EXISTENCE RESULTS FOR OPERATOR EQUATIONS INVOLVING DUALITY MAPPINGS VIA THE MOUNTAIN PASS THEOREM

INVOLVING DUALITY MAPPINGS VIA THE MOUNTAIN PASS THEOREM

JENIC ˘A CRˆINGANU

We derive existence results for operator equations having the form Jϕu=Nfu,

by using the Mountain Pass Theorem. HereJϕ is a duality mapping on a real reflexive and smooth Banach space, compactly imbedded in aL^q-space, andNf

is the Nemytskii operator generated by a Carath´eodory functionfwhich satisfies some appropriate growth conditions.

AMS 2000 Subject Classification: 35J20, 35J60.

Key words: Duality mapping, Mountain Pass Theorem, Nemytskii operator, p- Laplacian.

1. INTRODUCTION

In [9] the existence of the weak solution in W₀^1,p(Ω), 1 < p < +∞, for the Dirichlet problem

−∆_pu=f(x, u) in Ω⊂R^N, (1.1)

u= 0 on∂Ω (1.2)

was obtained using (among other methods) the Mountain Pass Theorem.

It is well known that the operator −∆_p : W₀^1,p(Ω) → W^−1,p0(Ω) is in fact the duality mapping on W₀^1,p(Ω) corresponding to the gauge function ϕ(t) = t^p−1. In this paper we generalize the results from [9] by considering operator equations of the form

(1.3) J_ϕu=N_fu,

where, instead of the special duality mapping (−∆_p), we consider the case of an arbitrary duality mapping. More precisely, we consider equation (1.1) in the functional framework below.

MATH. REPORTS11(61),1 (2009), 11–20

(H1) Xis a real reflexive and smooth Banach space, compactly imbedded in L^q(Ω), where 1< q <+∞and Ω⊂R^N,N ≥2, is a bounded domain with smooth boundary.

(H₂) J_ϕ :X→X^∗ is the duality mapping corresponding to the gauge func- tion ϕ:R+→R+; we assume that Jϕ is continuous and satisfies the (S₊) condition: ifx_n* x(weakly) inXand lim sup

n→∞

hJ_ϕx_n, x_n−xi ≤0, thenx_n→x (strongly) inX.

(H₃) N_f :L^q(Ω) → L^q0(Ω), ¹_q + _q¹0 = 1, defined by (N_fu)(x) =f(x, u(x)), for u ∈ L^q(Ω), x ∈ Ω, is the Nemytskii operator generated by the Carath´eodory function f : Ω×R → R, which satisfies the growth condition

(1.4) |f(x, s)| ≤c|s|^q−1+b(x), x∈Ω, s∈R, wherec >0 is constant and b∈L^q0(Ω).

By solution of equation (1.3) we mean an elementu∈X which satisfies

(1.5) J_ϕu= (i⁰N_fi)u,

where i is the compact imbedding of X into L^q(Ω) and i⁰ : L^q0(Ω)→ X^∗ is its adjoint.

Notice that by (1.4) the operator N_f is well-defined, continuous and bounded from L^q(Ω) into L^q0(Ω) (see, e.g., [8]), so that i⁰N_fi is also well- defined and compact from X intoX^∗.

Moreover, the functionalψ:L^q(Ω)→R, defined byψ(u) =R

ΩF(x, u)dx, whereF(x, s) =Rs

0 f(x, t) dtisC¹ onL^q(Ω), hence onX and ψ⁰(u) =Nfu for all u∈X.

Notice also thatF is a Carath´eodory function and there exists a constant c₁>0 and a function c∈L^q(Ω), c≥0, such that

(1.6) |F(x, s)| ≤c1|s|^q+c(x), x∈Ω, s∈R.

Let us remark that (1.5) is equivalent to (1.7) hJ_ϕu, vi=hN_f(iu), ivi_Lq(Ω),Lq0

(Ω)= Z

Ω

f(x, u)vdx, v∈X.

By Asplund’s theorem, J_ϕu = ∂Φ(u) for each u ∈ X, where Φ(u) = Rkuk

0 ϕ(t)dt and ∂Φ : X → P(X^∗) is the subdifferential of Φ in the sense of convex analysis, i.e.,

∂Φ(u) ={x^∗∈X^∗: Φ(v)−Φ(u)≥ hx^∗, v−ui for allv∈X}

(see, e.g., [4], [5] or [12]). Since Φ is convex and X is smooth, Φ is Gˆateaux differentiable on X and ∂Φ(u) = {Φ⁰(u)} for all u ∈ X. So, Jϕu = Φ⁰(u)

for all u ∈ X and, by continuity of Jϕ (see (H2)), we have Φ∈ C¹(X,R).

Consequently, the functional F :X →R defined by F(u) = Φ(u)−Ψ(u) =

Z kuk

0

ϕ(t)dt− Z

Ω

F(x, u)dx, u∈X,

is C¹ on X and F⁰(u) = Φ⁰(u)−Ψ⁰(u) =Jϕu−N_fu for allu∈X. Therefore, u ∈X is a solution of equation (1.3) if and only if u is a critical point forF, i.e., F⁰(u) = 0.

To prove thatF has at least a critical point in X we use the Mountain Pass Theorem, which we recall below.

Theorem 1.1. Let X be a real Banach space and assumeI ∈ C¹(X,R) satisfies the Palais-Smale (PS) condition. Suppose I(0) = 0 and that

(i) there are constantsρ, α >0 such that I|_kuk=ρ≥α;

(ii)there is an element e∈X, kek> ρ such thatI(e)≤0.

Then I possesses a critical value c≥α. Moreover, c= inf

g∈Γ max

u∈g([0,1])I(u)

whereΓ ={g∈ C([0,1], X) :g(0) = 0, g(1) =e}. (It is obvious that each criti- cal point u at level c (I⁰(u) = 0, I(u) =c) is a nontrivial one.)

For theproof see, e.g., [2], [11] or [13].

Let us recall that the functional I ∈ C¹(X,R) is said to satisfy the (PS) condition if any sequence (un) ⊂ X for which (I(un)) is bounded and I⁰(un)→0 as n→ ∞, possesses a convergent subsequence.

2. THE MAIN EXISTENCE RESULT

First, we establish some preliminary results.

Proposition 2.1. If (u_n) ⊂X is bounded and F⁰(u_n)→ 0 as n→ ∞, then (un) has a convergent subsequence.

Proof. This result is proved in [9], Lemma 2.

Theorem 2.1. Assume that there are θ >1 and s₀>0 such that (2.1) θF(x, s)≤sf(x, s), x∈Ω, |s| ≥s₀,

(2.2) lim inf

t→∞

Rt

0ϕ(s)ds tϕ(t) > 1

θ. Then F satisfies the (PS) condition.

Proof. According to Proposition 2.1 it is sufficient to show that any sequence (u_n)⊂Xfor which (F(u_n)) is bounded andF⁰(u_n)→0, is bounded.

As in the proof of Theorem 15 in [9], there exists a constant k > 0 such that

(2.3) Φ(un)−1

θ Z

Ω

f(x, un) undx≤k.

By F⁰(u_n) → 0, as in the proof of the same theorem, there exists n₀ ∈ N such that

hJ_ϕu_n, u_ni − Z

Ω

f(x, u_n)u_n

≤ ku_nk, (∀) n≥n₀, which implies

(2.4) −1

θϕ(ku_nk)ku_nk+1 θ

Z

Ω

f(x, u_n)u_ndx≤ 1

θku_nk, (∀) n≥1.

From (2.3) and (2.4) we obtain Φ(un)− 1

θϕ(ku_nk)ku_nk −1

θku_nk ≤k, (∀) n≥n0, so that

Z ku_nk

0

ϕ(s)ds−1

θϕ(ku_nk)ku_nk −1

θku_nk ≤k, (∀) n≥n0

or, equivalently,

(2.5) ϕ(ku_nk)ku_nk

" Rkunk

0 ϕ(s)ds ϕ(ku_nk)ku_nk−1

θ − 1

θϕ(ku_nk)

#

≤k.

By condition (2.2) there are constants γ > ¹_θ and t0 >0 such that Rt

0ϕ(s)ds

tϕ(t) ≥γ, t≥t0.

If ku_nk → ∞there existsn1 ∈Nsuch that ku_nk ≥t0 forn≥n1 and then Rku_nk

0 ϕ(s)ds

ku_nkϕ(ku_nk) ≥γ, n≥n1. From (2.5) we obtain

ϕ(ku_nk)ku_nk

γ−1

θ − 1

θϕ(ku_nk)

≤k, n≥n1.

a contradiction (because γ > ¹_θ and ϕ(ku_nk)→ ∞).

Consequently, (u_n) is bounded inX.

Theorem 2.2. Assume that there exists θ >1 such that

(2.6) lim

t→∞

Rt

0ϕ(s)ds t^θ = 0 and either

(i) there existss₁ >0 such that

(2.7) 0< θF(x, s)≤sf(x, s), x∈Ω, s≥s1, or

(ii)there exists s1<0 such that

(2.8) 0< θF(x, s)≤sf(x, s), x∈Ω, s≤s1. ThenF is unbounded from below.

Proof. We are going to prove the sufficiency of condition (i) (similar arguments may be used if (ii) holds).

Let u ∈ X, u > 0 (in fact i(u) > 0) be such that meas (M₁(u)) > 0, where

M₁(u) ={x∈Ω :u(x)≥s₁} (in fact i(u)(x)≥s₁).

We shall prove that F(λu)→ −∞ asλ→ ∞.

Forλ≥1, from the proof of Theorem 16 in [9] we have Z

Ω

F(x, λu)dx≥λ^θk₁(u)−k₂, where k1(u)>0, k2>0 are constants. Therefore,

F(λu) = Φ(λu)− Z

Ω

F(x, λu)dx≤Φ(λu)−λ^θk₁(u) +k₂

=λ^θ

" Rλkuk 0 ϕ(t)dt

λ^θ −k₁(u)

# +k₂.

By condition (2.6) there are constants α∈(0, k1(u)) andt0>0 such that Rt

0ϕ(s)ds

t^θ kuk^θ≤α, t≥t₀, and then

F(λu)≤λ^θ

" Rλkuk 0 ϕ(t)dt

(λkuk)^θ kuk^θ−k₁(u)

#

+k₂≤λ^θ(α−k₁(u)) +k₂ for λkuk ≥t₀. Sinceα−k₁(u)<0, we haveF(λu)→ −∞asλ→ ∞.

Remark 2.1. By Theorem 2.2, sinceF is unbounded from below, for each ρ >0 there existse∈X with kek> ρsuch that F(e)≤0.

Now we are in a position to prove the main result of this section.

Theorem2.3. Assume that hypotheses(H1), (H2)and(H3)hold. More- over, assume that

(i) there areθ >1 and s0 >0 such that

(2.9) 0< θF(x, s)≤sf(x, s), x∈Ω,|s| ≥s₀; (ii) lim inf

t→∞

Rt

0ϕ(s)ds tϕ(t) > 1

θ; (iii) lim

t→∞

Rt

0ϕ(s)ds t^θ = 0;

(iv)there are r₁, r₂ ∈(1, ∞), r₁ < r₂, andc₀ >0 such that F(u)≥c2kuk^r_X¹ −c3kuk^r_X² (∀) u∈X with kuk_X < c0, where c2, c3 >0 are constants.

Then equation (1.3)has at least a nontrivial solution in X.

Proof. Let us notice that condition (iv) is suggested by [10].

It is sufficient to show that F has at least a nontrivial critical point u∈X. To do it, we shall use Theorem 1.1. Clearly,F(0) = 0. By (i), (ii) and Theorem 2.1, F satisfies the (PS) condition. By (iv) we have

F(u)≥ kuk^r_X¹ c2−c3kuk^r_X^2−r1

for allu∈X,withkuk_X < c0,hence there are constantsρ, α >0,ρ < c0 such that F(u)≥α >0 provided thatkuk_X =ρ is small enough.

Finally, from (i), (iii) and Theorem 2.2 (see also Remark 2.1) there is an element e∈X,kek> ρ, such thatF(e)≤0, and the proof is complete.

Remark 2.2. A sufficient condition in order to have (iv) is the existence of real numbers r1, r2, 1< r1< r2,such that

(iv₁) lim

s→0

R_s

0 ϕ(t)dt

|s|^r1 >0, (iv₂) lim

s→0

f(x, s)

|s|^r2−1 <∞.

Indeed, by (iv1), Φ(u)≥c2kuk^r1 provided thatkuk>0 is small enough while, by (iv₂),

Z

Ω

F(x, u)dx

≤c3kuk^r2,

provided that kuk>0 is small enough (here, c₂, c₃ >0 are constants). Con- sequently, F(u)≥c2kuk^r1 −c3kuk^r2 forkuk>0 sufficiently small.

3. EXAMPLES

Example 3.1 Consider the Dirichlet problem (1.1), (1.2), where 1< p <

∞, Ω ⊂ R^N, N ≥ 2, is a bounded domain with smooth boundary, ∆_pu = div(|∇u|^p−2∇u) = PN

i=1 ∂

∂xi

|∇u|^p−2_∂x^∂u

i

is the so-called p-Laplacian and f : Ω×R→Ris a Carath´eodory function which satisfies the growth condition (3.1) |f(x, s)| ≤c |s|^q−1+ 1

, x∈Ω, s∈R, with c≥0 constant, 1< q < p^∗=

( _{N p}

N−p ifN > p, +∞ ifN ≤p.

Let us remark that thep-Laplacian operator−∆_p :W₀^1,p(Ω)→W^−1,p0(Ω) defined by

−∆_pu=−

N

X

i=1

∂

∂xi

|∇u|^p−2 ∂u

∂xi

, u∈W₀^1,p(Ω), or, equivalently,

h−∆_pu, vi= Z

Ω

|∇u|^p−2∇u ∇v, u, v∈W₀^1,p(Ω), is the duality mapping

Jϕ :W₀^1,p(Ω)→W^−1,p0(Ω)

corresponding to the gauge function ϕ(t) =t^p−1 (see, e.g., [9] or [12]).

On the other hand, the Nemytskii operatorNf is continuous and bounded from L^q(Ω) intoL^q0(Ω).

By solution of the Dirichlet problem (1.1), (1.2) we mean an element u∈W₀^1,p(Ω) which satisfies

(3.2) −∆_pu= i⁰N_fi

u in W^−1,p0(Ω), or, equivalently,

Z

Ω

|∇u|^p−2∇u∇v= Z

Ω

f(x, u)v, v∈W₀^1,p(Ω),

where i is the compact imbedding of W₀^1,p(Ω) into L^q(Ω) and i⁰ : L^q0(Ω) → W^−1,p0(Ω) is its adjoint. Consequently, (3.2) may be equivalently written as

(3.3) J_ϕu=N_fu

(here, by N_f we mean i⁰N_fi).

We shall formulate sufficient conditions for equation (3.3) to admit a non- trivial solution, via Theorem 2.3. Take X=W₀^1,p(Ω) with Ω⊂R^N,N ≥2, a bounded domain with smooth boundary, 1< p <∞,q∈(1, p^∗), ϕ(t) =t^p−1,

t≥0,f : Ω×R→R a Carath´eodory function which satisfies the conditions below.

•The growth condition (3.1).

•There are θ > p ands0 >0 such that

(3.4) 0< θF(x, s)≤sf(x, s), x∈Ω, |s| ≥s₀.

•

(3.5) lim sup

s→0

f(x, s)

|s|^p−2s < λ1 uniformly inx∈Ω, where λ₁ = inf

(kvk^p_1,p

kvk^p_0,p :v∈W₀^1,p(Ω), v6= 0 )

is the first eigenvalue of −∆_p in W₀^1,p(Ω).

SinceW₀^1,p(Ω) is a reflexive and smooth Banach space, compactly imbed- ded in L^q(Ω), hypothesis (H₁) of Theorem 2.3 is satisfied. Since J_ϕ =−∆_p : W₀^1,p(Ω)→W^−1,p0(Ω) is continuous and satisfies condition (S₊) (see, e.g., [9]) hypothesis (H₂) of Theorem 2.3 is satisfied, too. By the growth condition (3.1), hypothesis (H3) of Theorem 2.3 is also satisfied. Since θ > p, conditions (ii) and (iii) are satisfied. Finally, condition (3.5) implies condition (iv) in Theorem 2.3 (see, e.g., [9]).

Consequently, Theorem 2.3 applies and gives the already known result (see, e.g., [9]) on the existence of a nontrivial solution for problem (1.1), (1.2).

Example 3.2. Consider the Neumann problem

−∆_pu+|u|^p−2u=f(x, u) in Ω, (3.6)

|∇u|^p−2∂u

∂n = 0 on ∂Ω, (3.7)

where 1 < p < ∞, Ω ⊂ R^N, N ≥ 2, is a bounded domain with smooth boundary and f : Ω×R→ R is a Carath´eodory function which satisfies the growth condition (3.1).

By solution of the Neumann problem (3.6), (3.7) we mean an element u∈W^1,p(Ω) which satisfies

(3.8) Z

Ω

|∇u|^p−2∇u∇v+ Z

Ω

|u|^p−2uv = Z

Ω

f(x, u)v, v ∈W^1,p(Ω).

AssumeX =W^1,p(Ω) is endowed with the norm

|||u|||^p_1,p =kuk^p_0,p+k |∇u| k^p_0,p, u∈W^1,p(Ω),

which is equivalent to the standard norm on the space W^1,p(Ω) (see [6]).

In this case, the duality mappingJϕ on W^1,p(Ω),||| · |||_1,p

corresponding to the gauge function ϕ(t) =t^p−1 is defined (see [6]) by

J_ϕ : W^1,p(Ω),||| · |||_1,p

→ W^1,p(Ω),||| · |||_1,p∗

, Jϕu=−∆_pu+|u|^p−2u, u∈W^1,p(Ω).

It is easy to see that u ∈W^1,p(Ω) is a solution of problem (3.6), (3.7) in the sense of (3.8) if and only if

(3.9) Jϕu=N_fu

(here, byN_f we also meani⁰N_fi, whereiis the compact imbedding of (W^1,p(Ω),

||| · |||_1,p) intoL^q(Ω) andi⁰ :L^q0(Ω)→ W^1,p(Ω),||| · |||_1,p∗

is its adjoint).

We shall formulate sufficient conditions for equation (3.9) to admit a nontrivial solution, via Theorem 2.3.

Take X = W^1,p(Ω) with Ω ⊂ R^N, N ≥ 2, a bounded domain with smooth boundary, 1< p <∞,q∈(1, p^∗),ϕ(t) =t^p−1,t≥0 andf : Ω×R→ R a Carath´eodory function which satisfies the conditions below.

•The growth condition (3.1).

•There are θ > p ands0 >0 such that

(3.10) 0< θF(x, s)≤sf(x, s), x∈Ω, |s| ≥s0.

•

(3.11) lim sup

s→0

f(x, s)

|s|^p−2s < λ1 uniformly inx∈Ω, where λ₁ = inf

(|||v|||^p_1,p

kvk^p_0,p :v∈W^1,p(Ω), v6= 0 )

. Since W^1,p(Ω),||| · |||_1,p

is a reflexive and smooth Banach space, com- pactly imbedded inL^q(Ω) (see [6]), hypothesis (H1) of Theorem 2.3 is satisfied.

Since Jϕ : W^1,p(Ω),||| · |||_1,p

→ W^1,p(Ω),||| · |||_1,p∗

is continuous and satisfies condition (S₊) (see, e.g., [6]), hypothesis (H₂) of Theorem 2.3 is satisfied, too.

By the growth condition (3.1), hypothesis (H3) of Theorem 2.3 is also satisfied.

Since θ > p, conditions (ii) and (iii) are satisfied. Finally, condition (3.11) im- plies condition (iv) in Theorem 2.3 (see, e.g., [6]). Consequently, Theorem 2.3 applies and gives the already known result (see, e.g., [6]) on the existence of a nontrivial solution for problem (3.6), (3.7).

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Received 14 November 2008 University of Galat¸i

Department of Mathematics 800008 Galat¸i, Romania

jcringanu@ugal.ro