Local Integral Estimates for Quasilinear Equations with Measure Data

Local integral estimates as well as local nonexistence results for a class of quasilinear equations −Δp u = σP(u) + ω for p > 1 and Hessian equations F k[−u] = σP(u) + ω were established, where σ is a nonnegative locally integrable function or, more generally, a locally finite measure, ω is a positive Radon measure, and P(u) ~ exp⁡αu β with α > 0 and β ≥ 1 or P(u) = u p−1.


Introduction and Main Results
Let Ω ⊂ R be a bounded domain, let be a nonnegative locally integrable function or, more generally, a locally finite measure on Ω, and let be a nonnegative Borel measure. In this paper, we consider the following nonlinear partial differential equations with measure data: where ( ) = ℓ, , ( ) ∈ 1 ,loc (Ω) is defined, following [1,2], as ℓ, , ( ) = ℓ ( ) (2) and ℓ-truncated exponential function ℓ ( ) is given by Here, Δ is the -Laplacian of defined by Δ fl div(| | −2 ) ( > 1). For convenience, here and elsewhere in the paper, we assume that ℓ > − 1. We will understand (1) in the following potential-theoretic sense using -superharmonic functions (see Section 2).
Recently, Quoc-Hung and Véron [2] obtained twosided estimates on the solutions in terms of -truncated -fractional maximal potential of , which is suitable for dealing with exponential nonlinearities: The Scientific World Journal Some analogous estimates for Hessian equations also are given in this paper. In this paper, firstly, we will establish a priori estimates of (1) with exponential reaction ℓ, , ( ), defined by (2) and (3). One of our main results are the following theorems. (1) in Ω with > 1 and ℓ > − 1. Suppose that 4 ( 0 ) ⊂ Ω. Then, there exists a constant = ( , , , , ℓ) such that for all 0 < ≤ /4.
As a consequence of Theorem 1, we have the following nonexistence results of local solutions to quasilinear equations.

Theorem 2.
Let be a solution of (1) in an open connected set Ω ⊂ R . Suppose that 1 < < , = | | − with > , and 0 ∈ Ω. Then, ≡ 0. Now, we consider (1) with natural growth terms; that is, the ( ) term in (1) is replaced by −1 . It is worthwhile to point out that this problem turns out to be more complex than the supercritical case. The interaction between the differential operator −Δ and the lower order term was investigated by Jaye and Verbitsky [6,7].
Similarly, we have the following.
The plan of the paper is as follows. In Section 2, we collect some elements notions and potential estimates forsuperharmonic. Theorems 1 and 2 will be proved in Section 3. In this section, we also discuss the relations of and provided that there exist solutions of (1). After this, Section 4 presents the proof of Theorems 3 and 4 by a new iteration scheme. Section 5 is devoted to considering fully nonlinear analogues of the Dirichlet problem (1) for Hessian equations without proof.

Preliminaries
In this section, we first recall some notations and definitions. In the following, we denote by a general constant, possibly varying from line to line, to indicate a dependence of on the real parameters , , , , ℓ; we will write = ( , , , , ℓ). We also denote by ( 0 , ) = { ∈ R : | − 0 | < } the open ball with center 0 and radius > 0; when it is not important or clear from the context, we shall omit denoting the center as = ( 0 , ).
For > 0, > 1, such that < , the -truncated Wolff 's potential W 1, ( ) of a nonnegative Borel measure on R is defined by We also denote by W 1, [ ]( ) the ∞-truncated Wolff 's potential.
In this paper, all solutions are understood in the potentialtheoretic sense. A lower semicontinuous function : Ω → (−∞, +∞] is called -superharmonic if is not identically infinite in each component of Ω, and if for all open sets such that ⊂ Ω, and all functions ℎ ∈ ( ), -harmonic in , the implication holds: ℎ ≤ on implies ℎ ≤ in . Note that -superharmonic function does not necessarily The Scientific World Journal 3 belong to 1, loc (Ω), but its truncation ( ) = min{ , } does for every integer ; therefore, we will need the generalized gradient of a -superharmonic function defined by = lim →∞ ∇( ( )). For more properties of -superharmonic, see [12].
The following lower pointwise estimates for superharmonic functions play an important role in our estimate.
The following lemma was also proved in [13].
The following theorem is an analogue of the above theorems for -Hessian equations. For more details, see [14].

Proof of Theorems 1 and 2
In this section, we will give the proof of our main theorem. Firstly, we prove the following integral estimate for solutions of quasilinear equations (1), which shows that if (1) has a nontrivial -superharmonic supersolution, then is absolutely continuous with respect to . The fact can be used to obtain a characterization of removable singularities for the homogeneous quasilinear equation: for all balls such that 2 ⊂ Ω.
Proof. Define Integrating both sides of (21) against over , we find which combined with (20) implies that This inequality is equivalent to which, together with (20), leads to (19).
Proof of Theorem 1. For fixed 0 ∈ Ω, let > 0 be such that 4 The Scientific World Journal view of the lower pointwise potential estimate (14), we find that, for all ∈ ( 0 ), where depends on , .
Restrict the integration on ( 0 ) and let = ; thus, taking into account (25), we obtain in view of which combined with (26) leads to the fact that, for all ∈ ( 0 ), where ( 0 , ) is defined as Thus, taking into account (26) and (28) and arguing by induction, we find where M is a nonlinear integral operator defined by M = W 1, ( ℓ ). The iterates of M are denoted by M = M(M −1 ). It is then easy to see from Proposition 8 that, for all > 0, where ( ) appears in Proposition 8. Consequently, and this yields lim sup Here, we use the fact that In the following, we will divide the proof into two cases.
The proof of inequality (8) is completely similarly and more details are omitted.

Proof of Theorems 3 and 4
In this section, we will prove Theorem 3. It is interesting to note that, in order to prove this theorem, we should give a new iterative process.
The Scientific World Journal 5 Proof of Theorem 3. This proof will be divided into two parts according to the value of .

Case 1 (1 < ≤ 2). For nonnegative measurable functions , define
Obviously, N is a homogeneous superlinear operator acting on nonnegative functions. Assume that is a solution of (1); then, for all ∈ ( 0 ), where depends on , . Iterating (39) times yields Here, we use the fact that N is a homogeneous superlinear operator and th iterate of N is defined by N ( ) = N(N −1 ( )) for > 1.
In the following, we will estimate the iterates of N. Recall = ( 0 ) ; thus, in view of where ( 0 , ) is defined in (29). Consequently, for all ∈ ( 0 ), where ( ) appears in Proposition 8. Obviously, Here, the following fact has been used in this inequality: Note that here is arbitrary; this fact, together with (40) and (44), leads to which, combined with (36), leads to (9) provided that ( 0 ) < ∞. In a similar way, we can prove (9) if ( 0 ) = ∞; more details are omitted.
Case 2 ( > 2). A point worth emphasizing is that the operator N defined by (38) does not fall within this framework since it is not a superlinear operator. Therefore, define In this case, we have Thus, by Minkowski's inequality, where depends on , . It is clear that where 0 , ( 0 , ) appears in (29). Using (49) and (50), we find The Scientific World Journal Therefore, By reverse Hölder inequality, we get The following proof is similar to that of (46), so it is clear. This finishes the proof of Theorem 3.
The proof of Theorem 4 is standard and will be omitted.

A Fully Nonlinear Analogue: The -Hessian
We now move to -Hessian operator and present fully nonlinear counterparts of the results obtained in the previous theorems. More precisely, consider fully nonlinear -Hessian operator , introduced by Trudinger and Wang [15][16][17]: where [ ] denotes the -Hessian ( = 1, 2, . . . , ), The proof of the following theorems is completely analogous to that of (1). One only needs to use Propositions 9 and 10 in place of Propositions 6 and 7, respectively, and argue as in Sections 3 and 4 with W 2 /( +1), +1 in place of W 1, . Therefore, the proof is omitted.