Let Ω be a smooth bounded domain in () with boundary condition. In this paper, we are dealing with the following problem:
where λ is a positive parameter, satisfying , with , as usual, , and
and is almost everywhere positive in Ω. In the sequel, X will denote the Sobolev space . Associated to problem (P-+λ), we have the singular functional given by
If for all ,
then is called a weak solution of (P-+λ).
The operator is called the -biharmonic operator of fourth order, where p is a continuous non-constant function. This differential operator is a natural generalization of the p-biharmonic operator , where is a real constant. However, the -biharmonic operator possesses more complicated non-linearity than the p-biharmonic operator, due to the fact that is not homogeneous. This fact implies some difficulties; for example, we can not use the Lagrange multiplier theorem in many problems involving this operator.
The study of this kind of operators occurs in interesting areas such as electrorheological fluids (see ), elastic mechanics (see ), stationary thermo-rheological viscous flows of non-Newtonian fluids, image processing (see ) and the mathematical description of the processes filtration of an ideal barotropic gas through a porous medium (see ).
Problem (P-+λ) is a new variant of -biharmonic equations due to the singular term and the indefinite one. Note that results for -Laplace equations with singular non-linearity are rare. Meanwhile, elliptic and singular elliptic equations involving the -Laplace and the -biharmonic operators can be found in [1, 2, 5, 8, 9, 10, 13, 14, 17, 18, 21, 20, 22, 23].
Recently, Ayoujil and Amrouss  studied the following problem:
In the case when , they proved that the energy functional associated to problem (P) has a nontrivial minimum for any positive λ; see [4, Theorem 3.1]. In the case when and has a subcritical growth, they used Ekeland’s variational principle in order to prove the existence of a continuous family of eigenvalues which lies in a neighborhood of the origin. Finally, when
they showed (see [4, Theorem 3.8]) that for every the energy functional corresponding to (P) has a Mountain Pass-type critical point which is nontrivial and nonnegative, and hence , where Λ is the set of the eigenvalues. The same problem, for , is studied by Ayoujil and Amrouss in . They established the existence of infinitely many eigenvalues for problem (P) by using an argument based on the Ljusternik–Schnirelmann critical point theory. They showed that , and they pointed out that only under special conditions, which are somehow connected with a kind of monotony of the function , one has (this is in contrast with the case when is a constant, where one always has ).
Later, Ge, Zhou and Wu  considered the following problem:
where V is an indefinite weight and λ is a positive real number. They considered different situations concerning the growth rates, and they proved, using the mountain pass lemma and Ekeland’s principle, the existence of a continuous family of eigenvalues. A recent paper concerning this type of problems is .
Inspired by the above-mentioned papers, we study problem (P-+λ), which contains a singular term and indefinite many more general terms than the one studied in . In this new situation, we will show the existence of a weak solution for problem (P-+λ). The paper is organized as follows: In Section 2, we recall some definitions concerning variable exponent Lebesgue spaces, , as well as Sobolev spaces, . Moreover, some properties of these spaces will also be exhibited to be used later. Our main results are stated in Section 3. The proofs of our results will be presented in Section 4 and Section 5.
2 Notations and preliminaries
Let p be a Lipschitz continuous function on . We set and
We recall the following so-called Luxemburg norm on this space defined by the formula
Clearly, when , a positive constant, the space reduces to the classical Lebesgue space , and the norm reduces to the standard norm
For any positive integer k, as in the constant exponent case, let
where is a multi-index, and
Then is a separable and reflexive Banach space equipped with the norm
Furthermore, is the closure of in . Let be the conjugate space of with . Then the Hölder-type inequality
The modular on the space is the map defined by
and it satisfies the following propositions.
Proposition 2.1 (see ).
For all , we have the following assertions:
(resp. , ) if and only if (resp. , ).
if and only if .
Proposition 2.2 (see ).
Let p and q be two measurable functions such that and for a.e. . Let , . Then
Assuming that E and F are Banach spaces, we define the norm on the space as .
Zang and Fu , proved the equivalence of the norms, and they even proved that the norm is equivalent to the norm (see [24, Theorem 4.4]). Let us choose on X the norm defined by . Note that is also a separable and reflexive Banach space and that the modular is defined as by and satisfies the same properties as in Proposition 2.1. Hereafter, let
If and for any , by [4, Theorem 3.2] we deduce that X is continuously and compactly embedded in .
Throughout this paper, the letters , denote positive constants which may change from line to line.
3 Hypotheses and main results
Let us impose the following hypotheses on the non-linearity :
f is a function such that .
There exists with , and a nonnegative function on such that with
There exists a positive function h on Ω such that and
where and are such that for all .
There exists such that
for all and all , where C is a positive constant, and are such that for all we have .
There exists with such that in .
Some remarks regarding the hypotheses are in order.
Under assumptions (f3) and (f4), we have the following assertions.
for any , so the injection of is compact and continuous.
for any , where , so is compact and continuous.
for any , where , so is compact and continuous.
Moreover, under conditions (f5) and (f6), we remark the following.
There exists such that for all .
Due to condition (f5), there exists such that
Conditions (f5) and (f6) assure that for all .
Put . Then the first condition in the remark is satisfied.
Here we state our main results asserted in the following two theorems.
Assume that hypotheses (f1), (f2), (f3) and (f4) are fulfilled. Then for all problem (P) has at least one nontrivial weak solution with negative energy.
Assume that hypotheses (f5) and (f6) are fulfilled. Then for all problem (P) has at least one nontrivial weak solution with negative energy.
4 Proof of Theorem 3.3
The study of the existence of solutions to problem (P) is done by looking for critical points to the functional defined by
Now, we are in a position to show that possesses a nontrivial global minimum point in X.
Under assumptions (f2), (f3) and (f4), the functional is coercive on X.
Since , we infer that as ; in other words, is coercive on X. The proof of Lemma 4.1 is now completed. ∎
Suppose assumptions (f2) and (f3) are fulfilled. Then there exists such that , and for small enough.
Let such that in a subset and in . Using assertions on the functions g and F and assumption (f2), we have
Finally, we point out, using the hypothesis on φ and the definition of the modular on X, that
In fact, if
then , and consequently , which contradicts the choice of φ and gives the proof of Lemma 4.2. ∎
In the sequel, we put . Then we have the following lemma.
Let , ,
with for almost every , and assume that hypothesis (f1), (f2), (f3) and (f4) are fulfilled. Then reaches its global minimizer in X, that is, there exists such that .
Let be a minimizing sequence, that is to say . Suppose is not bounded, so as . Since is coercive, we have
This contradicts the fact that is a minimizing sequence, so is bounded in X, and therefore, up to a subsequence, there exists such that
Since is sequentially weakly lower semi-continuous (see ), we have
On the other hand, by Vital’s theorem (see [16, p. 113]), we can claim that
Indeed, we only need to prove that
there exist such that
Consequently, by the Hölder inequality and Proposition 2.1 one has
Since , we have
Since is bounded, claim (4.3) is valid.
In what follows, we remark, using assumptions (f2) and (f3), that for all there exists such that
Then by the Hölder inequality one has
Besides, if in X, then we have strong convergence in and . So the Lebesgue dominated convergence theorem and Proposition 2.2 enable us to state the following assertion: If
is weakly continuous, then
The proof of Lemma 4.3 is now completed. ∎
Proof of Theorem 3.3.
Now, let us show that the weak limit is a weak solution of problem (P) if is sufficiently large. Firstly, observe that . So, in order to prove that the solution is nontrivial, it suffices to prove that there exists such that
For this purpose, we consider the variational problem with constraints
From above we have
Therefore, the above remarks show that and that problem (P) has a solution for all .
We now argue that problem (P) has a solution for all . Fixing , by the definition of we can take such that has a nontrivial critical point . Since , we obtain that is a sub-solution of problem (P). We now want to construct a super-solution of problem (P) which dominates . For this purpose, we introduce the constrained minimization problem
By using the previous arguments to treat (4.5), follows that the above minimization problem has a solution . Moreover, is also a weak solution of problem (P) for all . With the arguments developed in , we deduce that problem (P) has a solution if .
Now, it remains to show that on . Due to the above arguments, one has
Relation (4.6) implies that
Let ζ be the unique solution of the problem
Relation (4.7) yields
Using the Green formula, we have
On the other hand, for all there exists a unique such that in Ω. Thus, relation (4.8) can be rewritten as
Applying the fundamental lemma of the calculus of variations, we deduce that
Since on , we conclude that on . Thus, is a nontrivial weak solution of problem (P) such that . This completes the proof of Theorem 3.3. ∎
5 Proof of Theorem 3.4
The proof of Theorem 3.4 is organized in several lemmas. Firstly, we show the existence of a local minimum for in a small neighborhood of the origin in X.
Under assumption (f5), the functional is coercive on X.
Since , we infer that as and is coercive on X. This ends the proof of Lemma 5.1. ∎
Under assumptions (f5) and (f6), there exists such that and for small enough.
Let such that in a subset and in . Using assertions on the functions g and F, we have
Since , we have for with
Finally, we point out that . In fact, if , then , and consequently in Ω, which is a contradiction. ∎
In the sequel, put . As a last proposition, we have the following.
Let , ,
with for almost every , and assume that assertions (f5) and (f6) hold. Then reaches its global minimizer in X, that is, there exists such that .
Proof of Theorem 3.4.
Let and . We define by , where . Since is a local minimizer for , one has
Since the measure of the domain of integration tends to zero as , it follows as that
Dividing by ϵ and letting , we get
Since the equality holds if we replace ϕ by , which implies that is a positive solution of problem (P), this completes the proof of Theorem 3.3. ∎
6 An example
In this section, we give an example to illustrate our results.
Let Ω be a smooth bounded domain in (), let p be a Lipschitz continuous function on with and , let and be continuous functions on such that for all , let be a continuous function, let
and let h and be two positive functions such that and . Put
with and for all . Then conditions (I1), (I2) and (I3) are satisfied, so for any problem (P) has a weak solution.
Moreover, if we suppose that for all , then assumptions (f5) and (f6) hold, and consequently, for any , problem (P) has at least one nontrivial weak solution in .
The authors would like to thank the anonymous referees for their suggestions and helpful comments which improved the presentation of the original manuscript.
S. N. Antontsev and S. I. Shmarev, A model porous medium equation with variable exponent of nonlinearity: Existence, uniqueness and localization properties of solutions, Nonlinear Anal. 60 (2005), no. 3, 515–545. CrossrefGoogle Scholar
M. Avci, Existence of weak solutions for a nonlocal problem involving the -Laplace operator, Univer. J. Appl. Math. 3 (2013), 192–197. Google Scholar
A. Ayoujil and A. R. El Amrouss, Continuous spectrum of a fourth order nonhomogeneous elliptic equation with variable exponent, Electron. J. Differential Equations 2011 (2011), Paper No. 24. Google Scholar
X. Fan, J. Shen and D. Zhao, Sobolev embedding theorems for spaces , J. Math. Anal. Appl. 262 (2001), no. 2, 749–760. Google Scholar
A. Ghanmi and K. Saoudi, A multiplicity results for a singular problem involving the fractional p-Laplacian operator, Complex Var. Elliptic Equ. 61 (2016), no. 9, 1199–1216. CrossrefWeb of ScienceGoogle Scholar
S. Heidarkhani, G. A. Afrouzi, S. Moradi, G. Caristi and B. Ge, Existence of one weak solution for -biharmonic equations with Navier boundary conditions, Z. Angew. Math. Phys. 67 (2016), no. 3, Article ID 73. Web of ScienceGoogle Scholar
R. A. Mashiyev, B. Cekic, M. Avci and Z. Yucedag, Existence and multiplicity of weak solutions for nonuniformly elliptic equations with nonstandard growth condition, Complex Var. Elliptic Equ. 57 (2012), no. 5, 579–595. CrossrefWeb of ScienceGoogle Scholar
V. D. Rădulescu and D. D. Repovš, Combined effects in nonlinear problems arising in the study of anisotropic continuous media, Nonlinear Anal. 75 (2012), no. 3, 1524–1530. Web of ScienceCrossrefGoogle Scholar
V. D. Rădulescu and D. D. Repovš, Partial Differential Equations with Variable Exponents. Variational Methods and Qualitative Analysis, Monogr. Res. Notes Math., CRC Press, Boca Raton, 2015. Google Scholar
M. Růžička, Electrortheological Fluids: Modeling and Mathematical Theory, Springer, Berlin, 2000. Google Scholar
K. Saoudi, Existence and non-existence of solution for a singular nonlinear Dirichlet problem involving the -Laplace operator, J. Adv. Math. Stud. 9 (2016), no. 2, 291–302. Google Scholar
K. Saoudi, M. Kratou and S. Alsadhan, Multiplicity results for the -Laplacian equation with singular nonlinearities and nonlinear Neumann boundary condition, Int. J. Differ. Equ. 2016 (2016), Article ID 3149482. Google Scholar
V. V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory (in Russian), Izv. Akad. Nauk SSSR Ser. Mat. 50 (1986), no. 4, 675–710, 877; translation in Math. USSR-Izv. 29 (1987), no. 1, 33-66. Google Scholar
About the article
Published Online: 2018-03-13
Citation Information: Advances in Nonlinear Analysis, Volume 8, Issue 1, Pages 1171–1183, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2016-0260.
© 2019 Walter de Gruyter GmbH, Berlin/Boston. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0