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Three solutions for equations involving nonhomogeneous operators of p-Laplace type in
Journal of Inequalities and Applications volume 2014, Article number: 427 (2014)
In this paper, we are concerned with the following elliptic equation
where the function is of type and satisfies a Carathéodory condition. We establish the existence of at least three weak solutions for the problem above which is based on an abstract three critical points theory due to Ricceri. Moreover, we determine precisely the intervals of λ’s for which the given problem possesses either only the trivial solution or at least two nontrivial solutions.
MSC:35D30, 35D50, 35J15, 35J60, 35J62.
In this paper, we establish the existence of at least three solutions for equations of the p-Laplace type
(P λ ) in ,
where the function is of type and satisfies a Carathéodory condition. A Ricceri-type three critical points theorem has been extensively studied by many researchers (see [1–5] and the references therein), but the results on the localization of the interval for the existence of three solutions are rare. The authors in [3, 4] investigated the existence of multiple solutions for quasilinear nonhomogeneous problems with Dirichlet boundary conditions by applying an abstract three critical points theorem which is the extension of the famous result of Ricceri [6, 7].
Ricceri’s theorems in [6–8] gave no further information on the size and location of an interval of values for the existence of at least three critical points. However, further information concerning these points was given in . Also the authors in  investigated the localization of the interval for the existence of three solutions for the Dirichlet problem involving the p-Laplace type operators which was motivated by the work of Arcoya and Carmona . It is well known that the first eigenvalue of the p-Laplacian plays a decisive role in obtaining these results in [3, 9]. Hence, by using the positivity of the principal eigenvalue of the p-Laplacian in , which was given in [10–12], we localize a three critical points interval for the problem above as in [3, 9]. Especially, the main aim of this paper is to determine precisely the intervals of λ’s for which problem (P λ ) admits only the trivial solution and for which problem (P λ ) has at least two nontrivial solutions, following the basic idea in . To do this, we consider some of the basic properties for the integral operator corresponding to problem (P λ ) in the setting of weighted Sobolev spaces.
Let be the Euclidean scalar product on or the usual pairing of and X, where denotes the dual space of X. Let and set . Let ω be a weight function defined by
a belongs to and there is a positive constant such that
Let X be the completion of with respect to the norm
From Hardy’s inequality and assumption (A), it follows that
which implies that on X, the norm is equivalent to the other norm given by
Note that there exist positive constants and such that
for all . The following Sobolev inequality will be used in the sequel:
for some positive constant (see ).
This paper is organized as follows. We first present some properties of the corresponding integral operators. Then we give and prove our main results in Theorem 2.12 and Theorem 2.14.
2 Main results
Definition 2.1 We say that is a weak solution of problem (P λ ) if
for all .
We assume that is a continuous derivative with respect to v of the mapping , , that is, . Suppose that φ and satisfy the following assumptions:
(J1) The following equalities
hold for all and for all .
(J2) satisfies the following conditions: is measurable for all and is continuous for almost all .
(J3) There are a function and a positive constant d such that
for almost all and for all .
(J4) is strictly convex in for all .
(J5) The following relations
hold for all and , where is a positive constant.
Let us define the functional by
for any . Under assumptions (J1)-(J3) and (J5), it follows from [, Lemma 3.2] that the functional Φ is well defined on X, and its Fréchet derivative is given by
for any .
Next, taking inspiration from the argument given in , we will show that the operator is a mapping of type which plays an important role in obtaining our main results.
Lemma 2.2 Assume that (A) and (J1)-(J5) hold. Then the functional is convex and weakly lower semicontinuous on X. Moreover, the operator is a mapping of type , i.e., if in X as and , then in X as .
Proof From assumption (J4), the operator Φ is strictly convex and thus is strictly monotone (see [, Proposition 25.10]), namely
for . The convexity of also implies that Φ is weakly lower semicontinuous in X, that is, implies
Now we claim that the operator is a mapping of type . Let be a sequence in X such that in X as and
From relations (2.2) and (2.4), we have
that is, the sequence converges to 0 in as . Hence the sequence has a subsequence such that
for almost all . Thus there exists such that
for almost all . It follows from conditions (A), (J3) and (J5) that
for almost all . By using Young’s inequality, we deduce that
for almost all . These together with relation (2.6) imply that
for almost all . Since and are positive constants, the above inequality implies that the sequence is bounded, and so is bounded in for almost all . By passing to a subsequence, we can suppose that as for some and for almost all . Then we have as for almost all . It follows from (2.5) that
for almost all . Since φ is strictly monotone by (J4), this means , that is, as for almost all . The arguments above hold for any subsequence of the sequence . Hence we obtain as for almost all . Then it implies that
Since the functional Φ is convex, it is obvious that
and so we get . Therefore, it is derived from (2.3) that
Consider the sequence in defined pointwise by
Then for all by (J1) and (J4). Since is continuous for almost all , we obtain that as for almost all . Therefore, by the Fatou lemma and relation (2.8), we have
in other words, by (J5). Since by (1.1), in conclusion, , as claimed. □
Corollary 2.3 Assume that (A) and (J1)-(J5) hold. Then the operator is bounded homeomorphism onto .
Proof It is immediate that the operator is strictly monotone, coercive, and hemicontinuous. Hence the Browder-Minty theorem implies that the inverse operator exists and is bounded; see Theorem 26.A in . Since the operator is a mapping of type by Lemma 2.2, it is easy to prove that the inverse operator is continuous and is omitted here. □
Before dealing with our main results in this section, we need the following assumptions for f. Let us put .
(F1) satisfies the Carathéodory condition in the sense that is measurable for all and is continuous for almost all .
(F2) f satisfies the following growth condition: for all ,
where , such that , with .
(F3) There exist a real number and a positive constant so small that
and for almost all with and for all , where .
Then we define the functionals by
for any . It is easy to check that and its Fréchet derivative is
for any .
Lemma 2.4 Assume that (A), and (F1)-(F2) hold. Then Ψ and are weakly-strongly continuous on X.
Proof The analogous arguments as in Lemma 4.4 of  imply that functionals Ψ and are weakly-strongly continuous on X. □
Lemma 2.5 Assume that (A), (J1)-(J3), (J5), and (F1)-(F2) hold. Then we have
for all .
Proof If is large enough and , then it follows from (J5), (F2) and Hölder’s inequality that
for some positive constants and . Since , we get that
for all . □
Now we will localize the interval for which problem (P λ ) has at least three solutions as the application of three critical points theorems given in  and , respectively. To do this, we consider the following eigenvalue problem:
Assume that (A) and (J1)-(J5) hold. Moreover, suppose that
for all such that , and , where
Denote the quantity
Then the eigenvalue problem (E) has a pair of a principal eigenvalue and an eigenfunction with and . Moreover, is simple and decays uniformly as .
Definition 2.7 Let X be a real Banach space. We call that is the class of all functionals satisfying the following property: if is a sequence such that in X as and , then has a subsequence and in X as .
The following lemma is three critical points theory which was introduced by Ricceri .
Lemma 2.8 ()
Let X be a separable and reflexive real Banach space; let be a coercive, sequentially weakly lower semicontinuous -functional, belonging to , bounded on each bounded subset of X and whose derivative admits a continuous inverse on . Let be a -functional with compact derivative. Assume that Φ has a strict local minimum with . Finally, set
Assume that . Then, for each compact interval (with the conventions , ), there exists with the following property: for every , the equation has at least three solutions whose norms are less than R.
In order to apply the above lemma to (P λ ), we have to show that the functional Φ belongs to . To do this, we need the following additional assumption:
(J6) The following relation holds for all :
To consider some examples that satisfy hypothesis (J6), we observe the following argument which is given in .
Remark 2.9 If is a continuous, strictly increasing function for with and
then the following estimate
holds for all .
Example 2.10 Let us consider
for all . If , then we obtain a Clarkson-type inequality for the function , i.e.,
for all . Therefore assumption (J6) holds.
Example 2.11 Let . Suppose that and there exists a positive constant such that for almost all . Let us consider
for all . Set for . Then it is easy to calculate that ϕ satisfies all the assumptions of Remark 2.9 and therefore condition (J6) is verified.
Combining with Proposition 2.6 and Lemma 2.8, we derive the following consequence.
Theorem 2.12 Assume that conditions (A), (J1)-(J6), (F1)-(F3) and (M) hold. Moreover, suppose that
(F4) for uniformly.
(F5) for uniformly.
(F6) For all compact , there exists a function such that
for almost all and for all .
Assume also that the condition is removed and replaced by the more general condition in assumption (F2). Set . Then, for each compact interval , there exists with the following property: for every , problem (P λ ) has at least three solutions whose norms are less than R.
Proof It is obvious that the functional Φ is coercive, sequentially weakly lower semicontinuous of class , bounded on each subset of X, and whose derivative is a homeomorphism by Corollary 2.3. Moreover, the functional has a compact derivative due to Lemma 2.4.
First of all, let us claim that the functional Φ belongs to . It follows from the same argument as in the proof of Theorem 3.1 in . For the sake of convenience, we give the proof. Let be a sequence in X that converges weakly to u in X as and . By Lemma 2.2, Φ is sequentially weakly lower semicontinuous, namely . Thus there exists a subsequence of , still denoted by , such that . Since as , the sequence also converges weakly to u in X as , and we get
If does not converge to u as n approaches infinity, the sequence also does not converge to 0 as . So we can choose and a subsequence of such that for all . By assumption (J5) and (1.1), we deduce that
for all . From (J6), we know
Thus we deduce that the following relation
holds for all . From (2.11) and (2.12), we have as , a contradiction. Therefore, we conclude that as and so .
Observe now that for every . Then 0 is a strict local (even global) minimum with . By assumptions (F4) and (F6), for every , we get
for almost all and for all , where . It implies that
by Proposition 2.6. Then it follows from (2.13), (2.14) and (J5) that
Hence we have
Since ε is arbitrary, the following inequality holds:
On the other hand, by conditions (F4) and (F5), we have that for every , there exists verifying that
for almost all and for all . From (2.14), (2.16) and (J5), we deduce
for some positive constant . Then it follows that
Hence we obtain
for all , which leads to
Taking now assumption (F3) into account, it follows from (2.15) and (2.17) that
Therefore, all the conditions of Lemma 2.8 are fulfilled and thus the proof is completed. □
In the rest of this section, we determine precisely the intervals of λ’s for which problem (P λ ) possesses either only the trivial solution or at least two nontrivial solutions. To do this, we assume that
(F7) uniformly for almost all .
Then we get that uniformly for almost all by the L’Hôspital’s rule. Let us consider that two functions
for every . Also we consider the following crucial value:
Then the same arguments in  imply that is a positive constant. From this fact, we obtain
The next lemma represents the differentiable form of the Arcoya and Carmona Theorem 3.4 in .
Lemma 2.13 Let Φ and Ψ be two functionals on X such that Φ and Ψ are weakly lower semicontinuous and continuously Gâteaux differentiable in X, and Ψ is nonconstant. Let also have the property, and that is a compact operator. Assume that there exists an interval such that the one parameter family of functionals is coercive in X for all . Let us assume that there exists
then the following properties hold.
The functional admits at least one critical point for every .
If furthermore , then
has at least three critical points for every .
has at least two critical points provided that .
has at least two critical points provided that .
Theorem 2.14 Assume that (A), (J1)-(J5), (F1)-(F3) and (M) hold. Then we have
If , where , then problem (P λ ) has only the trivial solution, where is the principal eigenvalue of problem (E), is a positive constant in (J5), and both of and are positive constants from (1.1).
If furthermore f satisfies condition (F7), then there exists a positive constant with such that problem (P λ ) has at least two nontrivial solutions for all .
Proof By Lemma 2.2, the functional is a sequentially weakly lower semicontinuous -functional and the operator is a mapping of type . It follows from Lemma 2.4 that the functional Ψ is also sequentially weakly lower semicontinuous -functional and the operator is compact. Due to Lemma 2.5, we have
for all and for all .
First we claim the assertion (i). Let be a nontrivial weak solution of problem (P λ ), that is,
for all . If we put , then it follows from (J5) that
Thus if u is a nontrivial weak solution of problem (P λ ), then necessarily , as required.
Next let us prove assertion (ii). Let be from (F3). For , define
Then it is obvious that for all and . From condition (F3),
It follows that the crucial number
is well defined. Let u be in X with . Using (J5) and (2.20), we have
Hence we get . To employ Lemma 2.13, we have to verify assumption (2.21). For all , we have that
for all , and hence
for all . Then it implies that
By assumption (F7), there exists a positive real number such that
for almost all and for all . Indeed, denote
Then there exists such that for almost all and for all with . Let s be fixed with . According to (2.20),
for almost all . Put . Then relation (2.23) holds.
Hence we deduce that
for some positive constant . If and , then we obtain by (J5) that
Since , by using (2.19), we have
and so since . Therefore, we conclude
It means that there exists a negative sequence such that as , so that for all integers n with . By Lemma 2.5, we put . Since is a critical point of , according to the part (a) of (ii) in Lemma 2.13, problem (P λ ) admits at least two nontrivial solutions for all
as claimed. □
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The authors would like to thank the referees for useful comments and remarks.
The authors declare that they have no competing interests.
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
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Choi, E.B., Kim, Y. Three solutions for equations involving nonhomogeneous operators of p-Laplace type in . J Inequal Appl 2014, 427 (2014). https://doi.org/10.1186/1029-242X-2014-427
- weighted Lebesgue-Sobolev spaces
- three critical points theorem
- multiple solutions