Ni-Serrin type equations arising from capillarity phenomena with non-standard growth
© Avci; licensee Springer 2013
Received: 28 December 2012
Accepted: 28 February 2013
Published: 18 March 2013
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
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.
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 , .
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.
where is a conjugate space of such that .
for all .
in measure in Ω and .
for all .
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 .
Proposition 6 
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.
where and .
3 Main results
Theorem 8 Assume the following conditions hold:
for all , where and are positive real numbers;
for all and , where such that . Then (P) has a weak solution.
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 9 Assume the following conditions hold:
for all , where , and α real numbers such that and ;
for all ;
for all and , where such that for all and ;
(f2) , uniformly for ;
(f3) There exists such that for and all ;
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 10 Suppose (M1), (M2), (AR) and (f1) hold. Then I satisfies the (PS) condition.
Since the functional (3.4) is of type (see Proposition 3.1 in ), we get in . We are done. □
There exist two positive real numbers γ and a such that , with ;
There exists such that , .
providing that . Since and , there exist two positive real numbers γ and a such that , with .
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. □
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 13 Let 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 16 Suppose that M and f satisfy the following conditions:
for all , where , , δ and α are real numbers such that and ;
for all and , where such that for all ;
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.
J is bounded from below and even;
There is a compact set such that and .
Then J possesses at least k pairs of distinct critical points and their corresponding critical values are less than .
I is bounded from below;
I satisfies the (PS) condition.
for large enough. Hence, I is bounded from below.
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. □
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. □
Theorem 19 Suppose (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.
Obviously, , so . By Krasnoselskii’s genus, each is a critical value of I, hence there is a sequence of solutions such that . □
The author would like to thank the referee for some valuable comments and helpful suggestions.
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