THE REGULARITY OF WEAK SOLUTIONS TO NONLINEAR SCALAR FIELD ELLIPTIC EQUATIONS CONTAINING p&q-LAPLACIANS

Mathematica

Volumen 33, 2008, 337–371

THE REGULARITY OF WEAK SOLUTIONS TO NONLINEAR SCALAR FIELD ELLIPTIC EQUATIONS CONTAINING p&q-LAPLACIANS

Chengjun He^a and Gongbao Li^a,b

Chinese Academy of Sciences, Wuhan Institute of Physics and Mathematics Wuhan, 430071, China; cjhe@wipm.ac.cn

Central China Normal University, School of Mathematics and Statistics Wuhan, 430079, China; ligb@mail.ccnu.edu.cn

Abstract. In this paper, we consider the regularity of weak solutions u ∈ W^1,p(R^N)∩ W^1,q(R^N)of the elliptic partial differential equation

−∆pu−∆qu=f(x), x∈R^N,

where1< q < p < N. We prove that these solutions are locally inC^1,α and decay exponentially at infinity. Furthermore, we prove the regularity for the solutionsu∈W^1,p(R^N)∩W^1,q(R^N)of the following equations

−∆pu−∆qu=f(x, u), x∈R^N,

where N ≥ 3, 1 < q < p < N, and f(x, u) is of critical or subcritical growth about u. As an application, we can show that the solution we got in [8] has the same regularity.

1. Introduction

In this paper, we study the regularity of weak solutions to the following nonlinear elliptic equations with p&q-Laplacians:

(1.1)

(−∆_pu+m|u|^p−2u−∆_qu+n|u|^q−2u=g(x, u), x∈R^N, u∈W^1,p(R^N)∩W^1,q(R^N),

where m, n > 0, N ≥ 3, 1< q < p < N, ∆_tu = div(|∇u|^t−2∇u) is the t-Laplacian of ufor t >1.

The p&q-Laplacian problem (1.1) comes, for example, from a general reaction diffusion system

(1.2) u_t = div[D(u)∇u] +c(x, u),

where D(u) = (|∇u|^p−2 +|∇u|^q−2). This system has a wide range of applications in physics and related sciences, such as biophysics, plasma physics, and chemical reaction design. In such applications, the function u describes a concentration, the

2000 Mathematics Subject Classification: Primary 35B65, 35D10.

Key words: Regularity, weak Solutions, p&q-Laplacians.

aPartially supported by the NSFC NO:10571069 and NSFC NO:10631030 and the Lab of Mathematical Sciences, CCNU, Hubei Province, China.

bCorresponding author: ligb@mail.ccnu.edu.cn.

first term on the right-hand side of (1.2) corresponds to the diffusion with a diffusion coefficient D(u), whereas the second one is the reaction and relates to source and loss processes. Typically, in chemical and biological applications, the reaction term c(x, u)has a polynomial form with respect to the concentration u.

Recently, the eigenvalue problem for a p&q-Laplacian type equation with p= 2 was studied by Bence [1] and the stationary solution of (1.2) was studied by Cherfils and Il’yasov in [4] on a bounded domain Ω⊂R^N with D(u) = (|∇u|^p−2+|∇u|^q−2) and c(x, u) = −p(x)|u|^p−2u−q(x)|u|^q−2u+λg(x)|u|^γ−2u for 1 < p < γ < q and γ < p^∗, where p^∗ = _n−p^np if p < n, andp^∗ = +∞, ifp≥n.

In [8], using the concentration compactness principle and Mountain Pass The- orem, we proved the existence of a nontrivial solution to (1.1) under suitable as- sumptions on g(x, u)(see (C₁)–(C₅) in [8]). It is natural to study the regularity of weak solutions of (1.1). To this end, we consider the following equation

(1.3) −∆_pu−∆_qu=f(x),

where f ∈ L^∞_loc(R^N). By a weak solution u to (1.3), we mean a function u ∈ W^1,p(R^N)∩W^1,q(R^N) (or W_loc^1,p(R^N)) such that

Z

R^N

£|∇u|^p−2∇u∇ϕ+|∇u|^q−2∇u∇ϕ−f(x)ϕ¤

dx= 0 for any ϕ∈C₀^∞(R^N).

It is obvious that (1.1) is a special case of (1.3) if we take f(x) = g(x, u(x))− m|u(x)|^p−2u(x)−n|u(x)|^q−2u(x).

For degenerate elliptic equations

(1.4) −∆_pu=f(x, u)

and systems with some special structure, theC^1,α regularity of weak solutions was proved in [7] whenp= 2, and in [11, 17, 18] and [6] when p≥2. The existence and integrability of second-order derivatives of weak solutions to (1.4) were studied in [13, 15, 19] for all 1< p <+∞, from which theC^1,α regularity of weak solutions to (1.4) is obtained.

With an extra assumption that u ∈ L^∞(Ω), [5] and [16] proved the local C^1,α regularity of the solutionsu to a general class of quasilinear elliptic equations (1.5)

Z

Ω

XN

j=1

©a_j(x, u,∇u)·ϕ_xjª

−h(x, u,∇u)ϕ dx= 0, ϕ∈C₀^∞(Ω),

wherea_j belongs toC⁰(Ω×R×R^N)∩C¹(Ω×R×R^N−{0})andhis a Caratheodory function, i.e., for each(t, p)∈R^N+1,h(x, t, p)is measurable in xand continuous in t and p for a.e. x∈R^N. It was shown that their results can be applied to (1.4) for all1< p <∞.

The decay of the solution u of p-Laplacian type equations were considered by many authors. When p = 2, [2] showed that under some conditions on f, if u is a

radially symmetric solution of (1.6)

(

−∆u=f(u) inR^N, u∈H¹(R^N), u 6= 0, then u∈C²(R^N) and

(1.7) |D^αu(x)| ≤Ce^−δ|x|, x∈R^N,

for someC, δ >0and for |α| ≤2. By introducing exponential weighted spaces, [3]

showed that positive solutions of (1.8)

(−∆u+f(x, u) = 0 in R^N,

u→0 at infinity,

decay exponentially at infinity.

Under suitable assumptions on V(x)andf, the existence andC^1,α regularity of weak solutions of thep-Laplacian type Schrödinger equations

(1.9)

(−∆_pu+V(x)|u|^p−2u=f(x, u), u∈W^1,p(R^N), 1< p <+∞,

were proved in [11]. Furthermore, it was shown in [11] that the solutions decay exponentially in x when |x| ≥ R for some R > 0. We extend this result to p&q- Laplacian type equations, too.

Our main results are as follows:

Theorem 1. Suppose that f ∈ L^∞_loc(R^N) and u ∈ W_loc^1,p(R^N)∩L^∞_loc(R^N) is a weak solution of (1.3) where p >1. Then

(i) |∇u| ∈L^∞_loc(R^N)and for every compact K ⊂R^N, there exists a constantC depending only on N, p, q, ess sup

K

|u| and ess sup

K

|f| such that

(1.10) k∇uk_L^∞_(K) ≤C;

(ii) x→ ∇u(x)is locally Hölder continuous inR^N, i.e., there exists anα∈(0,1) and a constant C depending only uponN,p,q,ess sup

K |u|and ess sup

K |f|for every compact K ⊂R^N, such that

(1.11) |∇u(x)− ∇u(y)| ≤C|x−y|^α, x, y ∈K.

Theorem 2. Suppose that f(x, t) satisfy:

(A1) f(x, t) : R^N ×R¹ → R¹ satisfies the Caratheodory conditions, i.e., for a.e.

x ∈ R^N, f(x, t) is continuous in t ∈ R¹ and for each t ∈ R¹, f(x, t) is Lebesgue measurable with respect to x∈R^N.

(A2) f(x, t) is of critical or subcritical growth about u at infinity, i.e., for any ε > 0, there is a C_ε > 0 such that |f(x, t)| ≤ ε|t|^q−1 +C_ε|t|^p∗−1 for all (x, t)∈R^N ×R¹,where p^∗ = _N^{N P}_−p if N > p, 0< p^∗ <+∞ if N ≤p.

Ifu∈W^1,p(R^N)∩W^1,q(R^N), 1< q < p < N, is a weak solution of

(1.12) −∆pu−∆qu=f(x, u),

then there is anα >0and a constantC depending only on N,p,q, ess sup

BR(x0)

|u|for any R >0, such that

|∇u(x)| ≤C, (1.13)

|∇u(x)− ∇u(y)| ≤C|x−y|^α (1.14)

for all x, y ∈B_R(x₀) and any x₀ ∈R^N.

In [8] the existence of a weak solution of (1.1) was obtained under the following assumptions:

(C₁) g: R^N ×R¹ → R¹ satisfies the Caratheodory conditions; g(x, t) ≥ 0, for t ≥0 and g(x, t)≡0, fort <0 and allx∈R^N,

(C2) lim

t→0⁺ g(x,t)

t^p−1 = 0 uniformly in x∈R^N; lim

s→+∞

g(x,t)

t^p−1 =` uniformly in x∈R<sup>N</sup> for some ` ∈(0,+∞),

and some extra technical conditions.

By Theorem 1 and 2, it is easy to see that weak solutions of (1.1) are locally in C^1,α. We also get the exponential decay of weak solutions at infinity under the hypotheses (C₁) and (C₂).

In fact, we have the following result:

Theorem 3. Supposeg(x, t)satisfies (A1), (A2) of Theorem 2 and uis a weak solution of (1.1). Then

(i) u is bounded on R^N, i.e., kuk_L^∞_(R^N₎ <+∞ and lim

R→+∞kuk_L^∞_(|x|>R)= 0;

(ii) u(x) decays exponentially as |x| → +∞, i.e., ∃C > 0, ε > 0, R > 0 such that

(1.15) |u(x)| ≤Ce^−ε|x| when |x| ≥R.

One cannot obtain Theorem 1 by the results in [5, 16] or [11], since the p&q- Laplace equations do not satisfy the assumptions in [5, 16] and [11]. Our results are new to our knowledge; they are the generalization of the results of [5, 16] and [11].

Theorem 2 is an application of Theorem 1, which may be applied to more cases.

To prove Theorem 1, we mainly use the frame works of [5, 16, 11], respectively, to different steps. Since the main purpose of [5, 16] and [11] is to consider the reg- ularity of weak solutions for p-Laplacian type equations, the ellipticity and growth conditions imposed on a_j are homogeneous about ∇u. For example, in [16], it is

required that

XN

i,j=1

∂a_j

∂η_i(x, µ, η)·ξ_iξ_j ≥γ ·(κ+|η|)^p−2|ξ|², XN

i,j=1

¯¯

¯¯∂a_j

∂η_i(x, µ, η)

¯¯

¯¯≤Γ·(κ+|η|)^p−2 (1.16)

for some γ, Γ > 0 and κ ∈ [0,1]. It is obvious that p&q-Laplace equations do not satisfy the above conditions. Since p&q-Laplace equations can not be included in the frame works of [5, 16] or [11], much more careful analysis is needed in the proof.

We use the method of Proposition 1 in [16] to get a useful identity (see (2.5) in §2 below). Although in [16] only a similar inequality is required to show the boundedness of the gradiant∇uof any weak solutionuto (1.3), we expect that this identity can be used somewhere. After the local boundedness of|∇u| is proved, we follow the usual way (see [7, 9]) to obtain theC^1,α regularity of the weak solution.

To prove Theorem 2, we use Theorem 1. To apply Theorem 1, we need only to prove the local boundedness of the weak solutionsu, i.e., kuk_L^∞_(BR(x0))≤C(x₀)for any givenx₀ ∈R^N and then apply Theorem 1 with f(x) =f(x, u(x)). Usually, one uses the test functionϕ=η^pu⁺(u⁺_L)^p(β−1) with

u⁺_L =

(u⁺, u < L, L, u≥L,

to prove the local boundedness of u⁺ (see, e.g., [10, 12]). As one may see, this test function does not work in our case. We follow [14] to define u¯=u⁺+k,and

¯ u_L =

(u,¯ u⁺< L, L+k, u ≥L,

and ϕ(x) =η^p(¯u¯u^p(β−1)_L −k^p(β−1)+1) for some k >0as a test function. It turns out that this test function does work.

To prove Theorem 3, we mainly use the method of [11]. The key step is to get a decay estimate of the weak solution as in [10](see (5.25) below). However, as both pand q-Laplacian are involved, the test functions used in [10, 11, 14] do not work.

We overcome this difficulty by using two test functions separately, to get a couple of inequalities and then combine them to get (5.25). As soon as (5.25) is obtained, the exponential decay of the solutions will be obtained as in [11].

The paper is organized as follows: In §2, we prove Theorem 1(i); in §3, we prove Theorem 1(ii); in §4, we prove the boundedness of weak solutions and then apply Theorem 1 to prove Theorem 2. In §5, we give the proof of Theorem 3.

Our symbols are standard. For example, Br(x0)for x0 ∈R^N, r >0is the open ball {x∈R^N¯¯ |x−x₀|< r};L^p(Ω) is the usualL^p-space over the domain Ω⊂R^N with normk · k_L^p_(Ω);measE means theN-dimensional Lebesgue measure of the set E ⊂R^N, and so on.

2. The proof of Theorem 1(i)

In this section, we give the proof of Theorem 1(i). To this end, we consider the following equation

(2.1)

(−∆_pu−∆_qu=f(x), x∈R^N, u∈W_loc^1,p(R^N), 1< q < p.

Notice that we have by the assumptions that

(2.2) f ∈L^∞_loc(R^N), u∈L^∞_loc(R^N).

We will show that

(2.3) k∇ukL^∞(BR(x0))≤C,

whereC is a constant depending only onN,p,q, andkuk_L^∞_(BR(x0)). For simplicity, we give the proof on B ≡ B₁(x₀), the unit ball in R^N with centre x₀ for any given x₀ ∈R^N. Firstly, we prove an identity inspired by [16].

Proposition 2.1. Ifψ is a nonnegativeC²-function with compact support and G: R¹ →R¹ is a piecewise C¹-function with only finitely many breaks and

(2.4) 0≤G⁰ ≤c₀

for some constantc₀, then any weak solution u of (2.1) satisfies Z

B

XN

i,j=1

©(|∇u|^p−2+|∇u|^q−2)δ_ij + [(p−2)|∇u|^p−4+ (q−2)|∇u|^q−4]u_xiu_xjª

·u_xs,xiu_xs,xjG⁰(u_xs)ψ dx

= Z

B

XN

j=1

(|∇u|^p−2+|∇u|^q−2)u_xj · d

dx_s{G(u_xs)ψ_xj}dx

− Z

B

f d

dx_s{G(u_xs)ψ}dx, (2.5)

whereδ_ij are the Kronecker symbols.

Proof. The proof follows by multiplying equation (2.1) by _dx^d

s(G(u_xs)ψ) and

integrating by parts. ¤

Next we show the L^∞-estimate of the gradient of solutions u of (2.1). Before that we give the following result.

Lemma 2.2. ([16], Corollary 1) For any v ∈ W^1,p(BR), where BR = BR(x0) for any fixedx0 ∈R^N, suppose that

(2.6)

Z

BR

|v|dx ≤M ·R^N

and (2.7)

Z

Ak,r

|∇v|^pdx≤M^p·(r⁰ −r)^−p·R^{N α}·(measA_k,r⁰)^1−α

for some constantM, some α∈(0, p/N), all k ≥0and all r and r⁰ satisfying R/2< r < r⁰ ≤R,

where A_k,r ={x ∈ B_r(x₀)¯¯v(x) > k}. Then there is a constant C depending only onN, p, and α such that

(2.8) v ≤C·M in BR/2(x0).

For the proof of Theorem 1(i), it is enough to prove the following result.

Proposition 2.3. Suppose that (2.2) holds for the weak solution u of (2.1).

Then for any x₀ ∈ R^N, there exists a constant C depending only on N, p, q, ess sup

B

|u|and ess sup

B

|f| such that

(2.9) |∇u| ≤C inB_1/2(x₀),

where B =B₁(x₀).

Proof. Choose a nonnegative C^∞-functionρ having the properties

(2.10) ρ(t)







= 0, for t≥1,

∈(0,1), for t∈(0,1),

= 1, for t≤0.

ForR ∈(0,1/8) and i∈Z⁺∪ {0}, we set R_i = 2R+ 2⁻ⁱ⁻¹R, Bi =BRi(x0),

ϕi(x) = ρ(2ⁱ⁺¹R⁻¹(|x−x0| −Ri)).

(2.11)

In the following, C stands for a generic constant depending only on N, p, q, ess sup

B

|u| and ess sup

B

|f| and may differ in different spaces, where B = B₁(x₀).

In contrast toC, the generic constant C(R) may also depend on R, and C(ε) may depend on ε.

To prove (2.9), we will first show that there is an R₀ >0depending only on N, p, q,ess sup

B

|u| and ess sup

B

|f| such that (2.12)

Z

Bi

|∇u|^p+2idx≤C(R) fori= 0,1, . . . ,[Np] provided that

(2.13) R ≤R₀,

where [Np] is the integer part of Np. It can be seen that (2.12) is true for i = 0.

Hence we may suppose that (2.12) holds for some i ∈ {1, . . . ,[Np]−1} and then we prove that it is true for i+ 1.

We pick an M > 0and define for t∈R¹ that g(t) =







t−1, if t≥1, 0, if t∈[−1,1], t+ 1, if t≤ −1,

g_M(t) =







M, if g(t)≥M, g(t), if g(t)∈[−M, M],

−M, if g(t)≤ −M, and

G(t) =g(t)|g_M(t)|²ⁱ.

It is obvious that G(t) satisfies the assumption of Proposition 2.1. Then for any s∈ {1,2, . . . , N}, we define

us=g(uxs) =







u_xs −1, if u_xs ≥1, 0, if u_xs ∈[−1,1], u_xs + 1, if u_xs ≤ −1,

us,M =gM(uxs) =







M, if u_s≥M, u_s, if u_s∈[−M, M],

−M, if u_s≤ −M.

Inserting

G(u_xs) =u_s|u_s,M|²ⁱ, ψ =ϕ²_i+1

into the left hand of (2.5) and noting that G⁰(u_xs)≥u²ⁱ_s,M ≥0, we have Z

B

XN

i,j=1

©(|∇u|^p−2+|∇u|^q−2)δ_ij + [(p−2)|∇u|^p−4 + (q−2)|∇u|^q−4]u_xiu_xjª

·u_xs,xiu_xs,xjG⁰(u_xs)ψ dx

= Z

B

©(|∇u|^p−2+|∇u|^q−2)|∇u_xs|²+£

(p−2)|∇u|^p−4+ (q−2)|∇u|^q−4¤

· |∇u· ∇u_xs|²ª

G⁰(u_xs)ψ dx

≥ Z

B

© £|∇u|^p−2|∇u_xs|²+ (p−2)|∇u|^p−4|∇u· ∇u_xs|²¤

+£

|∇u|^q−2|∇uxs|²+ (q−2)|∇u|^q−4|∇u· ∇uxs|²¤ ª

u²ⁱ_s,Mϕ²_i+1dx

≥ min{1, p−1}

Z

B

|∇u|^p−2|∇u_xs|²u²ⁱ_s,Mϕ²_i+1dx.

(2.14)

On the other hand, by the definition ofu_s,M, we have that |∇u| ≥1 on the support of us,M. Hence

Z

B

XN

j=1

(|∇u|^p−2+|∇u|^q−2)uxj· d

dx_s{G(uxs)ψxj}dx

= Z

B

XN

j=1

(|∇u|^p−2+|∇u|^q−2)uxj ·G⁰(uxs)uxsxsψxjdx

+ Z

B

XN

j=1

(|∇u|^p−2+|∇u|^q−2)uxj ·G(uxs)ψxsxjdx

≤ C Z

B

(|∇u|^p−2+|∇u|^q−2)|∇u|u²ⁱ_s,M|∇us|ϕi+1|∇ϕi+1|dx

+C(R) Z

B

XN

j=1

(|∇u|^p−1 +|∇u|^q−1)u²ⁱ_s,M|∇u|dx

≤ C Z

B

|∇u|^p−1u²ⁱ_s,M|∇u_s| ·ϕ_j+1|∇ϕ_j+1|dx+C(R) Z

B

|∇u|^p+2idx

≤ ε Z

B

|∇u|^p−2|∇u_s|²u²ⁱ_s,Mϕ²_i+1dx+C(ε) Z

B

|∇u|^pu²ⁱ_s,M|∇ϕ_i+1|²dx+C(R), (2.15)

and by (2.2) and the fact that|∇u| ≥1on the support of us,M, we have that Z

B

(−f) d

dx_s{G(u_xs)ψ}dx

≤C Z

B

|f|u²ⁱ_s,M|∇u_xs|ϕ²_i+1dx+C Z

B

|fku_s|u²ⁱ_s,Mϕ_i+1|∇ϕ_i+1|dx

≤C Z

B

|∇u|^p−1u²ⁱ_s,M|∇u_s|ϕ²_i+1dx+C Z

B

|∇u|^pu²ⁱ_s,Mϕ_i+1|∇ϕ_i+1|dx

≤ε Z

B

|∇u|^p−2|∇us|²u²ⁱ_s,Mϕ²_i+1dx+C(ε) Z

B

|∇u|^pu²ⁱ_s,Mϕ²_i+1dx +C(R)

Z

B

|∇u|^p+2idx.

(2.16)

Thus by (2.5), (2.14), (2.15) and (2.16), we have that min{1, p−1}

Z

B

|∇u|^p−2|∇u_xs|²u²ⁱ_s,Mϕ²_i+1dx

≤ Z

B

XN

j=1

(|∇u|^p−2+|∇u|^q−2)u_xj · d

dx_s{G(u_xs)ψ_xj}dx− Z

B

f d

dx_s{G(u_xs)ψ}dx

≤2ε Z

B

|∇u|^p−2|∇u_s|²u²ⁱ_s,Mϕ²_i+1dx+C(ε) Z

B

|∇u|^pu²ⁱ_s,M|∇ϕ_i+1|²dx +C(ε)

Z

B

|∇u|^pu²ⁱ_s,Mϕ²_i+1dx+C(R)

≤2ε Z

B

|∇u|^p−2|∇us|²u²ⁱ_s,Mϕ²_i+1dx+C(ε, R) Z

B

|∇u|^p+2idx+C(R)

≤2ε Z

B

|∇u|^p−2|∇u_s|²u²ⁱ_s,Mϕ²_i+1dx+C(ε, R).

Thenε can be chosen such that (2.17)

Z

B

|∇u|^p−2|∇u_xs|²u²ⁱ_s,Mϕ²_i+1dx≤C(R).

Now, we prove (2.12) fori+ 1. Notice that (2.18)

XN

s=1

|∇u|^p+2u²ⁱ_s,M = XN

s=1

( XN

j=1

u²_xj)^(p+2)/2u²ⁱ_s,M ≤ XN

s=1

XN

j=1

|u_xj|^p+2u²ⁱ_s,M

and the fact that

|uxj|^p+2u²ⁱ_s,M ≤ |uxs|^p+2u²ⁱ_s,M ≤ XN

s=1

|uxs|^p+2u²ⁱ_s,M, if |uxj| ≤ |uxs|, as well as

|uxj|^p+2u²ⁱ_s,M ≤ |uxj|^p+2u²ⁱ_j,M ≤ XN

s=1

|uxs|^p+2u²ⁱ_s,M, if |uxj| ≥ |uxs|.

Thus we have (2.19)

XN

s=1

XN

j=1

|uxj|^p+2u²ⁱ_s,M ≤N² XN

s=1

|uxs|^p+2u²ⁱ_s,M.

Hence with the help of (2.18) and (2.19), we have that XN

s=1

Z

B

|∇u|^p+2u²ⁱ_s,Mϕ²_i+1dx

≤C XN

s=1

Z

B

|u^p+2_xs u²ⁱ_s,Mϕ²_i+1dx by (2.18), (2.19)

≤C XN

s=1

Z

B

|us|^pusu²ⁱ_s,Mϕ²_i+1·uxsdx+C XN

s=1

Z

B

u²ⁱ_s,Mϕ²_i+1dx

≤C XN

s=1

Z

B

|u_s|^pu_s,xsu²ⁱ_s,Mϕ²_i+1u dx+C XN

s=1

Z

B

|u_s|^p−2u_su_s,xsu²ⁱ_s,Mϕ²_i+1u dx

+C XN

s=1

Z

B

|u_s|^pu_su²ⁱ⁻²_s,Mu_s,Mu_s,M,xsϕ²_i+1u dx

+C XN

s=1

Z

B

|u_s|^pu_su²ⁱ_s,Mϕ_i+1|∇ϕ_i+1|u dx+C(R) Z

Bi

|∇u|²ⁱdx

≤C XN

s=1

Z

B

|∇u|^p|∇u_s|u²ⁱ_s,Mϕ²_i+1dx+C XN

s=1

Z

B

|∇u|^pu²ⁱ_s,Mϕ²_i+1dx

+C XN

s=1

Z

B

|∇u|^p+1u²ⁱ_s,Mϕ_i+1|∇ϕ_i+1|dx+C(R)

≤2ε XN

s=1

Z

B

|∇u|^p+2u²ⁱ_s,Mϕ²_i+1dx+C(ε, R) XN

s=1

Z

B

|∇u|^pu²ⁱ_s,Mdx

+C(ε) XN

s=1

Z

B

|∇u|^p−2|∇u_s|²u²ⁱ_s,Mϕ²_i+1dx+C(R).

(2.20)

Here, integration by parts and Young’s inequality are used. Then, by virtue of (2.12) fori and (2.17), (2.20) implies that

(2.21)

XN

s=1

Z

B

|∇u|^p+2u²ⁱ_s,Mϕ²_i+1dx≤C(R).

Set i= 0 in (2.21). We get (2.22)

XN

s=1

Z

B

|∇u|^p+2ϕ²₁dx≤C(R), and lettingM →+∞in (2.21), we get

(2.23)

XN

s=1

Z

B

|∇u|^p+2u²ⁱ_sϕ²_i+1dx≤C(R).

So by (2.22) and (2.23) we get Z

Bi+1

|∇u|^p+2(i+1)dx≤ Z

B

|∇u|^p+2|∇u|²ⁱϕ²_i+1dx

≤C XN

s=1

Z

B

|∇u|^p+2|uxs|²ⁱϕ²_i+1dx

≤C XN

s=1

Z

B

|∇u|^p+2|u_s|²ⁱϕ²_i+1dx+C XN

s=1

Z

B

|∇u|^p+2ϕ²_i+1dx ≤C(R).

Thus (2.12) is proved.

Now, we use (2.12) to prove (2.9). From now on, we fix R by taking

(2.24) R=R0

for some given R0 ∈ R¹. As the dependence on R of the generic constant C does not matter any more, we do not indicate it in the following. For k≥0 and

R≤r ≤r⁰ ≤2R, we set

ϕ(x) =ρ((r⁰ −r)⁻¹·(|x−x₀| −r)), A_k,r ={x∈B_r(x₀)|u_s(x)> k}.

Fort∈R¹, we define

g(t) =







t−1, if t≥1, 0, if t∈[−1,1], t+ 1, if t≤ −1, and

G(t) = max{g(t)−k,0}.

It is obvious that G(t) satisfies the assumption of Proposition 2.1. Then we define u_s=g(u_xs)and insert

G(uxs) = max{us−k,0}, ψ =ϕ²

into (2.5), and following in the same way which leads to (2.17), we get (2.25)

Z

A_k,r0

|∇u|^p−2|∇u_s|²ϕ²dx≤C·(r⁰−r)⁻² Z

Ak,r

|∇u|^pdx.

Noticing that (2.12) gives that (2.26)

Z

BN p

|∇u|^{N p}dx≤C, and the fact that r⁰ < R_i implies that

(2.27) Br⁰(x0)⊂Bi(x0)

for any i∈ {0,1, . . . ,[Np]}, we have by (2.26) and (2.27) that (2.28)

ÃZ

A_k,r0

|∇u|^{N p}dx

!_1/N

≤ ÃZ

BN p

|∇u|^{N p}dx

!_1/N

≤C.

Then, (2.28) and Hölder’s inequality show that Z

A_k,r0

|∇u|^pdx≤( Z

A_k,r0

|∇u|^{N p}dx)^1/N ·(measA_k,r⁰)^N−1N

≤C·(measA_k,r⁰)^N−1N . (2.29)

Thus, by (2.25), (2.29), Young’s and Hölder’s inequalities, we get that (2.30)

Z

A_k,r0

|∇u|^p−2|∇u_s|²ϕ²dx≤C·(r⁰ −r)⁻²(measA_k,r⁰)^1−N1, and then

(2.31)

Z

Ak,r

|∇u|^p−2|∇u_s|²dx≤C·(r⁰−r)⁻²(measA_k,r⁰)^1−N1. Ifp≥2,(2.31) implies that

(2.32) Z

Ak,r

|∇u_s|²dx≤ Z

Ak,r

|∇u|^p−2|∇u_s|²dx≤C·(r⁰−r)⁻²(measA_k,r⁰)^1−N1. Ifp≤2,we additionally use (2.29), Hölder’s and Young’s inequalities to obtain that

Z

Ak,r

|∇u_s|^pdx

≤ ÃZ

Ak,r

(r⁰−r)^2−p|∇u|^p−2|∇us|²dx

!_p/2

· ÃZ

Ak,r

(r⁰−r)^−p|∇u|^pdx

!_(2−p)/2

≤ p

2(r⁰−r)^2−p · Z

Ak,r

|∇u|^p−2|∇u_s|²dx+ 2−p

2 (r⁰ −r)^−p· Z

Ak,r

|∇u|^pdx

≤C(r⁰ −r)^−p(measA_k,r⁰)^1−N1. (2.33)

If we choose R₀ ∈(1/2,1) in (2.24) at first, we have Z

B2R

|u_xs|dx≤ µZ

B2R

|∇u|^pdx

¶_1/p

·(measB_2R)^(p−1)p

≤C·£

κ_N ·(2R)^N¤^(p−1)

p

≤CR^N, (2.34)

whereκ_N denotes the volume of the unit ball in R^N. So (2.32), (2.33), (2.34) and Lemma 2.2 show that

u_s≤C inB_R(x₀).

As−u satisfies all the same estimates above as u does, we have shown that Propo-

sition 2.3 is true. Hence Theorem 1(i) is proved. ¤

3. The proof of Theorem 1(ii)

We will prove Theorem 1(ii) in this section. To this end, it is enough to prove the following result:

Proposition 3.1. Suppose that u is a weak solution of (2.1) and u, f(x) and

|∇u| are locally bounded. Then there is an α >0and a constantC depending only onN, p, q,ess sup

B

|u| and ess sup

B

|f| such that

(3.1) |∇u(x)− ∇u(x₀)| ≤C· |x−x₀|^α, ∀x∈B_1/2(x₀), whereB =B₁(x₀) for any given x₀ ∈R^N.

In the following, ρ is defined as in (2.10). By C, we denote a positive generic constant depending only on N, p, q, ess sup

B1(x0)

|u| and ess sup

B1(x0)

|f|. We pick an R∈(0,1/2)and set

(3.2) M = max

s ess sup

BR(x0)

|u_xs|.

Before we prove Proposition 3.1, we give the following results:

Lemma 3.2. ([[9], Lemma 3.9) There is a C depending only on N, such that (l−k)·(measA_l,ρ)^1−N1 ≤βρ^N meas⁻¹{B_ρ(x₀)\A_k,ρ} ·

Z

Al,k,ρ

|∇v|dx

for all l > k and v ∈ W^1,1(B_ρ(x₀)), where A_k,ρ = {x ∈ B_ρ(x₀)|v(x) > k} and A_l,k,ρ ={x∈B_ρ(x₀)|k < v(x)≤l}.

Lemma 3.3. ([9], Lemma 4.7) If a nonnegative sequence {y_h}, h = 0,1,2, . . ., satisfies

y_h+1 ≤cb^hy_h^1+ε, h= 0,1, . . . , wherec, ε and b >1are positive constants, then

yh ≤c^(1+ε)

h−1 ε b^(1+ε)

h−1

ε2 −^h_εy₀^(1+ε)h. Especially, ify₀ ≤θ =c^−1/εb^−1/ε2, then

y_h ≤θb^−1/ε and

y_h →0, ash→ ∞.

Lemma 3.4. ([9], Lemma 4.8) Suppose u(x) is measurable and bounded on B_ρ0(x₀). Considering B_ρ(x₀) and B_bρ(x₀), where b > 1 is a constant, if for all ρ≤b⁻¹ρ₀, u(x)satisfies one of the following inequalities

osc{u;B_ρ(x₀)} ≤cρ¯ ^ε,

osc{u;B_ρ(x₀)} ≤θosc{u;B_bρ(x₀)},

wherec,¯ ε≤1and θ < 1are positive constants, then osc{u;Bρ(x0)} ≤cρ^−α₀ ρ^α whenever ρ≤ρ₀,where

α= min{ε,−log_bθ}, c=b^αmax{¯cρ^ε₀,osc{u; B_ρ0(x₀)}}.

Lemma 3.5. ([5], Proposition 4.1) Suppose that u is a weak solution of (2.1) and u, f(x) and |∇u| are locally bounded. Then for any given x₀ ∈R^N, there is a µ >0 depending only on N, p, q, M, ess sup

B1(x0)

|u| and ess sup

B1(x0)

|f|, such that if for some 1≤s≤N

(3.3) meas{x∈B_R(x₀)|u_xs(x)≤M/2} ≤µR^N, then

uxs(x)≥M/8, ∀x∈BR/2(x0), whereM is defined in (3.2). Analogously, if

(3.4) meas{x∈B_R(x₀)|u_xs(x)≥ −M/2} ≤µR^N, then

u_xs(x)≤ −M/8, ∀x∈B_R/2(x₀).

Now, we begin to prove Proposition 3.1.

We have shown in §2 that the gradient of a weak solution u of (2.1) is locally bounded under the condition of Proposition 3.1. Therefore, by Lemma 3.5 there are two cases: Case I: Either (3.3) or (3.4) is satisfied; Case II: Neither (3.3) nor (3.4) is satisfied. We follow [5] to consider these two cases to prove Proposition 3.1.

Case I: Either (3.3) or (3.4) is satisfied. Notice that if either (3.3) or (3.4) holds, we have by Lemma 3.5 that

|u_xs(x)| ≥M/8, ∀x∈B_R/2(x₀).

Moreover, by the definition ofM (see (3.2)) we have

(3.5) M/8≤ |∇u| ≤M in BR/2(x0).

For l > k ≥ 0 and r, r⁰ ∈ R satisfying 0 < r < r⁰ ≤ R, we set for a solution u of (2.1) that

ϕ(x) =ρ((r⁰ −r)⁻¹·(|x−x₀| −r)), Ak,r ={x∈Br(x0)|uxs(x)> k}

and

A_l,k,r ={x∈B_r(x₀)|k < u_xs(x)≤l}.

Fort∈R¹, we define

g(t) =







t−1, if t≥1, 0, if t∈[−1,1], t+ 1, if t≤ −1,