A Multiplicity Result for Quasilinear Problems with Nonlinear Boundary Conditions in Bounded Domains

Abstract

We study the following quasilinear problem with nonlinear boundary condition −Δ_𝑝𝑢−𝜆𝑎(𝑥)𝑢|𝑢|^𝑝−2=𝑏(𝑥)𝑢|𝑢|^𝛾−2, in Ω and (1−𝛼)|∇𝑢|^𝑝−2(𝜕𝑢/𝜕𝑛)+𝛼𝑢|𝑢|^𝑝−2=0, on 𝜕Ω, where Ω⊆𝑅^𝑁 is a connected bounded domain with smooth boundary 𝜕Ω, the outward unit normal to which is denoted by 𝑛. Δ_𝑝 is the 𝑝-Laplcian operator defined by Δ_𝑝𝑢=div(|∇𝑢|^𝑝−2∇𝑢), the functions 𝑎 and 𝑏 are sign changing continuous functions in Ω, 1<𝑝<𝛾<𝑝^∗, where 𝑝^∗=𝑁𝑝/(𝑁−𝑝) if 𝑁>𝑝 and ∞ otherwise. The properties of the first eigenvalue 𝜆⁺₁(𝛼) and the associated eigenvector of the related eigenvalue problem have been studied in (Khademloo, In press). In this paper, it is shown that if 𝜆≤𝜆⁺₁(𝛼), the original problem admits at least one positive solution, while if 𝜆⁺₁(𝛼)<𝜆<𝜆^∗, for a positive constant 𝜆^∗, it admits at least two distinct positive solutions. Our approach is variational in character and our results extend those of Afrouzi and Khademloo (2007) in two aspects: the main part of our differential equation is the 𝑝-Laplacian, and the boundary condition in this paper also is nonlinear.

1. Introduction and Results

In this paper, we consider the problem−Δ_𝑝𝑢−𝜆𝑎(𝑥)𝑢|𝑢|^𝑝−2=𝑏(𝑥)𝑢|𝑢|^𝛾−2(|,𝑥∈Ω,1−𝛼)|∇𝑢^𝑝−2𝜕𝑢|𝑢|𝜕𝑛+𝛼𝑢^𝑝−2=0,𝑥∈𝜕Ω, (1.1) where Ω⊆𝑅^𝑁 is a connected bounded domain with a smooth boundary 𝜕Ω, the outward unit normal to which is denoted by 𝑛. Δ_𝑝 is the 𝑝-Laplcianian operator defined by Δ_𝑝𝑢=div(|∇𝑢|^𝑝−2∇𝑢). The functions 𝑎 and 𝑏 are assumed to be sign changing in Ω. Here, we say a function 𝑎(𝑥) changes sign if the measure of the sets {𝑥∈Ω;𝑎(𝑥)>0} and {𝑥∈Ω;𝑎(𝑥)<0} are both positive. 𝜆≥0 is a real parameter and exponent 𝛾 is assumed to satisfy the condition 1<𝑝<𝛾<𝑝^∗, where 𝑝^∗=𝑁𝑝/(𝑁−𝑝) if 𝑁>𝑝 and ∞ otherwise.

A host of literature exists for this type of problem when 𝑝=2. For the works concerning with problems similar to (1.1) in the case 𝑝=2, we refer to [1–3] and references therein.

The growing attention in the study of the 𝑝-Laplace operator is motivated by the fact that it arises in various applications, for example, non-Newtonian fluids, reaction diffusion problems, flow through porus media, glacial sliding, theory of superconductors, biology, and so forth (see [4, 5] and the references therein).

In this paper, we obtain new existence results by using a variational method based on the properties of eigencurves, that is, properties of the map 𝜆→𝜇(𝛼,𝜆), where 𝜇(𝛼,𝜆) denotes the principal eigenvalue of the problem−Δ_𝑝𝑢−𝜆𝑎(𝑥)𝑢|𝑢|^𝑝−2=𝜇𝑢|𝑢|^𝑝−2(|,𝑥∈Ω,1−𝛼)|∇𝑢^𝑝−2𝜕𝑢|𝑢|𝜕𝑛+𝛼𝑢^𝑝−2=0,𝑥∈𝜕Ω. (1.2) Similar to [6] our method works provided that the eigenvalue problem−Δ_𝑝𝑢=𝜆𝑎(𝑥)𝑢|𝑢|^𝑝−2(|,𝑥∈Ω,1−𝛼)|∇𝑢^𝑝−2𝜕𝑢|𝑢|𝜕𝑛+𝛼𝑢^𝑝−2=0,𝑥∈𝜕Ω, (1.3) has principal eigenvalues and it can be shown that this occurs on an interval [𝛼₀,1] where 𝛼₀≤0. Thus, we are able to obtain existence results for problem (1.2) even in the case of nonlinear Neumann boundary conditions where 𝛼 is small and negative. Our method depends on using eigencurves to produce an equivalent norm on 𝑊^1,𝑝(Ω); such an equivalent norm is also introduced in [2]. The results that we obtain in this paper are generalization of the previous results obtained by Pohozaev and Veron [7].

It can be shown that 𝜇(𝛼,𝜆) has the variational characterization:𝜇(𝛼,𝜆)=inf_Ω|∇𝑢|^𝑝𝛼𝑑𝑥+1−𝛼_𝜕Ω|𝑢|^𝑝𝑑𝜎−𝜆_Ω𝑎|𝑢|^𝑝𝑑𝑥;𝑢∈𝑊^1,𝑝(Ω),_Ω|𝑢|^𝑝𝑑𝑥=1 (1.4) from whence it follows that(i)𝜆 is a principal eigenvalue of (1.2) if and only if 𝜇(𝛼,𝜆)=0,(ii)𝛼→𝜇(𝛼,𝜆) is an increasing function,(iii)𝜆→𝜇(𝛼,𝜆) is a concave function with a unique maximum such that 𝜇(𝛼,𝜆)→−∞ as 𝜆→±∞ [6].

If 𝛼∈(0,1], then 𝜇(𝛼,𝜆)>0, and so 𝜆→𝜇(𝛼,𝜆) has exactly one negative zero 𝜆⁻₁(𝛼) and one positive zero 𝜆⁺₁(𝛼). Thus, 𝜆⁻₁(𝛼) and 𝜆⁺₁(𝛼) are principal eigenvalues for (1.2).

If 𝛼=0, then 𝜇(0,0)=0. If 𝑎(𝑥)>0, then 𝜇(0,𝜆) is decreasing, and if 𝑎(𝑥)<0, then 𝜇(0,𝜆) is increasing. Assume now that 𝑎(𝑥) changes sign in Ω: if ∫_Ω𝑎(𝑥)𝑑𝑥<0, there exists a unique 𝜆⁺₁(0)>0 such that 𝜇(0,𝜆⁺(0))=0 and 𝜇(0,𝜆)>0 for 𝜆∈(0,𝜆⁺₁(0)). If ∫_Ω𝑎(𝑥)𝑑𝑥=0, then 𝜇(0,0)=0 and 𝜇(0,𝜆)<0 for 𝜆≠0. If ∫_Ω𝑎(𝑥)𝑑𝑥>0, then there exists a unique 𝜆⁻₁(0)<0 such that 𝜇(0,𝜆⁻₁(0))=0 and 𝜇(0,𝜆)>0 for 𝜆∈(𝜆⁻₁(0),0).

Suppose now that ∫_Ω𝑎(𝑥)𝑑𝑥<0 and that 𝛼 is small and negative. Then, since 𝛼→𝜇(𝛼,𝜆) is increasing, it follows that there still exist principal eigenvalues 𝜆⁻₁(𝛼)<𝜆⁺₁(𝛼) of (2.5), but now both of them are positive.

It can be shown that there exists 𝛼₀<0 such that the above is true for all 𝛼∈(𝛼₀,0), but for 𝛼<𝛼₀, 𝜇(𝛼,𝜆)<0 for all 𝜆 so that principal eigenvalues no longer exist.

Similar considerations show that when ∫_Ω𝑎(𝑥)𝑑𝑥>0, there exists 𝛼₀<0 such that there are principal eigenvalues 𝜆⁻₁(𝛼)<𝜆⁺₁(𝛼)<0 for 𝛼₀<𝛼<0 but when ∫_Ω𝑎(𝑥)𝑑𝑥=0, there are no principal eigenvalues for 𝛼<0 (see [6]).

It is easy to see that if 𝜆⁻₁(𝛼) and 𝜆⁺₁(𝛼) exist, 𝜇(𝛼,𝜆)>0 for all 𝜆∈(𝜆⁻₁(𝛼),𝜆⁺₁(𝛼)).

Thus, we assume that the following conditions hold:(𝒜₁)𝛼∈(0,1)orthat∫_Ω𝑎(𝑥)𝑑𝑥≠0and𝛼∈(𝛼₀,1], (𝒜₂)𝑎(𝑥)∈𝐿^∞(Ω), (ℬ₁)𝑏(𝑥)∈𝐿^∞(Ω), (ℬ₂)𝑏⁺≢0, (ℬ₃)∫_Ω𝑏(𝑥)(𝑢⁺₁)^𝛾𝑑𝑥<0,

where 𝑢⁺₁ is the positive principal eigenfunction corresponding to 𝜆⁺₁(𝛼).

With these constructions we have the following.

Proposition 1.1. Assume (𝒜₁), then for every 𝜆∈(0,𝜆⁺₁(𝛼)), ‖𝑢‖_𝛼,𝜆∶=_Ω|∇𝑢|^𝑝−𝜆𝑎(𝑥)|𝑢|^𝑝𝛼𝑑𝑥+1−𝛼_𝜕Ω|𝑢|^𝑝𝑑𝜎^1/𝑝 (1.5) defines a norm in 𝑊^1,𝑝(Ω) which is equivalent to the usual norm of 𝑊^1,𝑝(Ω), that is, ‖𝑢‖=_Ω|∇𝑢|^𝑝𝑑𝑥+_Ω|𝑢|^𝑝𝑑𝑥^1/𝑝. (1.6)

Proof. See [2].

Now we can state our main results.

Theorem 1.2. Assume (𝒜₁), (𝒜₂), (ℬ₁), and (ℬ₂). Then, for every 𝜆∈(0,𝜆⁺₁(𝛼)), problem (1.2) admits at least one positive solution 𝑢∈𝑊^1,𝑝(Ω)∩𝐿^∞(Ω).

Theorem 1.3. Assume (𝒜₁), (𝒜₂), (ℬ₁), (ℬ₂), and (ℬ₃). Then, problem (1.2) has at least one positive solution 𝑢∈𝑊^1,𝑝(Ω)∩𝐿^∞(Ω) for 𝜆=𝜆⁺₁(𝛼).

Theorem 1.4. Assume (𝒜₁), (𝒜₂), (ℬ₁), (ℬ₂), and (ℬ₃). Then, there exists 𝜆^∗>𝜆⁺₁(𝛼) such that problem (1.2) admits at least two distinct positive weak solutions in 𝑊^1,𝑝(Ω)∩𝐿^∞(Ω), whenever 𝜆⁺₁(𝛼)<𝜆<𝜆^∗.

When 𝜆=0 and 𝛼=0, we have the following.

Corollary 1.5. Assume ∫_Ω𝑏(𝑥)𝑑𝑥<0. Then, the problem −Δ_𝑝𝑢=𝑏(𝑥)𝑢|𝑢|^𝛾−2||,𝑥∈Ω,∇𝑢^𝑝−2𝜕𝑢𝜕𝑛=0,𝑥∈𝜕Ω, (1.7) has a positive solution.

Throughout this paper, c denotes a positive constant. We will use fibrering method in a similar way to those in [8]. A brief description of the method and the proof of Theorem 1.2 are presented in Section 2. We then study the cases 𝜆=𝜆⁺₁(𝛼) and 𝜆>𝜆⁺₁(𝛼) in Sections 3 and 4.

2. The Case When 𝜆<𝜆⁺₁(𝛼)

In this section, motivated by Pohozaev [8], we will introduce the fibrering map as our framework for the study of problem (1.2).

For a (weak) solution of problem (1.2), we mean a function 𝑢∈𝑋=𝑊^1,𝑝(Ω) such that for every 𝑣∈𝑊^1,𝑝(Ω), there holds_Ω|∇𝑢|^𝑝−2∇𝑢∇𝑣−𝜆𝑎(𝑥)|𝑢|^𝑝−2𝛼𝑢𝑣𝑑𝑥+1−𝛼_𝜕Ω|𝑢|^𝑝−2𝑢𝑣𝑑𝜎=_Ω𝑏(𝑥)|𝑢|^𝛾−2𝑢𝑣𝑑𝑥. (2.1)

Now let us define the variational functional corresponding to problem (1.2). We set 𝐼_𝜆∶𝑋→𝑅 as𝐼_𝜆(𝑢)=1𝑝_Ω|∇𝑢|^𝑝−𝜆𝑎(𝑥)|𝑢|^𝑝𝛼𝑑𝑥+𝑝(1−𝛼)_𝜕Ω|𝑢|^𝑝1𝑑𝜎−𝛾_Ω𝑏(𝑥)|𝑢|^𝛾𝑑𝑥. (2.2)

It is easy to see that 𝐼_𝜆∈𝐶¹(𝑋,𝑅), and for 𝑣∈𝑋, there holds𝐼^′_𝜆(𝑢),𝑣=_Ω|∇𝑢|^𝑝−2∇𝑢∇𝑣−𝜆𝑎(𝑥)|𝑢|^𝑝−2𝛼𝑢𝑣𝑑𝑥+1−𝛼_𝜕Ω|𝑢|^𝑝−2𝑢𝑣𝑑𝜎=_Ω𝑏(𝑥)|𝑢|^𝛾−2𝑢𝑣𝑑𝑥. (2.3)

Since 𝐶^∞₀(Ω)⊂𝑋, we know that critical points of 𝐼_𝜆 are weak solutions of the problem (1.2). Thus, to prove our main theorems, it suffices to show that 𝐼_𝜆 admits critical point. We will do this in this paper. Our main tool is the theory of the fibrering maps. First, we will introduce this map for 𝐼_𝜆∶𝑋→𝑅.

Let 𝜙_𝑢(𝑡)=𝐼_𝜆(𝑡𝑢)(𝑡>0). We refer to such maps as fibrering maps. It is clear that if 𝑢 is a local minimizer of 𝐼_𝜆, then 𝜙_𝑢 has a local minimum at 𝑡=1.

Lemma 2.1. Let 𝑢∈𝑋−{0} and 𝑡>0. Then, (𝐼^′_𝜆(𝑡𝑢),𝑡𝑢)=0 if and only if 𝜙^′_𝑢(𝑡)=0.

Proof. The result is an immediate consequence of the fact that 𝜙^′_𝑢(𝑡)=𝐼^′_𝜆()=1𝑡𝑢,𝑢𝑡𝐼^′_𝜆(𝑡𝑢),𝑡𝑢. (2.4)

Thus critical points of 𝐼_𝜆 correspond to stationary points of the maps 𝜙_𝑢 that can be given by𝜙_𝑢(𝑡)=𝐼_𝜆()=𝑡𝑡𝑢^𝑝𝑝𝐴_𝜆(𝑢)−𝑡^𝛾𝛾𝐵(𝑢), (2.5) where𝐴_𝜆(𝑢)=_Ω|∇𝑢|^𝑝−𝜆𝑎(𝑥)|𝑢|^𝑝𝛼𝑑𝑥+1−𝛼_𝜕Ω|𝑢|^𝑝𝐵(𝑢)𝑑𝜎,=_Ω𝑏(𝑥)|𝑢|^𝛾𝑑𝑥. (2.6) Hence,𝜙^′_𝑢(𝑡)=𝑡^𝑝−1𝐴_𝜆(𝑢)−𝑡^𝛾−1𝐵(𝑢). (2.7) Thus, if 𝐴_𝜆(𝑢) and 𝐵(𝑢) have the same sign, 𝜙_𝑢 has exactly one turning point at𝑡(𝑢)=𝐴_𝜆(𝑢)𝐵(𝑢)^{1/(𝛾−𝑝)}, (2.8) provided that 𝐵(𝑢)≠0, and if 𝐴_𝜆(𝑢) and 𝐵(𝑢) have opposite signs, 𝜙_𝑢 has no turning points. Now substituting (2.8) into (2.5), we get𝐼_𝜆(𝑡(𝑢)𝑢)=1𝑝𝐴_𝜆(𝑢)𝐵(𝑢)^{𝑝/(𝛾−𝑝)}𝐴_𝜆(𝑢)−1𝛾𝐴_𝜆(𝑢)𝐵(𝑢)^{𝛾/(𝛾−𝑝)}𝐵(𝑢)=1𝑝𝐴_𝜆(𝑢)^{𝛾/(𝛾−𝑝)}𝐵(𝑢)^{𝑝/(𝛾−𝑝)}−1𝛾𝐴_𝜆(𝑢)^{𝛾/(𝛾−𝑝)}𝐵(𝑢)^{𝑝/(𝛾−𝑝)}=1𝑝−1𝛾𝐴_𝜆(𝑢)^{𝛾/(𝛾−𝑝)}𝐵(𝑢)^{𝑝/(𝛾−𝑝)}∶=𝐽_𝜆(𝑢). (2.9)

Lemma 2.2. Suppose that 𝑢₀ is a critical point of 𝐽_𝜆, where 𝐴_𝜆(𝑢₀) and 𝐵(𝑢₀) have the same sign. Then, 𝐼^′_𝜆(𝑡(𝑢₀)𝑢₀)=0.

Proof. Let 𝑢₀∈𝑋, 𝐵(𝑢₀)≠0, then 𝐼^′_𝜆𝑡𝑢₀𝑢₀,𝑣=_Ω∇𝑡𝑢₀𝑢₀^𝑝−2∇𝑡𝑢₀𝑢₀∇𝑣−𝜆𝑎(𝑥)𝑡𝑢₀𝑢₀^𝑝−2𝑡𝑢₀𝑢₀𝑣−𝑑𝑥_Ω𝑏(𝑥)𝑡𝑢₀𝑢₀^𝛾−2𝑡𝑢₀𝑢₀𝛼𝑣𝑑𝑥+1−𝛼_𝜕Ω𝑡𝑢₀𝑢₀^𝑝−2𝑡𝑢₀𝑢₀𝑢𝑣𝑑𝜎=𝑡₀^𝑝−1𝐴^′_𝜆𝑢₀,𝑣−𝑡𝑢₀^𝛾−1𝐵^′𝑢₀=1,𝑣𝑡𝑢₀𝐽^′_𝜆𝑢₀,𝑣=0 (2.10) for all 𝑣∈𝑋. This completes the proof.

The following lemma shows that the critical points of the functional 𝐼_𝜆 can be found by using the conditional variational problem associated with 𝐼_𝜆.

Lemma 2.3. Suppose that 𝐻 is a well-defined functional on 𝑋 and 𝑢₀ is a minimizer of 𝐽_𝜆 on 𝑆∶=𝑢∈𝑋,𝐻(𝑢)=𝑐,𝐻′(𝑢),𝑢≠0 (2.11) for some 𝑐≠0. Then,𝐽^′_𝜆(𝑢₀)=0.

Proof. If 𝑢₀ is a local minimizer of 𝐽_𝜆 on 𝑆, then 𝑢₀ is a solution of the optimization problem: minimize𝐽_𝜆(𝑢)subjectto𝛾(𝑢)=0, (2.12) where 𝛾(𝑢)=𝐻(𝑢)−𝑐. Hence, by the theory of the Lagrange multipliers, there exists 𝜇∈𝑅 such that 𝐽^′_𝜆(𝑢₀)=𝜇𝛾′(𝑢₀). Thus, 𝐽^′_𝜆𝑢₀,𝑢₀=𝜇𝛾′𝑢₀,𝑢₀. (2.13) Note that the functional 𝐽_𝜆 is 0-homogeneous and the Gateaux derivative of 𝐽_𝜆 at the point 𝑣∈𝑋, 𝐵(𝑣)≠0, in direction 𝑣, is zero, that is, (𝐽^′_𝜆(𝑢₀),𝑢₀)=0. It then follows that 𝜇=0 due to 𝑢₀∈𝑆. Hence, the proof is complete.

The following scheme for the investigation of the solvability of (1.2) is based on previous lemmas. First, we will prove the existence of nonzero critical points of 𝐽_𝜆 under the constraint given by suitable functional 𝐻. This will be an actual critical point of 𝐽_𝜆 and it will generate critical point of the Euler functional 𝐼_𝜆 which will coincide with the weak solution of problem (1.2).

Proof of Theorem 1.2. Suppose that (𝒜₁), (𝒜₂), (ℬ₁), (ℬ₂) satisfy. It follows from variational characterization of 𝜇(𝛼,𝜆) that 𝐴_𝜆(𝑢)≥0 for all 𝑢∈𝑋 and 0≤𝜆<𝜆⁺₁(𝛼). Moreover, we have 𝐴^′_𝜆(𝑢),𝑢=𝑝𝐴_𝜆(𝑢),𝑢∈𝑋. (2.14) Hence, for using Lemma 2.3, it is sufficient to consider the case 𝐴_𝜆(𝑢)=𝑐≠0, for example 𝑐=1. In this case, we have 𝐽_𝜆(𝑢)=1𝑝−1𝛾1𝐵(𝑢)^{𝑝/(𝛾−𝑝)}. (2.15) From the necessary condition for the existence of 𝑡(𝑢), we have 𝐵(𝑢)>0. It follows that we must consider a critical point with 𝐵(𝑢)>0.
The functional 𝐽_𝜆(𝑢) is nonnegative and so bounded below, hence we can look for positive local minimizer for 𝐽_𝜆(𝑢) on 𝑋.
Let us consider variational problem: ℳ_𝜆=sup𝐵(𝑢);𝐴_𝜆(𝑢)=1,𝐵(𝑢)>0. (2.16) Note that for 0≤𝜆<𝜆⁺₁(𝛼), this set is not empty, and from Lemma 2.3, the solution of this problem is a minimizer of 𝐽_𝜆(𝑢) on 𝑋.
Suppose {𝑢_𝑛} is the maximizing sequence of this problem. It follows from the equivalent property in Proposition 1.1, {𝑢_𝑛} is bounded and so we may assume that 𝑢_𝑛⇀𝑢₀ in 𝑋. Since 𝑋 may be compactly embedded in 𝐿^𝑝(Ω), 𝐿^𝛾(Ω), and 𝐿^𝑝(𝜕Ω), we have 𝑢_𝑛→𝑢₀ in 𝐿^𝑝(Ω), 𝐿^𝛾(Ω) and 𝐿^𝑝(𝜕Ω). Hence, 𝐵𝑢_𝑛⟶𝐵𝑢₀=ℳ_𝜆>0. (2.17) Moreover, we have 𝐴_𝜆𝑢₀≤liminf𝐴_𝜆𝑢_𝑛=1. (2.18) Here, the weak lower semicontinuity of the equivalent norm was used.
Assume that 𝐴_𝜆(𝑢₀)<1. By using the map: 𝐿(𝑡)=𝐴_𝜆𝑡𝑢₀, (2.19) we have 𝐿(1)<1 and lim_𝑡→∞𝐿(𝑡)=∞ and so 𝐿(𝑡₀)=1 for some 𝑡₀>1, that is, 𝐴_𝜆(𝑡₀𝑢₀)=1. So we derive 𝐵𝑡₀𝑢₀=𝑡^𝛾₀𝐵𝑢₀>ℳ_𝜆, (2.20)
which is a contradiction. Hence, 𝐴_𝜆(𝑢₀)=1 and so 𝑢₀ is a maximizer. As Lemma 2.2, the result would follow by considering 𝑢₁=𝑡(𝑢₀)𝑢₀∈𝑋. Then, 𝑢₁ is a weak solution of (1.2) and 𝑢₁≥0 in Ω. Now, following the bootstrap argument (used, e.g., in [9]), we prove 𝑢₁∈𝐿^∞(Ω). Then, we can apply the Harnack inequality due to Trudinger [5] in order to get 𝑢₁>0 in Ω (cf. [9]).

3. The Case When 𝜆=𝜆⁺₁(𝛼)

Proof of Theorem 1.3. If 𝜆=𝜆⁺₁(𝛼), It is easy to see that the set {𝑢∈𝑋;𝐴_𝜆(𝑢)=1} is unbounded and we are forced to require an additional assumption (ℬ₃). In this case, again we are looking for a maximizing of the problem: ℳ_𝜆⁺_1(𝛼)=sup𝐵(𝑢);𝐴_𝜆⁺_1(𝛼)(𝑢)=1,𝐵(𝑢)>0. (3.1) Suppose {𝑢_𝑛} is a maximizing sequence of this problem. First, we investigate the case when {𝑢_𝑛} is unbounded. Then, we may assume without loss of generality that ‖𝑢_𝑛‖→∞. So, we obtain 1=𝐴_𝜆⁺_1(𝛼)𝑢_𝑛=𝑢_𝑛^𝑝𝐴_𝜆⁺_1(𝛼)𝑣_𝑛, (3.2) where 𝑣_𝑛=𝑢_𝑛/‖𝑢_𝑛‖. Thus, we have 𝐴_𝜆⁺_1(𝛼)𝑣_𝑛=𝐴_𝜆⁺_1(𝛼)𝑢_𝑛𝑢_𝑛^𝑝=1𝑢_𝑛^𝑝⟶0as𝑛⟶∞. (3.3) Since ‖𝑣_𝑛‖=1, we may assume that 𝑣_𝑛⇀𝑣₀ in 𝑋. Again, since 𝑋 may be compactly embedded in 𝐿^𝑝(Ω), 𝐿^𝛾(Ω) and 𝐿^𝑝(𝜕Ω), we have lim_𝑛→∞Ω𝑎(𝑥)𝑣_𝑛^𝑝𝑑𝑥=_Ω𝑎(𝑥)𝑣₀^𝑝𝑑𝑥,lim_𝑛→∞Ω𝑏(𝑥)𝑣_𝑛^𝛾𝑑𝑥=_Ω𝑎(𝑥)𝑣₀^𝛾𝑑𝑥,lim_{𝑛→∞𝜕Ω}𝑣_𝑛^𝑝𝑑𝜎=_𝜕Ω𝑣₀^𝑝𝑑𝜎. (3.4) So, _Ω𝑎(𝑥)𝑣₀^𝑝1𝑑𝑥≥𝜆⁺₁(𝛼), (3.5) And, therefore, 𝑣₀≢0. Returning to 𝐴_𝜆⁺_1(𝛼)(𝑣_𝑛), we have also 0≤𝐴_𝜆⁺_1(𝛼)𝑣₀≤liminf_𝑛→∞𝐴_𝜆⁺_1(𝛼)𝑣_𝑛=0. (3.6) Due to simplicity of 𝜆⁺₁(𝛼), there exists 𝑐≠0 such that 𝑣₀(𝑥)=𝑐𝑢⁺₁(𝑥). Therefore, we have 0<ℳ_𝜆⁺_1(𝛼)=lim_𝑛→∞𝐵𝑢_𝑛=lim_𝑛→∞𝑢_𝑛^𝛾𝐵𝑣_𝑛, (3.7) that implies |𝑐|^𝛾𝐵(𝑢⁺₁)≥0, which contradicts (ℬ₃). Therefore, {𝑢_𝑛} is bounded and so 𝑢_𝑛⇀𝑢₀ in 𝑋 and 𝑢_𝑛→𝑢₀ in 𝐿^𝑝(Ω), 𝐿^𝛾(Ω) and 𝐿^𝑝(𝜕Ω). Hence, 𝐵(𝑢_𝑛)→𝐵(𝑢₀)=ℳ_𝜆⁺_1(𝛼)>0, and so 𝑢₀≠0.
Moreover, we obtain 0≤𝐴_𝜆⁺_1(𝛼)(𝑢₀)≤1. Here, the variational characteristic of 𝜆⁺₁(𝛼) and the weak lower semicontinuity of the norm were used. Let us now consider the case 𝐴_𝜆⁺_1(𝛼)(𝑢₀)=0. Again, we obtain 𝑢₀=𝑐𝑢⁺₁ for some 𝑐≠0, and so 𝐵(𝑢₀)=|𝑐|^𝛾𝐵(𝑢⁺₁)=ℳ_𝜆⁺_1(𝛼)>0 which contradicts (ℬ₃).
Now we prove that 𝐴_𝜆⁺_1(𝛼)(𝑢₀)=1. Suppose otherwise, then by using 𝐿(𝑡) in (2.20), we obtain some 𝑡₀>1 such that 𝐴_𝜆⁺_1(𝛼)(𝑡₀𝑢₀)=1. By direct calculation, we get a contradiction like (4.1). This yields that 𝑢₀ is a nonnegative solution of problem (1.2). Using the same ideas as the proof of Theorem 1.2, we have proved Theorem 1.3.

4. The Case When 𝜆>𝜆⁺₁(𝛼)

As it is proved in [10], we will show that for 𝜆>𝜆⁺₁(𝛼) but close enough to 𝜆⁺₁(𝛼), we have two distinct positive solutions for problem (1.2). The existence of one of them is obtained by using the following lemma.

Lemma 4.1. Under the assumption of Theorem 1.4, there exists a maximizer 𝑢₁ of the problem: sup𝐵(𝑢);𝐴_𝜆(𝑢)=1,𝐵(𝑢)>0, (4.1) whenever 𝜆∈(𝜆⁺₁,𝜆⁺₁(𝛼)+𝛿), for some 𝛿>0. Moreover, 𝑢₁∈𝑋∩𝐿^∞(Ω) is a positive weak solution of problem (1.2).

Proof. First note that using 𝐿(𝑡) in (2.20), it is easy to see that 𝑢₁ is a maximizer of problem (4.2) if and only if 𝑢₁ is a maximizer of the problem: ℳ_𝜆=sup𝐵(𝑢);𝐴_𝜆(𝑢)≤1,𝐵(𝑢)>0. (4.2) Now suppose that the result is false. Then, there exists a sequence {𝛿_𝑘} such that 𝛿_𝑘→0 and problem (4.1) has no solution for 𝜆⁺₁(𝛼)+𝛿_𝑘. For simplicity, we use 𝜆_𝑘=𝜆⁺₁(𝛼)+𝛿_𝑘. Let {𝑢^𝑘_𝑛} be a maximizing sequence of this problem, that is, 𝐴_𝜆𝑘𝑢^𝑘_𝑛≤1,𝐵𝑢^𝑘_𝑛⟶ℳ_𝜆𝑘>0. (4.3) We prove that if {𝑢^𝑘_𝑛} be bounded or unbounded, we arrive at a contradiction and so the lemma is proved.
The first case: {𝑢^𝑘_𝑛} is bounded. Thus, 𝑢^𝑘_𝑛⇀𝑢^𝑘₀ in 𝑋 for some 𝑢^𝑘₀∈𝑋 and 𝑢^𝑘_𝑛→𝑢^𝑘₀ in 𝐿^𝑝(Ω), 𝐿^𝛾(Ω), and 𝐿^𝑝(𝜕Ω). Hence, 𝐵(𝑢^𝑘₀)=ℳ_𝜆𝑘>0 and the weak lower semicontinuity of the norm gives 𝐴_𝜆𝑘(𝑢^𝑘₀)≤1. Therefore, 𝑢^𝑘₀ is a solution of problem (4.5), which is a contradiction.
The case {𝑢^𝑘_𝑛} is unbounded. Then we may assume that ‖𝑢^𝑘_𝑛‖→∞. Let 𝑣^𝑘_𝑛=𝑢^𝑘_𝑛/‖𝑢^𝑘_𝑛‖. Then, 𝑣^𝑘_𝑛⇀𝑣^𝑘₀ in 𝑋 for some 𝑣^𝑘₀∈𝑋 and 𝑣^𝑘_𝑛→𝑣^𝑘₀ in 𝐿^𝑝(Ω), 𝐿^𝛾(Ω) and 𝐿^𝑝(𝜕Ω). This implies that 𝐵𝑢^𝑘_𝑛=𝑢^𝑘_𝑛^𝛾𝐵𝑣^𝑘_𝑛⟶ℳ_𝜆𝑘>0. (4.4) Therefore, 𝐵(𝑣^𝑘₀)≥0. Furthermore, ‖𝑢^𝑘_𝑛‖^𝑝𝐴_𝜆𝑘(𝑣^𝑘_𝑛)≤1 and so 𝐴_𝜆𝑘(𝑣^𝑘_𝑛)≤1/‖𝑣^𝑘_𝑛‖^𝑝. Thus, we arrive at 𝐴_𝜆𝑘𝑣^𝑘₀≤liminf_𝑛→∞𝐴_𝜆𝑘𝑣^𝑘_𝑛⟶0,as𝑛⟶∞, (4.5) which implies _Ω∇𝑣^𝑘₀^𝑝𝜆𝑑𝑥≤1,_𝑘Ω𝑎(𝑥)𝑣^𝑘₀^𝑝𝑑𝑥≥1. (4.6)(4.7) It is a direct consequent of the comactly embedding of 𝑋 in 𝐿^𝑝(Ω), 𝐿^𝛾(Ω) and 𝐿^𝑝(𝜕Ω). Now, we pass to the limit for 𝑘→∞. Then, 𝜆_𝑘→𝜆⁺₁(𝛼) and since {𝑣^𝑘₀} is a bounded sequence, we may assume that 𝑣^𝑘₀⇀𝑣₀ in 𝑋 for some 𝑣₀∈𝑋 and 𝑣^𝑘₀→𝑣₀ in 𝐿^𝑝(Ω), 𝐿^𝛾(Ω), and 𝐿^𝑝(𝜕Ω).
It follows from (4.10) that 𝜆⁺₁(𝛼)_Ω𝑎(𝑥)𝑣₀^𝑝𝑑𝑥≥1, (4.8) and from the variational characteristic of 𝜆⁺₁(𝛼) and (4.6) that 0≤𝐴_𝜆⁺_1(𝛼)𝑣₀≤0, (4.9) which contradicts (ℬ₃).
Hence, for some 𝛿>0, problem (4.5) has at least one nonnegative solution 𝑢₁ for any 𝜆∈(𝜆⁺₁(𝛼),𝜆⁺₁(𝛼)+𝛿).

In order to find the second positive solution, we consider the minimizing problem:𝒩_𝜆=inf𝐴_𝜆(𝑢);𝐵(𝑢)=−1. (4.10)

Note that this set is empty for 𝜆<𝜆⁺₁(𝛼) (because of the variational characterization of 𝜆⁺₁(𝛼)). Hence, this second problem does not have a solution for 𝜆<𝜆⁺₁(𝛼).

Lemma 4.2. Under the assumption of Theorem 1.4, there exists 𝜀>0 such that for 𝜆∈(𝜆⁺₁,𝜆⁺₁(𝛼)+𝜀), problem (1.1) has a nonnegative solution 𝑢₂ satisfying 𝐴_𝜆(𝑢₂)<0.

Proof. First note that using the auxiliary function 𝐿(𝑡)=|𝑡|^𝛾𝐵(𝑢) and the assumption (ℬ₃), it is easy to see that 𝐵(𝑡₁𝑢⁺₁)=−1 for some 𝑡₁<1. Thus, the set {𝑢∈𝑋;𝐵(𝑢)=−1} is nonempty. Also, for this 𝑡₁, we have 𝐴_𝜆𝑡₁𝑢⁺₁=𝑡₁^𝑝𝜆⁺₁(𝛼)−𝜆_Ω𝑎(𝑥)𝑢⁺₁^𝑝𝑑𝑥<0, (4.11) for 𝜆>𝜆⁺₁(𝛼).
Again, we assume that the result is not true. Then, there exists 𝜀_𝑘→0 such that for 𝜆_𝑘=𝜆⁺₁(𝛼)+𝜀_𝑘, problem (1.1) has no solution.
Let {𝑢^𝑘_𝑛} be a minimizing sequence of this problem, that is,𝐵(𝑢^𝑘_𝑛)=−1,𝐴_𝜆𝑘(𝑢^𝑘_𝑛)→𝒩_𝜆𝑘<0.
Assume that {𝑢^𝑘_𝑛} is bounded. Using similar argument as in the proof of Lemma 4.1, we find a solution of problem (1.1) which is a contradiction. Let us assume {𝑢^𝑘_𝑛} is bounded. Again, we consider 𝑣^𝑘_𝑛=𝑢^𝑘_𝑛𝑢^𝑘_𝑛,with𝑢^𝑘_𝑛⟶∞, (4.12){𝑣^𝑘_𝑛} is bounded, and so 𝑣^𝑘_𝑛⇀𝑣^𝑘₀ in 𝑋 for some 𝑣^𝑘₀∈𝑋. Thus,_{𝐵(𝑣𝑘𝑛)=}𝑢^𝑘_𝑛‖𝑢^𝑘_𝑛‖^𝛾=−1‖𝑢^𝑘_𝑛‖^𝛾→0.
Letting 𝑛→∞, we arrive at 𝐵(𝑣^𝑘₀)=0. It follows easily that 𝐴_𝜆𝑘(𝑢^𝑘_𝑛)=‖𝑢^𝑘_𝑛‖^𝑝𝐴_𝜆𝑘(𝑣^𝑘_𝑛)≤0, and so 𝐴_𝜆𝑘(𝑣^𝑘_𝑛)≤0. As might be expected, we arrive at a contradiction with assumption (ℬ₃), by using similar steps as in the proof of Theorem 1.3. This completes the proof.

Proof of Theorem 1.4. Let 𝜂=min{𝜀,𝛿}, 𝜆^∗=𝜆⁺₁(𝛼)+𝜂 and 𝑤₁=𝑡(𝑢₁)𝑢₁ and 𝑤₂=𝑡(𝑢₂)𝑢₂, where 𝑡(𝑢_𝑖) is defined by (2.16).
It easy to see that 𝐴_𝜆(𝑤₁)>0 and 𝐴_𝜆(𝑤₂)≤0, and so 𝑤₁≢𝑤₂. Thus, 𝑤₁ and 𝑤₂ are two distinct non negative weak solutions to (1.2). Other properties of 𝑤₁ and 𝑤₂ follow in the same way as in Section 2.