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:
where is a bounded domain with smooth boundary ∂Ω, such that for any and .
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 , .
We note that if we choose the functional as in (P), then we get the problem
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
Let and denote
For any , we define the variable exponent Lebesgue space by
then endowed with the norm
becomes a Banach space.
For any and , we have
where is a conjugate space of such that .
The modular of , which is the mapping , is defined by
for all .
If ( ), then the following statements are equivalent:
(iii) in measure in Ω and .
If ( ), we have
(ii) ; ;
(iii) ; .
The variable exponent Sobolev space is defined by
with the norm
for all .
The space is defined as the closure of in with respect to the norm . For , we can define an equivalent norm
since the Poincaré inequality holds, i.e., there exists a positive constant such that
If , then the spaces , and are separable and reflexive Banach spaces.
Let . If for all , then the embedding is compact and continuous, where if and if .
LetXbe a Banach space and let define the functional . Then is convex. The mapping is a strictly monotone, bounded homeomorphism of type, namely
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.
We say that is a weak solution of (P) if
for any . The energy functional corresponding to the problem (P) is
where and .
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
for any , where
3 Main results
Theorem 8Assume the following conditions hold:
(M0) is a continuous function and satisfies the condition
for all , where and are positive real numbers;
(f0) satisfies the Carathéodory condition and there exist positive constants and such that
for all and , where such that . Then (P) has a weak solution.
Proof By the assumptions (M0) and (f0), we have
Therefore, by Proposition 3 and Proposition 5, it follows
By the assumption , I is coercive. Since I is weakly lower semicontinuous, I has a minimum point u in and u is a weak solution of (P). □
Theorem 9Assume the following conditions hold:
(M1) is a continuous function and satisfies the condition
for all , where , andαreal numbers such that and ;
for all ;
(f1) satisfies the Carathéodory condition and there exist positive constants and such that
for all and , where such that for all and ;
(f2) , uniformly for ;
(f3) There exists such that for and all ;
(AR) Ambrosetti-Rabinowitz’s condition holds, i.e., , such that
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.
Proof Let us assume that there exists a sequence in such that
Since , we have . Therefore,
By the above inequalities and assumptions (M1), (M2) and (AR), we get
This implies that is bounded in . Passing to a subsequence if necessary, there exists such that . Therefore, by Proposition 5, we have
By (3.2), we have . Thus
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:
(i) There exist two positive real numbersγandasuch that , with ;
(ii) There exists such that , .
Proof (i) Let . Then by (M1) and Proposition 3, we have
Since , by Proposition 5 we have the continuous embeddings and , and also there are positive constants , and such that
From (f1) and (f2), we get for all and , where is small enough and . Therefore, by (M1), Proposition 3 and (3.5), (3.6), it follows
providing that . Since and , there exist two positive real numbers γ and a such that , with .
(ii) From (AR) and (f3), one easily deduces
for all and . Therefore, for and nonnegative such that , we get
(recall that and almost everywhere). On the other hand, when , from (M1) we obtain that
Since , it is obvious . Hence, for , we have
From the assumption on θ (see (AR)), we conclude as . □
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
Definition 12 Let and . The genus of E is defined by
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 13Let and∂Ω be the boundary of an open, symmetric and bounded subset with . Then .
Corollary 14 .
Remark 15 If X is of an infinite dimension and separable and S is the unit sphere in X, then .
Theorem 16Suppose thatMandfsatisfy the following conditions:
(M3) is a continuous function and satisfies the condition
for all , where , , δandαare real numbers such that and ;
(f4) is a continuous function and there exist positive constants , , and such that
for all and , where such that for all ;
(f5) fis an odd function according tot, that is,
for all and .
If for all and , then the problem (P) has infinitely many solutions.
The following result obtained by Clarke in  is the main idea which we use in the proof of Theorem 16.
Theorem 17Let be a functional satisfying the (PS) condition. Furthermore, let us suppose that:
(i) Jis bounded from below and even;
(ii) There is a compact set such that and .
ThenJpossesses at leastkpairs of distinct critical points and their corresponding critical values are less than .
Lemma 18Suppose (M3), (f4) and the inequality hold.
(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
for large enough. Hence, I is bounded from below.
(ii) Let us assume that there exists a sequence in such that
From (3.8) we have . This fact combined with (3.7) implies that
where . Since , we obtain that is bounded in .
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 .
Set . By the compactness of and the condition (f4), there exists a constant such that
for all . If we consider (M3) and (f4), for and , we have
providing that . Since , we can find and such that
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. □
Theorem 19Suppose (M3), (f4) and (f5) hold. If for all , then the problem (P) has a sequence of solutions such that .
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
Obviously, , so . By Krasnoselskii’s genus, each is a critical value of I, hence there is a sequence of solutions 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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