Nontrivial Solutions for a Modified Capillary Surface Equation

Zhanping Liang School of Mathematical Sciences, Shanxi University, Taiyuan, Shanxi 030006, China Correspondence should be addressed to Zhanping Liang; lzp@sxu.edu.cn 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. A negative solution and a positive solution are obtained for a modified capillary surface equation by variational methods.

In the case that  = 1, (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  > 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][4][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 .
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  1 > 0 be the first eigenvalue of the problem (5).It is known that  1 is characterized by where  1, 0 (Ω) is the reflexive Banach space defined as the completion of  ∞ 0 (Ω) with respect to the norm ‖‖ := (∫ Ω |∇|  ) 1/ .Also,  1 is single and has an associated eigenfunction  1 > 0 in Ω and ‖ 1 ‖ = 1.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 () ≥ 0 in Ω and  ̸ = 0, respectively, () ≤ 0 in Ω and  ̸ = 0, is a positive, respectively, negative, solution. Define where (, ) = ∫  0 (, ).From a variational stand point, finding solutions of (1) in  1, 0 (Ω) is equivalent to finding critical points of the  1 functional .As to the differentiability of the functional , one can consult [2] for details.Since  satisfies the subcritical growth condition ( 0 ), stand proofs show that  is weakly continuous.Since the function () = √ 1 +  2 is convex, the functional  is also convex.In addition,  belongs to  1 .Hence,  is weakly lower semicontinuous.Thus, we have shown that  is weakly lower semicontinuous.Now, let us state the main results of this paper.
Theorem 1.Let () hold.Furthermore, assume that  satisfies the following conditions.
( 0 ) There is some  > 0 small such that 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).
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 ( 0 )-( 3 ), Liu and Su in [8] have studied the existence of solutions to p-Laplacian quasilinear elliptic equation Under the conditions ( 0 ) and ( 1 ), (10) may be resonant at the eigenvalue  1 near the origin.With the conditions ( 2 ) and ( 3 ), it may be resonant at  1 both near the origin and near infinity.In fact, the condition ( 0 ) allows (10) to be resonant near the origin from the right side of  1 , while the conditions ( 2 ) and ( 3 ) allow it to be resonant at infinity from the left side of  1 .Before concluding this section, we explain some notations used in the paper.|Ω| is the Lebesgue measure of Ω.   ( ∈ N) is always a positive constant independent of functions.⟨⋅, ⋅⟩ is the duality between ( 1, 0 (Ω)) * and  1, 0 (Ω).In addition, we use | ⋅ | to denote the usual norm of R  .

The Proof of the Main Results
In this section, we prove Theorems 1 and 2.
(ii) To obtain a negative solution, we only need to replace f with Similar to step (i), it is shown that (1) has a negative solution  2 with ( 2 ) < 0.
The proof is completed.
Integrating the equality Letting  → +∞, we show that Ĝ(, ) ≤ −/,  ≥   .Suppose that {  } ⊂  1, 0 (Ω) satisfies ‖  ‖ → ∞ and Î(  ) ≤  for some constant  ∈ R. Let V  =   /‖  ‖.Up to subsequence if necessary, we may assume that there exists which contradicts the fact that  ∈ R. From the fact that Î is weakly low semi-continuous, we know that it has a global minimizer  1 .As in the proof of Theorem 1,  1 is a positive solution of (1) with ( 1 ) < 0. In a similar way, we can obtain a negative solution with negative critical value.
The proof is completed.

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