In the present paper, in view of the variational approach, we discuss a Ni-Serrin type equation involving non-standard growth condition and arising from the capillarity phenomena. Establishing some suitable conditions, we prove the existence and multiplicity of solutions.
MSC: 35D05, 35J60, 35J70.
Keywords:-Laplacian; variable exponent Sobolev space; mountain pass theorem; genus theory; variational method; capillarity phenomena
We study the existence and multiplicity of solutions for a Ni-Serrin type equation involving non-standard growth condition and arising from capillarity phenomena of the following type:
Capillarity can be briefly explained by considering the effects of two opposing forces: adhesion, i.e., the attractive (or repulsive) force between the molecules of the liquid and those of the container; and cohesion, i.e., the attractive force between the molecules of the liquid. The study of capillary phenomena has gained some attention recently. This increasing interest is motivated not only by fascination in naturally-occurring phenomena such as motion of drops, bubbles and waves but also its importance in applied fields ranging from industrial and biomedical and pharmaceutical to microfluidic systems.
The study of ground states for equations of the form
where is the Kirchhoff stress term and the source term f was very general, was initiated by Ni and Serrin [1,2]. Moreover, radial solutions of the problem (1.1) have been studied in the context of the analysis of capillarity surfaces for a function of the form , (see [3-5]). Recently, in  Rodrigues studied a version of the problem (P) for the case and , .
which is called the -Kirchhoff type equation [7-9]. In this case, the problem (1.2) indicates a generalization of a model, the so-called Kirchhoff equation, introduced by Kirchhoff in . To be more precise, Kirchhoff established a model given by the equation
where ρ, , h, E, l are constants, which extends the classical D’Alambert wave equation by considering the effects of the changes in the length of the strings during the vibrations. A distinguishing feature of Kirchhoff equation (1.3) is that the equation contains a nonlocal coefficient which depends on the average of the kinetic energy on , and hence the equation is no longer a pointwise identity.
The nonlinear problems involving the -Laplacian operator, that is, , are extremely attractive because they can be used to model dynamical phenomena which arise from the study of electrorheological fluids or elastic mechanics, in the modeling of stationary thermo-rheological viscous flows of non-Newtonian fluids and in the mathematical description of the processes filtration of an ideal barotropic gas through a porous medium [11-15]. The detailed application backgrounds of the -Laplacian can be found in [16-20] and references therein.
2 Abstract framework and preliminary results
becomes a Banach space.
with the norm
Definition 7 Let X be a Banach space and be a -functional. We say that a functional J satisfies the Palais-Smale condition ((PS) for short) if any sequence in X, such that is bounded and as , admits a convergent subsequence.
Thanks to the conditions (M0) and (f0) (see below), the functional I is well defined and of class . Since the problem (P) is in the variational setting, the critical points of I are weak solutions of (P). Moreover, the derivative of I is the mapping given by the formula
3 Main results
Theorem 8Assume the following conditions hold:
Proof By the assumptions (M0) and (f0), we have
Therefore, by Proposition 3 and Proposition 5, it follows
Theorem 9Assume the following conditions hold:
Then (P) has at least one nontrivial weak solution.
To obtain the result of Theorem 9, we need to show that Lemma 10 and Lemma 11 hold.
Lemma 10Suppose (M1), (M2), (AR) and (f1) hold. ThenIsatisfies the (PS) condition.
By the above inequalities and assumptions (M1), (M2) and (AR), we get
From (f1) and Proposition 1, it follows
If we consider the relations given in (3.3), we get
From (M1), we get
Since the functional (3.4) is of type (see Proposition 3.1 in ), we get in . We are done. □
Lemma 11Suppose (M1), (AR) and (f1)-(f3) hold. Then the following statements hold:
(ii) From (AR) and (f3), one easily deduces
Proof of Theorem 9 From Lemma 10, Lemma 11 and the fact that , I satisfies the mountain pass theorem (see [25,26]). Therefore, I has at least one nontrivial weak solution. The proof of Theorem 9 is completed. □
Let X be a real Banach space and set
If such a mapping does not exist for any , we set . Note also that if E is a subset which consists of finitely many pairs of points, then . Moreover, from the definition, . A typical example of a set of genus k is a set which is homeomorphic to a dimensional sphere via an odd map.
Now, we will give some results of Krasnoselskii’s genus which are necessary throughout the present paper.
Theorem 16Suppose thatMandfsatisfy the following conditions:
(f5) fis an odd function according tot, that is,
The following result obtained by Clarke in  is the main idea which we use in the proof of Theorem 16.
(i) Jis bounded from below and even;
(i) Iis bounded from below;
(ii) Isatisfies the (PS) condition.
Proof (i) By the assumptions (M3) and (f4), we have
By Proposition 3 and Proposition 5, we get
Hence, we may extract a subsequence and such that in . In the rest of the proof, if we consider similar relations given in (3.3) and growth conditions assumed on f and apply the same processes which we used in the proof of Lemma 10, we can see that I satisfies the (PS) condition. □
then we have
Now, we will show that for every . Since is a reflexive and separable Banach space, for any , we can choose a k-dimensional linear subspace of such that . As the norms on are equivalent, there exists such that with implies .
It is clear that , so . Finally, by Lemma 18 above, we can apply Theorem 17 to obtain that the functional I admits at least k pairs of distinct critical points, and since k is arbitrary, we obtain infinitely many critical points of I. The proof is completed. □
Proof In the beginning, we will show that I is coercive. If we follow the same processes applied in the proof of Theorem 8 and consider the fact , it is easy to get the coerciveness of I. Since I is weak lower semi-continuous, I attains its minimum on , i.e., (P) has a solution. By help of coerciveness, we know that I satisfies the (PS) condition on . Moreover, from the condition (f5), I is even.
In the rest of the proof, since we develop the same arguments which we used in the proof of Theorem 16, we omit the details. Therefore, if we follow similar steps to those in (3.9) and (3.10) and consider the inequalities , we can find and such that
The author declares that he has no competing interests.
The author would like to thank the referee for some valuable comments and helpful suggestions.
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