ON THE EXISTENCE OF SOLUTIONS TO A CLASS OF p-LAPLACE ELLIPTIC EQUATIONS

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Radial Solution and Nonradial Solutions Hui-mei He and Jian-qing Chen

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ON THE EXISTENCE OF SOLUTIONS TO A CLASS OF p-LAPLACE ELLIPTIC EQUATIONS

HUI-MEI HE AND JIAN-QING CHEN

Fujian Normal University Fuzhou, 350007 P.R. China

EMail:hehuimei20060670@163.com jqchen@fjnu.edu.cn

Received: 19 September, 2008

Accepted: 07 December, 2008

Communicated by: C. Bandle 2000 AMS Sub. Class.: 35J20.

Key words: p-Laplace elliptic equations, Radial solutions, Nonradial solutions.

Abstract: We study the equation−∆pu+|x|^a|u|^p−2u = |x|^b|u|^q−2u with Dirichlet boundary condition onB(0, R)or onR^N. We prove the existence of the radial solution and nonradial solutions of this equation.

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1. Introduction and Main Result

Equations of the form (1.1)

( −∆_tu+g(x)|u|^s−2u=f(x, u) inΩ

u= 0, on∂Ω

have attracted much attention. Many papers deal with the problem (1.1) in the case of t= 2, Ω =R^N, s= 2, glarge at infinity andf superlinear, subcritical and bounded inx, see e.g. [1], [2] and [4]. The problem (1.1) witht = 2, Ω =B(0,1), g(x) = 0 and f(x, u) = |x|^bu^l−1 was studied in [9]; in particular, it was proved that under some conditions the ground states are not radial symmetric. The caset = 2, Ω = B(0,1) or on R^N, g(x) = 1 and f(x, u) = |x|^b|u|^l−2u was studied in [7]. The problem (1.1) witht =s, Ω =R^N, g(x) = V(|x|)andf(x, u) =Q(|x|)|u|^l−2uwas studied by J. Su., Z.-Q. Wang and M. Willem ([11], [12]). They proved embedding results for functions in the weightedW^1,p(R^N)space of radial symmetry. The results were then used to obtain ground state and bound state solutions of equations with unbounded or decaying radial potentials.

In this paper, we consider the nonlinear elliptic problem

(1.2)







−∆_pu+|x|^a|u|^p−2u=|x|^b|u|^q−2u inΩ u >0, u∈W^1,p(Ω),

u= 0, on∂Ω

and prove the existence of the radial and the nonradial solutions of the problem (1.2).

Here, ∆_pu = div(|∇u|^p−2∇u)is the p-Laplacian operator, and 1 < p < N, a ≥ 0, b≥0.

We denote byW_r^1,p(R^N)the space of radially symmetric functions in W^1,p(R^N) =

u∈L^p(R^N) :∇u∈L^p(R^N) .

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W_r,a^1,p(R^N)is denoted by the space of radially symmetric functions in W_a^1,p(R^N) =

u∈W^1,p(R^N) : Z

R^N

|x|^a|u|^p <∞

. We also denote byD_r^1,p(R^N)the space of radially symmetric functions in

D^1,p(R^N) = n

u∈L^N−p^{N p} R^N

:∇u∈L^p R^No . Our main results are:

Theorem 1.1. Ifa ≥0, b≥0, 1< p < N and p < q <q˜= N p

N −p+ bp N −p, pb−a

p+ (p−1)(q−p) p

<(q−p)(N −1), then the problem (1.2) has a radial solution.

Remark 1. In [8], Sirakov proves that the problem (1.2) with p = 2 has a solution for

2< q < q^#= 2N

N −2− 4b a(N −2).

In [6], P. Sintzoff and M. Willem proved the existence of a solution of the problem (1.2) with

p= 2, q≤2^∗, 2b−a 1 + q

2

<(N −1)(q−2).

Theorem1.1 extends the results of [6] to the general equation with ap−Laplacian operator.

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Theorem 1.2. Suppose thata≥0, b≥0, 1< p < N and p < q < N p

N −p, pb−a

p+(p−1)(q−p) p

<(q−p)(N −1), aq < pb,

then for everyR, problem (1.2) withΩ =B(0, R), Rlarge enough has a radial and a nonradial solution.

This paper is organized as follows: In Section 2, we study (1.2) in the case of Ω =R^N. We prove the existence of a radial least energy solution of (1.2) when

1< p < N, p < q <q,˜ pb−a

p+(p−1)(q−p) p

<(q−p)(N−1).

In Section 3, we consider the existence of nonradial solutions of (1.2) with Ω = B(0, R), R large enough. Finally, in Section 4, we consider necessary conditions for the existence of solutions of (1.2).

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2. Radial Solution

In this paper, unless stated otherwise, all integrals are understood to be taken over all ofR^N. Also, throughout the paper, we will often denote various constants by the same letter.

Lemma 2.1. Suppose that 1 < p < N. There existAN > 0,such that, for every u∈W_r,a^1,p(R^N), u ∈C(R^N\{0}), fora ≥ _p−1^p (1−N),we have that

|x|^N−1^p ⁺

a(p−1)

p2 |u(x)| ≤A_N Z

|x|^a|u|^p ^p−1

p2 Z

|∇u|^p ¹

p2

. Proof. Since

d dr

|u|^pr^a·

p−1 p r^N−1

= p

2 |u|²^p₂−1

·2u·du drr^a·

p−1 p r^N⁻¹ +|u|^p

a·p−1

p +N −1

r^a·^p−1^p ⁻¹r^N−1, and

a≥ p

p−1(1−N), we get that

d dr

|u|^pr^a·^p−1^p r^N−1

≥pu|u|^p−2du

drr^a·^p−1^p r^N⁻¹ and obtain

r^a·^p−1^p r^N−1|u(r)|^p ≤A_N Z +∞

r

|u|^p−1

du dr

S^N−1S^a·^p−1^p dS

≤A_N Z

|u|^p−1

du dr

|x|^a·^p−1^p dx

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≤A_N Z

|x|^a|u|^p

^p−1_p Z

|∇u|^p ¹_p

. It follows that

|x|^N−1+a·^p−1^p |u(x)|^p ≤AN

Z

|x|^a|u|^p

^p−1_p Z

|∇u|^p ¹_p

, and we have

|x|^N−1^p ⁺

a(p−1)

p2 |u(x)| ≤A_N Z

|x|^a|u|^p ^p−1

p2 Z

|∇u|^p ¹

p2

.

Lemma 2.2. If1 < p < N, p ≤ r < _N−p^pN , then for anyu ∈ W^1,p(R^N), we have that

Z

|u|^rdx ≤C Z

|∇u|^p

^N(r−p)

p2 Z

|u|^p

^{N p+r(p−N)}

p2

.

Proof. The proof can be adapted directly from the Gagliardo-Nirenberg inequality.

The following inequality extends the results of [5] to the general equation with thep−Laplacian operator.

Lemma 2.3. For

1< p < N, p < q < pN

N −p+ c

N−1

p +^a(p−1)_p2

, a≥ p

p−1(1−N),

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there existB_N,p,csuch that for everyu∈D_r^1,p(R^N), we have Z

|x|^c|u|^qdx ≤B_N,p,c Z

|∇u|^p

p(N−1)+a(p−1)^c +^N

p2

q−p−p(N−1)+a(p−1)^cp²

. Proof. Using Lemma2.1and Lemma2.2, we have

Z

|x|^c|u|^qdx

= Z

|x|^N−1^p ⁺

a(p−1) p2

(N−1)/p+a(p−1)p^c −2

|u|

(N−1)/p+a(p−1)p^c −2

|u|

^q−(N−1)/p+a(p−1)p^c −2

dx

≤ Z

|x|^a|u|^p ^p−1

p2 · ^c

(N−1)/p+a(p−1)p−2 Z

|∇u|^p ¹

p2· ^c

(N−1)/p+a(p−1)p−2

· Z

|∇u|^p ^N

p2(q−p− ^c

(N−1)/p+a(p−1)p−2)Z

|u|^p ^{N p}

p2+^p−N

p2

q− ^c

(N−1)/p+a(p−1)p−2

= Z

|x|^a|u|^pdx

p(N−1)+a(p−1)^c(p−1) Z

|u|^p ^{N p}

p2+^p−N

p2

q−p(N−1)+a(p−1)^cp²

· Z

|∇u|^p

p(N−1)+a(p−1)^c +^N

p2

q−p−p(N−1)+a(p−1)^cp²

≤B_N,p,c Z

|∇u|^p

p(N−1)+a(p−1)^c +^N

p2

q−p−p(N−1)+a(p−1)^cp²

.

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Next, to prove Theorem1.1, we consider the following minimization problem m =m(a, b, p, q) = inf

u∈W_r,a^1,p(R^N) R|x|^b|u|^qdx=1

Z

|∇u|^p+|x|^a|u|^p dx.

Theorem 2.4. Ifa ≥0, b≥0, 1< p < N and p < q <q˜= N p

N −p+ bp N −p, pb−a

p+ (p−1)(q−p) p

<(q−p)(N −1), thenm(a, b, p, q)is achieved.

Proof. Let(u_n)⊂W_r,a^1,p(R^N)be a minimizing sequence form=m(a, b, p, q) : Z

|x|^b|un|^qdx= 1, Z

(|∇u_n|^p +|x|^a|u_n|^p)dx→m.

By going (if necessary) to a subsequence, we can assume thatu_n * uinW_r,a^1,p(R^N).

Hence, by weak lower semicontinuity, we have Z

(|∇u|^p +|x|^a|u|^p)dx≤m, Z

|x|^b|u|^qdx≤1.

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Ifcis defined byq = _N^pN_−p +_N−p^pc , thenc < band it follows from Lemma2.3that Z

|x|≤ε

|x|^b|u_n|^qdx≤ε^b−c Z

|x|^c|u_n|^qdx≤Cε^b−c. Since(u_n)is bounded inW_r,a^1,p(R^N). We deduce from Lemma2.1that

Z

|x|≥¹

ε

|x|^b|u_n|^qdx= Z

|x|≥¹

ε

|x|^b−a|u_n|^q−p|x|^a|u_n|^pdx

≤ 1

ε

b−a−(q−p)(^N−1_p +^a(p−1)

p2 )

C Z

|x|^a|u|^pdx

≤Cε^a(

q+1 p −^q

p2)−b+^{(q−p)(N−1)}_p

.

So we get that, for everyt <1, there existsε >0, such that for everyn, Z

ε≤|x|≤¹

ε

|x|^b|u_n|^qdx≥t.

By the Rellich theorem and Lemma2.1, 1≥

Z

|x|^b|u_n|^qdx≥ Z

ε≤|x|≤¹_ε

|x|^b|u_n|^qdx≥t.

FinallyR

|x|^b|u|^qdx= 1andm =m(a, b, p, q)is achieved atu.

Now we will prove Theorem1.1.

Proof. By Theorem 2.4, m is achieved. Then by the Lagrange multiplier rule, the symmetric criticality principle (see e.g. [13]) and the maximum principle, we obtain

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a solution of







−∆_pυ+|x|^a|υ|^p−2υ =λ|x|^b|υ|^q−2υ, υ >0, υ ∈W^1,p(R^N).

Henceu=λ^q−p¹ υ is a radial solution of (1.2), withλ= ^p_qm.

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3. Nonradial Solutions

In this section, we will prove Theorem1.2. We use the preceding results to construct nonradial solutions of problem (1.2) in the caseΩ =B(0, R).

Consider

M =M(a, b, p, q) = inf

u∈Wa^1,p(R^N) R|x|^b|u|^qdx=1

Z

|∇u|^p+|x|^a|u|^p dx.

It is clear thatM ≤m,and using our previous results, we prove thatM is achieved under some conditions.

Theorem 3.1. Ifa ≥0, b≥0, 1< p < N and p < q < q^#= pN

N −p− p²b a(N −p), thenM(a, b, p, q)is achieved.

Proof. Let(u_n)⊂W_a^1,p(R^N)be a minimizing sequence forM =M(a, b, p, q) : Z

|x|^b|u_n|^qdx= 1, Z

|∇un|^p+|x|^a|un|^p

dx→M.

By going (if necessary) to a subsequence, we can assume thatu_n * uinW_a^1,p(R^N).

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Hence, by weak lower semicontinuity, we have Z

|∇u|^p+|x|^a|u|^p

dx≤M, Z

|x|^b|u|^qdx≤1.

Ifcis defined byq = _N^pN_−p −_a(N−p)^p²^c , thenc > band r = a

c, s=

aN p N−p

aq−pc

are conjugate. It follows from the Hölder and Sobolev inequalities that Z

|x|≥¹_ε

|x|^b|u_n|^qdx≤ 1

ε b−cZ

|x|^c|u_n|^qdx

= 1

ε b−cZ

|x|^c|u_n|^pc^a|u_n|^q−^pc^adx

≤ε^c−b Z

|x|^a|u_n|^pdx ¹_r Z

|u_n|^N−p^{N p} dx ¹_s

≤Cε^c−b.

As in Theorem2.4, for everyt <1, there existsε >0such that, for everyn, Z

|x|≤¹_ε

|x|^b|u_n|^qdx≥t.

By the compactness of the Sobolev theorem in the bounded domain, for 1 < p <

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N, p < q < _N^{N p}_−p, 1≥

Z

|x|^b|u|^qdx≥ Z

|x|≤¹_ε

|x|^b|u_n|^qdx≥t.

HenceR

|x|^b|u|^qdx= 1andM =M(a, b, p, q)is achieved atu.

Now we will prove Theorem1.2.

Proof. By Theorem2.4,m(a, b, p, q)is positive. Sincepb > aq, it is easy to verify thatM(a, b, p, q) = 0.Let us define

M(a, b, p, q, R) = inf

u∈W_a^1,p(B(0,R)) R

B(0,R)|x|^b|u|^qdx=1

Z

B(0,R)

|∇u|^p+|x|^a|u|^p dx,

m(a, b, p, q, R) = inf

u∈W_r,a^1,p(B(0,R)) R

B(0,R)|x|^b|u|^qdx=1

Z

B(0,R)

|∇u|^p+|x|^a|u|^p dx.

It is clear that, for every R > 0, M(a, b, p, q, R) and m(a, b, p, q, R) are achieved and

R→∞lim M(a, b, p, q, R) =M(a, b, p, q) = 0,

R→∞lim m(a, b, p, q, R) =m(a, b, p, q)>0.

Then from Theorem 1.1, we know that problem (1.2) with B(0, R) has a radial solution.

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On the other hand, by the Lagrange multiplier rule, the symmetric criticality principle (see e.g.[13]) and the maximum principle, we obtain a solution of







−∆_pυ +|x|^a|υ|^p−2υ =λ|x|^b|υ|^q−2υ inB(0, R) υ >0, u∈W^1,p(B(0, R)),

υ = 0, on∂B(0, R).

Hence u = λ^q−p¹ υ is a solution of (1.2), with λ = ^p_qM(a, b, p, q, R). Thus, Prob- lem (1.2) has a nonradial solution.

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4. Necessary Conditions

In this section we obtain a nonexistence result for the solution of problem (1.2) using a Pohozaev-type identity. The Pohozaev identity has been derived for very general problems by H. Egnell [3].

Lemma 4.1. Letu∈W^1,p(R^N)be a solution of (1.2), thenusatisfies N −p

p Z

|∇u|^pdx+ N +a p

Z

|x|^a|u|^pdx−N +b q

Z

|x|^b|u|^qdx= 0.

Theorem 4.2. Suppose that

˜

q= N p

N −p+ pb N −p ≤q

or N +a

p ≤ N +b q . Then there is no solution for problem (1.2).

Proof. Multiplying (1.2) byuand integrating, we see that Z

|x|^b|u|^qdx= Z

|∇u|^p+|x|^a|u|^p dx.

On the other hand, using Lemma4.1, we obtain N −p

p −N +b q

Z

|∇u|^pdx+

N +a

p −N +b q

Z

|x|^a|u|^pdx= 0.

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So, ifuis a solution of problem (1.2), we must have N −p

p < N +b

q , N +a

p > N +b q .

Remark 2. The second assumption of Theorem2.4,

pb−a

p+(q−p)(p−1) p

<(q−p)(N−1) implies that

N +b

q < N+a p .

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