Some representation formulae of the minimizers are also stated

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu

MINIMIZATION OF ENERGY INTEGRALS ASSOCIATED WITH THE p-LAPLACIAN IN R^N FOR REARRANGEMENTS

SHANMING JI, JINGXUE YIN, RUI HUANG

Abstract. In this article, we establish the existence of minimizers for energy integrals associated with thep-Laplacian inR^Nwith the admissible set being a rearrangement class of a given function. Some representation formulae of the minimizers are also stated.

1. Introduction

In this article, we study the optimization problems of minimizing the energy integrals associated with thep-Laplacian equation

−∆pu=f −h, x∈R^N, (1.1)

lim

|x|→+∞u(x) = 0. (1.2)

Here 1< p < N, ∆_pstands for the usualp-Laplacian; i.e., ∆_pu= div(|∇u|^p−2∇u).

Letf₀, h∈L^∞(R^N) be fixed nonnegative functions with compact supports, and let Rbe the class of rearrangements of f₀ with compact support; that is,R={f ∈ L^∞(R^N);|{x;f(x) ≥ α}| = |{x;f₀(x) ≥ α}|,∀α ∈ R,suppf is bounded}, where

| · |is the Lebesgue measure. For λ≥0 andf varying inR, we define the energy functional with (λ >0) or without (λ= 0) penalty as

Ψλ(f) = Z

R^N

|∇uf|^pdx+λ Z

R^N

gf dx, (1.3)

whereu_fis the solution of problem (1.1)–(1.2),g∈C²(R^N) is the penalty function, lim_|x|→+∞g(x) = +∞and ∆_pg≥σ^p−1for some constantσ >0. The optimization problem (1.4) is to find the minimizer for energy integral Ψ_λ(f), namely,

minf∈RΨλ(f). (1.4)

The optimization problems of maximizing or minimizing convex functionals over the set of rearrangements of a given function have been investigated by many au- thors. In such problems, the theory of rearrangements and functionals on rear- rangements established by Burton [2, 3] has proved to be a crucial tool in addressing questions such as existence, uniqueness and symmetry of optimal solutions. This

2000Mathematics Subject Classification. 35A01 35J15 35Q99.

Key words and phrases. Optimization problem; rearrangements; energy integral; penalty;

p-Laplacian.

c

2014 Texas State University - San Marcos.

Submitted April 11, 2014. Published June 11, 2014.

1

theory has already been applied to shape optimization problems of membranes, solid and fluid mechanics, eigenvalue optimization problems of some differential operators and so on, see [7] and references therein.

In recent years, a great deal of attention has been devoted to optimization prob- lems where the cost functionals are the energy integrals associated with elliptic equations. For problems in bounded domains, numerous elliptic operators have been studied, including the Laplacian [2, 5, 6], p-Laplacian [4, 10] and some semi- linear operators [8]. For example, Marras [10] studied the minimization problem of energy integral Ψ0(f) associated with thep-Laplacian on bounded domain

−∆pu=f, x∈Ω, u= 0, x∈∂Ω,

where p >1,f ∈ R. There are also some works dealing with elliptic operators in unbounded domains. Bahrami and Fazli [1] considered the minimization problem of energy integral

Φλ(f) =1 2

Z

R³

f ufdx+λ Z

R³

gf dx,

whereuf is the solution of Poisson’s equation

−∆u=f−2h, x∈R³, lim

|x|→+∞u(x) = 0, (1.5)

where f ∈ R, h∈L^∞(R³), g ∈C²(R³), lim_|x|→+∞g(x) = +∞, ∆g ≥c > 0 and λ≥0.

We mention here some details of the previous works. The weakly sequentially continuity of the functional Ψ_λ(f) on spaceL^q(Ω) forq≥1 and bounded domain Ω, is essential in the proof of [10] and other works when applying Burton’s theory of rearrangements. However, the continuity is generally not true on unbounded domains due to the general loss of compact imbedding of Sobolev spaces on un- bounded domains, especially on the whole space. Thus the authors in [1] investigate the problem on bounded domains to solve the optimization problem on unbounded domains.

We are interested in the extension of the work of Bahrami and Fazli [1] to the nonlinear diffusion case. As a matter of fact, the p-Laplacian arises in various physical contexts: non-Newtonian fluids, reaction diffusion problems, nonlinear elasticity, electrochemical machining, elastic-plastic torsional creep, etc., see [4]

and references therein.

We state here our main results of existence and representation formulae of min- imizers for problem (1.4) in the caseλ >0 andλ= 0 respectively.

Theorem 1.1. The optimization problem (1.4)has a solution for λ > λ₀≡ p⁰

σkf₀k

1

∞p−1.

Moreover, iffλ∈ Ris a solution of (1.4)anduf_λ is the solution of problem (1.1)–

(1.2)corresponding tofλ, then there exists a decreasing functionϕλ such that f_λ=ϕ_λ◦(p⁰u_fλ+λg),

almost everywhere inR^N.

Theorem 1.2. Let f0 andh be as introduced above. There exists a constantκ= κ(N, p) ∈ (0,¹₂] depending only on N and p, such that if kf0k∞ < khk−∞;supph, supph⊂Br_h withrh>0appropriately large, and

|suppf0| ≤κkhk_−∞;supph

khk_∞

_p−1^p |supph|

|B_rh|

_N(p−1)^N−p

|supph|, (1.6) then the optimization problem (1.4)with λ= 0has a solution. Moreover, iffˆ∈ R is a solution of (1.4) with λ = 0 and u_fˆ is the solution of problem (1.1)–(1.2) corresponding tofˆ, then there exists a decreasing function ϕˆ such that

fˆ= ˆϕ◦ufˆ, almost everywhere inR^N.

The crucial point of the proofs, compared with the linear diffusion case, lies in the estimates on the different contributions of the two opposed-signed functions f and −h to the solution. In the previous work [1], the classical theory of linear elliptic equations was applied, namely, the explicit expression of solutions based on the superposition principle is feasible and effective in deriving the estimates on solutions of linear elliptic equations. However, such a method is inapplicable in the current problem due to the nonlinearity of thep-Laplacian. It turns out to be more difficult as we attempt to estimate the different contributions of the two opposed- signed functions. To overcome those difficulties, we use the De Giorgi and Moser iteration techniques to derive estimates in quasilinear case and we take advantage of the representation formulas of the problem on bounded domains since they provide additional correlation between the solution and the free term.

The organization of this paper is as follows. Section 2 is devoted to the basic notations and some preliminary results, especially some fundamental estimates.

Then we will discuss the case with (λ >0) and without (λ= 0) penalty in Section 3 and Section 4 respectively.

2. Definitions and preliminary results

Throughout this paper, we assume that 1 < p < N, where N is the spatial dimension,p⁰= _p−1^p the conjugate exponent ofp,p∗= _N^{N p}_−p the Sobolev conjugate exponent ofpandp⁰_∗=_p^p∗

∗−1. The measure of a Lebesgue measurable setA⊂R^N is denoted by |A|. ByBr(x) we denote the ball centered atx∈R^N with radius r;

if the center is the origin, we writeBr for simplicity. Constant ωN stands for the measure of the unit ball inR^N.

Let us begin with the usual concept of rearrangement. Denote by Lα(f) the level set of a measurable function f at heightα; that isLα(f) ={x∈R^N;f(x) = α}. The strong support of a nonnegative function f is defined as suppf ={x∈ R^N;f(x)>0}. Furthermore, we define

kfk_−∞;suppf= sup{M ≥0;f(x)≥M, a.e.in suppf}.

When f and g are nonnegative measurable functions that vanish outside sets of finite measure inR^N, we sayf is a rearrangement ofgwhenever

|{x∈R^N;f(x)≥α}|=|{x∈R^N;g(x)≥α}|, ∀α≥0.

Now fixf0∈L^∞(R^N) being a measurable nonnegative function vanishing outside a set of finite measure. As defined in Section 1,Rdenotes the set of all rearrange- ments on R^N off0 with bounded support. The subset of Rcontaining functions vanishing outside the ballB_ris denoted byR(r); here we assumeω_Nr^N ≥ |suppf₀| in order that R(r) is nonempty. The weak closure in L^p0∗(Br) of R(r) is denoted byR(r)^w.

Henceforth we may regard a function f ∈ L^q(R^N) as a function in L^q(Br) by restricting its domain; we can also regard a functionf ∈L^q(Br) as its zero extension inL^q(R^N) when necessary for 1≤q≤+∞.

To solve the optimization problems (1.4), we first need to consider the similar problems whose admissible sets are nested subsets of R. We define minimizing problems (2.1) as follows:

min

f∈R(r)Ψλ(f). (2.1)

The sets of solutions of (1.4) and (2.1) are denoted bySλ andSλ(r) respectively.

In the following part of this section we state and prove some lemmas which are essential in our analysis. We begin with some results about properties of rearrange- ment classes and the relative variational problems proved by Burton in [3].

Lemma 2.1. Forr >0 satisfyingωNr^N ≥ |suppf0| andq≥1, we have (i) kfkq =kf0kq, for f ∈ R(r);

(ii) R(r)^w is weakly sequentially compact in L^q(Br);

(iii) R(r)^w = {f ∈ L¹(B_r);Rs

0 f⁴dt ≤ Rs

0f₀⁴dt,0 < s ≤ ω_Nr^N,R

Brf dx = R

Brf₀dx}, where f⁴ is a decreasing function on the interval (0, ω_Nr^N) satisfying

|{s∈(0, ω_Nr^N);f⁴(s)≥α}|=|{x∈B_r;f(x)≥α}|, ∀α >0.

RemarkFrom the representation ofR(r)^win (iii), we find that the weak closure of R(r) inL^p0∗(Br) is actually the weak closure inL^q(Br) for 1≤q <+∞. Combining (i) and the property of weak closure we have

kfkq ≤ kf0kq, ∀f ∈ R(r)^w, 1≤q≤+∞. (2.2) The following two lemmas are simple variations of [3, Lemma 2.15 and Theorem 3.3].

Lemma 2.2 ([3, Lemma 2.15]). Let T : L^p0(B_r) → R be the linear functional defined as T(f) =R

B_rf vdx forr >0,ωNr^N ≥ |suppf0| andv ∈L^p(Br). If fˆis a minimizer of T relative toR(r)^w and

|Lα(v)∩supp ˆf|= 0, ∀α∈R,

we have fˆ∈ R(r) andfˆ=ϕ◦v a.e. i Br, for a decreasing function ϕ.

Lemma 2.3 ([3, Theorem 3.3]). Let Ψ : L^p0(B_r) → R be a weakly sequentially continuous and Gˆateaux differentiable functional.

(i) There exists a minimizer forΨrelative toR(r)^w.

(ii) Iff^∗ is a minimizer forΨrelative toR(r)^w and the Gˆateaux differential of Ψatf^∗isΨ⁰(f^∗)∈L^p(B_r),f^∗is a minimizer for the functionalh·,Ψ⁰(f^∗)i relative toR(r)^w.

The following Sobolev’s inequality plays an important role in our estimates. For more details, see [9].

Lemma 2.4 ([9, Theorem 7.10]). For1< p < N andp_∗= _N^{N p}_−p, we have

kukp∗ ≤C0k∇ukp, ∀u∈W^1,p(R^N), (2.3) whereC0=C0(N, p)is a constant depending only onN andp.

Remark Invoking usual approximations, we see that this estimate is also valid providedu∈L^p∗(R^N),∇u∈L^p(R^N;R^N).

Henceforth, we assumer >0,ωNr^N ≥ |suppf0|and λ≥0. Forf ∈L^p0∗(R^N), we consider the problem (1.1)–(1.2). It is a classical result of variational the- ory that such a problem has a unique solution u ∈ W ≡ {w ∈ W_loc^1,p(R^N);w ∈ L^p∗(R^N),∇w∈L^p(R^N;R^N)} satisfying

sup

v∈W

Z

R^N

p(f−h)v− |∇v|^p dx=

Z

R^N

p(f−h)u− |∇u|^p dx

= (p−1) Z

R^N

|∇u|^pdx,

(2.4)

Z

R^N

(f−h)vdx= Z

R^N

|∇u|^p−2∇u· ∇vdx, ∀v∈W. (2.5) Lemma 2.5. Let u be the solution of problem (1.1)–(1.2) corresponding to f ∈ L^p0∗(R^N). We have

k∇uk_p≤C

1 p−1

0 (kf₀k_p⁰_∗+khk_p⁰_∗)^p−11 , (2.6) kukp∗≤C

p p−1

0 (kf0kp⁰_∗+khkp⁰_∗)^p−11 , (2.7) whereC0 is the constant in (2.3).

Proof. From (2.5), we apply H¨older’s inequality to find Z

R^N

|∇u|^pdx= Z

R^N

(f−h)udx≤ kf−hk_p⁰_∗kuk_p∗.

Combining this inequality with Sobolev’s inequality (2.3), we get the results.

Lemma 2.6. The functionalΨλ defined in (1.3)is weakly sequentially continuous and Gˆateaux differentiable in L^p0(Br) with derivative p⁰uf +λg ∈L^p(Br) at f ∈ L^p0(Br), whereuf is the solution of problem (1.1)–(1.2)corresponding tof. Proof. It suffices to prove that the functional I(f) ≡R

R^N|∇uf|^pdx is weakly se- quentially continuous and Gˆateaux differentiable inL^p0(B_r) with derivativep⁰u_f ∈ L^p(B_r) atf ∈L^p0(B_r). Letf_n * f inL^p0(B_r) andu_fn,u_f be the solutions of the problem (1.1)–(1.2) corresponding tofn,f respectively. Using (2.4), we have

(p−1)I(f) + Z

R^N

p(fn−f)ufdx

= Z

R^N

p(fn−h)uf− |∇uf|^p

dx≤(p−1)I(fn)

= Z

R^N

p(f−h)uf_n− |∇uf_n|^p dx+

Z

R^N

p(fn−f)uf_ndx

≤(p−1)I(f) + Z

R^N

p(fn−f)uf_ndx.

(2.8)

By assumption, we have

n→∞lim Z

R^N

(fn−f)ufdx= lim

n→∞

Z

Br

(fn−f)ufdx= 0. (2.9) Let us prove that

n→∞lim Z

R^N

(fn−f)uf_ndx= lim

n→∞

Z

B_r

(fn−f)uf_ndx= 0. (2.10) From (2.6), (2.7) andkfk_p0

∗;R^N =kfkp⁰_∗;B_r ≤ |Br|^N1kfkp⁰;B_r forf ∈L^p0(Br), we see that the normsk∇uf_nk_p;RN,kuf_nk_p∗;RN andkuf_nk1,p;B_r are bounded by constants independent of n. Therefore, we can choose a subsequence of {ufn} denoted by {uf_nk} and a functionw∈W, such that{uf_nk}converges weakly inL^p∗(R^N) and strongly inL^p(B_r) tow,{∇u_fnk} converges weakly inL^p(R^N;R^N) to∇w. From

Z

R^N

(fn_k−f)uf_nkdx= Z

R^N

(fn_k−f)wdx+ Z

R^N

(fn_k−f)(uf_nk −w)dx, and

Z

R^N

(fn_k−f)(uf_nk −w)dx

≤ kfn_k−fkp⁰;B_rkuf_nk −wkp;B_r,

the limit (2.10) is valid for a subsequence{nk}. Combining this with (2.8)–(2.9), we deduce

k→∞lim I(fn_k) =I(f). (2.11) We claim that the functionwis actuallyu_f , which is a fixed function independent of the choice of subsequence{nk}, to show that the sequence{uf_n}itself converges and equality (2.10) is valid. Indeed, from

(p−1)I(fn_k) = Z

R^N

p(fn_k−h)uf_nk − |∇uf_nk|^p dx,

k→∞lim Z

R^N

(fn_k−h)uf_nkdx= Z

R^N

(f−h)w dx, and the classical result

lim inf

k→∞

Z

R^N

|∇uf_nk|^pdx≥ Z

R^N

|∇w|^pdx, (2.12)

using (2.11) and (2.4), we get (p−1)I(f)≤

Z

R^N

p(f−h)w− |∇w|^p

dx≤(p−1)I(f). (2.13) By the uniqueness of the maximizer of R

R^N p(f−h)v− |∇v|^p

dx in W, we have w=uf. Thus (2.8)–(2.10) yield the weak continuity.

Let z ∈ L^p0(B_r) and {tn} be a positive sequence such that lim_n→∞t_n = 0.

Takingf_n=f+t_nz in the inequality (2.8), we find Z

R^N

p⁰u_fzdx≤I(f+t_nz)−I(f) tn

≤ Z

R^N

p⁰u_fnz dx.

As already observed,{u_fn} converges tou_f strongly in L^p(B_r). Therefore,

n→∞lim

I(f+tnz)−I(f) tn

= Z

R^N

p⁰ufz dx.

Since the sequence {tn} and the function z are arbitrary, it follows that I(f) is

Gˆateaux differentiable with derivativep⁰uf.

Note that the functional Ψλ is not weakly sequentially continuous inL^p0(R^N).

Lemma 2.7. Let u be the solution of the problem (1.1)–(1.2) corresponding to f ∈ R(r)^w. We have

kuk_∞;RN ≤C1(N, p)kf−hk

N−p N p−N+p

∞ kf−hk

p2 (p−1)(N p−N+p)

p⁰_∗ , (2.14)

whereC1(N, p) is a constant depending only onN andp.

Proof. For anyk >0, takev= (u−k)₊ ∈W in (2.5). We deduce Z

R^N

|∇v|^pdx≤ Z

R^N

|f −h||v|dx.

By Sobolev’s inequality and H¨older’s inequality, we have kvk^p_p

∗;A(k)≤C₀^p Z

A(k)

|f−h||v|dx≤C₀^pkvk_p∗;A(k)kf−hk_p0

∗;A(k),

whereA(k) ={x∈R^N;u(x)> k}. Therefore, kvk^p−1_p

∗;A(k)≤C₀^pkf −hkp⁰_∗;A(k)≤C₀^pkf−hk∞|A(k)|^1/p0∗. Combining this with

kvk_p∗;A(k)≥ kvk_p∗;A(h)≥(h−k)|A(h)|^1/p∗, ∀h > k >0, we have

|A(h)| ≤C₀^p0kf−hk

1

∞p−1

h−k

p∗

|A(k)|^{p∗ −1p−1} , ∀h > k >0. (2.15) By iteration, we see that|A(k₀+d)|= 0 fork₀>0,

d=C₀^p0kf−hk

1

∞p−12

(p∗ −1)(N−p)

p2 |A(k0)|^p

0 N. From estimate (2.7), we see that

k0|A(k0)|^p∗1 ≤ kukp∗≤C₀^p0kf−hk

1 p−1

p⁰_∗ . Hence

u≤k₀+d≤k₀+ 2^NC^p

0+^p02_N^p∗

0 kf−hk

1 p−1

∞

kf −hk

p0p∗

N(p−1)

p⁰_∗

k

p0p∗

N

0

.

Let α = ^p0_N^p∗ = _{(N−p)(p−1)}^p2 , A = 2^NC₀^p02p∗kf −hk

1

∞p−1kf −hk

p0p∗

N(p−1)

p⁰_∗ and k0 = (αA)^α+11 . We get u≤(α^α+11 +α^−α+1α )A^α+11 . By considering−uinstead of u, we

complete the proof.

In Section 4, the caseλ= 0, more precise estimates are required to demonstrate our result. We begin with an estimate on the lower bound of the energy functional R

R^N|∇u|^pdx.

Lemma 2.8. Let u be the solution of the problem (1.1)–(1.2) corresponding to f ∈ R(r)^w andsupph⊂Br_h,kf0k1≤ khk1. We have

k∇ukp ≥C2(N, p)r⁻

N−p p(p−1)

h (khk1− kf0k1)^p−11 , (2.16) whereC₂(N, p) is a constant depending only onN andp.

Proof. From (2.4), it suffices to prove that there existsv∈W such that Z

R^N

p(f−h)v− |∇v|^p

dx≥C(N, p)r⁻

N−p p−1

h (khk1− kf0k1)^p−1p ,

for a constant C(N, p) depending only on N and p. We verify that the function v(x) ≡ −kmin{(^rh+a−|x|_a )₊,1} ∈ W fulfills the conditions for some specially se- lected positive constants k and a. Indeed, noticing the signs of f, h and v, we have

Z

R^N

p(f−h)v− |∇v|^p

dx≥kp(khk1− kf0k1)−ωN

k a

p

(rh+a)^N

=kp(khk1− kf0k1)−ωNk^p N^Nr^N_h^−p p^p(N−p)^N−p

= (p−1)p^p0(N−p)^N−pp−1 ω

1 p−1

N N^p−1N

r⁻

N−p p−1

h (khk1− kf0k1)^p−1p , fora=_N^p_−prh andk^p−1=^{pp(N−p)N−p(khk1−kf0k1)}

ω_NN^Nr^N−p_h .

Next we deduce the local boundedness of solutions by the Moser iteration tech- nique.

Lemma 2.9. Let u be the solution of the problem (1.1)–(1.2) corresponding to nonnegative functionf ∈L^p0∗(B_r),v= (−u)+ andsupph⊂B_rh. There holds

kvk_∞;BR/2(x0)≤C3(N, p) 1 R^N

Z

BR(x0)

|v|^p∗dx^1/p∗

, (2.17)

for any x0 ∈ R^N and R > 0 provided BR(x0)∩Br_h = ∅, where C3(N, p) is a constant depending only onN andp.

Proof. For 0 < ρ < ρ⁰ ≤ R, let η(x) be a cut-off function η ∈ C₀^∞(Bρ⁰(x0)), satisfying 0≤η≤1,η(x) = 1 onB_ρ(x₀),η(x) = 0 onR^N\B_ρ⁰(x₀) and|∇η(x)| ≤

2

ρ⁰−ρ. We writeBR=BR(x0) in this proof for the sake of convenience.

Chooseη^pv^s as a test function in (2.5) fors≥1 and setq=s+p−1. We have Z

R^N

|∇u|^p−2∇u· ∇(η^pv^s)dx= Z

R^N

(f−h)η^pv^sdx= Z

BR

f η^pv^sdx≥0, or

− Z

BR

η^p|∇v|^p−2∇v· ∇(v^s)dx− Z

BR

v^s|∇v|^p−2∇v· ∇(η^p)dx≥0.

Therefore, using Young’s inequality, we deduce Z

BR

η^p|∇(v^qp)|^pdx

= q^p sp^p

Z

B_R

η^p|∇v|^p−2∇v· ∇(v^s)dx≤ q^p sp^p

Z

B_R

|∇v|^p−1|∇(η^p)|v^sdx

≤ q^p sp^p−1

Z

BR

η^p−1|∇η||∇v|^p−1v^sdx= q s Z

BR

η^p−1|∇(v^qp)|^p−1|∇η|v^qpdx

≤p−1 p

Z

B_R

η^p|∇(v^qp)|^pdx+ q^p ps^p

Z

B_R

|∇η|^pv^qdx.

Hence we obtain Z

BR

η^p|∇(v^qp)|^pdx≤q^p s^p

Z

BR

|∇η|^pv^qdx≤p^p Z

BR

|∇η|^pv^qdx, ∀q≥p.

Combining this with Sobolev’s inequality (2.3), we have Z

BR

η^{N−pN p} v^{N−pN q} dx^N−p_N

≤C₀^p Z

BR

|∇(ηv^qp)|^pdx≤(4pC0)^p Z

BR

|∇η|^pv^qdx.

It follows that 1 R^N

Z

B_ρ

v^{N−pN q} dx^N−p_N

≤(8pC₀)^p 1 R^N−p(ρ⁰−ρ)^p

Z

B_ρ0

v^qdx .

Denoteρk =^R₂(1 + ₂¹_k),k= 0,1, . . . and chooseq=p_∗(_N^N_−p)^k, ρ=ρk+1,ρ⁰ =ρk. Since _N−p^N >1, invoking iterations we see that (2.17) is valid.

There are difficulties in carrying out an estimate independent ofron the corre- sponding solution of (1.1)–(1.2) due to the fact thatf varies inR(r)^w. Hence we introduce the following comparison principle.

Lemma 2.10. Let u_f and u₀ be the solutions of the problem (1.1)–(1.2) corre- sponding to f ∈ R(r)^w and f = 0respectively. There holds

uf(x)≥u0(x), a.e. in R^N. Proof. From (2.5), we see that

Z

R^N

(|∇uf|^p−2∇uf− |∇u0|^p−2∇u0)· ∇ϕdx= Z

R^N

f ϕdx, ∀ϕ∈W.

Choosingϕ= (u0−uf)+∈W, we obtain Z

A

(|∇uf|^p−2∇uf− |∇u0|^p−2∇u0)·(∇uf− ∇u0)dx=− Z

A

f ϕdx≤0, where A = {x ∈ R^N;uf(x) ≤ u0(x)}. Thus ∇ϕ ≡ 0 andϕ ≡ 0 from Sobolev’s

inequality.

Now we could give a locally lower bound of the solution independent off andr.

Lemma 2.11. Let u be the solution of the problem (1.1)–(1.2) corresponding to f ∈ R(r)^w. For any ε >0, there exists r_ε>0 depending only on N,p,h andε, such that

u(x)≥ −ε, ∀x∈R^N\Br_ε.

Proof. Letu₀be as defined in Lemma 2.10, which is independent off andr. Using the similar method in the proof of Lemma 2.10, we demonstrate u0 ≤ 0 in R^N. Utilizing Lemma 2.9, we find

ku₀k_∞;B1/2(x0)≤C₃(N, p)ku₀k_p∗;B1(x0),

provided B1(x0)∩supph = ∅. Let r ≥ rh+ 1 and |x0| ≥ r, where rh > 0 satisfies supph⊂Br_h. From (2.7), we see thatku0k_p∗;RN ≤C₀^p0khk

1 p−1

p⁰_∗ . It follows

ku0k_∞;B1/2(x₀) ≤ ε provided |x0| is large enough. By applying Lemma 2.11, we

complete the proof.

3. The caseλ >0

First we are concerned with the existence of minimizers for the energy functional in bounded domains, then we will show that a solution valid in a sufficiently large bounded domain is in fact valid in the whole space.

Lemma 3.1. Let λ≥0,r >0 andωNr^N ≥ |suppf0|.

(i) The functional Ψλ attains its minimum relative toR(r)^w.

(ii) If fr,λ is a minimizer for Ψλ relative to R(r)^w, fr,λ is a solution of the variational problem

min

f∈R(r)^w

Z

R^N

f(p⁰u_fr,λ+λg)dx,

whereu_fr,λ is the solution of (1.1)–(1.2)corresponding to f_r,λ. The above lemma is a simple consequence of Lemma 2.3 and Lemma 2.6.

Lemma 3.2. Let λ > λ0 ≡ ^p_σ⁰kf0k

1

∞p−1. If fr,λ is a minimizer of Ψλ relative to R(r)^w andψ_r,λ=p⁰u_fr,λ+λg, we have

|Lα(ψr,λ)∩suppfr,λ|= 0, ∀α∈R.

Proof. We argue by contradiction. Suppose there exists ˆα∈Rsuch that|Sαˆ|>0, S_αˆ = L_αˆ(ψ_r,λ)∩suppf_r,λ ⊂B_r. We have ψ_r,λ = p⁰u_fr,λ +λg = ˆα, a.e. in S_αˆ. Therefore,

kfk∞≥f−h=−∆pu_fr,λ = ∆_p λ p⁰g

≥ λ p⁰

p−1

σ^p−1>kf0k∞, a.e. inS_αˆ, in the sense of distribution, which contradicts to (2.2). This completes the proof.

Lemma 3.3. Let λ0 be as defined in the lemma above and λ > λ0, ωNr^N ≥

|suppf0|. The set of solutions of the variational problem (2.1)denoted bySλ(r)is nonempty. If fr,λ∈Sλ(r), we have

fr,λ=ϕr,λ◦(p⁰uf_r,λ+λg), (3.1) almost everywhere inBr for a decreasing functionϕr,λ.

Proof. From Lemma 3.1, there exists fr,λ ∈ R(r)^w, which is a minimizer of Ψλ

relative toR(r)^w. By Lemma 3.2, the level sets ofψr,λ=p⁰uf_r,λ+λgon suppfr,λ

have zero measure. Utilizing Lemma 3.1 (ii) and Lemma 2.2, we see thatfr,λ∈ R(r) solves the variational problem (2.1) and has the representation (3.1). As already shown in the proof, the minimum for Ψ_λ relative to R(r)^w actually equals the minimum relative to R(r) under the assumption of this lemma. Thus for any fr,λ∈Sλ(r),fr,λ has a representation as (3.1) for someϕr,λ. We have proved that the variational problem (2.1) has a solution forλ > λ0and ω_Nr^N ≥ |suppf₀|. Now we will show that ifr is chosen large enough, it ceases to have any influence on the variational problem (2.1).

Lemma 3.4. Let λ > λ0. There exists rλ >0 satisfying ωNr^N_λ ≥ |suppf0| such that for anyr≥rλ andfr,λ∈Sλ(r), we have suppfr,λ⊂Br_λ.

Proof. Let r_a > 0, ω_Nr^N_a > |suppf₀| = |suppf_r,λ|. From estimates (2.2) and (2.14), we see thatkuf_r,λk_∞;RN is bounded by a constant depending onN,p,kf0k_∞, khk∞,|suppf₀|,|supph| and independent ofr,λ. Sinceλ > λ₀,g∈C²(R^N) and lim_|x|→+∞g(x) = +∞, we can find a constantr_λ≥r_a such that

p⁰uf_r,λ(x) +λg(x)≥p⁰uf_r,λ(z) +λg(z), ∀x∈R^N\Br_λ, z∈Br_a. (3.2) Using the representation (3.1) of fr,λ in Br, the decreasing property of ϕr,λ and the fact suppfr,λ⊂Br, we deduce

0≤f_r,λ(x)≤ inf

|z|≤ra

f_r,λ(z), ∀x∈R^N\Brλ.

By the assumption ofra, we get inf_|z|≤rafr,λ(z) = 0 since|Br_a\suppfr,λ|>0. It

follows suppfr,λ⊂Br_λ.

Now we are ready to prove Theorem 1.1.

Proof of Theorem 1.1. Letλ > λ0, r≥rλ andfr,λ∈Sλ(r). From Lemma 3.4, we have suppfr,λ⊂Br_λ. Therefore,fr,λ∈ R(rλ)⊂ R(r). It shows that the minimum of Ψλ relative toR(r) is attained at and only at some points in subset R(rλ) for r ≥ rλ. Since R = S

r≥rλR(r), we obtain Sλ = Sλ(rλ) = Sλ(r) for r ≥ rλ. It follows (1.4) has a solution. To prove the last part of this theorem, for anyr≥rλ

andfλ∈Sλ=Sλ(r), we have by applying Lemma 3.3 fλ=ϕr,λ◦(p⁰uf_λ+λg), a.e. inBr,

for a decreasing functionϕ_r,λ. We can use the similar method in the proof of (3.2) to chooser≥r_λandC_λ∈Rsuch that

p⁰u_fλ(x) +λg(x)≥C_λ= sup

z∈B_rλ

(p⁰u_fλ(z) +λg(z)), ∀x∈R^N\Br.

Noticing that suppfλ ⊂ Br_λ, we have that ϕr,λ(t) = 0 for t ∈ [Cλ, C_λ⁰], and C_λ⁰ = sup_z∈B

r(p⁰uf_λ(z) +λg(z))≥Cλ. Now define ϕλ(t) =

(ϕr,λ(t), t≤Cλ, 0, t > C_λ.

Clearlyϕλ is a decreasing function andfλ=ϕλ◦(p⁰uf_λ+λg) a.e. inR^N. 4. The caseλ= 0

To derive the existence result in this case, we need some additional conditions onf andh. Similarly, we first deduce the following lemma in bounded domains.

Lemma 4.1. Suppose kf0k_∞ <khk_−∞;supph,r > 0 andωNr^N ≥ |suppf0|. Let fr be a minimizer ofΨ0 relative toR(r)^w anduf_r be the solution of the problem (1.1)–(1.2)corresponding to fr. We have

|L_α(u_fr)∩suppf_r|= 0, ∀α∈R.

Proof. We argue by contradiction. Suppose there exists ˆα∈Rsuch that|Aαˆ|>0, Aαˆ =Lαˆ(uf_r)∩suppfr ⊂ Br. We have uf_r = ˆαa.e. in Aαˆ. Hence −∆puf_r = f −h = 0 a.e. inAαˆ, in the sense of distributions. SinceAαˆ ⊂suppfr, we find f >0 inA_αˆ, which followsh >0 a.e. inA_αˆandA_αˆ ⊂supph. Thuskhk−∞;supph≤ kfk∞≤ kf0k∞ from (2.2), which is a contradiction to our assumption.

Lemma 4.2. The set of solutions of the variational problem (2.1) with λ = 0 denoted byS0(r)is nonempty under the assumption of the lemma above. Moreover, if fr∈S0(r), we have

f_r=ϕ_r◦u_fr, a.e. inB_r, (4.1) for a decreasing functionϕr.

Proof. Utilizing Lemma 3.1, Lemma 4.1 and Lemma 2.2, we obtain the required results by using the similar method in the proof of Lemma 3.3.

Lemma 4.3. There exists a constant κ =κ(N, p)∈ (0,¹₂] depending only on N andp, such that ifkf0k_∞<khk_−∞;supph,supph⊂Br_h and

|suppf0| ≤κkhk_−∞;supph

khk_∞

_p−1^p |supph|

|B_rh|

_N(p−1)^N−p

|supph|, (4.2) we have suppfr⊂Br0 for any r≥r0 andfr∈S0(r), wherer0≥rh is a constant independent ofr andf_r.

Proof. From the representation offrin (4.1) and the decreasing property ofϕr, we see that

sup

x∈suppfr

uf_r(x) =s0≤ inf

z∈Br\suppfr

uf_r(z). (4.3)

Using (2.5), we calculate Z

R^N

|∇uf_r|^pdx

= Z

R^N

(fr−h)uf_rdx

= Z

suppfr\supph

fruf_rdx− Z

supph\suppfr

huf_rdx+ Z

suppfr∩supph

(fr−h)uf_rdx

≤s₀kfrk1;suppf_r\supph−s₀khk1;supph\suppf_r+kufrk∞kfr−hk∞|suppf_r|.

(4.4) By assumption, for anyκ≤¹₂, utilizing (2.2) and (2.14), we have

kfrk_{1;suppfr\supph}≤ kfrk1≤ kf0k1≤ kf0k_∞|suppf0|

<khk_−∞;supph(|supph| − |suppf_r|)≤ khk_{1;supph\suppfr}, (4.5) kf0k1≤ kf0k∞|suppf0|< 1

2khk−∞;supph|supph| ≤ 1

2khk1, (4.6) kuf_rk_∞kfr−hk_∞|suppfr|

≤C₁kf_r−hk

N p N p−N+p

∞ kf_r−hk

p2 (p−1)(N p−N+p)

p⁰_∗ |suppf_r|

≤C12 ^p

2

(p−1)(N p−N+p)khk

N p N p−N+p

∞ khk

p2 (p−1)(N p−N+p)

p⁰_∗ |suppf0|

≤C₁2 ^p

2

(p−1)(N p−N+p)khk

p p−1

∞ |supph|^N(p−1)p |suppf₀|,

(4.7)

where C1 = C1(N, p) is the constant in (2.14). From the assumption and the estimates (2.16), (4.6), we deduce

Z

R^N

|∇uf_r|^pdx≥C₂^pr⁻

N−p p−1

h (khk1− kf0k1)^p−1p ≥C₂^pr⁻

N−p p−1

h

1 2khk1

_p−1^p

≥2^−p0C₂^pr⁻

N−p p−1

h (khk−∞;supph|supph|)^p−1p ,

(4.8)

whereC2=C2(N, p) is the constant in (2.16). Let

κ= min{1

2, C₂^pω

N−p N(p−1)

N

2·2^p0·2 ^p

2

(p−1)(N p−N+p)C1

}.

Combining (4.2), (4.4)–(4.5), (4.7)–(4.8), we obtain

s0≤ − C₂^pkhk^p₁⁰r⁻

N−p p−1

h

2·2^p0(khk_{1;supph\suppfr}− kf_rk_{1;suppfr\supph})

≤ − C₂^p 2·2^p0khk

1 p−1

1 r⁻

N−p p−1

h ≡ −δ, whereδ >0 is a constant independent ofrandfr.

Forε= ¹₂δ, applying Lemma 2.14, we find that there existsr0≥rhindependent ofrandfrsuch that

uf_r(x)≥ −1

2δ >−δ≥s0= sup

x∈suppf_r

uf_r(x), ∀x∈R^N\Br₀.

It follows suppf_r⊂B_r0.

Proof of Theorem 1.2. The first part of this theorem can be proved by using the similar method in the proof of Theorem 1.1. It follows S0 = S0(r0) =S0(r) for r≥r0. To prove the last part of this theorem, for any ˆf ∈S0 =S0(r0), we have from (4.1)

fˆ=ϕ_r0◦u_fˆ, a.e. inB_r0,

for a decreasing functionϕ_r0. Combining this with (4.3), we obtainϕ_r0(t)≥0 for t≤s₀ andϕ_r0(t) = 0 fort∈[s₀, s⁰₀],s⁰₀= sup_x∈B

r0u_fˆ(x). Now define ˆ

ϕ(t) =

(ϕr₀(t), t≤s0, 0, t > s₀.

Clearly, ˆϕis a decreasing function and ˆf = ˆϕ◦ufˆa.e. inR^N. Acknowledgments. The first author was supported in part by the Scientific Re- search Foundation of Graduate School of South China Normal University. The second author and the third author were supported in part by National Natural Scientific Foundation of China and Specialized Research Fund for the Doctoral Program of Higher Education.

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Shanming Ji

School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China

E-mail address:[email protected]

Jingxue Yin

School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China

E-mail address:[email protected]

Rui Huang

School of Mathematical Sciences, South China Normal University, Guangzhou 510631.

Department of Mathematics, South China University of Technology, Guangzhou 510640, China

E-mail address:[email protected]