Ni-Serrin type equations arising from capillarity phenomena with non-standard growth

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.

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 \mathrm{\Omega }\subset {\mathbb{R}}^N is a bounded domain with smooth boundary ∂ Ω, p\in C(\overline{\mathrm{\Omega }}) such that for any x\in \overline{\mathrm{\Omega }} and L(u):={\int }_{\mathrm{\Omega }}\frac{|\mathrm{\nabla }u{|}^{p(x)}+\sqrt{1+|\mathrm{\nabla }u{|}^{2p(x)}}}{p(x)}dx.

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 G(u)=\frac{\mathrm{\nabla }u}{\sqrt{1+|\mathrm{\nabla }u{|}^2}} 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 f(u)=ku, k>0 (see [3–5]). Recently, in [6] Rodrigues studied a version of the problem (P) for the case M(L(u))\equiv 1 and f(x,u)\equiv \lambda f(x,u), \lambda >0.

We note that if we choose the functional L(u) as {\int }_{\mathrm{\Omega }}\frac{|\mathrm{\nabla }u{|}^{p(x)}}{p(x)}dx in (P), then we get the problem

which is called the p(x)-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 [10]. To be more precise, Kirchhoff established a model given by the equation

where ρ, P_0, 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 \frac{P_0}{h}+\frac{E}{2l}{\int }_0^l|\frac{\partial u}{\partial x}{|}^{2}dx which depends on the average \frac{E}{2l}{\int }_0^l|\frac{\partial u}{\partial x}{|}^{2}dx of the kinetic energy \frac{1}{2}|\frac{\partial u}{\partial x}{|}^2 on [0,l], and hence the equation is no longer a pointwise identity.

The nonlinear problems involving the p(x)-Laplacian operator, that is, div(|\mathrm{\nabla }u{|}^{p(x)-2}\mathrm{\nabla }u), 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 p(x)-Laplacian can be found in [16–20] and references therein.

We state some basic properties of the variable exponent Lebesgue-Sobolev spaces L^{p(x)}(\mathrm{\Omega }) and W^{1,p(x)}(\mathrm{\Omega }), where \mathrm{\Omega }\subset {\mathbb{R}}^N is a bounded domain (for details, see [21–24]).

Set

Let p\in C_{+}(\overline{\mathrm{\Omega }}) and denote

For any p\in C_{+}(\overline{\mathrm{\Omega }}), we define the variable exponent Lebesgue space by

then L^{p(x)}(\mathrm{\Omega }) endowed with the norm

becomes a Banach space.

Proposition 1 [22, 24]

For any u\in L^{p(x)}(\mathrm{\Omega }) and v\in L^{p^{\mathrm{\prime }}(x)}(\mathrm{\Omega }), we have

where L^{p^{\mathrm{\prime }}(x)}(\mathrm{\Omega }) is a conjugate space of L^{p(x)}(\mathrm{\Omega }) such that \frac{1}{p(x)}+\frac{1}{p^{\mathrm{\prime }}(x)}=1.

The modular of L^{p(x)}(\mathrm{\Omega }), which is the mapping \rho :L^{p(x)}(\mathrm{\Omega })\to \mathbb{R}, is defined by

for all u\in L^{p(x)}(\mathrm{\Omega }).

Proposition 2 [22, 24]

If u,u_n\in L^{p(x)}(\mathrm{\Omega }) (n=1,2,\dots), then the following statements are equivalent:

1. (i)

   {lim}_{n\to \mathrm{\infty }}{|u_n-u|}_{p(x)}=0;

2. (ii)

   {lim}_{n\to \mathrm{\infty }}\rho (u_n-u)=0;

3. (iii)

   u_n\to u in measure in Ω and {lim}_{n\to \mathrm{\infty }}\rho (u_n)=\rho (u).

Proposition 3 [22, 24]

If u,u_n\in L^{p(x)}(\mathrm{\Omega }) (n=1,2,\dots), we have

1. (i)

   ;

2. (ii)

   {|u|}_{p(x)}>1\Rightarrow {|u|}_{p(x)}^{p^{-}}\le \rho (u)\le {|u|}_{p(x)}^{p^{+}}; ;

3. (iii)

   {lim}_{n\to \mathrm{\infty }}{|u_n|}_{p(x)}=0\iff {lim}_{n\to \mathrm{\infty }}\rho (u_n)=0; {lim}_{n\to \mathrm{\infty }}{|u_n|}_{p(x)}=\mathrm{\infty }\iff {lim}_{n\to \mathrm{\infty }}\rho (u_n)=\mathrm{\infty }.

The variable exponent Sobolev space W^{1,p(x)}(\mathrm{\Omega }) is defined by

with the norm

for all u\in W^{1,p(x)}(\mathrm{\Omega }).

The space W_0^{1,p(x)}(\mathrm{\Omega }) is defined as the closure of C_0^{\mathrm{\infty }}(\mathrm{\Omega }) in W^{1,p(x)}(\mathrm{\Omega }) with respect to the norm {\parallel u\parallel }_{1,p(x)}. For u\in W_0^{1,p(x)}(\mathrm{\Omega }), we can define an equivalent norm

since the Poincaré inequality holds, i.e., there exists a positive constant C_1>0 such that

for all u\in W_0^{1,p(x)}(\mathrm{\Omega }) [18, 24].

Proposition 4 [22, 24]

If , then the spaces L^{p(x)}(\mathrm{\Omega }), W^{1,p(x)}(\mathrm{\Omega }) and W_0^{1,p(x)}(\mathrm{\Omega }) are separable and reflexive Banach spaces.

Proposition 5 [22, 24]

Let q\in C_{+}(\overline{\mathrm{\Omega }}). If for all x\in \overline{\mathrm{\Omega }}, then the embedding W^{1,p(x)}(\mathrm{\Omega })\hookrightarrow L^{q(x)}(\mathrm{\Omega }) is compact and continuous, where p^{\ast }(x)=\frac{Np(x)}{N-p(x)} if and p^{\ast }(x)=+\mathrm{\infty } if p(x)\ge N.

Proposition 6 [18]

Let X be a Banach space and let define the functional \mathrm{\Lambda }={\int }_{\mathrm{\Omega }}\frac{|\mathrm{\nabla }u{|}^{p(x)}}{p(x)}dx. Then \mathrm{\Lambda }:X\to \mathbb{R} is convex. The mapping {\mathrm{\Lambda }}^{\mathrm{\prime }}:X\to X^{\ast } is a strictly monotone, bounded homeomorphism of (S_{+}) type, namely

Definition 7 Let X be a Banach space and J:X\to \mathbb{R} be a C^1-functional. We say that a functional J satisfies the Palais-Smale condition ((PS) for short) if any sequence \{u_n\} in X, such that \{J(u_n)\} is bounded and J^{\mathrm{\prime }}(u_n)\to 0 as n\to \mathrm{\infty }, admits a convergent subsequence.

We say that u\in W_0^{1,p(x)}(\mathrm{\Omega }) is a weak solution of (P) if

for any v\in W_0^{1,p(x)}(\mathrm{\Omega }). The energy functional I:W_0^{1,p(x)}(\mathrm{\Omega })\to \mathbb{R} corresponding to the problem (P) is

where \mathcal{M}(t)={\int }_0^{t}M(\xi )d\xi and F(x,u)={\int }_0^{u}f(x,\phi )d\phi.

Thanks to the conditions (M₀) and (f₀) (see below), the functional I is well defined and of class C^1. 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 I^{\mathrm{\prime }}:W_0^{1,p(x)}(\mathrm{\Omega })\to {(W_0^{1,p(x)}(\mathrm{\Omega }))}^{\ast } given by the formula

for any u,v\in W_0^{1,p(x)}(\mathrm{\Omega }), where

Theorem 8 Assume the following conditions hold:

(M₀) M:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+} is a continuous function and satisfies the condition

for all t>0, where m_0 and \alpha >1 are positive real numbers;

(f₀) f:\overline{\mathrm{\Omega }}\times \mathbb{R}\to \mathbb{R} satisfies the Carathéodory condition and there exist positive constants C_2 and C_3 such that

for all x\in \overline{\mathrm{\Omega }} and t\in \mathbb{R}, where p,q\in C_{+}(\overline{\mathrm{\Omega }}) 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 W_0^{1,p(x)}(\mathrm{\Omega }) and u is a weak solution of (P). □

Theorem 9 Assume the following conditions hold:

(M₁) M:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+} is a continuous function and satisfies the condition

for all t>0, where m_1, m_2 and α real numbers such that and \alpha >1;

(M₂) M satisfies

for all t>0;

(f₁) f:\overline{\mathrm{\Omega }}\times \mathbb{R}\to \mathbb{R} satisfies the Carathéodory condition and there exist positive constants C_5 and C_6 such that

for all x\in \overline{\mathrm{\Omega }} and t\in \mathbb{R}, where \beta \in C_{+}(\overline{\mathrm{\Omega }}) such that for all x\in \overline{\mathrm{\Omega }} and ;

(f₂) f(x,t)=o(|t{|}^{\alpha p^{+}-1}), t\to 0 uniformly for x\in \overline{\mathrm{\Omega }};

(f₃) There exists t_{\ast }>0 such that F(x,t)>0 for x\in \overline{\mathrm{\Omega }} and all t\ge t_{\ast };

(AR) Ambrosetti-Rabinowitz’s condition holds, i.e., \mathrm{\exists }{t}_{\ast }>0, \theta >\frac{m_2}{m_1}\alpha p^{+} 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 10 Suppose (M₁), (M₂), (AR) and (f₁) hold. Then I satisfies the (PS) condition.

Proof Let us assume that there exists a sequence \{u_n\} in W_0^{1,p(x)}(\mathrm{\Omega }) such that

Then

Since \sqrt{1+|\mathrm{\nabla }u{|}^{2p(x)}}\ge |\mathrm{\nabla }u{|}^{p(x)}, we have L(u_n)\ge \frac{2}{p^{+}}{\int }_{\mathrm{\Omega }}|\mathrm{\nabla }{u}_n{|}^{p(x)}dx. Therefore,

By the above inequalities and assumptions (M₁), (M₂) and (AR), we get

This implies that \{\parallel u_n\parallel \} is bounded in W_0^{1,p(x)}(\mathrm{\Omega }). Passing to a subsequence if necessary, there exists u\in W_0^{1,p(x)}(\mathrm{\Omega }) such that u_n\rightharpoonup u. Therefore, by Proposition 5, we have

(3.3)

By (3.2), we have 〈I^{\mathrm{\prime }}(u_n),u_n-u〉\to 0. Thus

From (f₁) and Proposition 1, it follows

If we consider the relations given in (3.3), we get

Hence,

From (M₁), we get

Since the functional (3.4) is of type (S_{+}) (see Proposition 3.1 in [6]), we get u_n\to u in W_0^{1,p(x)}(\mathrm{\Omega }). We are done. □

Lemma 11 Suppose (M₁), (AR) and (f₁)-(f₃) hold. Then the following statements hold:

1. (i)

   There exist two positive real numbers γ and a such that I(u)\ge a>0, u\in W_0^{1,p(x)}(\mathrm{\Omega }) with \parallel u\parallel =\gamma;

2. (ii)

   There exists u\in W_0^{1,p(x)}(\mathrm{\Omega }) such that \parallel u\parallel >\gamma, .

Proof (i) Let . Then by (M₁) and Proposition 3, we have

Since , by Proposition 5 we have the continuous embeddings W_0^{1,p(x)}(\mathrm{\Omega })\hookrightarrow L^{\alpha p^{+}}(\mathrm{\Omega }) and W_0^{1,p(x)}(\mathrm{\Omega })\hookrightarrow L^{{\beta }^{+}}(\mathrm{\Omega })\hookrightarrow L^{{\beta }^{-}}(\mathrm{\Omega }), and also there are positive constants C_7, C_8 and C_9 such that

and

From (f₁) and (f₂), we get F(x,t)\le \epsilon |t{|}^{\alpha p^{+}}+C_{\epsilon }|t{|}^{\beta (x)} for all x\in \overline{\mathrm{\Omega }} and t\in \mathbb{R}, where \epsilon >0 is small enough and C_{\epsilon }>0. 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 I(u)\ge a>0, u\in W_0^{1,p(x)}(\mathrm{\Omega }) with \parallel u\parallel =\gamma \in (0,1).

(ii) From (AR) and (f₃), one easily deduces

for all x\in \overline{\mathrm{\Omega }} and t\ge t_{\ast }. Therefore, for \delta >1 and nonnegative u\in W_0^{1,p(x)}(\mathrm{\Omega }) such that \{x\in \overline{\mathrm{\Omega }}:u(x)\ge t_{\ast }\}, we get

(recall that F\ge 0 and F(\cdot ,t_{\ast })>0 almost everywhere). On the other hand, when t>t_{\ast }, from (M₁) we obtain that

Since t>1, it is obvious L(t\omega )\le t^{p^{+}}L(\omega ). Hence, for \omega \in W_0^{1,p(x)}(\mathrm{\Omega })\mathrm{\setminus }\{0\}, we have

From the assumption on θ (see (AR)), we conclude I(t\omega )\to -\mathrm{\infty } as t\to +\mathrm{\infty }. □

Proof of Theorem 9 From Lemma 10, Lemma 11 and the fact that I(0)=0, 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. □

In the sequel, using Krasnoselskii’s genus theory (see [25, 27]), we show the existence of infinitely many solutions of the problem (P). So, we recall some basic notations of Krasnoselskii’s genus.

Let X be a real Banach space and set

Definition 12 Let E\in \mathfrak{R} and X={\mathbb{R}}^k. The genus \gamma (E) of E is defined by

If such a mapping does not exist for any k>0, we set \gamma (E)=\mathrm{\infty }. Note also that if E is a subset which consists of finitely many pairs of points, then \gamma (E)=1. Moreover, from the definition, \gamma (\mathrm{\varnothing })=0. A typical example of a set of genus k is a set which is homeomorphic to a (k-1) 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 X={\mathbb{R}}^N and ∂ Ω be the boundary of an open, symmetric and bounded subset \mathrm{\Omega }\subset {\mathbb{R}}^N with 0\in \mathrm{\Omega }. Then \gamma (\partial \mathrm{\Omega })=N.

Corollary 14 \gamma (S^{N-1})=N.

Remark 15 If X is of an infinite dimension and separable and S is the unit sphere in X, then \gamma (S)=\mathrm{\infty }.

Theorem 16 Suppose that M and f satisfy the following conditions:

(M₃) M:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+} is a continuous function and satisfies the condition

for all t>0, where m_3, m_4, δ and α are real numbers such that and ;

(f₄) f:\overline{\mathrm{\Omega }}\times \mathbb{R}\to \mathbb{R} is a continuous function and there exist positive constants C_{11}, C_{12}, C_{13} and C_{14} such that

for all x\in \overline{\mathrm{\Omega }} and t\ge 0, where s,q\in C(\overline{\mathrm{\Omega }}) such that for all x\in \overline{\mathrm{\Omega }};

(f₅) f is an odd function according to t, that is,

for all x\in \overline{\mathrm{\Omega }} and t\in \mathbb{R}.

If for all x\in \overline{\mathrm{\Omega }} and , then the problem (P) has infinitely many solutions.

The following result obtained by Clarke in [28] is the main idea which we use in the proof of Theorem 16.

Theorem 17 Let J\in C^1(X,\mathbb{R}) be a functional satisfying the (PS) condition. Furthermore, let us suppose that:

1. (i)

   J is bounded from below and even;

2. (ii)

   There is a compact set K\in \mathfrak{R} such that \gamma (K)=k and .

Then J possesses at least k pairs of distinct critical points and their corresponding critical values are less than J(0).

Lemma 18 Suppose (M₃), (f₄) and the inequality hold.

1. (i)

   I is bounded from below;

2. (ii)

   I satisfies the (PS) condition.

Proof (i) By the assumptions (M₃) and (f₄), we have

By Proposition 3 and Proposition 5, we get

for \parallel u\parallel large enough. Hence, I is bounded from below.

(ii) Let us assume that there exists a sequence \{u_n\} in W_0^{1,p(x)}(\mathrm{\Omega }) such that

From (3.8) we have |I(u_n)|\le C_{16}. This fact combined with (3.7) implies that

where \parallel u_n\parallel >1. Since , we obtain that \{\parallel u_n\parallel \} is bounded in W_0^{1,p(x)}(\mathrm{\Omega }).

Hence, we may extract a subsequence \{u_n\}\subset W_0^{1,p(x)}(\mathrm{\Omega }) and u\in W_0^{1,p(x)}(\mathrm{\Omega }) such that u_n\rightharpoonup u in W_0^{1,p(x)}(\mathrm{\Omega }). 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. □

Proof of Theorem 16 Set (see [7, 25])

then we have

Now, we will show that for every k\in \mathbb{N}. Since W_0^{1,p(x)}(\mathrm{\Omega }) is a reflexive and separable Banach space, for any k\in \mathbb{N}, we can choose a k-dimensional linear subspace X_k of W_0^{1,p(x)}(\mathrm{\Omega }) such that X_k\subset C_0^{\mathrm{\infty }}(\mathrm{\Omega }). As the norms on X_k are equivalent, there exists r_k\in (0,1) such that u\in X_k with \parallel u\parallel \le r_k implies |u{|}_{L^{\mathrm{\infty }}}\le \delta.

Set S_{r_k}^{(k)}=\{u\in X_k:\parallel u\parallel =r_k\}. By the compactness of S_{r_k}^{(k)} and the condition (f₄), there exists a constant {\eta }_k>0 such that

(3.9)

for all u\in S_{r_k}^{(k)}. If we consider (M₃) and (f₄), for u\in S_{r_k}^{(k)} and t\in (0,1), we have

providing that C_{18}\ge \frac{m_4}{\alpha {(p^{-})}^{\alpha }}{2}^{\alpha -1}. Since , we can find t_k\in (0,1) and {\epsilon }_k>0 such that

i.e.,

It is clear that \gamma (S_{t_k{r}_k}^{(k)})=k, 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 x\in \overline{\mathrm{\Omega }}, then the problem (P) has a sequence of solutions \{\pm u_k:k=1,2,\dots \} 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 W_0^{1,p(x)}(\mathrm{\Omega }), i.e., (P) has a solution. By help of coerciveness, we know that I satisfies the (PS) condition on W_0^{1,p(x)}(\mathrm{\Omega }). 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 t_k\in (0,1) and {\epsilon }_k>0 such that

Obviously, \gamma (S_{t_k{r}_k}^{(k)})=k, so . By Krasnoselskii’s genus, each c_k is a critical value of I, hence there is a sequence of solutions \{\pm u_k:k=1,2,\dots \} such that . □

The author would like to thank the referee for some valuable comments and helpful suggestions.

Affiliations

1. Department of Mathematics, Faculty of Science, Dicle University, Diyarbakir, 21280, Turkey

   • Mustafa Avci

Corresponding author

Correspondence to Mustafa Avci.

Competing interests

The author declares that he has no competing interests.

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