , and we have as :
Consequently,
Proof. The statement is due to ([1], Theorem 2.2, Remark 2.3), [5], Proposition A.3, [6].
Proposition 2.34 (see [1], Theorems 3.1, 3.2). Under assumption (2.129) (resp. (2.130)), system (2.131) with and sufficiently small has at least two different weak solutions ), and we have even if .
Remark 2.35. If (within a set with measure equal to zero), with the same reasoning used about (2.132), we get that
hence, even if .
3. Neumann Problems
Let be an open, bounded, and connected set. Let and as in Section 2, the measure on the outward unit normal to if if . Let us assume
We note that for each we set where is the trace operator from into . Morever we consider the functionals (as in ()) such that
It is easy to verify the following.
Proposition 3.1. Let as . Then,
Let us set and for each
Let us introduce the conditions:()there exists ;()there exist and and .
Proposition 3.2. Let () holds with . Let . Let as . Then with and as : () holds if as .
Proof. Reasoning by contradiction, for each there exist , with , and such that
then, set , we have
Since , there exists such that (within a subsequence)
Consequently, from (3.6), passing to limit as , we get
from which , and then the contradiction .
Proposition 3.4. Let () holds with . Let as . Then,
Moreover, if as , we have with and as holds if and for some .
Proof. The first statement is evident. Let us prove the second one. Reasoning by contradiction, for each there exist , with and for some , and a sequence such that
Let be a strictly increasing sequence such that as . Let . Then, and
moreover, there exists such that (within a subsequence)
Consequently,
then , and the contradiction .
Proposition 3.5. Let () holds with . Let as . Then,
Remark 3.6. It is suitable to make some clarifications.(i)The assumption ββ (see Propositions 3.1, 3.2, and 3.4) can be replaced by β do not change sign.β In this case we can choose and such that and .(ii)The assumption ββ (see Propositions 3.4 and 3.5) can be replaced by ββ. In this case, we can choose and such that and + for some , with instead of .(iii)When for each do not change sign, then the conclusion of the Proposition 3.2 [resp. Proposition 3.3] holds even if and as (resp. as ). In order to simplify the presentation of the applications, we suppose in the next and, while the additional assumptions on and the assumptions on (the same of Propositions 3.1, 3.2, 3.4, and 3.5) will be pointed out case by case.
Application 3.7. Let for each :
where
Let us consider the system:
Let us introduce the conditions:
Evidently (3.20) . Moreover (3.21) (Proposition A.2). Hence (Propositions 3.3 and 3.5)
Proposition 3.8 (see ([1], Theorems 2.1 and 4.1; Remarks 2.1 and 4.1); Proposition A.4; [5, 6]). Under assumption (3.18), we have:(i)When (3.20) and (3.21) hold, with as in (3.23) system (3.19) has at least two weak solutions and , and it results in
(ii)When (3.20) and (3.22) hold, with as in (3.24) system (3.19) has at least two weak solutions and , and it results in
Consequently, when (3.20)β(3.22) hold, with as in (3.24) and instead of system (3.19) has at least four different weak solutions.
Proposition 3.9. If , then as.
Proof. It is sufficient (Remark 1.1) to verify that
Let . Let, for example, . Since , there exists such that
Let a compact set and an open set such that
Since Propositions A.1 and A.2, there exist a compact set , with , and such that
where is the characteristic function of . Let us choose such that
and we set . Then,
Proposition 3.10. If
then .
Proof. We recall that ([1], Theorem 4.1)
Reasoning by contradiction let, for example, . As and for all , we have and since (3.33) . Then, it is possible to choose such that . Let us add that there exist and such that . Let now and . Since is necessarily bounded. Then (within a subsequence) . Consequently, from the inequality:
as and from (3.34), we get the contradiction:
Remark 3.11. Let us note that the conditions (3.20), (3.21), and (3.33) are compatible.
Application 3.12. Let for each :
where
Let us consider the system:
Pointing out that , we advance the conditions
Taking into account that
we have (Propositions 3.2 and 3.4)
Proposition 3.13 . (see ([1], Theorems 2.1 and 4.1; Remark 2.1); Proposition A.4; [5, 6]). Under assumption (3.39), we have(i)When (3.41) and (3.43) hold, with as in (3.45) system (3.40) has at least one weak solution (), and it results in
(ii)When (3.41)β(3.43) hold, with as in (3.46) system (3.40) has at least one weak solution , and it results in as . Consequently, when (3.41)β(3.43) hold, with as in (3.46) and instead of system (3.40) has at least two different weak solutions.
About the properties of and expressed by Proposition 3.13, it is necessary to remark that if (3.40), then as . In fact,
Application 3.14. Let and for any :
where
Let us consider the system:
Let us introduce the conditions:
we have (Propositions 3.3 and 3.5)
Proposition 3.15 (see ([1], Theorems 2.2 and 4.2; Remarks 2.3 and 4.4); Proposition A.4; [5]). Under assumption (3.50), we have(i)When (3.52) and (3.53) hold, with as in (3.55) system (3.51) has at least two weak solutions and , and it results in
(ii)When (3.53) and (3.54) hold, with as in (3.56) system (3.51) has at least two weak solutions and , and it results in
Consequently, when (3.52)β(3.54) hold, with as in (3.56), and instead of system (3.51) has at least four different weak solutions.
Proposition 3.16. Under the assumption and as , we have(i)if for some , then as ;(ii)if for some , thenas.
Proof. First of all is a weak supersolution to the equation:
Also, has a similar property. Then [6] it is sufficient to verify that
About (3.60), let us prove (Remark 1.1) that
Let . Let, for example, . Let
Since Propositions A.1 and A.2, there exists , with and , such that
Then with , we have
Passing to (3.61), let us introduce the function , and let us remember that ([1], Theorem 4.2)
where . Reasoning by contradiction, let us set, for example, and set . Since
as in Proposition 2.24, we get the contradiction:
Application 3.17. Let and set for each :
where
Let us consider the system:
Let us make the assumptions:
About Neumannβs problem (3.71), we have an existence result similar to the one of Proposition 3.15 related to system (3.51). About the positive sign of the components of the weak solutions and to system (3.71), as in Proposition 3.16, we show.Proposition 3.18. Under the assumption as and as with , we have(i)if either or for all for some , then as ;(ii)if for all for some , then as .The following remark deals also with Application 3.14.Remark 3.19. Making in (3.50) (resp. (3.70)) the change
system (3.51) (resp. (3.71)) has at least the two weak solutions and ([1], Theorem 4.2; Remark 4.4). The components of keep the properties that Propositions 3.15 and 3.16 (Proposition 3.15 and Proposition 3.18 resp.) underline.
Application 3.20. Let for each :
where
Let us consider the system:
Let us introduce the conditions:
We have (Propositions 3.2 and 3.4)
Proposition 3.21 (see ([1], Theorems 2.2 and 4.2; Remarks 2.3 and 4.4); Proposition A.4; [5, 6]). Under assumption (3.75), we have(i)When (3.77)β(3.79) hold, with as in (3.81), system (3.76) has at least two weak solutions and ), and it results in , . If , then(ii)When (3.78)β(3.80) hold, with as in (3.82), system (3.76) has at least two weak solutions and ), and it results in , . If , then
Consequently, when (3.77)β(3.80) hold, with as in (3.82), and instead of system (3.76) has at least four different weak solutions. Obviously, and if .
The following proposition gives a sufficient condition to
Proposition 3.22. Let . If and as , then (3.85) and (3.86) hold.
Proof. Since
using Propositions A.1 and A.2, we can verify that
from which (Remark 1.1) we get (3.85). Let us prove (3.86). Reasoning by contradiction, let us set, for example, . If , we have
Then as in Proposition 3.16, we get a contradiction.
Remark 3.23. Making in (3.75) the change:
system (3.76) has at least the two weak solutions and ([1], Theorem 4.2; Remark 4.4). The components of , all bounded, are locally HΓΆlderian with their first derivatives. If as , then (3.86) holds.
Application 3.24. Let for each :
where
Let us consider the system:
We advance the condition:
and we note that (Proposition 3.1)
Proposition 3.25. Under conditions (3.92) and (3.94), with as in (3.95), system (3.93) has at least two weak solutions and ), and it results in
Proof. We recall that ([1], Section 2), set , we have
We introduce the functional which is in . We still remember that ([1], Theorem 2.3; Remark 2.5)
The property is due to Proposition A.4. Let us verify that as . Reasoning by contradiction, let us set, for example, and . As , we have
Then, since (there exists such that (, from which the contradiction:
Application 3.26. Let for each :
where
Let as , where and (dual space of ). Let . Let us consider the system:
Let us introduce the conditions:
and let us note that (Proposition 3.3)
Proposition 3.27. Under assumptions (3.102) and (3.104), if and is sufficiently small, then with as in (3.105), system (3.103) has at least one weak solution . When , it results in
Proof. The existence of is due to ([1], Theorem 3.2). About (3.106), it is sufficiently (Remark 1.1) to verify that
Let with, for example, . Let . Let be a compact set having positive measure such that
Proposition A.1 lets us choose satisfying the following conditions:
Then with , we have
Now we replace conditions (3.104) with the following:
Proposition 3.28. Under assumptions (3.102) and (3.111), if and is sufficiently small, then with and as system (3.103) has at least two different weak solution and (. When , it results in
Proof. The existence of and is due to ([1], Theorems 3.1, 3.2, and 3.3; Remark 3.1). Relation (3.112) is proved as in Proposition 3.27.
Appendix
In this appendix, we present some results used previously. The first one is trivial. The second one is easy to prove. It is possible to show the third one and the fourth one with the technique developed by Drabek in ([7, Lemma 3.2]). The symbols are the same introduced in Section 3.
Proposition A.1. Let be an open set of . Let be a compact set with . If is an open set such that , then there exists a family of functions such that
where is the characteristic function of .
Proposition A.2. Let be an open, bounded, connected and set. Let be an open neighborhood of . If is a subset of with , then there exist a compact set with and a family of functions such that
where is the characteristic function of .
Let be an open, bounded, connected and set. Let as be a Carathèodory function into defined for for and for such that
where , .
Proposition A.3. Let . If there exist and with such that
then .
Proposition A.4. Let . If there exist with , with such that
then .
Remark A.5. If , we can suppose .
Acknowledgment
This paper is supported by the Second University of Naples.
References
L. Toscano and S. Toscano, βOn the solvability of a class of general systems of variational equations with nonmonotone operators,β Journal of Interdisciplinary Mathematics, vol. 14, no. 2, pp. 123β147, 2011.
P. Drábek and S. I. Pohozaev, βPositive solutions for the -Laplacian: application of the fibering method,β Proceedings of the Royal Society of Edinburgh. Section A. Mathematics, vol. 127, no. 4, pp. 703β726, 1997.
S. I. Pohozaev and L. Véron, βMultiple positive solutions of some quasilinear Neumann problems,β Applicable Analysis, vol. 74, no. 3-4, pp. 363β390, 2000.
A. Anane, βSimplicité et isolation de la première valeur propre du -laplacien avec poids,β Comptes Rendus des Séances de l'Académie des Sciences. Série I. Mathématique, vol. 305, no. 16, pp. 725β728, 1987.
P. Tolksdorf, βRegularity for a more general class of quasilinear elliptic equations,β Journal of Differential Equations, vol. 51, no. 1, pp. 126β150, 1984.
N. S. Trudinger, βOn Harnack type inequalities and their application to quasilinear elliptic equations,β Communications on Pure and Applied Mathematics, vol. 20, pp. 721β747, 1967.