# Nontrivial Solutions for a Modified Capillary Surface Equation

Authors: Zhanping Liang
Journal: Journal of Function Spaces and Applications
 (2013)

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## Abstract

A negative solution and a positive solution are obtained for a modified capillary surface equation by variational methods.


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## Body

## 1. Introduction

In this paper, we study the existence of nontrivial solutions to the following quasilinear elliptic equation:(1)-div(|∇u|2p-2∇u1+|∇u|2p)=f(x,u)inΩ,u=0on∂Ω,
where p>1, and Ω is a bounded domain in ℝN with smooth boundary. The function f∈C(Ω-×ℝ,ℝ) with the subcritical growth (f)
(2)|f(x,t)|≤c(1+|t|q-1),t∈ℝ,x∈Ω,where q∈[1,Np/(N-p)) if 1<p<N or q∈[1,+∞) if 1<N≤p, and c is a positive constant.In the case thatp=1, (1) is the mean curvature equation or the capillary surface equation; when f(x,u)≡u, it describes the equilibrium shape of a liquid surface with constant surface tension in a uniform gravity field, and this is the shape of a pendent drop [1]. When p>1, one calls (1) a modified capillary surface equation which is also worth considering even though it is not exactly the capillary surface equation [2]. For the capillary surface equation, radially symmetric solutions in the case that Ω is a ball or entire space have been investigated precisely; See, for example, [3–5] and the references therein. In [2], by minimization sequence method and the Ambrosetti-Rabinowitz mountain pass lemma without Palais-Smale condition, positive solutions were obtained to nonlinear eigenvalue problem for the modified capillary surface equation which is of the form
(3)-div(|∇u|2p-2∇u1+|∇u|2p)=λf(x,u)inΩ,u≥0inΩ,u=0on∂Ω,
where λ is a positive parameter. In the proof of the main results of [2], λ is crucial not only to the existence of global or local minimizer but also to the construction of mountain pass geometry. In our paper, one object is to find existence conditions of solutions to (1) without the constraint of λ. Since
(4)1+|∇u|2p-1~|∇u|pas|∇u|→∞,
the other object is to investigate the probability to present the property of f by the eigenvalue of the problem
(5)-Δpu=λ|u|p-2uinΩ,u=0on∂Ω,
where Δpu=div(|∇u|p-2∇u).In the following, we recall some known facts about problem (5). Let λ1>0 be the first eigenvalue of the problem (5). It is known that λ1 is characterized by
(6)λ1:=inf{∫Ω‍|∇u|pdx:∫Ω‍|u|pdx=1,u∈W01,p(Ω)∖{0}},
where W01,p(Ω) is the reflexive Banach space defined as the completion of C0∞(Ω) with respect to the norm ∥u∥:=(∫Ω‍|∇u|pdx)1/p. Also, λ1 is single and has an associated eigenfunction φ1>0 in Ω and ∥φ1∥=1. The reader is referred to [6, 7] for details.By a solutionu of (1), we mean that u satisfies (1) in the weak sense; that is, for all φ∈W01,p(Ω),
(7)∫Ω‍|∇u|2p-2∇u∇φ1+|∇u|2pdx=∫Ω‍f(x,u)φdx.
A solution such that u(x)≥0 in Ω and u≠0, respectively, u(x)≤0 in Ω and u≠0, is a positive, respectively, negative, solution.Define(8)J(u)=1p∫Ω‍(1+|∇u|2p-1)dx,u∈W01,p(Ω),K(u)=∫Ω‍F(x,u)dx,u∈W01,p(Ω),I(u)=J(u)-K(u),u∈W01,p(Ω),
where F(x,t)=∫0t‍f(x,s)ds. From a variational stand point, finding solutions of (1) in W01,p(Ω) is equivalent to finding critical points of the C1 functional I. As to the differentiability of the functional I, one can consult [2] for details. Since f satisfies the subcritical growth condition (f0), stand proofs show that K is weakly continuous. Since the function φ(t)=1+t2p is convex, the functional J is also convex. In addition, J belongs to C1. Hence, J is weakly lower semicontinuous. Thus, we have shown that I is weakly lower semi-continuous.Now, let us state the main results of this paper.Theorem 1.
Let(f) hold. Furthermore, assume that f satisfies the following conditions.(f0) There is some r>0 small such that
(9)pF(x,t)≥λ1|t|p,|t|≤r,x∈Ω,(f1)limsup|t|→∞(pF(x,t)/|t|p)<λ1 uniformly for x∈Ω.
Then, (1) has at least a negative solution and a positive solution which correspond to negative critical values of the associated functional given by (8).Theorem 2.
Let(f) and (f0) hold. Furthermore, assume that f satisfies the following conditions.(f2)lim|t|→∞(pF(x,t)/|t|p)=λ1 uniformly for x∈Ω,(f3)lim|t|→∞(f(x,t)-pF(x,t))=+∞ uniformly for x∈Ω.
Then, (1) has at least a negative solution and a positive solution which correspond to negative critical values of the associated functional given by (8).Remark 3.
With the conditions(f0)–(f3), Liu and Su in [8] have studied the existence of solutions to p-Laplacian quasilinear elliptic equation
(10)-Δpu=f(x,u)inΩ,u=0on∂Ω.
Under the conditions (f0) and (f1), (10) may be resonant at the eigenvalue λ1 near the origin. With the conditions (f2) and (f3), it may be resonant at λ1 both near the origin and near infinity. In fact, the condition (f0) allows (10) to be resonant near the origin from the right side of λ1, while the conditions (f2) and (f3) allow it to be resonant at infinity from the left side of λ1.Remark 4.
Theorems1 and 2 have shown a new fact that the interaction between the first eigenvalue of -Δp with zero Dirichlet boundary data and nonlinearity f can influence the existence of nontrivial solutions to (1).Before concluding this section, we explain some notations used in the paper.|Ω| is the Lebesgue measure of Ω. ci(i∈ℕ) is always a positive constant independent of functions. 〈·,·〉 is the duality between (W01,p(Ω))* and W01,p(Ω). In addition, we use |·| to denote the usual norm of ℝN.


## 2. The Proof of the Main Results

In this section, we prove Theorems1 and 2.Proof of Theorem1.
The proof consists of two steps.
(i) To obtain a positive solution, cut-off techniques are used. Define(11)f^(x,t)={f(x,t),t≥0,0,t<0,F^(x,t)=∫0t‍f^(x,s)ds,I^(u)=J(u)-∫Ω‍F^(x,u)dx,u∈W01,p(Ω).
Since f∈C(Ω-×ℝ,ℝ) and (f1) holds, for any given ɛ>0, there exists c1>0 such that
(12)F^(x,t)≤1p(λ1-ɛ)|t|p+c1,t∈ℝ,x∈Ω.
By the Poincaré inequality, for u∈W01,p(Ω),
(13)I^(u)=1p∫Ω‍1+|∇u|2pdx-∫Ω‍F^(x,u)dx-|Ω|p≥1p∫Ω‍|∇u|pdx-1p(λ1-ɛ)∫Ω‍|u|pdx-(c1+1p)|Ω|≥1p(1-λ1-ɛλ1)∥u∥p-(c1+1p)|Ω|=ɛpλ1∥u∥p-(c1+1p)|Ω|.
Hence, I^ is coercive; that is, I^(u)→∞ as n→∞. In addition, since f^ also satisfies the condition (f), I^ is weakly lower semi-continuous. So, it has a global minimizer.
Take a numbert0>0 such that 0<t0φ1≤r in Ω. By the condition (f0), we have that
(14)I^(t0φ1)=1p∫Ω‍(1+t02p|∇φ1|2p-1)dx-∫Ω‍F(x,t0φ1)dx<1pt0p∫Ω‍|∇φ1|pdx-1pλ1t0p∫Ω‍φ1pdx=0.
Thus, the global minimizer of I^ is a nontrivial critical point, denoted by u1 which satisfies I^(u1)<0. Putting u1-(x)=min{u1(x),0}, we have that
(15)〈I^′(u1),u1-〉=∫Ω‍|∇u1-|2p1+|∇u1-|2pdx=0.
Hence, u1-=0. So, u1 is a positive solution of (1), and I(u1)<0.
(ii) To obtain a negative solution, we only need to replacef^ with
(16)f~(x,t)={0,t>0,f(x,t),t≤0.
Similar to step (i), it is shown that (1) has a negative solution u2 with I(u2)<0.
The proof is completed.Proof of Theorem2.
We adopt the notations in the proof of Theorem1.
First of all, we show that the functionalI^ is also coercive under the conditions (f2) and (f3). Write
(17)F^(x,t)=1pλ1(t+)p+G^(x,t),f^(x,t)=λ1|t|p-2t++g^(x,t),
where t+=max{t,0}. Given x∈Ω, we have that
(18)limt→+∞pG^(x,t)tp=0,limt→+∞(g^(x,t)t-pG^(x,t))=+∞.
Thus, for every M>0, there exists RM>0 such that
(19)g^(x,t)t-pG^(x,t)≥M,t≥RM,x∈Ω.
Integrating the equality
(20)ddt(G^(x,t)tp)=g^(x,t)t-pG^(x,t)tp+1
over the interval [t,T]⊂[RM,+∞),
(21)G^(x,T)Tp-G^(x,t)tp≥Mp(1tp-1Tp).
Letting T→+∞, we show that G^(x,t)≤-M/p, t≥RM.
Suppose that{un}⊂W01,p(Ω) satisfies ∥un∥→∞ and I^(un)≤C for some constant C∈ℝ. Let vn=un/∥un∥. Up to subsequence if necessary, we may assume that there exists v0∈W01,p(Ω) such that
(22)vn⇀v0inE,vn→v0inLp(Ω),vn(x)→v0(x)a.e.x∈Ω.
Given M=1 in (19), we have that
(23)G^(x,t)≤-1p,t≥R1.
Let c2=max(x,t)∈Ω-×[-R1,R1]|G^(x,t)|. Thus,
(24)C∥un∥p≥1p∥un∥p(∫Ω‍1+|∇u|2pdx-λ1∫Ω‍|un|pdx)-1∥un∥p∫Ω‍G^(x,un)dx-|Ω|p∥un∥p≥1p∫Ω‍(|∇vn|p-λ1|vn|p)dx-|Ω|p∥un∥p-1∥un∥p∫|un|≥R1‍G^(x,un)dx-1∥un∥p∫|un|≤R1‍G^(x,un)dx≥1p∫Ω‍(|∇vn|p-λ1|vn|p)dx-|Ω|p∥un∥p-c2|Ω|∥un∥p=1p∫Ω‍(|∇vn|p-λ1|vn|p)dx+c3∥un∥p,
where c3=(1/p+c2)|Ω|. It follows from (22) and the previous inequality that
(25)limsupn→∞∫Ω‍|∇vn|pdx≤λ1∫Ω‍|v0|pdx.
Because the norm is weakly lower semi-continuous, using Poincaré inequality, we get that
(26)limsupn→∞∫Ω‍|∇vn|pdx≤λ1∫Ω‍|v0|pdx≤∫Ω‍|∇v0|pdx≤liminfn→∞∫Ω‍|∇vn|pdx≤limsupn→∞∫Ω‍|∇vn|pdx.
Hence, ∫Ω‍|∇v0|pdx=λ1∫Ω‍|v0|pdx and vn→v0 in W01,p(Ω) with ∥v0∥=1. So, v0 is the corresponding eigenfunction to λ1. Without loss of generality, we may assume that v0=φ1. Thus, un→+∞ a.e. x∈Ω. Consequently, G^(x,un(x))→-∞ a.e. x∈Ω. Therefore,
(27)C≥-∫Ω‍G^(x,un)dx→+∞,
which contradicts the fact that C∈ℝ. From the fact that I^ is weakly low semi-continuous, we know that it has a global minimizer u1. As in the proof of Theorem 1, u1 is a positive solution of (1) with I(u1)<0. In a similar way, we can obtain a negative solution with negative critical value.
The proof is completed.



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*Source: 710592-2013-03-12.xml*