Existence and Multiplicity of Solutions for Semipositone Problems Involving p-Laplacian

Abstract

We prove existence and multiplicity of positive solutions for semipositone problems involving 𝑝-Laplacian in a bounded smooth domain of ℝ^𝑁 under the cases of sublinear and superlinear nonlinearities term.

1. Introduction

In this paper, we shall study the following semipositone problem involving the 𝑝-Laplacian: −div||∇𝑢𝑢||=^𝑝−20∇𝑢on=𝜕𝜆Ω𝑓,(𝑢)inΩ,(1.1) where Ω⊂ℝ^𝑁 is a smooth bounded domain, 𝜆 is a positive parameter, and 𝑓∶[0,+∞)→ℝ is a continuous function satisfying the condition(F0)𝑓(0)=−𝑎<0.

Such problems are usually referred in the literature as semipositone problems. We refer the reader to [1], where Castro and Shivaji initially called them nonpositone problems, in contrast with the terminology positone problems, coined by Keller and Cohen in [2], when the nonlinearity 𝑓 was positive and monotone.

Under the case of 𝑝≡2, a novel variational approach is presented by Costa et al. [3] to the question of existence and multiplicity of positive solutions to problem (1.1), where they consider both the sublinear and superlinear cases. The aim of this paper is to extend their results to the case of 𝑝-Laplacian. The main difficulty is in verifying (PS)_𝑐-condition because of the operator −Δ_𝑝 is not self-adjoint linear.

We define the discontinuous nonlinearity 𝑔(𝑠) by 𝑔(𝑠)=0𝑓,(𝑠)if,𝑠i≤f𝑠0,>0.(1.2) We shall consider the modified problem −div||∇𝑢𝑢||=^𝑝−20∇𝑢on=𝜕𝜆Ω𝑔.(𝑢)inΩ,(1.3) We note that the set of positive solutions of (1.1) and (1.3) do coincide.

Our main results concerning problems (1.1) and (1.3) are the following:

Theorem 1.1. Assume (F0) and the following assumptions:(F1)lim_𝑠→+∞(𝑓(𝑠)/𝑠^𝑝−1)=0 (the sublinear case);(F2)𝐹(𝛿)>0 for some 𝛿>0, where 𝐹(𝑢)=∫^𝑢0𝑓(𝑠)𝑑𝑠.Then, there exist 0<𝜆₀≤𝜆_∗ such that (1.3) has no nontrivial nonnegative solution for 0<𝜆<𝜆₀, and has at least two nontrivial nonnegative solutions 𝑢_𝜆, 𝑣_𝜆 for all 𝜆>𝜆_∗. Moreover, when Ω is a ball 𝐵_𝑅=𝐵_𝑅(0), these two solutions are non-increasing, radially symmetric and, if 𝑁≥2, at least one of them is positive, hence a solution of (1.1).

Theorem 1.2. Assume (F0), (F2) and the following assumptions:(F3)lim_𝑠→+∞(𝑓(𝑠)/𝑠^𝑝−1)=+∞, lim_𝑠→+∞(𝑓(𝑠)/𝑠^𝑝∗−2)=0 (the sublinear, subcritical case);(F4)𝜃𝐹(𝑠)≤𝑓(𝑠)𝑠for all 𝑠≥𝐾 and some 𝜃>𝑝 (the AR-condition).Then, (1.3) has at least one nonnegative solution 𝑣_𝜆 for all 𝜆>0. IfΩ=𝐵_𝑅 then 𝑣_𝜆 is nonincreasing, radially symmetric and one of the two alternatives occurs. (i)There exists 𝜆^∗>0 such that, for all 0<𝜆<𝜆^∗, 𝑣_𝜆 is a positive solution of (1.1) having negative normal derivative on 𝜕𝐵_𝑅. (ii)For some sequence 𝜇_𝑛→0, problem (1.1) with 𝜆≡𝜇_𝑛 has a positive solution 𝑤_𝜇𝑛 with zero normal derivative on 𝜕𝐵_𝑅.

2. Preliminaries

We start by recalling some basic results on variational methods for locally Lipschitz functionals. Let (𝑋,‖⋅‖) be a real Banach space and 𝑋^∗ is its topological dual. A function 𝑓∶𝑋→ℝ is called locally Lipschitz if each point 𝑢∈𝑋 possesses a neighborhood Ω_𝑢 such that |𝑓(𝑢₁)−𝑓(𝑢₂)|≤𝐿‖𝑢₁−𝑢₂‖ for all 𝑢₁,𝑢₂∈Ω_𝑢, for a constant 𝐿>0 depending on Ω_𝑢. The generalized directional derivative of 𝑓 at the point 𝑢∈𝑋 in the direction 𝑣∈𝑋 is 𝑓⁰(𝑢,𝑣)=limsup_{𝑤→𝑢,𝑡→0+}1𝑡(𝑓(𝑤+𝑡𝑣)−𝑓(𝑤)).(2.1) The generalized gradient of 𝑓 at 𝑢∈𝑋 is defined by 𝜕𝑓(𝑢)=𝑢^∗∈𝑋^∗∶⟨𝑢^∗,𝜑⟩≤𝑓⁰(𝑢;𝜑)∀𝜑∈𝑋,(2.2) which is a nonempty, convex, and 𝑤^∗-compact subset of 𝑋, where ⟨⋅,⋅⟩ is the duality pairing between 𝑋^∗ and 𝑋. We say that 𝑢∈𝑋 is a critical point of 𝑓 if 0∈𝜕𝑓(𝑢). For further details, we refer the reader to Chang [4].

We list some fundamental properties of the generalized directional derivative and gradient that will be possibly used throughout the paper.

Proposition 2.1 (see [4, 5]). (1)  Let 𝑗∶𝑋→ℝ be a continuously differentiable function. Then 𝜕𝑗(𝑢)={𝑗′(𝑢)}, 𝑗⁰(𝑢;𝑧) coincides with ⟨𝑗′(𝑢),𝑧⟩_𝑋, and (𝑓+𝑗)⁰(𝑢,𝑧)=𝑓⁰(𝑢;𝑧)+⟨𝑗′(𝑢),𝑧⟩_𝑋for all 𝑢,𝑧∈𝑋.
(2)  𝜕(𝜆𝑓)(𝑢)=𝜆𝜕(𝑢) for all 𝜆∈ℝ.
(3)  If 𝑓 is a convex functional, then 𝜕𝑓(𝑢) coincides with the usual subdifferential of 𝑓 in the sense of convex analysis.
(4)  If 𝑓 has a local minimum (or a local maximum) at 𝑢₀∈𝑋, then 0∈𝜕𝑓(𝑢₀).
(5)  ‖𝜉‖_𝑋^∗≤𝐿  for all 𝜉∈𝜕𝑓(𝑢).
(6)  𝑓⁰(𝑢,ℎ)=max{⟨𝜉,ℎ⟩∶𝜉∈𝜕𝑓(𝑢)} for all ℎ∈𝑋.
(7)  The function 𝑚(𝑢)=min_{𝑤∈𝜕𝑓(𝑢)}‖𝑤‖_𝑋^∗,(2.3) exists and is lower semicontinuous; that is, lim_𝑢→𝑢0inf𝑚(𝑢)≥𝑚(𝑢₀).

In the following we need the nonsmooth version of the Palais-Smale condition.

Definition 2.2. One says 𝜑 that nonsmooth satisfies the (PS)_𝑐-condition if any sequence {𝑢_𝑛}⊂𝑋 such that 𝜑(𝑢_𝑛)→𝑐 and 𝑚(𝑢_𝑛)→0.
In what follows we write the (PS)_𝑐-condition as simply as the PS-condition if it holds for every level 𝑐∈ℝ for the Palais-Smale condition at level 𝑐.
We note that property (4) above says that a local minimum (or local maximum) of 𝜑 is a critical point of 𝜑.Finally, we point out that many of the results of the classical critical point theory have been extended by Chang [4] to this setting of locally Lipschitz functionals. For example, one has the following celebrated theorem.

Theorem 2.3 (nonsmooth mountain pass theorem; see [4, 5]). If 𝑋 is a reflexive Banach space, 𝜑∶𝑋→ℝ is a locally Lipschitz function which satisfies the nonsmooth (PS)_𝑐-condition, and for some 𝑟>0 and 𝑒₁∈𝑋 with ‖𝑒₁‖>𝑟, max{𝜑(0),𝜑(𝑒₁)}≤inf{𝜑(𝑢)∶‖𝑢‖=𝑟}. Then 𝜑 has a nontrivial critical 𝑢∈𝑋 such that the critical value 𝑐=𝜑(𝑢) is characterized by the following minimax principle: 𝑐=inf_𝛾∈Γmax_[]𝑡∈0,1𝜑(𝛾(𝑡)),(2.4) where Γ={𝛾∈𝐶([0,1],𝑋)∶𝛾(0)=0,𝛾(1)=𝑒₁}.

We would like to point out that we can obtain the same results for problem (1.3) through approximation of the discontinuous nonlinearity by a sequence of continuous functions. Variational methods were then applied to the corresponding sequence of problems and limits were taken. For the rest of this paper, we write 𝑋=𝑊^01,𝑝(Ω) with the norm by ‖𝑢‖=(∫_Ω|∇𝑢|^𝑝𝑑𝑥)^1/𝑝 and denote by 𝑐 and 𝑐_𝑖 the generic positive constants for simplicity.

3. Proof of the Main Results

Now, having listed some basic results on critical point theory for the Lipschitz functionals, let us consider the functional 𝜑_𝜆(𝑢)=1𝑝_Ω||∇𝑢||𝑑𝑥−𝜆_Ω𝐺(𝑢)𝑑𝑥,(3.1) where 𝐺(𝑢)=∫^𝑢0𝑔(𝑠)𝑑𝑠 and 𝑔(𝑠) were defined in (1.2). Clearly 𝐺∶ℝ→ℝ is a locally Lipschitz continuous function and satisfies 𝐺(𝑠)=0 for 𝑠≤0. In view of [3, Theorems 2.1 and 2.2], the above formula for 𝜑_𝜆(𝑢) defines a locally Lipschitz functional on 𝑋 whose critical points are solutions of the differential inclusion −div||∇𝑢||^𝑝−2∇𝑢∈𝜕𝐺(𝑢)a.e.inΩ.(3.2) In our present case, it follows that 𝜕𝐺(𝑠)=𝑓(𝑠) for 𝑠>0, 𝜕𝐺(𝑠)=0 for 𝑠<0, and 𝜕𝐺(𝑠)=[−𝑎,0] for 𝑠=0.

We start with some preliminary lemmas.

Lemma 3.1. Assume (F0), (F1), and (F2). Then there exists 𝜆₀ such that problem (1.3) has no nontrivial solution 0≤𝑢∈𝑋 for 0<𝜆<𝜆₀.

Proof . If 𝑢≥0 is a solution of problem (1.3), then, multiplying the equation by 𝑢 and integrating over Ω yields ‖𝑢‖^𝑝=_Ω||∇𝑢||^𝑝𝑑𝑥=𝜆_Ω𝑔(𝑢)𝑢𝑑𝑥=𝜆_{{𝑥∈Ω∣𝑢(𝑥)≥𝛿0}}𝑢𝑔(𝑢)𝑑𝑥+_{{𝑥∈Ω∣𝑢(𝑥)≥𝛿0}}𝑢𝑔(𝑢)𝑑𝑥,(3.3) hence ‖𝑢‖^𝑝≤𝜆_{{𝑥∈Ω∣𝑢(𝑥)≥𝛿0}}𝑢𝑔(𝑢)𝑑𝑥,(3.4) where we have chosen 𝛿₀>0 so that 𝑔(𝑠)≤0 for 0≤𝑠≤𝛿₀ (such a 𝛿₀ exists in view of (F0)). Now, since (F1) implies the existence of 𝑐>0 such that 𝑠𝑔(𝑠)≤𝑐(1+𝑠^𝑝),(3.5) for all 𝑠>0, we obtain from (3.4) that ‖𝑢‖^𝑝≤𝜆𝑐_{{𝑥∈Ω∣𝑢(𝑥)≥𝛿0}}(1+𝑢^𝑝)𝑑𝑥≤𝜆𝑐1+1𝛿^𝑝0_{{𝑥∈Ω∣𝑢(𝑥)≥𝛿0}}𝑢^𝑝𝑑𝑥≤𝜆𝑐₁_Ω𝑢^𝑝𝑑𝑥,(3.6) so that ‖𝑢‖^𝑝≤𝜆𝑐₂‖𝑢‖^𝑝,(3.7) where this last constant 𝑐₂>0 is independent of both 𝑢 and 𝜆. Therefore we must have 𝜆≥1𝑐₂∶=𝜆₀>0.(3.8)

Lemma 3.2. Assume (F0) and either (F1) or (F3). Then 𝑢=0 is a strict local minimum of the functional 𝜑_𝜆.

Proof . Since (F1) or (F3) implies the existence of 𝑐₃>0 such that 𝐺(𝑠)≤𝑐₃1+|𝑠|^𝑝∗∀𝑠∈ℝ,(3.9) recall also that 𝐺(𝑠)=0 for 𝑠≤0. Then, with 𝛿₀>0 as in the proof of Lemma 3.1 and noticing that 𝐺(𝑠)≤0 for all −∞<𝑠≤𝛿₀, we can write for an arbitrary 𝑢∈𝑋, 𝜑_𝜆(𝑢)=1𝑝‖𝑢‖^𝑝≥1−𝜆_Ω𝐺(𝑢)𝑑𝑥𝑝‖𝑢‖^𝑝≥1−𝜆_{{𝑥∈Ω∣𝑢(𝑥)≥𝛿0}}𝐺(𝑢)𝑑𝑥𝑝‖𝑢‖^𝑝−𝜆𝑐₃≥1_{{𝑥∈Ω∣𝑢(𝑥)≥𝛿0}}1+𝑢^𝑝∗𝑑𝑥𝑝‖𝑢‖^𝑝−𝜆𝑐₃1+𝛿1^𝑝∗₀_{{𝑥∈Ω∣𝑢(𝑥)≥𝛿0}}𝑢^𝑝∗𝑑𝑥,(3.10) so that, using the Sobolev embedding theorem in the last inequality and with a constant 𝑐₄>0 independent of 𝑢 and Ω, we obtain 𝜑_𝜆(𝑢)≥1𝑝‖𝑢‖^𝑝−𝜆𝑐₄‖𝑢‖^𝑝∗=1𝑝‖𝑢‖^𝑝1−𝑝𝜆𝑐⁴‖𝑢‖^{𝑝∗−𝑝}.(3.11) Therefore, for each 0<𝜌<𝜌_𝜆∶=1/(𝑝𝜆𝑐₄)^{𝑝∗−𝑝}, it follows that 𝜑_𝜆(𝑢)≥𝛼_𝜌>0 if ‖𝑢‖=𝜌. This shows that 𝑢=0 is a strict local minimum of 𝜑_𝜆.

Lemma 3.3. Under the same assumptions as in Lemma 3.2, let 𝑢̂∈𝑋 be a critical point of 𝜑_𝜆. Then, 𝑢̂∈𝐶^1,𝛼(Ω) and 𝑢̂ is a nonnegative solution of problem (1.3), where 0<𝛼<1.

Proof . We shall follow some of the arguments in [3]. As mentioned earlier, if 𝑢̂ is a critical point of 𝜑_𝜆, then it is shown in [3] that 𝑢̂ is a solution of the differential inclusion −div||∇𝑢||̂^𝑝−2∇𝑢̂∈𝜕𝐺(𝑢)̂a.e.inΩ.(3.12) Since 𝑔 is only discontinuous at 𝑠=0, the above differential inclusion reduces to an equality, except possibly on the subset 𝐴⊂Ω where 𝑢̂=0. Since 𝑓 is subcritical, using the 𝐶^1,𝛼 regularity results for quasilinear elliptic equations with 𝑝-growth condition (see, for example, [6]), we have 𝑢̂∈𝐶^1,𝛼(Ω⧵𝐴). And, in this latter case, −div(|∇𝑢|̂^𝑝−2∇𝑢)̂ takes on values in the bounded interval [−𝑎,0]. Therefore, by the Michael selection theorem (see Theorem  1.2.5 of [5]), we see that −Δ_𝑝∶𝑢↦[−𝑎,0] admits a continuous selection. Using the 𝐶^1,𝛼 regularity results for quasilinear elliptic equations with 𝑝-growth condition again, it follows that 𝑢̂∈𝐶^1,𝛼(Ω), 0<𝛼<1.
Next, using a Morrey-Stampacchia theorem [7, Theorem  3.2.2, page 69], we have that −div(|∇𝑢|̂^𝑝−2∇𝑢)̂=0 a.e. in 𝐴. Therefore, since we defined 𝑔(0)=0, it follows that −div||∇𝑢||̂^𝑝−2∇𝑢̂=𝑔(𝑢)̂a.e.inΩ.(3.13) Replacing the inclusion (3.2) on 𝑢̂, we conclude that 𝑢̂∈𝐶^1,𝛼(Ω) is a solution of (1.3). Finally, recalling that 𝑔(𝑠)=0 for 𝑠≤0, it is clear that 𝑢̂≥0. The proof of Lemma 3.3 is complete.

Lemma 3.4. Assume either (F1) or (F3), (F4). Then 𝜑_𝜆 satisfies the nonsmooth (𝑃𝑆)_𝑐-condition at every 𝑐∈ℝ.

Proof. Let {𝑢_𝑛}_𝑛≥1⊆𝑋 be a sequence such that |𝜑_𝜆(𝑢_𝑛)|≤𝑐 for all 𝑛≥1 and 𝑚(𝑢_𝑛)→0 as 𝑛→∞. In the superlinear and subcritical case, from (F4), we have 𝑐≥𝜑_𝜆𝑢_𝑛=1𝑝_Ω≥‖‖𝑢||∇𝑢_𝑛||^𝑝𝑑𝑥−𝜆_Ω𝐺𝑢_𝑛𝑑𝑥_𝑛‖‖^𝑝≥1𝑝−𝜆𝜃_Ω𝜉𝑢_𝑛,𝑢_𝑛𝑑𝑥−𝑐𝑝−𝜃1‖‖𝑢^𝑛‖‖^𝑝+_Ω1𝜃‖‖𝑢^𝑛≥1‖‖^𝑝−𝜆𝜉𝑢_𝑛,𝑢_𝑛𝑑𝑥−𝑐𝑝−𝜃1‖‖𝑢^𝑛‖‖^𝑝−𝜆𝜃‖𝜉‖^𝑋∗‖‖𝑢_𝑛‖‖−𝑐,(3.14) where 𝜉(𝑢_𝑛)∈𝜕𝐺(𝑢_𝑛). Hence {𝑢_𝑛}_𝑛≥1⊆𝑋 is bounded.
Thus by passing to a subsequence if necessary, we may assume that 𝑢_𝑛⇀𝑢 in 𝑋 as 𝑛→+∞. We have 𝐽^𝑢_𝑛,𝑢_𝑛−𝑢−_Ω𝜉𝑢_𝑛𝑢_𝑛−𝑢𝑑𝑥≤𝜀_𝑛‖‖𝑢_𝑛−𝑢‖‖(3.15) with 𝜀_𝑛↓0, where 𝜉(𝑢_𝑛)∈𝜕Ψ(𝑢_𝑛) and 𝐽=(1/𝑝)∫_Ω|∇𝑢_𝑛|^𝑝𝑑𝑥. From (F3) and Chang [4] we know that 𝜉(𝑢_𝑛(𝑥))∈𝐿^{(𝑝∗−1)′}(Ω) ((𝑝^∗−1)^=(𝑝^∗−1)/(𝑝^∗−2)). Since 𝑋 is embedded compactly in 𝐿^𝑝∗−1(Ω), we have that 𝑢_𝑛→𝑢 as 𝑛→∞ in 𝐿^𝑝∗−1(Ω). So using the Hölder inequality, we have _Ω𝜉_𝑛(𝑥)𝑢_𝑛−𝑢𝑑𝑥⟶0as𝑛⟶+∞.(3.16)
Therefore, we obtain that lim_𝑛→∞sup⟨𝐽′(𝑢_𝑛),𝑢_𝑛−𝑢⟩≤0. But we know that 𝐽′ is a mapping of type (𝑆₊). Thus we have 𝑢_𝑛⟶𝑢in𝑋.(3.17) Using the similar method, we can more easily get the nonsmooth (PS)_𝑐-condition in the case of sublinear.

Remark 3.5. Note that we cannot directly apply the results of [4] because the operator −Δ_𝑝 is not self-adjoint linear.

Lemma 3.6. Under assumptions (F0), (F1), and (F2), let Ω=𝐵_𝑅⊂ℝ^𝑁 with 𝑁≥2, and let 𝑢∈𝐶¹(𝐵_𝑅) be a radially symmetric, nonincreasing function such that 𝑢≥0 and 𝑢 is a minimizer of 𝜑_𝜆(𝑢) with 𝜑_𝜆(𝑢)=𝑚<0. Then, 𝑢 does not vanish in 𝐵_𝑅; that is, 𝑢(𝑥)>0 for all 𝑥∈𝐵_𝑅.

Proof. Since 𝑔 is discontinuous at zero, we note that the conclusion does not follow directly from uniqueness of solution for the Cauchy problem with data at 𝑟=𝑅 (in fact, writing 𝑢=𝑢(𝑟), 𝑟=|𝑥|, we may have 𝑢(𝑅)=𝑢′(𝑅)=0 and 𝑢≢0).
Now, since 𝑢≢0 by assumption, 𝑅₀∶=inf{𝑟≤𝑅∶𝑢(𝑠)=0for𝑟≤𝑠≤𝑅} satisfies 0<𝑅₀≤𝑅. If 𝑅₀=𝑅 there is nothing to prove in view of the fact that 𝑢 is non-increasing. On the other hand, if 𝑅₀<𝑅 then 𝑢′(𝑅₀)=0 and 𝑢(𝑟)>0 for 𝑟∈[0,𝑅₀). It is not hard to prove that this contradicts that 𝑢 is a minimizer of 𝜑_𝜆. Indeed, if 𝑅₀<𝑅then 𝜑_𝜆(𝑢)=1𝑝_𝐵𝑅0||∇𝑢||^𝑝𝑑𝑥−𝜆_𝐵𝑅0𝐺(𝑢)𝑑𝑥=𝑚<0.(3.18) A simple calculation shows that the rescaled function 𝑣(𝑟)=𝑢(𝑅₀𝑟/𝑅)∈𝑊^01,𝑝(𝐵_𝑅)∩𝐶¹(𝐵_𝑅) satisfies 𝜑_𝜆(𝑣)=𝛿^𝑝−𝑁1𝑝_𝐵𝑅0||∇𝑢||^𝑝𝑑𝑥−𝛿^−𝑝𝜆_𝐵𝑅0𝐺(𝑢)𝑑𝑥,(3.19) where 𝛿∶=𝑅₀/𝑅 is less than 1. Therefore, since we are assuming 1<𝑝≤𝑁, we would reach the contradiction 𝜑_𝜆(𝑣)<𝑚.

Remark 3.7. Note that the condition (F2) is necessary because of which guarantee 𝐺 can have positive values.

Proof of Theorem 1.1. We observe that the functional 𝜑_𝜆 is weakly lower semicontinuous on 𝑋. Moreover, the sublinearity assumption (F1) on 𝑔(𝑢) implies that 𝜑_𝜆 is coercive. Therefore, the infimum of 𝜑_𝜆 is attained at some 𝑢_𝜆: inf_𝑢∈𝑋𝜑_𝜆(𝑢)=𝜑_𝜆𝑢_𝜆.(3.20) And, in view of Lemma 3.3, 𝑢_𝜆∈𝐶^1,𝛼(Ω) is a nonnegative solution of (1.3). We now claim that 𝑢_𝜆 is nonzero for all 𝜆>0 large.

Claim 1. There exists Λ>0 such that 𝜑_𝜆(𝑢_𝜆)<0 for all 𝜆≥Λ.
In order to prove the claim it suffices to exhibit an element 𝑤∈𝑋 such that 𝜑_𝜆(𝑤)<0 for all 𝜆>0 large. This is quite standard considering that 𝐺(𝛿)>0 by (F2). Indeed, letting Ω_𝜀∶={𝑥∈Ω∶dist(𝑥,𝜕Ω)>𝜀} for 𝜀>0 small, define 𝑤 so that 𝑤(𝑥)=𝛿 for 𝑥∈Ω_𝜀 and 0≤𝑤(𝑥)≤𝛿 for 𝑥∈Ω⧵Ω_𝜀. Then 𝜑_𝜆(𝑤)=1𝑝‖𝑤‖^𝑝≤1−𝜆_Ω𝜀𝐺(𝑤)𝑑𝑥+_Ω⧵Ω𝜀𝐺(𝑤)𝑑𝑥𝑝‖𝑤‖^𝑝−𝜆𝐺(𝛿)measΩ_𝜀−𝑐(1+𝛿^𝑝)measΩ⧵Ω_𝜀,(3.21) where we note that the expression in the above parenthesis is positive if we choose 𝜀>0 sufficiently small. Therefore, there exists Λ>0 such that 𝜑_𝜆(𝑤)<0 for all 𝜆≥Λ, which proves the claim.
On the other hand, when Ω=𝐵_𝑅, let 𝑢^∗𝜆 denote the Schwarz symmetrization of 𝑢_𝜆, namely, 𝑢^∗𝜆 is the unique radially symmetric, nonincreasing, nonnegative function in 𝑋 which is equi-measurable with 𝑢_𝜆. As is well known, _𝐵𝑅𝐺𝑢^∗𝜆𝑑𝑥=_𝐵𝑅𝐺𝑢_𝜆𝑑𝑥,(3.22) and ‖𝑢^∗𝜆‖≤‖𝑢_𝜆‖ (the Faber-Krahn inequality; see [8]), so that 𝜑_𝜆(𝑢^∗𝜆)≤𝜑_𝜆(𝑢_𝜆). Therefore, we necessarily have 𝜑_𝜆(𝑢^∗𝜆)=𝜑_𝜆(𝑢_𝜆) and may assume that 𝑢_𝜆=𝑢^∗𝜆. Moreover, 𝑢_𝜆>0 in Ω by Lemma 3.6. Therefore, 𝑢_𝜆 is a positive solution of both problems (1.1) and (1.3).
Next, we recall that 𝑢=0 is a strict local minimum of 𝜑_𝜆 by Lemma 3.2. Therefore, since 𝜑_𝜆 satisfies the nonsmooth Palais-Smale condition by Lemma 3.4, we can use the minima 𝑢=0 and 𝑢=𝑢_𝜆 of 𝜑_𝜆 to apply the nonsmooth mountain pass theorem and conclude that there exists a second nontrivial critical point 𝑣_𝜆 with 𝜑_𝜆(𝑣_𝜆)>0. Again, 𝑣_𝜆 is a nonnegative solution of problem (1.3) in view of Lemma 3.3. In addition, when Ω=𝐵_𝑅, arguments similar to above passage show that we may assume 𝑣_𝜆=𝑣^∗𝜆. The proof of Theorem 1.1 is complete.

Proof of Theorem 1.2. As is well-known, the AR condition (F4) readily implies the existence of an element 𝑒_𝜆∈𝑋 such that 𝜑_𝜆(𝑒_𝜆)≤0. On the other hand, Lemma 3.2 says that 𝑢=0 is a (strict) local minimum of 𝜑_𝜆 and Lemma 3.4 says that 𝜑_𝜆 satisfies nonsmooth (PS)_𝑐 for every 𝑐∈ℝ. Therefore, an application of the nonsmooth mountain pass theorem stated in section 2 yields the existence of a critical point 𝑣_𝜆 such that 𝜑_𝜆𝑣_𝜆>0.(3.23) In particular, 𝑣_𝜆≠0, and it follows that 𝑣_𝜆 is a nonnegative solution of problem (1.3) by Lemma 3.3. As in the proof of Theorem 1.1, we may assume that 𝑣_𝜆=𝑣^∗𝜆 in the case of a ball Ω=𝐵_𝑅.
Finally, still in the case of a ball Ω=𝐵_𝑅, we claim that there exists 𝜆^∗>0 such that, for all 0<𝜆<𝜆^∗, 𝑣_𝜆=𝑣^∗𝜆 is a positive solution of problem (1.3) (hence of problem (1.1)) having negative normal derivative on 𝜕𝐵_𝑅. Indeed, if that is not the case, then, for any given 𝜆>0, we can find 0<𝜇=𝜇(𝜆)<𝜆 such that the nonnegative solution 𝑣_𝜇=𝑣^∗𝜇 of problem (1.1) with 𝜆=𝜇 obtained a bove satisfies 𝑣_𝜇(𝑟)>0for𝑟∈0,𝑅₀,𝑣^𝜇𝑅₀=0,𝑣_𝜇(𝑟)=0for𝑟∈𝑅₀,𝑅,(3.24) for some 0<𝑅₀≤𝑅. Therefore, the rescaled function 𝑤_𝜇(𝑟)∶=𝑣_𝜇(𝑅₀𝑟/𝑅) is a positive solution of (1.1) with 𝜆=𝜇₀ (again in the ball 𝐵_𝑅), with 𝜇₀∶=𝜇𝑅^𝑝0/𝑅^𝑝≤𝜇. This shows that we can always construct a decreasing sequence 𝜇_𝑛>0 satisfying alternative (ii) of Theorem 1.2, in case alternative (i) does not hold.

Acknowledgments

The author is very grateful to an anonymous referee for his or her valuable suggestions. Research supported by NSFC (no. 11061030, no. 10971087).