Existence and Multiplicity of Nonnegative Solutions for Quasilinear Elliptic Exterior Problems with Nonlinear Boundary Conditions

Abstract

Existence and multiplicity results are established for quasilinear elliptic problems with nonlinear boundary conditions in an exterior domain. The proofs combine variational methods with a fibering map, due to the competition between the different growths of the nonlinearity and nonlinear boundary term.

1. Introduction

Consider the following quasilinear elliptic problem: −div𝑎(𝑥)|∇𝑢|^𝑝−2∇𝑢+𝑏(𝑥)|𝑢|^𝑝−2()𝑎(𝑥|𝑢=ℎ𝑥,𝑢,inΩ,)|∇𝑢^𝑝−2𝜕𝑢(𝑥𝑢|𝜕𝑛+𝑔)|^𝑝−2𝑢=0,onΓ=𝜕Ω, (1) where Ω is a smooth exterior domain in ℝ^𝑁, 1<𝑝<𝑁, and   𝑛 is the unit vector of the outward normal on the boundary Γ=𝜕Ω.

Equations of the type (1) arise in many and diverse contexts like differential geometry [1], nonlinear elasticity [2], non-Newtonian fluid mechanics [3], glaciology [4], and mathematical biology [5]. As a result, questions concerning the solvability of problem (1) have received great attention; see [6–10].

For ℎ(𝑥,𝑢)=𝑐(𝑥)|𝑢|^𝑞−2𝑢−𝑑(𝑥)|𝑢|^𝑠−2𝑢−𝑒(𝑥)|𝑢|^𝑡−2𝑢 with 𝑞,𝑠∈(1,𝑝^∗), 𝑡>𝑝^∗, by using the fibering method, Kandilakis and Lyberopoulos [6] studied the existence of nonnegative solutions for problem (1) in unbounded domains with a noncompact boundary. When ℎ(𝑥,𝑢)=𝑐(𝑥)|𝑢|^𝑞−2𝑢−𝑑(𝑥)|𝑢|^𝑠−2𝑢 with 𝑞,𝑠∈(1,𝑝^∗), Lyberopoulos [7] studied the existence versus absence of nontrivial weak solutions for problem (1). Similar consideration can be found in Kandilakis and Magiropoulos [8]. In [9], Filippucci et al. established existence and nonexistence results for problem (1) via variational methods combined with the geometrical feature, where ℎ(𝑥,𝑢)=𝜆𝑔(𝑥)|𝑢|^𝑟−2𝑢−|𝑢|^𝑞−2𝑢. Recently, Chen et al. [10] considered the existence and multiple of solutions for problem (1) by the variational principle and the mountain pass lemma.

Motivated by these findings, we consider the following quasilinear elliptic problem: −div𝑎(𝑥)|∇𝑢|^𝑝−2∇𝑢+𝑏(𝑥)|𝑢|^𝑝−2𝑢=𝑓(𝑥)|𝑢|^𝑞−2𝑎(𝑥|𝑢,inΩ,)|∇𝑢^𝑝−2𝜕𝑢(𝑥𝑢|𝜕𝑛+𝑔)|^𝑟−2𝑢=0,onΓ=𝜕Ω, (2) where Ω is a smooth exterior domain in ℝ^𝑁 and  𝑛 is the unit vector of the outward normal on the boundary Γ=𝜕Ω. Since 𝑟≠𝑝, problem (2) is essentially different from problem (1). Using the Nehari manifold and fibering map, Wu [11] considered problem (2) for 𝑝=2; Afrouzi and Rasouli [12] considered problem (2) for 1<𝑝<𝑁.

Throughout this paper, we make the following assumptions.(𝐻₁)1<𝑝<𝑁, 1<𝑞<𝑝^∗, and 1<𝑟<𝑝_∗, where 𝑝^∗=𝑁𝑝/(𝑁−𝑝) and   𝑝_∗=(𝑁−1)𝑝/(𝑁−𝑝).(𝐻₂) The function 𝑎(𝑥)≥𝑎₀>0 and 𝑎(𝑥)∈𝐿^∞(Ω).(𝐻₃) The function 𝑏(𝑥)≥𝑏₀>0 and 𝑏(𝑥)∈𝐿^∞(Ω).(𝐻₄) The function 𝑓(𝑥) satisfies Ω⁺_𝑓∶={𝑥∈Ω∶𝑓(𝑥)>0}≠∅ and 𝑓(𝑥)∈𝐿^𝛿(Ω) with 𝑝/(𝑝−𝑞)≥𝛿>𝑞₀=𝑝^∗/(𝑝^∗−𝑞).(𝐻₅) The function 𝑔(𝑥)≥0,  𝑔(𝑥)≢0 in Γ and 𝑔(𝑥)∈𝐿^𝜎(Γ) with 𝑝/(𝑝−𝑟)≥𝜎>𝑟₀=𝑝_∗/(𝑝_∗−𝑟).

The purpose of this paper is to find existence and multiplicity of nonnegative solutions to problem (2). Our proofs are based on the variational method. The main difficulty is the lack of compactness of the Sobolev embeddings in unbounded domains. To overcome this difficulty, we impose the integrality conditions (𝐻₄)-(𝐻₅) on 𝑓 and 𝑔 to establish compact Sobolev embedding theorems (see Lemmas 3 and 4).

The rest of the paper is organized as follows. In Section 2, we set up the variational framework of the problem and give some preliminaries. Section 3 is devoted to the existence results for problem (2). The multiplicity of nonnegative solutions for problem (2) is considered in the last section.

2. Variational Framework and Some Preliminaries

In this section, we set up the variational framework and give some preliminaries.

Define the weighted Sobolev space 𝐸 as the completion of 𝐶^∞₀(ℝ^𝑁) under the norm ‖𝑢‖^𝑝_𝐸=_Ω|𝑎(𝑥)∇𝑢|^𝑝+𝑏(𝑥)|𝑢|^𝑝𝑑𝑥, (3) which is equivalent to the standard one under assumptions (𝐻₂)-(𝐻₃). Moreover, denote by 𝐿^𝑞(Ω;|𝑓|) and 𝐿^𝑟(Γ;𝑔) the weighted Lebesgue spaces equipped with the norm: ‖𝑢‖^𝑞_{𝐿𝑞(Ω;|𝑓|)}=_Ω𝑓(𝑥)|𝑢|^𝑞‖𝑢‖𝑑𝑥,_{𝑟𝐿𝑟(Γ;𝑔)}=_Γ𝑔(𝑥)|𝑢|^𝑟𝑑𝑆, (4) respectively. The definition of the weak solution of problem (2) reads as follows.

Definition 1. One says 𝑢∈𝐸 is a weak solution of problem (2) if _Ω𝑎(𝑥)|∇𝑢|^𝑝−2∇𝑢∇𝑣𝑑𝑥+_Ω𝑏(𝑥)|𝑢|^𝑝−2+𝑢𝑣𝑑𝑥_Γ𝑔(𝑥)|𝑢|^𝑟−2𝑢𝑣𝑑𝑆=_Ω𝑓(𝑥)|𝑢|^𝑞−2𝑢𝑣𝑑𝑥 (5) holds for all 𝑣∈𝐸.

The energy functional corresponding to problem (2) is 𝐽(𝑢)=1𝑝‖𝑢‖^𝑝_𝐸+1𝑟𝐺(𝑢)−1𝑞𝐹(𝑢), (6) where 𝐺(𝑢)=∫_Γ𝑔(𝑥)|𝑢|^𝑟𝑑𝑆, 𝐹(𝑢)=∫_Ω𝑓(𝑥)|𝑢|^𝑞𝑑𝑥. It is well known that the weak solutions of (2) are the critical points of the energy functional 𝐽(⋅). If 𝑢 is a critical point of 𝐽(⋅), then necessarily 𝑢 belongs to the Nehari manifold: 𝒩=𝑢∈𝐸⧵{0}∶𝐽^′(𝑢),𝑢=0, (7) where 𝐽^′(𝑢),𝑢=‖𝑢‖^𝑝_𝐸+𝐺(𝑢)−𝐹(𝑢). (8) For all 𝑢∈𝒩, we have 𝑢∈𝐹₊, where 𝐹₊={𝐹(𝑢)>0}. (9)

Moreover 𝐽(𝑢)=1𝑝−1𝑞‖𝑢‖^𝑝_𝐸+1𝑟−1𝑞𝐺(𝑢)=1𝑝−1𝑟‖𝑢‖^𝑝_𝐸+1𝑟−1𝑞𝐹(𝑢)=1𝑝−1𝑞𝐹(𝑢)+1𝑟−1𝑝𝐺(𝑢). (10)

The variational framework that we adopt is based on the so-called one-dimensional fibering method proposed by Pohozaev [13]. The central idea of this strategy consists in embedding the original variational problem into the “wider” space 𝐸₁∶=ℝ×𝐸 and then investigating the conditional solvability of the new problem in 𝐸₁ under an appropriately imposed constraint. To this end, we define the extended functional 𝜙∶[0,∞)×𝐸→ℝ by setting for any 𝑣∈𝐸𝜙()()=𝑡𝑡,𝑣=𝐽𝑡𝑣^𝑝𝑝‖𝑣‖^𝑝_𝐸+𝑡^𝑟𝑟𝐺(𝑣)−𝑡^𝑞𝑞𝐹(𝑣),𝑡≥0. (11) If 𝑢=𝑡𝑣 is a critical point of 𝐽(⋅), then necessarily 𝜙_𝑡(𝑡,𝑣)=0; that is, 𝑡^𝑝−1‖𝑣‖^𝑝_𝐸+𝑡^𝑟−1𝐺(𝑣)−𝑡^𝑞−1𝐹(𝑣)=0. (12) In particular, if 𝑡≠0, then (12) is equivalent to 𝜓(𝑡,𝑣)=‖𝑣‖^𝑝_𝐸, (13) where 𝜓(𝑡,𝑣)=𝑡^𝑞−𝑝𝐹(𝑣)−𝑡^𝑟−𝑝𝐺(𝑣). (14)

Now, suppose that 𝑡=𝑡(𝑣)≥0 solves (13) for all 𝑣∈𝐸⧵{0}; then 𝑡∈𝐶¹(𝐸). Furthermore, if 𝑡(𝑣)>0 exists and is unique for all 𝑣∈𝐸⧵{0}, then (13) generates a bijection between 𝐸⧵{0} and 𝒩. Moreover, the following proposition holds.

Lemma 2 (see [13]). If 𝑣 is a conditional critical point of Φ(⋅), under the constraint ‖𝑣‖_𝐸=1, then 𝑢=𝑡(𝑣)𝑣 is a critical point of 𝐽(⋅), where Φ(𝑣)=𝐽(𝑡(𝑣)𝑣) and 𝑡(𝑣) is a nonnegative solution of (13).

In view of Lemma 2, the problem of finding solutions of (2) will be reduced to that of locating the critical point of Φ(⋅) under the constraint ‖𝑣‖_𝐸=1.

The following compact embedding theorems play an important role in the proof of our main results.

Lemma 3. Assume (𝐻₂)–(𝐻₄). Then the embedding 𝐸↪𝐿^𝑞(Ω;|𝑓|) is compact.

Proof. Let 𝑢∈𝐸. Since 𝑝/(𝑝−𝑞)≥𝛿>𝑞₀, it follows that 𝑝≤𝑞𝛿^′=𝑞𝛿/(𝛿−1)<𝑝^∗. Let 𝑊^1,𝑝(Ω) be the standard Banach space endowed with the norm ‖𝑢‖^𝑝_{𝑊1,𝑝(Ω)}=∫_Ω(|∇𝑢|^𝑝+|𝑢|^𝑝)𝑑𝑥. By assumptions (𝐻₂)-(𝐻₃), 𝐸∼𝑊^1,𝑝(Ω). Similar to the proof of [10, Lemma 2] (see also the proof of [14, Theorem 7.9]), we can prove that 𝑊^1,𝑝(Ω)↪𝐿^𝑞𝛿′(Ω) is compact and so is 𝐸↪𝐿^𝑞𝛿′(Ω). Let 𝑆₁ be the best trace embedding constant; that is, 𝑆₁=inf_{𝑢∈𝐸,𝑢≠0}‖𝑢‖^𝑝_𝐸‖𝑢‖^𝑝_{𝐿′𝑞𝛿(Ω)}. (15) By Hölder's inequality, we have _Ω𝑓(𝑥)|𝑢|^𝑞𝑑𝑥≤_Ω𝑓^𝛿𝑑𝑥^1/𝛿_Ω|𝑢|^𝑞𝛿′𝑑𝑥^1/𝛿′≤𝑆₁^−𝑞/𝑝𝑓_𝐿𝛿(Ω)‖𝑢‖^𝑞_𝐸. (16) This shows that the embedding 𝐸↪𝐿^𝑞(Ω;|𝑓|) is continuous.
Assume {𝑢_𝑛} is a bounded sequence in 𝐸. Then by the compact embedding 𝐸↪𝐿^𝑞𝛿′(Ω), there exist 𝑢∈𝐸 and a subsequence of {𝑢_𝑛} (not relabelled) such that 𝑢_𝑛→𝑢 strongly in 𝐿^𝑞𝛿′(Ω).
By Hölder's inequality again, we infer _Ω𝑓(𝑥)𝑢_𝑛−𝑢^𝑞≤𝑑𝑥_Ω𝑓^𝛿𝑑𝑥^1/𝛿_Ω𝑢_𝑛−𝑢^𝑞𝛿′𝑑𝑥^1/𝛿′⟶0as𝑛⟶∞. (17) This completes the proof.

Lemma 4. Assume (𝐻₂)-(𝐻₃) and (𝐻₅). Then the embedding 𝐸↪𝐿^𝑟(Γ;𝑔) is compact.

Proof. Let 𝑢∈𝐸. Since 𝑝/(𝑝−𝑟)≥𝜎>𝑟₀, it follows that 𝑝≤𝑟𝜎^′=𝑟𝜎/(𝜎−1)<𝑝_∗. Hence the embedding 𝐸↪𝐿^𝑟𝜎′(Γ) is compact (see [15, 16]). Let 𝑆₂ be the best trace embedding constant; that is, 𝑆₂=inf_{𝑢∈𝐸,𝑢≠0}‖𝑢‖^𝑝_𝐸‖𝑢‖^𝑝_{𝐿′𝑟𝜎(Γ)}. (18) By Hölder's inequality, we have _Γ𝑔(𝑥)|𝑢|^𝑟𝑑𝑆≤_Γ𝑔^𝜎𝑑𝑆^1/𝜎_Γ|𝑢|^𝑟𝜎′𝑑𝑆^1/𝜎′≤𝑆₂^−𝑟/𝑝𝑔_𝐿𝜎(Γ)‖𝑢‖^𝑟_𝐸. (19) This shows that the embedding 𝐸↪𝐿^𝑟(Γ;𝑔) is continuous.
Assume {𝑢_𝑛} is a bounded sequence in 𝐸. Then by the compact embedding 𝐸↪𝐿^𝑟𝜎′(Γ), there exist 𝑢∈𝐸 and a subsequence of {𝑢_𝑛} (not relabelled) such that 𝑢_𝑛→𝑢 strongly in 𝐿^𝑟𝜎′(Γ).
By Hölder's inequality again, we infer _Γ𝑔(𝑥)𝑢_𝑛−𝑢^𝑟≤𝑑𝑆_Γ𝑔^𝜎𝑑𝑆^1/𝜎_Γ𝑢_𝑛−𝑢^𝑟𝜎′𝑑𝑆^1/𝜎′⟶0as𝑛⟶∞. (20) This completes the proof.

We also need the following mountain pass lemma (see [17, 18]).

Lemma 5. Let 𝐸 be a real Banach space and 𝐽∈𝐶¹(𝐸,ℝ) with 𝐽(0)=0. Suppose(𝐴₁) there are 𝜌,𝛼>0 such that 𝐽(𝑢)≥𝛼 for ‖𝑢‖_𝐸=𝜌;(𝐴₂) there is 𝑒∈𝐸,‖𝑒‖_𝐸>𝜌 such that 𝐽(𝑒)<0.
Define Γ₀=𝛾∈𝐶¹([0,1],𝐸)∣𝛾(0)=0,𝛾(1)=𝑒. (21) Then 𝑐=inf_𝛾∈Γ0max_0≤𝑡≤1𝐽𝛾(𝑡)≥𝛼 (22) is finite and 𝐽(⋅) possess a (PS)_𝑐 sequence at level 𝑐. Furthermore, if 𝐽 satisfies the (PS) condition, then 𝑐 is a critical value of 𝐽.

To get multiplicity results, we need the following fountain theorem due to Bartsch [19] and a critical point theorem in [20, 21].

Let 𝑋 be a reflexive and separable Banach space. It is well known that there exist 𝑒_𝑗∈𝑋 and 𝑒^∗_𝑗∈𝑋^∗ (𝑗=1,2,…) such that(1)⟨𝑒_𝑖,𝑒^∗_𝑗⟩=𝛿_𝑖𝑗, where 𝛿_𝑖𝑗=1 for 𝑖=𝑗 and 𝛿_𝑖𝑗=0 for 𝑖≠𝑗,(2)𝑋=span{𝑒₁,𝑒₂,…} and 𝑋^∗=span{𝑒^∗₁,𝑒^∗₂,…}.

For convenience, we write 𝑋_𝑗=span𝑒_𝑗,𝑌_𝑘=^𝑘_𝑗=1𝑋_𝑗,𝑍_𝑘=^∞_𝑗=𝑘𝑋_𝑗,𝑗,𝑘=1,2,…. (23)

Lemma 6 (fountain theorem [19]). Assume 𝐽∈𝐶¹(𝑋,ℝ¹) is an even functional that satisfies the (PS)_𝑐 condition. If for every 𝑘∈ℕ there exist 𝜌_𝑘>𝑟_𝑘>0 such that(𝐵₁)𝑎_𝑘∶=inf_{𝑢∈𝑍𝑘,‖𝑢‖𝑋=𝑟𝑘}𝐽(𝑢)→∞ as 𝑘→∞,(𝐵₂)𝑏_𝑘∶=max_{𝑢∈𝑌𝑘,‖𝑢‖𝑋=𝜌𝑘}𝐽(𝑢)≤0, then J has a sequence of critical points {𝑢_𝑘} with 𝐽(𝑢_𝑘)→∞.

Lemma 7 (see [20, 21]). Let 𝐽∈𝐶¹(𝑋,ℝ¹), where 𝑋 is a Banach space. Assume that 𝐽 satisfies the (PS)_𝑐 condition and is even and bounded from below, and 𝐽(0)=0. If for any 𝑘∈ℕ there exists a 𝑘-dimensional subspace 𝑌_𝑘 and 𝜌_𝑘>0 such that sup_{𝑢∈𝑌𝑘,‖𝑢‖𝑋=𝜌𝑘}𝐽(𝑢)<0, (24) then 𝐽 has a sequence of critical values 𝑐_𝑘<0 satisfying 𝑐_𝑘→0 as 𝑘→∞.

3. Existence of Nonnegative Solutions

In this section, the existence results are established for problem (2). The proofs combine variational methods with a fibering map. Since 𝐽(𝑢)=𝐽(|𝑢|), we may suppose that the solution to problem (2) is nonnegative throughout this paper.

Theorem 8. Let (𝐻₁)–(𝐻₅) hold with either 𝑞<min{𝑝,𝑟} or 𝑞>max{𝑝,𝑟}. Then problem (2) admits a nonnegative nontrivial weak solution 𝑢∈𝐸⧵{0} which is also a ground state.

Proof. Suppose 𝑞<min{𝑝,𝑟}. Rewriting (13) as 𝑡^𝑝−𝑞‖𝑣‖^𝑝_𝐸+𝑡^𝑟−𝑞𝐺(𝑣)=𝐹(𝑣), (25) we immediately see that for every 𝑣∈𝐹₊ (where 𝐹₊ is defined by (9)) there exists a unique 𝑡(𝑣)>0 satisfying (25). Moreover, it can be easily checked that 𝜇𝑡𝜇𝑣=𝑡(𝑣),for𝜇>0,𝑣∈𝐹₊,Φ(𝑣)=1𝑝−1𝑞(𝑡(𝑣))^𝑝‖𝑣‖^𝑝_𝐸+1𝑟−1𝑞(𝑡(𝑣))^𝑟𝐺(𝑣)<0∀𝑣∈𝐹₊. (26)(27)
Consider now the variational problem 𝑀∶=inf_{𝑣∈𝐹+,‖𝑣‖𝐸=1}Φ(𝑣)<0. (28) Let {𝑣_𝑛} be a minimizing sequence in 𝐹₊ with ‖𝑣_𝑛‖_𝐸=1. Then there exists 𝑣₀∈𝐸 such that 𝑣_𝑛⇀𝑣₀ in 𝐸. By Lemmas 3 and 4, we have 𝐹(𝑣_𝑛)→𝐹(𝑣₀)≥0 and 𝐺(𝑣_𝑛)→𝐺(𝑣₀)≥0.
We first assert that 𝑣₀∈𝐹₊. Suppose the contrary; then 𝐹(𝑣₀)=0. In view of (25), 𝐹𝑣_𝑛≥𝑡(𝑣_𝑛)^𝑝−𝑞. (29)
Letting 𝑛→∞, it follows that 𝑡(𝑣_𝑛)→0. Thus Φ𝑣_𝑛=1𝑝−1𝑞𝑡𝑣_𝑛^𝑝𝑣_𝑛^𝑝_𝐸+1𝑟−1𝑞𝑡𝑣_𝑛^𝑟𝐺𝑣_𝑛⟶0,as𝑛⟶∞, (30) which contradicts 𝑀<0.
Next, we prove ‖𝑣₀‖_𝐸=1. If not, then ‖𝑣₀‖_𝐸<liminf_𝑛→∞‖𝑣_𝑛‖_𝐸=1. So, there exists 𝜇>1 such that ‖𝜇𝑣₀‖_𝐸=1. From (25), we have 𝑡(𝜇𝑣₀)^𝑝−𝑞+𝑡(𝜇𝑣₀)^𝑟−𝑞𝐺𝜇𝑣₀=𝐹𝜇𝑣₀. (31) This and (26) yield 𝜇^−𝑝𝑡(𝑣₀)^𝑝−𝑞+𝑡(𝑣₀)^𝑟−𝑞𝐺𝑣₀=𝐹𝑣₀. (32)
On the other hand, it follows from (29) that {𝑡(𝑣_𝑛)} is bounded and so there exists a subsequence (not relabelled) such that 𝑡(𝑣_𝑛)→𝑡₀>0. Thus by (25), we have 𝑡₀^𝑝−𝑞+𝑡₀^𝑟−𝑞𝐺𝑣₀=𝐹𝑣₀. (33)
Hence 𝑡₀<𝑡(𝑣₀). Notice that Ψ(𝑡)=1𝑝−1𝑞𝑡^𝑝𝑣₀^𝑝_𝐸+1𝑟−1𝑞𝑡^𝑟𝐺𝑣₀ (34) is strictly decreasing for all 𝑡>0; we have 𝑀=liminf_𝑛→∞Φ𝑣_𝑛≥Ψ𝑡₀>Ψ𝑡𝑣₀=Φ𝑣₀, (35) which is a contradiction. So, ‖𝑣₀‖_𝐸=1 and 𝑣₀ is a critical point of Φ(⋅). By Lemma 2, 𝑢=𝑡(𝑣₀)𝑣₀ is a nontrivial solution of problem (2). Since 𝑡=𝑡(𝑣) is a unique solution of (25), then (25) generates a bijection between 𝐸⧵{0} and 𝒩 and so the obtained solution is actually a ground state.
The case 𝑞>max{𝑝,𝑟} can be treated in a similar way.

Remark 9. Afrouzi and Rasouli [12] consider the following problem: −div|∇𝑢|^𝑝−2∇𝑢+𝑏(𝑥)|𝑢|^𝑝−2𝑢=𝜆𝑓(𝑥)|𝑢|^𝑞−2||𝑢,inΩ,∇𝑢^𝑝−2𝜕𝑢(𝑥𝑢|𝜕𝑛=𝑔)|^𝑟−2𝑢,onΓ=𝜕Ω, (36) where Ω is a bounded domain in ℝ^𝑁 and 1<𝑞<𝑝<𝑟<𝑝_∗=(𝑁−1)𝑝/(𝑁−𝑝). The functions 𝑓 and 𝑔 are continuous functions which change sign in Ω. Using the Nehari manifold and fibering map, they proved that problem (36) has at least two nontrivial nonnegative solutions if |𝜆| is sufficiently small. In fact, by slight modification, we can prove that the result they established is still true if Ω is a smooth exterior domain or the parameters satisfy 1<𝑟<𝑝<𝑞<𝑝^∗=𝑁𝑝/(𝑁−𝑝). But for 𝑔(𝑥)≥0, we can prove that problem (36) has at least one nontrivial nonnegative solution for |𝜆| is sufficiently small via the method used in [12]. Notice that our result (Theorem 8) does not need |𝜆| to be small.

Theorem 10. Let (𝐻₁)–(𝐻₅) hold with 𝑟<𝑞<𝑝. Then problem (2) has a nonnegative nontrivial weak solution.

Proof. From Lemma 3, we have 𝐹(𝑢)≤𝑆₁^−𝑞/𝑝𝑓_𝐿𝛿(Ω)‖𝑢‖^𝑞_𝐸. (37) Thus 𝐽(𝑢)≥1𝑝‖𝑢‖^𝑝_𝐸−1𝑞𝑆₁^−𝑞/𝑝𝑓_𝐿𝛿(Ω)‖𝑢‖^𝑞_𝐸. (38) So 𝐽(𝑢) is coercive by 𝑝>𝑞.
By Lemmas 3 and 4, it is easy to verify that 𝐽 is weakly lower semicontinuous. So 𝐽 has a minimum point 𝑢 in 𝐸 and 𝑢 is a weak solution of (2).
In the following, we prove inf_𝑢∈𝐸𝐽(𝑢)<0. Let 𝑣∈𝑊₀^1,𝑝(Ω)∩𝐹₊. Then 𝐽()=1𝑡𝑣𝑝𝑡^𝑝‖𝑣‖^𝑝_𝐸−1𝑞𝑡^𝑞𝐹(𝑣). (39) Thus 𝐽(𝑡𝑣)<0 for 𝑡>0 is sufficiently small. Notice that 𝑊₀^1,𝑝(Ω)∩𝐹₊⊂𝐸; we obtain inf_𝑣∈𝐸𝐽(𝑣)≤inf_{𝑣∈𝑊01,𝑝(Ω)∩𝐹+}𝐽(𝑣)<0. (40) Thus the minimum point of 𝐽 is nontrivial.

Theorem 11. Let (𝐻₁)–(𝐻₅) hold with 𝑝<𝑞<𝑟. Then problem (2) has a nonnegative nontrivial weak solution in 𝑊₀^1,𝑝(Ω)∩𝐹₊.

Proof. Let 𝑢∈𝑊₀^1,𝑝(Ω)∩𝐹₊. Then 𝐽()=𝑡𝑡𝑢^𝑝𝑝‖𝑢‖^𝑝_𝐸−𝑡^𝑞𝑞𝐹(𝑢). (41) Thus 𝐽(𝑡𝑢)≥𝛼>0 for 𝑡 small and 𝐽(𝑡𝑢)<0 for 𝑡 large.
Let {𝑢_𝑛}⊂𝑊₀^1,𝑝(Ω)∩𝐹₊ satisfy 𝐽(𝑢_𝑛)→𝑐 in 𝐸 and 𝐽^′(𝑢_𝑛)→0 in 𝐸^∗. Then 𝑐+1≥𝐽𝑢_𝑛−1𝑞𝐽^′𝑢_𝑛,𝑢_𝑛=1𝑝−1𝑞𝑢_𝑛^𝑝_𝐸. (42) This shows that ‖𝑢_𝑛‖_𝐸 is bounded in 𝐸. Up to a subsequence, we obtain 𝑢_𝑛⇀𝑢 in 𝐸. Thus _Ω𝑎(𝑥)|∇𝑢|^𝑝−2∇𝑢∇𝑢_𝑛+−𝑢𝑑𝑥_Ω𝑏(𝑥)|𝑢|^𝑝−2𝑢𝑢_𝑛−𝑢𝑑𝑥⟶0. (43) It follows from Lemmas 3 and 4 that 𝑢_𝑛⟶𝑢in𝐿^𝑞Ω;𝑓,𝐿^𝑟Γ;𝑔. (44)
Hence _Ω𝑓(𝑥)|𝑢|^𝑞−2𝑢𝑢_𝑛−𝑢𝑑𝑥⟶0,_Γ𝑔(𝑥)𝑢_𝑛^𝑟−2𝑢_𝑛𝑢_𝑛𝐵−𝑢𝑑𝑆⟶0,_𝑛∶=_Ω𝑓(𝑥)𝑢_𝑛^𝑞−2𝑢_𝑛−|𝑢|^𝑞−2𝑢𝑢_𝑛𝐶−𝑢𝑑𝑥⟶0,_𝑛∶=_Γ𝑔(𝑥)𝑢_𝑛^𝑟−2𝑢_𝑛−|𝑢|^𝑟−2𝑢𝑢_𝑛−𝑢𝑑𝑆⟶0. (45) Then (45) give that ⟨𝐽^′(𝑢),𝑢_𝑛−𝑢⟩→0. Notice that 𝐽^′(𝑢_𝑛)→0; we have 𝐽^′𝑢_𝑛−𝐽^′(𝑢),𝑢_𝑛−𝑢=𝐴_𝑛+𝐵_𝑛−𝐶_𝑛⟶0, (46) where 𝐴_𝑛=_Ω𝑎(𝑥)∇𝑢_𝑛^𝑝−2∇𝑢_𝑛−|∇𝑢|^𝑝−2∇𝑢∇𝑢_𝑛+−𝑢𝑑𝑥_Ω𝑏(𝑥)𝑢_𝑛^𝑝−2𝑢_𝑛−|𝑢|^𝑝−2𝑢𝑢_𝑛−𝑢𝑑𝑥. (47)
Using the standard inequality in ℝ^𝑁 given by 𝜉^𝑝−2𝜉−𝜂^𝑝−2𝜂,𝜉−𝜂≥𝐶_𝑝𝜉−𝜂^𝑝𝜉,𝑝≥2,_𝑝−2𝜉−𝜂^𝑝−2𝜂,𝜉−𝜂≥𝐶_𝑝𝜉−𝜂²𝜉+𝜂^𝑝−2,1<𝑝<2, (48) we have from (46) that 𝑢_𝑛→𝑢 in 𝐸. Thus 𝐽(⋅) satisfies (PS) condition. Then the assertion of this theorem follows from Lemma 5.

Next, we seek for a solution in 𝐸⧵𝑊₀^1,𝑝(Ω) with 𝑝<𝑞<𝑟. In this case, we find it necessary to strengthen our hypothesis by assuming that the function 𝑔(⋅) is positive. That is, (𝐻₅) will be replaced by(𝐻^′₅) the function 𝑔(𝑥)>0 and 𝑔(𝑥)∈𝐿^𝜎(Γ) with 𝑝/(𝑝−𝑟)≥𝜎>𝑟₀=𝑝_∗/(𝑝_∗−𝑟).

Theorem 12. Let (𝐻₁)–(𝐻₄) and (𝐻^′₅) hold with 𝑝<𝑞<𝑟. Suppose also ℳ∶=(𝐹(𝑣))^𝑟−𝑝>𝜉(𝐺(𝑣))^𝑞−𝑝‖𝑣‖_𝐸^{𝑝(𝑟−𝑞)}≠∅, (49) where 𝜉=(𝑟−𝑝)^𝑟−𝑝/((𝑞−𝑝)^𝑞−𝑝(𝑟−𝑞)^𝑟−𝑞)(𝑟/𝑝)^𝑟−𝑞. Then problem (2) has a nonnegative nontrivial weak solution in 𝐸/𝑊₀^1,𝑝(Ω).

Proof. Define ℒ_𝑣0=𝑣𝑣∈ℳ∶𝐺‖𝑣‖_𝐸𝑣≥𝐺⁰𝑣_0𝐸. (50) Fix 𝑣₀ with ℒ_𝑣0≠∅. In view of assumption (𝐻^′₅), we have 𝐺(𝑣₀)>0. For all 𝑢∈ℒ_𝑣0, there exist 𝑡>0 and 𝑣∈ℒ_𝑣0 with ‖𝑣‖_𝐸=1 such that 𝑢=𝑡𝑣. Moreover, 𝐺(𝑣)≥𝐺(𝑣₀/‖𝑣₀‖_𝐸)>0. Thus 𝐽(𝑢)=𝑡^𝑝𝑝‖𝑣‖^𝑝_𝐸+𝑡^𝑟𝑟𝐺(𝑣)−𝑡^𝑞𝑞𝐹(𝑣)⟶∞, (51) as 𝑡→∞. Hence 𝐽(⋅) is coercive in ℒ_𝑣0. By Lemmas 3 and 4, it is easy to verify that 𝐽 is weakly lower semicontinuous. So 𝐽 has a minimum point 𝑢 in ℒ_𝑣0 and 𝑢 is a weak solution of (2).
In the following, we prove that inf_{𝑢∈ℒ𝑣0}𝐽(𝑢)<0. Notice that 𝜓(𝑡,𝑣)=0 as 𝑡→0, 𝜓(𝑡,𝑣)=−∞ as 𝑡→+∞, and 𝜓^′_𝑡(𝑡,𝑣)=(𝑞−𝑝)𝑡^{𝑞−𝑝−1}𝐹(𝑣)−(𝑟−𝑝)𝑡^{𝑟−𝑝−1}𝐺(𝑣); we infer that 𝜓(𝑡,𝑣) attain its maximum at 𝑡_𝑀(𝑣), where 𝑡_𝑀(𝑣)=𝑞−𝑝𝑟−𝑝𝐹(𝑣)𝐺(𝑣)^{1/(𝑟−𝑞)}. (52)
If 0<‖𝑣‖^𝑝_𝐸<𝜓(𝑡_𝑀(𝑣),𝑣), then (13) has exactly two solutions 𝑡₁(𝑣) and 𝑡₂(𝑣) with 0<𝑡₁(𝑣)<𝑡_𝑀(𝑣)<𝑡₂(𝑣). Let 𝑡(𝑣)=𝑡₂(𝑣). We have from (11) and (12) that 𝐽(𝑡(𝑣)𝑣)=𝑞−𝑝(𝑡(𝑣𝑝𝑞))^𝑝‖𝑣‖^𝑝_𝐸−𝑟−𝑞(𝑡(𝑣𝑞𝑟))^𝑟𝐺(𝑣). (53)
Let 𝑣∈ℒ_𝑣0. It follows that (𝑡(𝑣))^𝑟−𝑝>𝑡_𝑀(𝑣)^𝑟−𝑝>𝑟𝑞−𝑝‖𝑣‖^𝑝_𝐸𝑝𝐺(𝑣),𝑟−𝑞 (54) which ensures 𝐽(𝑡(𝑣)𝑣)<0. Thus inf_{𝑢∈ℒ𝑣0}𝐽(𝑢)<0. This implies that the weak solution of (2) is nontrivial.

Remark 13. Condition (49) may be viewed as grading the “strength” of interaction induced by 𝐹(𝑢) and 𝐺(𝑢). Hence, qualitatively speaking, one may rephrase Theorem 12 as saying that if 𝑝<𝑞<𝑟, then problem (2) admits a nontrivial weak solution provided that 𝐹(⋅) “prevails” over 𝐺(⋅).

4. Multiplicity of Nonnegative Solutions

In this section, we establish multiplicity results for the cases 𝑞>max{𝑝,𝑟} and 𝑞<min{𝑝,𝑟} by Lemmas 6 and 7, respectively. To this purpose, the assumption (𝐻₄) will be replaced by the following:(𝐻^′₄) the function 𝑓(𝑥) satisfies 𝑓(𝑥)>0 in Ω and 𝑓(𝑥)∈𝐿^𝛿(Ω) with 𝑝/(𝑝−𝑞)≥𝛿>𝑞₀=𝑝^∗/(𝑝^∗−𝑞).

Theorem 14. Let (𝐻₁)–(𝐻₃), (𝐻^′₄), and (𝐻₅) hold with 𝑞>max{𝑝,𝑟}. Then problem (2) has a sequence of solutions 𝑢_𝑘 in E with 𝐽(𝑢_𝑘)→∞ as 𝑘→∞.

Proof. We will prove this theorem by fountain theorem. The proof is divided into three steps.
(1) Let 𝑌_𝑘 and 𝑍_𝑘 be defined by (23) and 𝛽_𝑘=sup_{𝑢∈𝑍𝑘,‖𝑢‖𝐸=1}‖𝑢‖_𝐿^𝑞_(Ω;𝑓). Then it follows that 𝛽_𝑘→0 (see [22]). Therefore, we have 𝐽(𝑢)≥1𝑝‖𝑢‖^𝑝_𝐸−1𝑞𝐹(𝑢)≥1𝑝‖𝑢‖^𝑝_𝐸−1𝑞𝛽^𝑞_𝑘‖𝑢‖^𝑞_𝐸. (55) Choosing 𝑟_𝑘=(𝑞/(2𝑝𝛽^𝑞_𝑘))^{1/(𝑞−𝑝)}, we obtain that if 𝑢∈𝑍_𝑘,‖𝑢‖_𝐸=𝑟_𝑘, then 𝐽(𝑢)≥1𝑟2𝑝^𝑝𝑘⟶+∞. (56) Thus (𝐵₁) in Lemma 6 is proved.
(2) Since in the finite dimensional space 𝑌_𝑘 all norms are equivalent, there exist 𝐶>0 such that 𝐹(𝑢)≥𝐶‖𝑢‖^𝑞_𝐸 hold for all 𝑢∈𝑌_𝑘. Thus by (19), 𝐽(𝑢)≤1𝑝‖𝑢‖^𝑝_𝐸+1𝑟𝑆₂^−𝑟/𝑝𝑔_𝐿𝜎(Γ)‖𝑢‖^𝑟_𝐸−1𝑞𝐶‖𝑢‖^𝑞_𝐸,∀𝑢∈𝑌_𝑘. (57) Therefore (𝐵₂) in Lemma 6 is satisfied for every 𝜌_𝑘∶=‖𝑢‖_𝐸>0 large enough.
(3) Let {𝑢_𝑛} be a (PS)_𝑐 sequence of 𝐽. Then we have 𝑐+1≥𝐽𝑢_𝑛−1𝑞𝐽^′𝑢_𝑛,𝑢_𝑛=1𝑝−1𝑞𝑢_𝑛^𝑝_𝐸+1𝑟−1𝑞𝐺(𝑢)≥1𝑝−1𝑞𝑢_𝑛^𝑝_𝐸. (58) Therefore {𝑢_𝑛} is bounded in 𝐸. Similar to the proof of Theorem 11, we can verify that 𝐽(⋅) satisfies (PS) condition.
Obviously, 𝐽(⋅) is an even functional and 𝐽(0)=0. Thus the assertion of Theorem 14 follows from Lemma 6.

Theorem 15. Let (𝐻₁)–(𝐻₃), (𝐻^′₄), and (𝐻₅) hold with 𝑞<min{𝑝,𝑟}. Then problem (2) has a sequence of solutions 𝑢_𝑘 in E with 𝐽(𝑢_𝑘)<0 and 𝐽(𝑢_𝑘)→0 as 𝑘→∞.

Proof. Since 𝐽(𝑢)=1𝑝‖𝑢‖^𝑝_𝐸+1𝑟𝐺(𝑢)−1𝑞𝐹(𝑢)≥1𝑝‖𝑢‖^𝑝_𝐸−1𝑞𝑆₁^−𝑞/𝑝𝑓_𝐿𝛿(Ω)‖𝑢‖^𝑞_𝐸 (59) and 𝑞<𝑝,  𝐽(⋅) is coercive and bounded from below. As before, we can verify that (PS) condition holds.
Let 𝑌_𝑘 be defined by (23) and 𝑢∈𝑌_𝑘. Since 𝑌_𝑘 is a finite dimensional space and 𝑞<min{𝑝,𝑟}, we can choose 𝜌_𝑘>0 small enough such that sup_{𝑢∈𝑌𝑘,‖𝑢‖𝐸=𝜌𝑘}𝐽(𝑢)≤1𝑝‖𝑢‖^𝑝_𝐸+1𝑟𝑆₂^−𝑟/𝑝𝑔_𝐿𝜎(Γ)‖𝑢‖^𝑟_𝐸−1𝑞𝐶‖𝑢‖^𝑞_𝐸<0. (60) We obtain a sequence of solutions 𝑢_𝑘 by Lemma 7.

Acknowledgment

This work is supported by the Fundamental Research Funds for the Central Universities (2013B09914).