Nontrivial Solutions for a Modified Capillary Surface Equation

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: ||−div∇𝑢^2𝑝−2∇𝑢1+|∇𝑢|^2𝑝=𝑓(𝑥,𝑢)inΩ,𝑢=0on𝜕Ω, (1) where 𝑝>1, and Ω is a bounded domain in ℝ^𝑁 with smooth boundary. The function 𝑓∈𝐶(Ω×ℝ,ℝ) with the subcritical growth (𝑓)𝑓(𝑥,𝑡)≤𝑐1+|𝑡|^𝑞−1,𝑡∈ℝ,𝑥∈Ω, (2)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 that 𝑝=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 ||−div∇𝑢^2𝑝−2∇𝑢1+|∇𝑢|^2𝑝=𝜆𝑓(𝑥,𝑢)inΩ,𝑢≥0inΩ,𝑢=0on𝜕Ω, (3) 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 1+|∇𝑢|^2𝑝−1∼|∇𝑢|^𝑝as|∇𝑢|→∞, (4) the other object is to investigate the probability to present the property of 𝑓 by the eigenvalue of the problem −Δ_𝑝𝑢=𝜆|𝑢|^𝑝−2𝑢inΩ,𝑢=0on𝜕Ω, (5) where Δ_𝑝𝑢=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 𝜆₁∶=inf_Ω|∇𝑢|^𝑝𝑑𝑥∶_Ω|𝑢|^𝑝𝑑𝑥=1,𝑢∈𝑊₀^1,𝑝(Ω)⧵{0}, (6) where 𝑊₀^1,𝑝(Ω) is the reflexive Banach space defined as the completion of 𝐶^∞₀(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 solution 𝑢 of (1), we mean that 𝑢 satisfies (1) in the weak sense; that is, for all 𝜑∈𝑊₀^1,𝑝(Ω), _Ω|∇𝑢|^2𝑝−2∇𝑢∇𝜑1+|∇𝑢|^2𝑝𝑑𝑥=_Ω𝑓(𝑥,𝑢)𝜑𝑑𝑥. (7) 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 𝐽(𝑢)=1𝑝_Ω1+|∇𝑢|^2𝑝−1𝑑𝑥,𝑢∈𝑊₀^1,𝑝(Ω),𝐾(𝑢)=_Ω𝐹(𝑥,𝑢)𝑑𝑥,𝑢∈𝑊₀^1,𝑝(Ω),𝐼(𝑢)=𝐽(𝑢)−𝐾(𝑢),𝑢∈𝑊₀^1,𝑝(Ω), (8) where 𝐹(𝑥,𝑡)=∫^𝑡₀𝑓(𝑥,𝑠)𝑑𝑠. From a variational stand point, finding solutions of (1) in 𝑊₀^1,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+𝑡^2𝑝 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 𝑝𝐹(𝑥,𝑡)≥𝜆₁|𝑡|^𝑝,|𝑡|≤𝑟,𝑥∈Ω, (9) (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 −Δ_𝑝𝑢=𝑓(𝑥,𝑢)inΩ,𝑢=0on𝜕Ω. (10) 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. Theorems 1 and 2 have shown a new fact that the interaction between the first eigenvalue of −Δ_𝑝 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. 𝑐_𝑖(𝑖∈ℕ) is always a positive constant independent of functions. ⟨⋅,⋅⟩ is the duality between (𝑊₀^1,𝑝(Ω))^∗ and 𝑊₀^1,𝑝(Ω). In addition, we use |⋅| to denote the usual norm of ℝ^𝑁.

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 𝑓()=𝑓()𝐹()=𝑥,𝑡𝑥,𝑡,𝑡≥0,0,𝑡<0,𝑥,𝑡^𝑡₀𝑓()𝐼̂(𝑢)(𝑢)−𝑥,𝑠𝑑𝑠,=𝐽_Ω𝐹(𝑥,𝑢)𝑑𝑥,𝑢∈𝑊₀^1,𝑝(Ω). (11) Since 𝑓∈𝐶(Ω×ℝ,ℝ) and (𝑓₁) holds, for any given 𝜀>0, there exists 𝑐₁>0 such that 𝐹()≤1𝑥,𝑡𝑝𝜆₁−𝜀|𝑡|^𝑝+𝑐₁,𝑡∈ℝ,𝑥∈Ω. (12) By the Poincaré inequality, for 𝑢∈𝑊₀^1,𝑝(Ω), 𝐼̂(𝑢)=1𝑝_Ω1+|∇𝑢|^2𝑝𝑑𝑥−_Ω𝐹()|Ω|𝑥,𝑢𝑑𝑥−𝑝≥1𝑝_Ω|∇𝑢|^𝑝1𝑑𝑥−𝑝𝜆₁−𝜀_Ω|𝑢|^𝑝−𝑐𝑑𝑥₁+1𝑝|Ω|≥1𝑝𝜆1−¹−𝜀𝜆₁‖𝑢‖^𝑝−𝑐₁+1𝑝|Ω|=𝜀𝑝𝜆₁‖𝑢‖^𝑝−𝑐₁+1𝑝|Ω|. (13) Hence, 𝐼̂ is coercive; that is, 𝐼(̂𝑢)→∞ as 𝑛→∞. In addition, since 𝑓 also satisfies the condition (𝑓), 𝐼̂ is weakly lower semi-continuous. So, it has a global minimizer.
Take a number 𝑡₀>0 such that 0<𝑡₀𝜑₁≤𝑟 in Ω. By the condition (𝑓₀), we have that 𝐼̂𝑡₀𝜑₁=1𝑝_Ω1+𝑡₀^2𝑝∇𝜑₁^2𝑝−−1𝑑𝑥_Ω𝐹𝑥,𝑡₀𝜑₁<1𝑑𝑥𝑝𝑡^𝑝_0Ω|∇𝜑₁|^𝑝1𝑑𝑥−𝑝𝜆₁𝑡^𝑝_0Ω𝜑^𝑝₁𝑑𝑥=0. (14) Thus, the global minimizer of 𝐼̂ is a nontrivial critical point, denoted by 𝑢₁ which satisfies 𝐼(̂𝑢₁)<0. Putting 𝑢⁻₁(𝑥)=min{𝑢₁(𝑥),0}, we have that 𝐼^′̂𝑢₁,𝑢⁻₁=_Ω|∇𝑢⁻₁|^2𝑝1+|∇𝑢⁻₁|^2𝑝𝑑𝑥=0. (15) Hence, 𝑢⁻₁=0. So, 𝑢₁ is a positive solution of (1), and 𝐼(𝑢₁)<0.
(ii) To obtain a negative solution, we only need to replace 𝑓 with 𝑓()=𝑓()𝑥,𝑡0,𝑡>0,𝑥,𝑡,𝑡≤0. (16) Similar to step (i), it is shown that (1) has a negative solution 𝑢₂ with 𝐼(𝑢₂)<0.
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 𝐼̂ is also coercive under the conditions (𝑓₂) and (𝑓₃). Write 𝐹()=1𝑥,𝑡𝑝𝜆₁𝑡^+𝑝+𝐺(),𝑓()𝑥,𝑡𝑥,𝑡=𝜆₁|𝑡|^𝑝−2𝑡⁺+𝑔(𝑥,𝑡), (17) where 𝑡⁺=max{𝑡,0}. Given 𝑥∈Ω, we have that lim_𝑡→+∞𝑝𝐺(𝑥,𝑡)𝑡^𝑝=0,lim_𝑡→+∞𝑔(𝑥,𝑡)𝑡−𝑝𝐺(𝑥,𝑡)=+∞. (18) Thus, for every 𝑀>0, there exists 𝑅_𝑀>0 such that 𝑔(𝑥,𝑡)𝑡−𝑝𝐺(𝑥,𝑡)≥𝑀,𝑡≥𝑅_𝑀,𝑥∈Ω. (19) Integrating the equality 𝑑𝐺()𝑑𝑡𝑥,𝑡𝑡^𝑝=𝑔(𝑥,𝑡)𝑡−𝑝𝐺(𝑥,𝑡)𝑡^𝑝+1 (20) over the interval [𝑡,𝑇]⊂[𝑅_𝑀,+∞), 𝐺(𝑥,𝑇)𝑇^𝑝−𝐺(𝑥,𝑡)𝑡^𝑝≥𝑀𝑝1𝑡^𝑝−1𝑇^𝑝. (21) Letting 𝑇→+∞, we show that 𝐺(𝑥,𝑡)≤−𝑀/𝑝, 𝑡≥𝑅_𝑀.
Suppose that {𝑢_𝑛}⊂𝑊₀^1,𝑝(Ω) satisfies ‖𝑢_𝑛‖→∞ and 𝐼(̂𝑢_𝑛)≤𝐶 for some constant 𝐶∈ℝ. Let 𝑣_𝑛=𝑢_𝑛/‖𝑢_𝑛‖. Up to subsequence if necessary, we may assume that there exists 𝑣₀∈𝑊₀^1,𝑝(Ω) such that 𝑣_𝑛⇀𝑣₀𝑣in𝐸,_𝑛→𝑣₀in𝐿^𝑝(Ω),𝑣_𝑛(𝑥)→𝑣₀(𝑥)a.e.𝑥∈Ω. (22) Given 𝑀=1 in (19), we have that 𝐺()1𝑥,𝑡≤−𝑝,𝑡≥𝑅₁. (23) Let 𝑐₂=max_{(𝑥,𝑡)∈Ω×[−𝑅1,𝑅1]}|𝐺(𝑥,𝑡)|. Thus, 𝐶𝑢_𝑛^𝑝≥1𝑝𝑢_𝑛^𝑝Ω1+|∇𝑢|^2𝑝𝑑𝑥−𝜆_1Ω𝑢_𝑛^𝑝−1𝑑𝑥𝑢_𝑛^𝑝Ω𝐺𝑥,𝑢_𝑛|Ω|𝑑𝑥−𝑝𝑢_𝑛^𝑝≥1𝑝_Ω∇𝑣_𝑛^𝑝−𝜆₁𝑣_𝑛^𝑝|Ω|𝑑𝑥−𝑝𝑢_𝑛^𝑝−1𝑢_𝑛^𝑝|𝑢_𝑛^|≥𝑅₁𝐺𝑥,𝑢_𝑛−1𝑑𝑥𝑢_𝑛^𝑝|𝑢_𝑛^|≤𝑅₁𝐺𝑥,𝑢_𝑛≥1𝑑𝑥𝑝_Ω∇𝑣_𝑛^𝑝−𝜆₁𝑣_𝑛^𝑝|Ω|𝑑𝑥−𝑝𝑢_𝑛^𝑝−𝑐₂|Ω|𝑢_𝑛^𝑝=1𝑝_Ω∇𝑣_𝑛^𝑝−𝜆₁𝑣_𝑛^𝑝𝑐𝑑𝑥+³𝑢_𝑛^𝑝, (24) where 𝑐₃=(1/𝑝+𝑐₂)|Ω|. It follows from (22) and the previous inequality that limsup_𝑛→∞Ω|∇𝑣_𝑛|^𝑝𝑑𝑥≤𝜆_1Ω|𝑣₀|^𝑝𝑑𝑥. (25) Because the norm is weakly lower semi-continuous, using Poincaré inequality, we get that limsup_𝑛→∞Ω|∇𝑣_𝑛|^𝑝𝑑𝑥≤𝜆_1Ω𝑣₀^𝑝𝑑𝑥≤_Ω∇𝑣₀^𝑝𝑑𝑥≤liminf_𝑛→∞Ω∇𝑣_𝑛^𝑝𝑑𝑥≤limsup_𝑛→∞Ω|∇𝑣_𝑛|^𝑝𝑑𝑥. (26) Hence, ∫_Ω|∇𝑣₀|^𝑝𝑑𝑥=𝜆₁∫_Ω|𝑣₀|^𝑝𝑑𝑥 and 𝑣_𝑛→𝑣₀ in 𝑊₀^1,𝑝(Ω) with ‖𝑣₀‖=1. So, 𝑣₀ is the corresponding eigenfunction to 𝜆₁. Without loss of generality, we may assume that 𝑣₀=𝜑₁. Thus, 𝑢_𝑛→+∞ a.e. 𝑥∈Ω. Consequently, 𝐺(𝑥,𝑢_𝑛(𝑥))→−∞ a.e. 𝑥∈Ω. Therefore, 𝐶≥−_Ω𝐺𝑥,𝑢_𝑛𝑑𝑥→+∞, (27) which contradicts the fact that 𝐶∈ℝ. From the fact that 𝐼̂ 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.

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).