Journal of Function Spaces and Applications

Volume 2013 (2013), Article ID 710592, 4 pages

http://dx.doi.org/10.1155/2013/710592

Research Article

Nontrivial Solutions for a Modified Capillary Surface Equation

Zhanping Liang

School of Mathematical Sciences, Shanxi University, Taiyuan, Shanxi 030006, China

Received 1 December 2012; Accepted 6 February 2013

Academic Editor: James H. Liu

Copyright © 2013 Zhanping Liang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

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

1. Introduction

In this paper, we study the existence of nontrivial solutions to the following quasilinear elliptic equation:

where , and is a bounded domain in with smooth boundary. The function with the subcritical growth

where if or if , and is a positive constant.

In the case that , (1) is the mean curvature equation or the capillary surface equation; when , 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 , 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 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

where is a positive parameter. In the proof of the main results of [2], 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 . Since

the other object is to investigate the probability to present the property of by the eigenvalue of the problem

where .

In the following, we recall some known facts about problem (5). Let be the first eigenvalue of the problem (5). It is known that is characterized by

where is the reflexive Banach space defined as the completion of with respect to the norm . Also, is single and has an associated eigenfunction in and . The reader is referred to [6, 7] for details.

By a solution of (1), we mean that satisfies (1) in the weak sense; that is, for all ,

A solution such that in and , respectively, in and , is a positive, respectively, negative, solution.

Define

where . From a variational stand point, finding solutions of (1) in is equivalent to finding critical points of the functional . As to the differentiability of the functional , one can consult [2] for details. Since satisfies the subcritical growth condition (), stand proofs show that is weakly continuous. Since the function is convex, the functional is also convex. In addition, belongs to . Hence, is weakly lower semicontinuous. Thus, we have shown that is weakly lower semi-continuous.

Now, let us state the main results of this paper.

Theorem 1. Let hold. Furthermore, assume that satisfies the following conditions.

  There is some small such that

  uniformly for .

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 and hold. Furthermore, assume that satisfies the following conditions.

  uniformly for ,  uniformly for .

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 –, Liu and Su in [8] have studied the existence of solutions to p-Laplacian quasilinear elliptic equation

Under the conditions and , (10) may be resonant at the eigenvalue near the origin. With the conditions and , it may be resonant at both near the origin and near infinity. In fact, the condition allows (10) to be resonant near the origin from the right side of , while the conditions and allow it to be resonant at infinity from the left side of .

Remark 4. Theorems 1 and 2 have shown a new fact that the interaction between the first eigenvalue of with zero Dirichlet boundary data and nonlinearity can influence the existence of nontrivial solutions to (1).

Before concluding this section, we explain some notations used in the paper. is the Lebesgue measure of . is always a positive constant independent of functions. is the duality between and . In addition, we use to denote the usual norm of .

2. The Proof of the Main Results

In this section, we prove Theorems 1 and 2.

Proof of Theorem 1. The proof consists of two steps.

(i) To obtain a positive solution, cut-off techniques are used. Define

Since and () holds, for any given , there exists such that

By the Poincaré inequality, for ,

Hence, is coercive; that is, as . In addition, since also satisfies the condition , is weakly lower semi-continuous. So, it has a global minimizer.

Take a number such that in . By the condition , we have that

Thus, the global minimizer of is a nontrivial critical point, denoted by which satisfies . Putting , we have that

Hence, . So, is a positive solution of (1), and .

(ii) To obtain a negative solution, we only need to replace with

Similar to step (i), it is shown that (1) has a negative solution with .

The proof is completed.

Proof of Theorem 2. We adopt the notations in the proof of Theorem 1.

First of all, we show that the functional is also coercive under the conditions and . Write

where . Given , we have that

Thus, for every , there exists such that

Integrating the equality

over the interval ,

Letting , we show that , .

Suppose that satisfies and for some constant . Let . Up to subsequence if necessary, we may assume that there exists such that

Given in (19), we have that

Let . Thus,

where . It follows from (22) and the previous inequality that

Because the norm is weakly lower semi-continuous, using Poincaré inequality, we get that

Hence, and in with . So, is the corresponding eigenfunction to . Without loss of generality, we may assume that . Thus, a.e. . Consequently, a.e. . Therefore,

which contradicts the fact that . From the fact that is weakly low semi-continuous, we know that it has a global minimizer . As in the proof of Theorem 1, is a positive solution of (1) with . In a similar way, we can obtain a negative solution with negative critical value.

The proof is completed.

Acknowledgments

The author would like to express sincere thanks to the anonymous referee whose careful reading and valuable comments improved the paper. This work is partially supported by the National Natural Science Foundation of China (Grant no. 11071149) and Science Council of Shanxi Province (2010011001-1 and 2012011004-2).

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