Minimal L^p-Solutions to Singular Sublinear Elliptic Problems

We solve the existence problem for the minimal positive solutions $u\in L^{p}(\Omega, dx)$ to the Dirichlet problems for sublinear elliptic equations of the form \[ \begin{cases} Lu=\sigma u^q+\mu\qquad \quad \text{in} \quad \Omega, \\ \liminf\limits_{x \rightarrow y}u(x) = 0 \qquad y \in \partial_{\infty}\Omega, \end{cases} \] where $0<q<1$ and $Lu:=-\text{div} (\mathcal{A}(x)\nabla u)$ is a linear uniformly elliptic operator with bounded measurable coefficients. The coefficient $\sigma$ and data $\mu$ are nonnegative Radon measures on an arbitrary domain $\Omega \subset \mathbb{R}^n$ with a positive Green function associated with $L$. Our techniques are based on the use of sharp Green potential pointwise estimates, weighted norm inqualities, and norm estimates in terms of generalized energy.


Introduction
Let Ω be a nonempty open connected set in R n (n ≥ 3) which possesses a positive Green function G, and M + (Ω) denotes the class of all nonnegative Radon measures in Ω.
Key words and phrases.sublinear elliptic equation, measure data, divergence form operator, Green function.
Here Lu := −div(A(x)∇u) with bounded measurable coefficients is assumed to be uniformly elliptic, i.e., A : Ω → R n×n is a real symmetric matrix-valued function and there exists positive constants m ≤ M so that m|ξ| 2 ≤ A(x)ξ • ξ ≤ M|ξ| 2 for almost every x ∈ Ω and for every ξ ∈ R n .
In this paper, a solution u to the problem (1.1) will be understood in the sense that u is an A-superharmonic function on Ω such that u ∈ L q loc (Ω, dσ) with u ≥ 0 dσ-a.e., and satisfies the corresponding integral equations (1.2) u = G(u q dσ) + Gµ in Ω.
Here, the Green potential of a measure σ ∈ M + (Ω), is defined by where a function G : Ω × Ω → (0, ∞] called a positive Green function associated with L in Ω.
In the classical case L := −∆, these sublinear equations are closely related to the study of porous medium equations, and were studied by Brezis and Kamin [4] under the assumption of the bounded domain.The reader can also see such a sublinear problem under various assumptions, for instance, [1,2,5,8,9,10,11,13], and the literature cited there.
There are many of the existing solutions theories to elliptic equations (1.1) involving measures.For instance, Véron [14], considered problem (1.1) with different boundary conditions: homogeneous Dirichlet boundary conditions (u = 0 on ∂Ω) and measure boundary conditions (u = µ on ∂Ω where µ is Radon measure) under a smooth bounded domain Ω.
The homogeneous case (µ = 0) of problem (1.1) was investigated by Seesanea and Verbitsky [11].Nevertheless, when it comes to the case µ ≥ 0, the relation between σ and µ seems to be nontrivial in the scale of Lebesgue space.
Furthermore, in [13], the author introduced the bilateral pointwise estimates in terms of the intrinsic nonlinear potentials which can be utilized to obtain the existence of a positive solution u ∈ L p (Ω, dx) to (1.1).Unfortunately, the definition of intrinsic nonlinear potential is defined in terms of the best localized constant of related sublinaer weighted norm inequality, which make it difficult to be verified.
In this present paper, we aims to provide a simple approach to overcome the difficulties in [11] and deduce useful sufficient conditions on measures σ and µ for the existence of the positive minimal A-superharmonic solution u ∈ L p (Ω, dx) to (1.1).
Our main results read as follows.
A sufficient condition for (1.3) and (1.4) with γ = p(n−2)−n n is given by where n n−2 < p < ∞.Therefore, the following corollary can be simply deduced from Theorem 1.1.
Observe that Corollary 1.2 was done by Boccardo and Orsina [3], with a different proof, when Ω is a bounded domain in R n .

Organization of the paper
In Section 2, we organize some definitions and well-known research that are relevant to our problem.In Section 3, we prove an estimate for p-th integrability of potentials in terms of generalized Dirichlet energy and the existence result of the problem by applying the previous estimate.Moreover, we provide a sufficient condition for the existence of a positive solution to (1.1).

Notation
We use the following notation in this paper.Let Ω be a connected open subset in R n .
• D:= a relatively compact open subset of Ω.
• H A (D):= the set of all continuous A-harmonic functions in D.
• L p (Ω, dx):= the L p space with respect to Lebesgue measure.

Preliminaries
Thoughout, let Ω be a domain (connected open set) in R n .
2.1.Function Spaces.Definition 2.1.For 1 ≤ p < ∞ and µ ∈ M + (Ω), we denote by L p (Ω, dµ) the space of all real-valued measurable functions f on Ω such that The set of A-harmonic functions on Ω is denoted by H A (Ω). Every A-harmonic function u has a continuous representative which coincides with u a.e.see [7,Theorem 3.70].
Let u be an A-superharmonic function in Ω.Then there exists a unique measure ω ∈ M + (Ω) such that The measure ω is called the Riesz measure associated with u, see [ The first theorem gives an auxiliary fact that will be used in the proof of the main lemma.The complete proof can be seen in [9, Lemma 3.3].
Theorem 2.5 ( See [6]).Let 0 < q < 1 and σ ∈ M + (Ω).Suppose G is a positive Green function associated with L on Ω.If u ∈ L q loc (Ω, dσ) is a positive supersolution to the sublinear integral equation We use the following pointwise iterated inequalities to derive the Green potential estimate, see [6, Lemma 2.5].
Theorem 2.6 (See [6]).Let σ ∈ M + (Ω) with σ ≡ 0, and let G be the positive Green function associated with L on Ω.Then the following estimates hold.(i) If t ≥ 1, then The argument of finding a solution depends on the following weighted norm inequalities of the (s, r)-type in the case where 0 < r < s and 1 < s < ∞, for operators G: where c is a positive constant independent of f, for an arbitrary measure σ ∈ M + (Ω), under certain assumptions on G, see [12, Theorem 1.1].
The next results are essential lemmas to prove positive solutions to (1.1) when µ = 0.The complete proofs of the following two lemmas can be found in [11,  Lemma 2.10 (See [11]).Let 0 < q < 1, and let σ ∈ M + (Ω) with σ ≡ 0. Suppose G is a positive Green function associated with L on Ω. Suppose that n n−2 < p < ∞ and the condition (2.10) where c is a positive constant independent of f and q and s = p(n−2)−n(1−q) nq .
3) with γ = p(n−2)−n n implies (2.10).In fact, where C is a positive constant depending on γ and q.

Construction of minimal L p -solutions
In this section, we prove our main result stated Theorem 1.1 and its consequence in Corollary 1.2.
The following lemma is one of the key ingredients in our approach.
where C is a positive constant depending on γ and q.
Applying the iterated inequality (2.3) with t = γ + q, together with Fubini's theorem and Hölder's inequality with the exponents γ γ+q−1 and γ 1−q , we obtain • Case:γ + q ≤ 1. Write where a = γ + q and F is a positive ω-measurable function to be determined later.Applying Hölder's inequality with the exponents 1 a and 1 1−a , we get (3.1) The right-hand side of (3.1) is estimated by using Fubini's theorem, followed by inequality (2.4) with t = γ 1−q This completes the proof of the lemma.
The following lemma gives Green potentials norm estimates in terms of generalized energy.Lemma 3.2.Let G be a positive Green function associated with L in where p = n(1+γ) n−2 and c is a positive constant depending on γ.Proof.Notice that w := (Gµ) 1−q with 0 < q < 1 is a positive Asuperharmonic function on Ω since Gµ ≡ +∞, and w := Gω, where ω ∈ M + (Ω) is the Riesz measure of w, see [9].
Applying the Lemma 3.1 together with Lemma 2.11, we get the desired estimate, where C and C are constants in Lemma 2.11 and Lemma 3.1 respectively.
We are now ready to prove the main theorem of this work.
Then, we find that L γ+q (Ω, dσ) < +∞.This shows that there exists a positive solution u ∈ L p (Ω, dx) to (1.1) We finish this paper by providing a proof of Corollary 1.2.The following proof is mainly influenced by Seesanea and Verbitsky [9] and by Boccardo and Orsina [3].