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

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 $1<p(x)<N$ 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.

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 $f(x,u)\equiv \lambda f(x,u)$, $\lambda >0$.

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]).

becomes a Banach space.

Proposition 1 [22, 24]

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

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

Proposition 2 [22, 24]

Proposition 3 [22, 24]

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

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

Proposition 4 [22, 24]

If $1<p^{-}\le p^{+}<\mathrm{\infty }$, 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 $q(x)<p^{\ast }(x)$ 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 $p(x)<N$ and $p^{\ast }(x)=+\mathrm{\infty }$ if $p(x)\ge N$.

Proposition 6 [18]

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.

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

Theorem 8 Assume the following conditions hold:

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

for all $x\in \overline{\mathrm{\Omega }}$ and $t\in \mathbb{R}$, where $p,q\in C_{+}(\overline{\mathrm{\Omega }})$ such that $q^{+}<\alpha p^{-}<p^{\ast }(x)$. Then (P) has a weak solution.

By the assumption $q^{+}<\alpha p^{-}$, 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:

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

for all $t>0$;

for all $x\in \overline{\mathrm{\Omega }}$ and $t\in \mathbb{R}$, where $\beta \in C_{+}(\overline{\mathrm{\Omega }})$ such that $\beta (x)<p^{\ast }(x)$ for all $x\in \overline{\mathrm{\Omega }}$ and $\alpha p^{+}<{\beta }^{-}$;

(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 $t\ge t_{\ast }$;

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.

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

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. □

providing that $\epsilon C_{10}^{\alpha p^{+}}<\frac{m_1}{2\alpha {(p^{+})}^{\alpha }}$. Since $\parallel u\parallel <1$ and $\alpha p^{+}<{\beta }^{-}$, 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 $\parallel u\parallel =\gamma \in (0,1)$.

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.

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:

for all $t>0$, where $m_3$, $m_4$, δ and α are real numbers such that $0<m_3\le m_4$ and $1<\delta \le \alpha$;

for all $x\in \overline{\mathrm{\Omega }}$ and $t\ge 0$, where $s,q\in C(\overline{\mathrm{\Omega }})$ such that $1<s(x)<q(x)<p^{\ast }(x)$ for all $x\in \overline{\mathrm{\Omega }}$;

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

If $p(x)<q(x)<p^{\ast }(x)$ for all $x\in \overline{\mathrm{\Omega }}$ and $q^{+}<\delta p^{-}$, 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.

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

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

where $\parallel u_n\parallel >1$. Since $q^{+}<\delta p^{-}$, 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])

Now, we will show that $c_k<0$ 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)