Existence of Positive Bounded Solutions of Semilinear Elliptic Problems

Abstract

This paper is concerned with the existence of bounded positive solution for the semilinear elliptic problem Δ𝑢=𝜆𝑝(𝑥)𝑓(𝑢) in Ω subject to some Dirichlet conditions, where Ω is a regular domain in ℝ^𝑛 (𝑛≥3) with compact boundary. The nonlinearity 𝑓 is nonnegative continuous and the potential 𝑝 belongs to some Kato class 𝐾(Ω). So we prove the existence of a positive continuous solution depending on 𝜆 by the use of a potential theory approach.

1. Introduction

In this paper, we study the existence of positive bounded solution of semilinear elliptic problem(𝑥)𝑓(𝑢)𝑢Δ𝑢=𝜆𝑝inΩ,𝜕Ω=𝛼𝜑,lim_{|𝑥|→+∞}𝑢(𝑥)=𝛽(ifΩisunbounded), (P) where Ω is a 𝒞^1,1-domain in ℝ^𝑛(𝑛≥3) with compact boundary, 𝛼 and 𝛽 are fixed nonnegative constants such that 𝛼+𝛽>0, and 𝛽=0 when Ω is bounded. The parameter 𝜆 is nonnegative, and the function 𝜑 is nontrivial nonnegative and continuous on 𝜕Ω.

Numerous works treated semilinear elliptic equations of the type()𝑢(𝑥)(𝑥)Δ𝑢=𝐹𝑥,𝑢in𝐷,=Φon𝜕𝐷,lim_{|𝑥|→+∞}𝑢(𝑥)=𝑏>0. (Q) For the case of nonpositive function 𝐹, many results of existence of positive solutions are established in smooth domains or in ℝ^𝑛, for instance, see [1–5] and the references therein.

In the case where 𝐹 changes sign, many works can be cited, namely, the work of Glover and McKenna [6], whose used techniques of probabilistic potential theory for solving semilinear elliptic and parabolic differential equations in ℝ^𝑛. Ma and Song [7] adapted the same techniques of those of Glover and McKenna to elliptic equations in bounded domains. More precisely, the hypotheses in [6, 7] require in particular that 𝐹(𝑥,𝑢)=𝐹(𝑢) and on each compact, there is a positive constant 𝐴 such that −𝐴𝑢≤𝐹(𝑢)≤0.

In [8], Chen et al. used an implicit probabilistic representation together with Schauder's fixed point theorem to obtain positive solutions of the problem (Q). In fact, the authors considered a Lipschitz domain 𝐷 in ℝ^𝑛(𝑛≥3), with compact boundary and imposed to the function 𝐹 to satisfy on 𝐷×(0,𝑏), 𝑏∈(0,+∞]−𝑈(𝑥)𝑡≤𝐹(𝑥,𝑡)≤𝑉(𝑥)𝑓(𝑡), (1.1) where 𝑓 is nonnegative Borel measurable function defined on (0,𝑏) and the potentials 𝑈,𝑉 are nonnegative Green-tight functions in 𝐷. In particular, the authors showed the existence of solutions of (Q) bounded below by a positive harmonic function.

In [9], Athreya studied (Q) with the singular nonlinearity 𝐹(𝑥,𝑡)=𝑔(𝑡)≤max(1,𝑡^−𝛼), 0<𝛼<1, in a simply connected bounded 𝐶²-domain 𝐷 in ℝ^𝑛,𝑛≥3. He showed the existence of solutions bounded below by a given positive harmonic function ℎ₀, under the boundary condition Φ≥(1+𝐴)ℎ₀, where 𝐴 is a constant depending on ℎ₀, 𝛼, and 𝐷.

Recently, Hirata [20] gave a Chen-Williams-Zhao type theorem for more general regular domains 𝐷. More precisely, the author imposed to the function 𝐹 to satisfy |𝐹(𝑥,𝑢)|≤𝑉(𝑥)𝑢^−𝜇,𝜇>0, (1.2) where the potential 𝑉 belongs to a class of functions containing Green-tight ones. We remark that the class of functions introduced by Hirata coincides with the classical Kato class introduced for smooth domains in [10, 11].

In this paper, we will consider 𝐹(𝑥,𝑢)=𝜆𝑝(𝑥)𝑓(𝑢). We impose to the potential 𝑝 to be in a new Kato class 𝐾(Ω) (see Definition 1.1 below), which contains the Green-tight functions and the classical Kato class used by Hirata. More precisely, we will prove using potential theory's tools, the existence of positive solution for (P). Moreover, we will give global behaviour for the solution.

So, in the remainder of this introduction, we will give some results related on potential theory, and we will prove others. In the second section, we will give the main theorem and some examples of applications.

Let us recall that ℬ(Ω) is the set of Borel measurable functions in Ω and 𝒞₀(Ω) is the set of continuous ones vanishing at 𝜕Ω∪{∞}. The exponent + means that only the nonnegative functions are considered.

We denote by 𝐻_Ω𝜑 the bounded continuous solution of the Dirichlet problemΔ𝑢=0inΩ,𝑢=𝜑on𝜕Ω,lim_{|𝑥|→+∞}𝑢(𝑥)=0,ifΩisunbounded, (1.3) where the function 𝜑 is nontrivial nonnegative continuous on 𝜕Ω. In the remainder of this paper, we denote ℎ=1−𝐻_Ω1, and we remark that ℎ=0 when Ω is bounded.

Let us recall some notations and notions concerning essentially the potential theory.(i)For 𝑓∈ℬ⁺(Ω), we denote by 𝑉𝑓 the potential defined in Ω by 𝑉𝑓(𝑥)=_Ω𝐺𝑥,𝑦𝑓𝑦𝑑𝑦, (1.4) where 𝐺 is the Green function of the Laplace operator Δ on Ω with Dirichlet conditions.(ii)We recall that if 𝑓∈𝐿¹_loc(Ω) and 𝑉𝑓∈𝐿¹_loc(Ω), then we have Δ(𝑉𝑓)=−𝑓 in Ω (in the sense of distributions), see [10, page 52].(iii)Let (𝑋_𝑡,𝑡>0) be the Brownian motion in ℝ^𝑛 and 𝑃^𝑥 be the probability measure on the Brownian continuous paths starting at 𝑥. For 𝑝∈ℬ⁺(Ω), we define the kernel 𝑉_𝑝 by 𝑉_𝑝𝑓(𝑥)=𝐸^𝑥𝜏Ω₀𝑒^{−∫𝑡0𝑝(𝑋𝑠)𝑑𝑠}𝑓𝑋_𝑡𝑑𝑡, (1.5) where 𝐸^𝑥 is the expectation on 𝑃^𝑥 and 𝜏_Ω=inf{𝑡>0∶𝑋_𝑡∉Ω}. If 𝑝∈ℬ⁺(Ω) is such that 𝑉𝑝<∞, the kernel 𝑉_𝑝 satisfies (see [10, 12]) 𝑉=𝑉_𝑝+𝑉_𝑝𝑝𝑉=𝑉_𝑝+𝑉𝑝𝑉_𝑝. (1.6) So for, each 𝑢∈ℬ(Ω) such that 𝑉(𝑝|𝑢|)<∞, we have 𝑢−𝑉_𝑝𝑝𝑢𝑢+𝑉𝑝𝑢=𝑢+𝑉𝑝𝑢𝑢−𝑉_𝑝𝑝𝑢=𝑢. (1.7)(iv)We recall that a function 𝑓∶[0,∞)→ℝ is called completely monotone if (−1)^𝑛𝑓^(𝑛)≥0, for each 𝑛∈ℕ. Moreover, if 𝑓 is completely monotone on [0,∞), then by [13, Theorem  12a], there exists a nonnegative measure 𝜇 on [0,∞) such that 𝑓(𝑡)=^∞₀exp(−𝑡𝑥)𝑑𝜇(𝑥). (1.8) So, using this fact and the Hölder inequality, we deduce that if 𝑓 is completely monotone from [0,∞) to (0,∞), then Log(𝑓) is a convex function.(v)Let 𝑓∈ℬ⁺(Ω) be such that 𝑉𝑓<∞. From (1.5), it is easy to see that for each 𝑥∈Ω, the function 𝑡→𝑉_𝑡𝑞𝑓(𝑥) is completely monotone on [0,∞).

Now, we recall some properties relating to the Kato class 𝐾(Ω).

Definition 1.1 (see [14, 15]). A Borel measurable function 𝑝 in Ω belongs to the class 𝐾(Ω) if 𝑝 satisfies

lim_𝛼→0sup_{𝑥∈Ω()Ω∩𝐵𝑥,𝛼}𝜌𝑦𝜌(𝑥)𝐺𝑥,𝑦𝑝𝑦𝑑𝑦=0,lim_𝑀→∞sup_{𝑥∈Ω𝑦Ω∩≥𝑀}𝜌𝑦𝜌(𝑥)𝐺𝑥,𝑦𝑞𝑦𝑑𝑦=0(ifΩisunbounded), (1.9) where 𝜌(𝑥)=min(1,𝛿(𝑥)) and 𝛿(𝑥) is the Euclidean distance between 𝑥 and 𝜕Ω.

Remark 1.2. When Ω is a bounded domain, then we can replace 𝜌(𝑥) by 𝛿(𝑥) and the condition (1.9) is superfluous.

Proposition 1.3 (See [14, 15]). Let 𝑝 be a nonnegative function in 𝐾(Ω). Then one has (i)‖𝑝‖_Ω∶=sup_𝑥∈Ω∫_Ω𝜌(𝑦)/𝜌(𝑥)𝐺(𝑥,𝑦)𝑝(𝑦)𝑑𝑦<∞,(ii)the potential 𝑉𝑝∈𝒞₀(Ω).

Proposition 1.4 (see [16, 17]). Let 𝑝 be a nonnegative function belonging to 𝐾(Ω). Then, one has (i)𝛼_𝑝=sup_{𝑥,𝑦∈ΩΩ}𝐺(𝑥,𝑧)𝐺𝑧,𝑦𝐺𝑝(𝑧)𝑥,𝑦𝑑𝑧<∞, (1.10)(ii) for any nonnegative superharmonic function 𝑣 in Ω, one has _Ω𝐺𝑥,𝑦𝑣𝑦𝑝𝑦𝑑𝑦≤𝛼_𝑝𝑣(𝑥),∀𝑥∈𝐷. (1.11)

Proposition 1.5. Let 𝑣 be a nonnegative superharmonic function in Ω and 𝑝 be a nonnegative function in 𝐾(Ω). Then, for each 𝑥∈Ω such that 0<𝑣(𝑥)<∞, one has exp−𝛼_𝑝𝑣(𝑥)≤𝑣(𝑥)−𝑉_𝑝𝑝𝑣(𝑥)≤𝑣(𝑥). (1.12)

Proof. Let 𝑣 be a nonnegative superharmonic function, then by [18, Theorem  2.1, page 164], there exists a sequence (𝑣_𝑘)_𝑘 of nonnegative measurable functions in Ω such that the sequence (𝑉𝑣_𝑘)_𝑘 defined in Ω by 𝑉𝑣_𝑘(𝑥)∶=_Ω𝐺𝑥,𝑦𝑣_𝑘𝑦𝑑𝑦 (1.13) increases to 𝑣.
Let 𝑥∈Ω such that 0<𝑣(𝑥)<∞. Then, there exists 𝑘₀∈ℕ such that 0<𝑉𝑣_𝑘(𝑥)<∞, for 𝑘≥𝑘₀.
Now, for a fixed 𝑘≥𝑘₀, we consider the function 𝜅(𝑡)=𝑉_𝑡𝑝𝑣_𝑘(𝑥). Since the function 𝜅 is completely monotone on [0,∞), then log(𝜅) is convex on [0,∞). Therefore, 𝜅(0)(1)−𝜅≤𝜅exp^′(0)𝜅(0), (1.14) which means 𝑉𝑣_𝑘(𝑥)≤𝑉_𝑝𝑣_𝑘(𝑥)𝑉exp𝑝𝑉𝑣_𝑘(𝑥)𝑉𝑣_𝑘(𝑥). (1.15) Hence, it follows from Proposition 1.4 (i) that exp−𝛼_𝑣𝑉𝑣_𝑘(𝑥)≤𝑉_𝑝𝑣_𝑘(𝑥). (1.16) Consequently, from (1.6), we obtain that exp−𝛼_𝑝𝑉𝑣_𝑘(𝑥)≤𝑉𝑣_𝑘(𝑥)−𝑉_𝑝𝑝𝑉𝑣_𝑘(𝑥)(𝑥)≤𝑉𝑣_𝑘(𝑥). (1.17) By letting 𝑘→+∞, we deduce the result.

2. Main Result

In this section, we will give an existence result for the problem (P). Assume the following assumptions.(A₁)  The function 𝑝 is nonnegative and belongs to 𝐾(Ω).(A₂)   The function 𝑓 is a nonnegative, continuous on [0,+∞) and satisfies ∀𝑐>0,∃𝑎≥0 such that, ∀0≤𝑠<𝑡≤𝑐, 𝑓(𝑡)−𝑓(𝑠)≤𝑎(𝑡−𝑠).(A₃)   𝜎₀=inf_𝑥∈Ω(𝛼𝐻_Ω𝜑(𝑥)+𝛽ℎ(𝑥))/𝑓(0)𝑉𝑝(𝑥)>0.

Remark 2.1. Let 𝑓 be in 𝐶¹([0,+∞)), then for 𝑎∶=max(sup_{𝑡∈[0,𝑐]}𝑓^′(𝑡),0), the function 𝑓 satisfies (A₂). In particular, if 𝑓 is nonincreasing, then 𝑎=0 holds.

Consider the function 𝜃∶𝜆→𝜆exp(𝜆𝑎𝛼_𝑝), where 𝛼_𝑝 is the constant associated to the potential 𝑝 defined by (1.10). It is obvious to see that 𝜃 is bijective from [0,+∞) to [0,+∞).

Theorem 2.2. Assume that the hypotheses (A₁)–(A₃) are satisfied. Then, for each 𝜆∈[0,𝜃⁻¹(𝜎₀)), the problem (P) has a positive continuous bounded solution satisfying 𝜃(𝜆)1−𝜎₀exp−𝜆𝑎𝛼_𝑝𝛼𝐻_Ω𝜑+𝛽ℎ≤𝑢≤𝛼𝐻_Ω𝜑+𝛽ℎ. (2.1)

Remark 2.3. We remark that if 𝑓 satisfies the hypothesis (A₂) and 𝑓(0)=0, we take 𝜎₀=+∞, in this case for each 𝜆∈ℝ₊, the problem (P) has a positive bounded solution satisfying exp−𝜆𝑎𝛼_𝑝𝛼𝐻_Ω𝜑+𝛽ℎ≤𝑢≤𝛼𝐻_Ω𝜑+𝛽ℎ. (2.2)

Now, let us give some examples of applications of the above theorem.

Example 2.4. Assume that (A₁) is satisfied. Let 𝜇≥1. Then, for each 𝜆∈ℝ₊, the following problem Δ𝑢=𝜆𝑝(𝑥)𝑢^𝜇𝑢(𝑥)(𝑥)inΩ,=𝛼𝜑on𝜕Ω,lim_{|𝑥|→+∞}𝑢(𝑥)=𝛽 (2.3) admits a positive continuous bounded solution. Indeed, for each 𝑐>0, one verifies that for 𝑎=𝜇𝑐^𝜇−1, the function 𝑓(𝑡)=𝑡^𝜇 satisfies (A₂).

Example 2.5. Let 𝜇≥0. Assume (A₁) and (A₃). Consider the following: Δ𝑢=𝜆𝑝(𝑥)(1+𝑢)^−𝜇𝑢(𝑥)(𝑥)inΩ,=𝛼𝜑on𝜕Ω,lim_{|𝑥|→+∞}𝑢(𝑥)=𝛽,𝑢>0. (2.4) Then, the function 𝑓(𝑡)=(1+𝑡)^−𝜇 is in 𝐶¹([0,+∞)) and decreasing. By Remark 2.1, the hypothesis (A₂) is satisfied for 𝑎=0. So that for each 𝜆∈[0,𝜎₀), (2.4) has a positive continuous bounded solution satisfying 𝜆1−𝜎₀𝛼𝐻_Ω𝜑+𝛽ℎ≤𝑢≤𝛼𝐻_Ω𝜑+𝛽ℎ. (2.5)

Example 2.6. Let Ω be a 𝐶^1,1-bounded domain and suppose that the hypothesis (A₂) is satisfied. Let 𝑔 be a nonnegative function in 𝐿^𝑞(Ω) such that 𝑞>𝑛/2 and suppose that 𝜇<1−𝑛/𝑞. Then, 𝑔(𝑥)Δ𝑢=𝜆𝛿(𝑥)^𝜇𝑓(𝑢)𝑢(𝑥)(𝑥)inΩ,=𝛼𝜑on𝜕Ω (2.6) has a positive continuous solution.
Let us verify the assumptions (A₁) and (A₃). From [16, Proposition 2.3], the function 𝑝=𝑔/𝛿(⋅)^𝜇∈𝐾(Ω), and so the hypothesis (A₁) is satisfied. From [16, Proposition 2.7(iii)], there exists a constant 𝑐₁>0 such that we have for each 𝑥∈Ω𝑉𝑝(𝑥)≤𝑐₁𝛿(𝑥). (2.7) Now, since 𝜑 is nontrivial continuous function at 𝜕Ω, then there exists 𝑐₂>0, such that one has on Ω𝐻_Ω𝜑(𝑥)≥𝑐₂𝜑₁𝛿(𝑥). (2.8) Thus, 𝜎₀=inf_𝑥∈Ω(𝐻_Ω𝜑(𝑥)/𝑓(0)𝑉𝑝(𝑥))>0 and so the assumption (A₃) is satisfied.

Example 2.7. Let Ω=𝐵(0,1)^c be the exterior of the unit ball in ℝ^𝑛(𝑛≥3). Suppose that the hypothesis (A₂) is satisfied. Let 𝛾,𝜇∈ℝ such that 𝛾<1<2<𝑛<𝜇. Then, 1Δ𝑢=𝜆|𝑥|^𝜇−𝛾(|𝑥|−1)^𝛾𝑓(𝑢)𝑢(𝑥)(𝑥)inΩ,=𝛼𝜑in𝜕Ω,lim_{|𝑥|→+∞}𝑢(𝑥)=𝛽 (2.9) has a positive continuous solution.
From [14], the function 𝑝(𝑥)=1/|𝑥|^𝜇−𝛾(|𝑥|−1)^𝛾∈𝐾(Ω) and so the assumption (A₁) is satisfied. Moreover, from [14, Proposition 3.5], there exists a constant 𝑐₁>0 such that one has 𝑉𝑝(𝑥)≤𝑐₁|𝑥|−1|𝑥|^𝑛−1. (2.10) Now, from [19, page 258], there exists a constant 𝑐₂>0 such that one has on Ω𝛼𝐻_Ω𝜑(𝑥)+𝛽ℎ(𝑥)≥𝑐₂|𝑥|−1|𝑥|^𝑛−1. (2.11) Thus, 𝜎₀=inf_𝑥∈Ω((𝛼𝐻_Ω𝜑(𝑥)+𝛽ℎ(𝑥))/𝑓(0)𝑉𝑝(𝑥))≥𝑐₂/𝑐₁>0 and so the assumption (A₃) is satisfied.

In the next, we will give the proof of Theorem 2.2.

Proof. Let 𝑝∈𝐾⁺(Ω) and put 𝑤∶=𝛼𝐻_Ω𝜑+𝛽ℎ. Let 𝑐=‖𝜔‖_∞>0, then from (A₂), there exists 𝑎≥0, such that the function 𝜓∶𝑡→𝑎𝑡+𝑓(0)−𝑓(𝑡) is a nondecreasing function on [0,𝑐]. Let 𝜎₀ be the constant given by (A₃), and let 𝜆∈[0,𝜃⁻¹(𝜎₀)) where 𝜃(𝜆)∶=𝜆exp(𝜆𝑎𝛼_𝑝). Put 𝑞∶=𝜆𝑎𝑝. Consider the nonempty bounded convex set given by Λ∶=𝑢∈ℬ⁺(Ω)∶𝜃(𝜆)1−𝜎₀exp−𝜆𝑎𝛼_𝑝𝑤≤𝑢≤𝑤. (2.12) Let 𝑇 be the operator defined on Λ by 𝑇𝑢∶=𝑤−𝑉_𝑞𝑞𝑤−𝜆𝑓(0)𝑉_𝑞𝑝+𝜆𝑉_𝑞𝑝𝜓(𝑢). (2.13) We claim that the operator 𝑇 maps Λ to itself. Indeed, by (A₂) and using the monotony of the function 𝜓, we have for 𝑢∈Λ𝑇𝑢=𝑤−𝑉_𝑞𝜆𝑝𝑎𝑤+𝑓(0)+𝜆𝑉_𝑞𝑝𝜓(𝑢)=𝑤−𝑉_𝑞𝜆𝑝𝜓(𝑤)+𝑓(𝑤)+𝜆𝑉_𝑞𝑝𝜓(𝑢)≤𝑤−𝜆𝑉_𝑞𝑝𝑓(𝑤)+𝜆𝑉_𝑞𝑝𝜓(𝑢)−𝜓(𝑤)≤𝑤. (2.14) On the other hand, by using Proposition 1.5 and (A₃), we have 𝑇𝑢≥exp−𝜆𝑎𝛼_𝑝𝑤−𝜆𝑓(0)𝑉_𝑞𝑝+𝜆𝑉_𝑞𝑝𝜓(𝑢)≥exp−𝜆𝑎𝛼_𝑝𝑤−𝜆𝑓(0)𝑉𝑝≥exp−𝜆𝑎𝛼_𝑝𝜆𝑤−𝜎₀𝑤=𝜃(𝜆)1−𝜎₀exp−𝜆𝑎𝛼_𝑝𝑤. (2.15) Hence, 𝑇Λ⊂Λ.
Next, we prove that the operator 𝑇 is nondecreasing on Λ. Let 𝑢₁,𝑢₂∈Λ such that 𝑢₁≤𝑢₂,then by hypothesis (A₂), we obtain𝑇𝑢₂−𝑇𝑢₁=𝜆𝑉_𝑞𝑝𝜓𝑢₂−𝜓𝑢₁≥0. (2.16) Now, consider the sequence (𝑢_𝑘)_𝑘 defined by 𝑢₀=𝑤−𝜆𝑎𝑉_𝑞𝑝𝑤−𝜆𝑓(0)𝑉_𝑞𝑝, (2.17) and 𝑢_𝑘+1=𝑇𝑢_𝑘 for 𝑘∈ℕ.
It is obvious to see that 𝑢₀∈Λ and 𝑢₁=𝑇𝑢₀≥𝑢₀. Thus, using the fact that Λ is invariant under 𝑇 and the monotony of 𝑇, we deduce that𝜃(𝜆)1−𝜎₀exp−𝜆𝑎𝛼_𝑝𝑤≤𝑢₀≤𝑢₁≤⋯≤𝑢_𝑘≤𝑤. (2.18) Hence, the sequence (𝑢_𝑘)_𝑘 converges to a function 𝑢∈Λ.
Therefore, from the monotone convergence theorem and the fact that 𝜓 is continuous, the sequence (𝑇𝑢_𝑘)_𝑘 converges to 𝑇𝑢. So,𝑢∶=𝑤−𝑉_𝑞𝑞𝑤−𝜆𝑓(0)𝑉_𝑞𝑝+𝜆𝑉_𝑞𝑝𝜓(𝑢), (2.19) or equivalently 𝑢−𝑉_𝑞𝑞𝑢=𝑤−𝑉_𝑞𝑞𝑤−𝑉_𝑞𝜆𝑝𝑓(𝑢). (2.20) Applying the operator (𝐼+𝑉(𝑞⋅)) to both sides of the above equality and using (1.6) and (1.7), we conclude that 𝑢 satisfies 𝑢=𝛼𝐻_Ω𝜑+𝛽ℎ−𝜆𝑉𝑝𝑓(𝑢). (2.21) Finally, let us verify that 𝑢 is a solution of the problem (P). Using the fact that 𝑝∈𝐾⁺(Ω) and 𝑓(𝑢) is bounded on [0,𝑐], we obtain 𝑝𝑓(𝑢)∈𝐾⁺(Ω). So, Proposition 1.3 (ii) yields 𝑉(𝑝𝑓(𝑢))∈𝒞₀(Ω) which implies with the continuity of the harmonic continuous function 𝑤 that 𝑢 is continuous on Ω. This completes the proof.

Acknowledgment

The author express her sincere gratitude to Professor Habib Msâagli for his guidance and the useful discussions. Thanks go to the referees for valuable comments and useful remarks on the paper.