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 ℝⁿ (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

()

where Ω is a 𝒞^1,1-domain in ℝⁿ (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

()

For the case of nonpositive function F, many results of existence of positive solutions are established in smooth domains or in ℝⁿ, 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 ℝⁿ. 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 ℝⁿ (n ≥ 3), with compact boundary and imposed to the function F to satisfy on D × (0, b), b ∈ (0, +∞]

()

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 ℝⁿ, 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

()

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 problem

()

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
()
where G is the Green function of the Laplace operator Δ on Ω with Dirichlet conditions.

(ii)
We recall that if and , 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 ℝⁿ 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
()
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])
()
So for, each u ∈ ℬ(Ω) such that V(p|u|) < ∞, we have
()

(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
()
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

()

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)
(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)

()
(ii)
for any nonnegative superharmonic function v in Ω, one has
()

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

()

Proof. Let v be a nonnegative superharmonic function, then by [18, Theorem 2.1, page 164], there exists a sequence of nonnegative measurable functions in Ω such that the sequence defined in Ω by

()

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,

()

which means

()

Hence, it follows from Proposition 1.4 (i) that

()

Consequently, from (1.6), we obtain that

()

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) .

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 → λ exp (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

()

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

()

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

()

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:

()

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

()

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,

()

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∈Ω

()

Now, since φ is nontrivial continuous function at ∂Ω, then there exists c₂ > 0, such that one has on Ω

()

Thus,

and so the assumption (A₃) is satisfied.

Example 2.7. Let be the exterior of the unit ball in ℝⁿ (n ≥ 3). Suppose that the hypothesis (A₂) is satisfied. Let γ, μ ∈ ℝ such that γ < 1 < 2 < n < μ. Then,

()

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

()

Now, from [19, page 258], there exists a constant c₂ > 0 such that one has on Ω

()

Thus,

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

()

Let T be the operator defined on Λ by

()

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 ∈ Λ

()

On the other hand, by using Proposition 1.5 and (A₃), we have

()

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 obtain

()

Now, consider the sequence

defined by

()

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 that

()

Hence, the sequence

converges to a function u ∈ Λ.

Therefore, from the monotone convergence theorem and the fact that ψ is continuous, the sequence converges to Tu. So,

()

or equivalently

()

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

()

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.

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Existence of Positive Bounded Solutions of Semilinear Elliptic Problems

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Existence of Positive Bounded Solutions of Semilinear Elliptic Problems

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