Local Integral Estimates for Quasilinear Equations with Measure Data

Abstract

Local integral estimates as well as local nonexistence results for a class of quasilinear equations for and Hessian equations were established, where is a nonnegative locally integrable function or, more generally, a locally finite measure, is a positive Radon measure, and with and or .

1. Introduction and Main Results

Let be a bounded domain, let be a nonnegative locally integrable function or, more generally, a locally finite measure on , and let be a nonnegative Borel measure. In this paper, we consider the following nonlinear partial differential equations with measure data:where is defined, following [1, 2], asand -truncated exponential function is given byHere, is the -Laplacian of defined by . For convenience, here and elsewhere in the paper, we assume that . We will understand (1) in the following potential-theoretic sense using -superharmonic functions (see Section 2).

Quasilinear equations (1) where term is replaced by for superlinear case are well studied (see [3–5] and references therein, see [6, 7] for natural growth terms , and see [8, 9] for sublinear problems ). In particular, it was shown in [3] that if is a solution of (1) with nonlinear source terms in and , then there is a constant such that for all ; (4) improves the related results of [10, 11].

Recently, Quoc-Hung and Véron [2] obtained two-sided estimates on the solutions in terms of -truncated -fractional maximal potential of , which is suitable for dealing with exponential nonlinearities: where 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 , defined by (2) and (3). One of our main results are the following theorems.

Theorem 1. Let be a solution of (1) in with and . Suppose that . Then, there exists a constant such thatfor all .

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 . Suppose that , with , and . Then, .

Now, we consider (1) with natural growth terms; that is, the term in (1) is replaced by . 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 and the lower order term was investigated by Jaye and Verbitsky [6, 7].

Theorem 3. Let be a solution of (1) in with and . Suppose that . Then, there exists a constant such thatfor all .

Similarly, we have the following.

Theorem 4. Let be a solution of (1) with in an open connected set . Suppose that , , and . Then, .

Remark 5. The four previous theorems are particular case of the more general class of nonlinear Wolff integral equations: where , which includes fractional Laplacian . Therefore, we also can obtain similar results of these integral equations.

The plan of the paper is as follows. In Section 2, we collect some elements notions and potential estimates for -superharmonic. 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.

2. 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 ; we will write . We also denote by the open ball with center and radius ; when it is not important or clear from the context, we shall omit denoting the center as .

Let and be a nonnegative Borel measures in which are finite on compact subsets of . The -measure of a measurable set is denoted by . We denote by , resp.) the space of measurable functions such that is integrable (locally integrable) with respect to . When , we write (, resp.).

For ,, such that , the -truncated Wolff’s potential of a nonnegative Borel measure on is defined by We also denote by the -truncated Wolff’s potential.

In this paper, all solutions are understood in the potential-theoretic sense. A lower semicontinuous function is called -superharmonic if is not identically infinite in each component of , and if for all open sets such that , and all functions , -harmonic in , the implication holds: on implies in . Note that -superharmonic function does not necessarily belong to , but its truncation does for every integer ; therefore, we will need the generalized gradient of a -superharmonic function defined by . For more properties of -superharmonic, see [12].

The following lower pointwise estimates for -superharmonic functions play an important role in our estimate.

Proposition 6 (see [13]). There exists a positive constant such that if is -superharmonic on and , thenfor all balls such that .

The following lemma was also proved in [13].

Proposition 7. Let be a ball such that . Then, there exists a positive constant such that if is -superharmonic on and , thenwhere is the Wolff potential of .

Given we consider a ball and shrinking balls , where with is an integer.

Proposition 8 (see Lemma in [3]). Let be locally finite nonnegative measures on . Then, there exists a constant such that for any we havewhere with .

The following theorem is an analogue of the above theorems for -Hessian equations. For more details, see [14].

Proposition 9. If is such that is -convex in , thenfor all balls , provided that ; here, is the corresponding -Hessian measure associated with the -convex function .

Proposition 10. Let be such that is -convex in . Then, there is a constant such that if then whenever the ball .

3. 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: in terms of Hausdorff measures. For more details, see Theorem  2.18 in [4] and Theorem  3.1 in [3].

Lemma 11. Let and be locally finite nonnegative measures on and . There exists a constant such that if is a solution to (1), thenfor all balls such that .

Proof. DefineAccording to Proposition 6 and the definitions of , we know that, for all ,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 , let be such that . Suppose that is a positive solution of (1). In view of the lower pointwise potential estimate (14), we find that, for all ,where depends on .
Restrict the integration on and let ; thus, taking into account (25), we obtainin view of which combined with (26) leads to the fact that, for all ,where is defined asThus, taking into account (26) and (28) and arguing by induction, we findwhere is a nonlinear integral operator defined by . The iterates of are denoted by . It is then easy to see from Proposition 8 that, for all , where appears in Proposition 8. Consequently, and this yieldsHere, we use the fact that In the following, we will divide the proof into two cases.
Case  1 (). In this case, (33), combined with (30), implies that Recalling that,which leads to (7).

Case  2 (). According to [12], we know that a.e. in . Therefore, choose a sequence , such that and . Then, (7) holds with in place of , for all . Then, (7) holds by the lower semicontinuity of Wolff potentials inequality.

The proof of inequality (8) is completely similarly and more details are omitted.

Proof of Theorem 2. Let be a nonnegative -superharmonic of (1). Then, satisfies (7), while it is well known that Provided that with , which contradicts (7).

4. 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.

Proof of Theorem 3. This proof will be divided into two parts according to the value of .
Case  1 (). For nonnegative measurable functions , defineObviously, is a homogeneous superlinear operator acting on nonnegative functions. Assume that is a solution of (1); then, for all ,where depends on .
Iterating (39) times yieldsHere, we use the fact that is a homogeneous superlinear operator and th iterate of is defined by for .
In the following, we will estimate the iterates of . Recall ; thus, in view of where is defined in (29). Consequently, for all , 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 towhich, combined with (36), leads to (9) provided that . In a similar way, we can prove (9) if ; more details are omitted.
Case  2 (). A point worth emphasizing is that the operator 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 thatwhere appears in (29). Using (49) and (50), we find 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.

5. 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 [15–17]:where denotes the -Hessian ,where are the eigenvalues of the Hessian matrix ; that is, is the sum of the principal minors of , which coincides with the Laplacian if , and the Monge-Ampère operator if .

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 in place of . Therefore, the proof is omitted.

Theorem 12. Let be a solution of (54) in with and . Suppose that . Then, there exists a constant such that for all .

Theorem 13. Let be a solution of (54) in with and . Suppose that . Then, there exists a constant such that for all .

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

This work was partially supported by the National Natural Science Foundation of China (no. 11401473 and no. 11326100), Natural Science Foundation of Gansu Province (no. 145RJZA214 and no. 1506RJYA272), Fundamental Research Funds for the Central Universities (no. 31920160059), Science and Humanity Foundation of the Ministry of Education (no. 15YJA880085), Foundation of State Nationalities Affairs Commission (no. 14XBZ016), and Innovative Team Subsidize of Northwest University for Nationalities.