Critical point result of Schechter type in a Banach space

Critical point result of Schechter type in a Banach space

Hannelore Lisei

and Orsolya Vas

Babes,–Bolyai University, Faculty of Mathematics and Computer Science, Kog˘alniceanu Str. 1, Cluj-Napoca, RO – 400084 , Romania

Received 11 June 2015, appeared 22 March 2016 Communicated by Gennaro Infante

Abstract. Using Ekeland’s variational principle we give a critical point theorem of Schechter type for extrema on a sublevel set in a Banach space. This result can be applied to localize the solutions of PDEs which contain nonlinear homogeneous opera- tors.

Keywords: critical point, Ekeland’s variational principle, Palais–Smale type compact- ness condition,p-Laplacian.

2010 Mathematics Subject Classification: 47J30, 47J05, 58E05, 34B15.

1 Introduction

Finding the set of solutions of certain PDEs is closely related to the investigation of the critical points of a certain functional defined on an appropriate Hilbert or Banach space. Mountain pass theorems, saddle point theorems, linking theorems, mountain cliff theorems give suffi- cient conditions for the existence of a minimizer for a certain differentiable functional defined on the whole space or on a bounded region (for example, see [3,16–19,21]).

In [13–15] R. Precup studies critical point theorems of Schechter type forC¹functionals on a closed ball and also on a closed conical shell in a Hilbert space by using Palais–Smale type compactness conditions and also Leray–Schauder conditions on the boundary. These results can be used successfully to localize the solutions of PDEs involving the Laplace operator.

In our paper we improve the above mentioned Schechter type results (on a ball) for sub- level sets in locally uniformly convex Banach spaces and then apply our result for localizing the solutions for p-Laplace type equations on bounded, and also on unbounded domains.

The paper is structured as follows: Section 2 contains certain preliminaries concerning duality mappings on Banach spaces and the assumptions for the critical point problem which we are investigating. Section 3 states the main result of our paper. Section 4 presents two examples of localizing the solutions for problems containing the p-Laplacian.

BCorresponding author. Email: hanne@math.ubbcluj.ro

2 Preliminaries

Let X be a real Banach space, X^∗ its dual, h·,·idenotes the duality between X^∗ and X. The norm onXand onX^∗ is denoted by k · k.

A continuous function ϕ : R+ → _R+ is called a normalization function if it is strictly increasing, ϕ(0) =0 andϕ(r)→_∞forr→_∞.

The duality mapping corresponding to the normalization function ϕis the set valued operator Jϕ: X→ P(X^∗)defined by

J_ϕx =ⁿx^∗ ∈ X^∗ :hx^∗,xi= ϕ(kxk)kxk,kx^∗k= ϕ(kxk)^o, x ∈X.

Assumption (A1): XandX^∗ are locally uniformly convex reflexive Banach spaces.

Observe that X^∗ is strictly convex, because a locally uniformly convex Banach space is also strictly convex, see [5, Theorem 3, p. 31]. Then it follows that card(J_ϕx) = 1 by [6, Proposition 1, p. 342]. Hence, Jϕ :X→X^∗

hJϕx,xi= ϕ(kxk)kxk and kJϕxk= ϕ(kxk). The following result holds.

Theorem 2.1. [6, Theorem 5, p. 345]Let X be a reflexive, locally uniformly convex Banach space and J_ϕ: X→X^∗. Then J_ϕis bijective and its inverse J⁻_ϕ¹is bounded, continuous and monotone. Moreover, it holds J_ϕ⁻¹ = χ⁻¹J^∗_ϕ−1, whereχ : X → X^∗∗ is the canonical isomorphism between X and X^∗∗ and J^∗

ϕ⁻¹ :X^∗ →X^∗∗is the duality mapping on X^∗ corresponding to the normalization function ϕ⁻¹. We consider ¯J : X^∗ → Xdefined by ¯J = J⁻_ϕ¹. By Theorem2.1 it follows that ¯J is bounded, continuous and monotone. Forw∈ X^∗denote v= J_ϕ⁻¹wand compute

hw, ¯Jwi=hJ_ϕv,vi= ϕ(kvk)kvk=kJ_ϕvkkvk=kwkkJ⁻_ϕ¹wk and

kJw^¯ k=kJ⁻_ϕ¹wk=kχ⁻¹J^∗_ϕ−1wk=kJ_ϕ^∗−1wk_X^∗∗ = ϕ⁻¹(kwk). We conclude

hw, ¯Jwi= ϕ⁻¹(kwk)kwk and kJw^¯ k= ϕ⁻¹(kwk) for eachw∈ X^∗. (2.1) Fix 0<R.

Assumption (A2): Let H:X →_Rbe of classC¹ such that the level set N_R={u∈ X: H(u) =R}is non-void and bounded ,

uinf∈N_RhH⁰(u),ui>0, and the operatorH⁰ maps bounded sets into bounded sets.

We denote

X_R= {u∈X: H(u)≤R}.

Assumption (A3): Let F:X_R→_Rbe of classC¹ such thatFis bounded by below onX_R.

We introduce some auxiliary mappings:

D: N_R →X^∗, D(u) =F⁰(u)− hF⁰(u),ui hH⁰(u),ui^H

0(u), E: N_R →X, E(u) = JD^¯ (u)− hH⁰(u), ¯JDui

hH⁰(u),ui ^u.

Lemma 2.2. Assume that (A1)holds, and that F : X_R → _Rand H : X → _Rare C¹functions. For all u∈ N_Rthe following properties hold:

(1) hH⁰(u),E(u)i=0,

(2) hF⁰(u),E(u)i= ϕ⁻¹(kD(u)k)kD(u)k. Proof. Letu∈ N_R be arbitrary. We compute

hH⁰(u),E(u)i=

H⁰(u), ¯JD(u)−hH⁰(u), ¯JDui hH⁰(u),ui ^u

=0.

Observe that

hD(u),ui=

F⁰(u)− hF⁰(u),ui hH⁰(u)_,ui^H

0(u),u

=0. (2.2)

By using the statement (1) of this lemma, by (2.2) and (2.1) we have hF⁰(u),E(u)i=

F⁰(u)− hF⁰(u),ui hH⁰(u),ui^H

0(u),E(u)

=hD(u),E(u)i

=

D(u), ¯JD(u)i − hH⁰(u), ¯JDui

hH⁰(u),ui hD(u),u

= hD(u)_{, ¯}JD(u)i= ϕ⁻¹(kD(u)k)kD(u)k_.

3 Main result

Theorem 3.1. Assume that(A1),(A2)and(A3)are satisfied. Then, there exists a sequence(x_n)_n ⊂ X_Rsuch that F(x_n)→infF(X_R)and one of the following statements hold

(a) F⁰(x_n)→0as n→_∞;

(b) for each n∈_Nwe have H(xn) =R,hH⁰(xn), ¯JF⁰(xn)i ≤0and F⁰(x_n)− hF⁰(x_n),x_ni

hH⁰(xn),xni^H

0(x_n)→0 as n→_∞. If, in addition, there exists a∈_R+such that

hH⁰(x), ¯JF⁰(x)i ≥ −a for each x∈ N_R,

and F satisfies a Palais–Smale type compactness condition (i.e. any sequence satisfying(a)or(b)has a convergent subsequence) and the following boundary condition holds

F⁰(x) +µH⁰(x)6=0 for anyµ>0, x∈ N_R, (3.1) then there exists x ∈X_R such that

F(x) =infF(X_R) and F⁰(x) =0.

Proof. By Ekeland’s variational principle, see [8, Theorem 1, p. 444], applied for X_R (we use here thatHis continuous, henceXRis a closed set), the distanced(x,y) =kx−yk, the function F(which is continuous and bounded by below, see (A3)),ε= _n¹ and foru∈ X_Rsuch that

F(u)≤infF(X_R) + ¹ n, it follows that there exists a sequence(x_n)_nin X_Rsuch that

F(x_n)≤F(u)≤infF(X_R) + ¹ n, and

F(xn)<F(y) + ¹

nkxn−yk for eachy ∈XR\ {xn}. (3.2) This yieldsF(x_n)→_infF(X_R)_.

Since(xn)_nbelongs toX_R, we distinguish two cases:

(1) there exists a subsequence of(xn)_n, still denoted by(xn)_n, such that H(xn)< Rfor each n∈_N;

(2) there exists a subsequence of(xn)_n, still denoted by(xn)_n, such that H(xn) =Rfor each n∈_N.

Case (1)Fix n∈ _{N. Let}t >0 and z∈ X such thatkzk=1. Since His a continuous function and H(x_n)< R, we have that there exits δ > 0 (small enough) such that H(x_n−tz) < Rfor eacht ∈(0,δ). Hencexn−tz ∈XR\ {xn}for eacht∈ (0,δ)and by (3.2) it holds

F(xn)−F(xn−tz)< ^t n. By takingt &0 it follows

hF⁰(xn),zi ≤ ¹ n.

Butz ∈ X with kzk = 1 was arbitrary chosen, hence kF⁰(xn)k ≤ _n¹, which yields F⁰(xn) → 0 as n → ∞. Hence we constructed a sequence (x_n)_n which satisfies the statement (a) of this theorem.

Case (2)Fix n∈ _{N. We have} H(x_n) = R. Letz ∈ Xsuch that kzk= 1. We use the definition of the Fréchet derivative of H: for eachε >0 there exists δ_ε >0 such that for eacht ∈ (0,δ_ε) we have

−εt< H(x_n−tz)−H(x_n) +hH⁰(x_n),tzi<εt.

Hence,

R−εt−thH⁰(xn),zi< H(xn−tz)<R+εt−thH⁰(xn),zi for eacht ∈(0,δε). (3.3)

•_IfhH⁰(x_n)_,zi>0: by takingε=hH⁰(x_n)_,ziin (3.3) we get H(x_n−tz)< R for each t∈(0,δ_ε).

Hencex_n−tz∈ X_R\ {x_n}fort ∈(0,δ_ε). By (3.2) it follows that fort ∈(0,δ_ε) F(x_n)−F(x_n−tz)< ^t

n,

then byt&0 we get hF⁰(x_n),zi ≤ ¹

n for eachz∈ Xwithkzk=1 and hH⁰(x_n),zi>0. (3.4)

• If hH⁰(xn),zi = 0: we approximate z by a sequence (z_k)_k such that kz_kk = 1 and hH⁰(x_n),z_ki > 0 for each k ∈ _{N, while} kz−z_kk → 0 as k → _{∞. Since} hH⁰(x_n),·i is con- tinuous, we havehH⁰(x_n),z_ki → hH⁰(x_n),zi=0 ask→_∞.

Let k ∈ _N be fixed. By considering (3.3) for z_k instead of z and ε = hH⁰(xn),z_ki for t∈(0,δε)we get

H(x_n−tz_k)< R.

Then, x_n−tz_k ∈ X_R\ {x_n}_fort ∈(_0,δ_ε). By (3.2) we obtain fortsufficiently small F(xn)−F(xn−tz_k)< ^t

n, which yields

hF⁰(xn),z_ki ≤ ¹ n. Butkz−z_kk →0 as k→_{∞, hence}

hF⁰(x_n),zi ≤ ¹ n. This inequality and (3.4) imply

hF⁰(x_n),zi ≤ ¹

n for eachz∈ Xwithkzk=1 and hH⁰(x_n),zi ≥0. (3.5) Further we have two possible cases.

Case (2a) There exists a subsequence of (xn)_n, which we still denote by (xn)_n, such that hH⁰(x_n), ¯JF⁰(x_n)i>0: by takingz = _k ¹

JF⁰(xn)kJF¯ ⁰(x_n)in (3.5) we get hF⁰(x_n), ¯JF⁰(x_n)i ≤ ¹

nkJF⁰(x_n)k. By the property (2.1) of ¯J it follows that

hF⁰(xn), ¯JF⁰(xn)i= ϕ⁻¹(kF⁰(xn)k)kF⁰(xn)k and kJF^{¯ 0}(xn)k= ϕ⁻¹(kF⁰(xn)k) which yields

kF⁰(x_n)k ≤ ¹ n,

hence F⁰(xn)→ 0 asn → _∞ and we obtained a sequence(xn)_n which satisfies the statement (a) of this theorem.

Case (2b) There exists a subsequence of (xn)_n, which we still denote by (xn)_n, such that hH⁰(x_n), ¯JF⁰(x_n)i ≤0: by takingz= _kE(¹_x

n)kE(x_n)in (3.5) (ifkE(x_n)k=0, then by Lemma2.2(2) we getkD(xn)k=0) and by Lemma2.2(2) we get

ϕ⁻¹(kD(xn)k)kD(xn)k=hF⁰(xn),E(xn)i ≤ ¹

nkE(xn)k. Hence,

ϕ⁻¹(kD(x_n)k)kD(x_n)k ≤ ¹

nkE(x_n)k. (3.6)

Denote the kernel ofH⁰(x_n)byK_n={x ∈X:hH⁰(x_n),xi=0}and the projection mapping Pn: X→ K_n by Pnv=v− hH⁰(x_n)_,vi

hH⁰(xn),xni^xn.

Since v ∈ X 7→ hH⁰(xn),vi is linear and continuous, it follows that Pn is also linear and continuous.

Since (x_n)_n ⊂ N_R and the level set N_R is bounded, it follows that (x_n)_n is a bounded sequence. By the assumption onH⁰ it follows that(H⁰(xn))_nis also bounded and there exists

0< β_R := inf

u∈NR

hH⁰(u),ui ≤ hH⁰(x_n),x_ni for eachn∈_N. We write

kP_nvk ≤ kvk+ kH⁰(x_n)kkvk

infn∈_NhH⁰(xn),xnikx_nk ≤

1+ kH⁰(x_n)kkx_nk β_R

kvk for eachv ∈X.

Hence there existsα_R>0 (independent ofn) such that

kP_nvk ≤α_Rkvk for eachv∈ X.

We takev = JD^¯ (x_n)to get P_nJD¯ (x_n) =E(x_n)and

kE(xn)k ≤α_RkJD^¯ (xn)k=α_Rϕ⁻¹(kD(xn)k) for each n∈_N.

Then by (3.6) we have

ϕ⁻¹(kD(xn)k)kD(xn)k ≤ ^αR

n ϕ⁻¹(kD(xn)k) for each n∈_N.

This yields D(xn) → 0 as n → ∞. Hence we constructed a sequence (xn)_n which satisfies statement (b) of this theorem.

If, in addition, Fsatisfies the (PS) type compactness condition.

Case (a) F⁰(x_n) → 0 as n → _∞ and there exist x ∈ X_R and a subsequence (x_nk)_{k such that} kxn_k−xk →0 ask→_{∞. Since} Fis aC¹function, we get F⁰(x) =0 and by the construction of (x_n)_n we haveF(x_n)→infF(X_R), henceF(x) =infF(X_R).

Case (b)We haveD(x_n)→ _{0 as}n → _∞, H(x_n) = R andhH⁰(x_n)_{, ¯}JF⁰(x_n)i ≤ _{0 for all} n ∈ _N and there exist x ∈ X_R (X_R is a closed set, since H is continuous) and a subsequence (xn_k)_k such that kx_nk −xk → 0 as k → _{∞. Hence} F(x) = lim_k→∞F(x_nk) = infF(X_R), F⁰(x) = lim_k→_∞F⁰(x_nk)_andx∈ N_R, i.e. H(x) = R. ButD(x_nk)→_{0 as}k →∞, which implies

F⁰(x)− hF⁰(x),xi hH⁰(x),xi^H

0(x) =0. (3.7)

Applying the operator ¯J we get

JF¯ ⁰(x)− hF⁰(x),xi hH⁰(x),xi^JH¯

0(x) =0, which yields

hH⁰(x), ¯JF⁰(x)i − hF⁰(x),xi

hH⁰(x)_,xihH⁰(x), ¯JH⁰(x)i=0,

hH⁰(x), ¯JF⁰(x)i − hF⁰(x),xi hH⁰(x),xi^ϕ

−1(kH⁰(x)k)kH⁰(x)k=0. (3.8) Since we are investigating Case (b), it follows that(hH⁰(xn), ¯JF⁰(xn)i)_n is a bounded sequence inR, hence there existb∈_R,b≤0, and a subsequence, denoted again by(x_nk)_k, such that

hH⁰(x_nk), ¯JF⁰(x_nk)i →b.

Butkx_nk−xk →0 ask →_{∞, hence}

hH⁰(x), ¯JF⁰(x)i=b≤0.

Using (3.8) it follows

hF⁰(x),xi hH⁰(x),xi^ϕ

−1(kH⁰(x)k)kH⁰(x)k= b≤0.

Since hH⁰(x)_,xi > 0 (assumption on H⁰ and the fact that x ∈ N_R as the limit of (x_nk)_{k), we} obtain

hF⁰(x),xi ≤0.

•IfhF⁰(x),xi=0, then (3.7) impliesF⁰(x) =0.

•IfhF⁰(x),xi<0, then (3.7) implies

F⁰(x) +µH⁰(x) =0 whereµ=−hF⁰(x),xi

hH⁰(x)_,xi >0 andx∈ N_R, which contradicts the assumption (3.1) from the statement of this theorem.

4 Applications

4.1 Example 1

Consider the Sobolev space W₀^1,p(_Ω)_{, where Ω} is a bounded domain in R^k with Lipschitz continuous boundary and 1 < p<∞, equipped with the norm

kuk_1,p = Z

Ω|∇u(x)|^pdx ¹_p

, where

∇u= ∂u

∂x₁, . . . , ∂u

∂x_k

, |∇u|=

∑

∂u

∂x_i 2!¹₂

.

The Banach space (W₀^1,p(_Ω),k · k_1,p)is uniformly convex, see [1, Theorem 3.6]. Moreover, it is also uniformly smooth (which is proved by using Clarkson’s inequalities [1, 2.38 Theorem, p. 44] and [4, Definition 2.4., p. 13]). The dual space(W₀^1,p(_Ω))^∗ will be denoted byW^−1,p0(_Ω), where ¹_p+ _p¹0 =1, and by [4, Theorem 2.10] it follows that it is uniformly convex.

The Rellich–Kondrachov Theorem states that the embeddingW₀^1,p(_Ω),→L^q(_Ω)is compact for q∈ (1,p^∗)(where p^∗ = _k^kp_−p if p < k and p^∗ = _{∞, if} p ≥ k) and there existsCq > 0 such that

kuk_Lq(_Ω)≤C_qkuk_{1,p for each}u∈W₀^1,p(_Ω)_{. (4.1)}

In the context of our paper we choose (X,k · k) = (W₀^1,p(_Ω),k · k_1,p), ¯J = J⁻_ϕ¹, where ϕ(t) =t^p−1fort ∈_R+and let H :W₀^1,p(_Ω)→_Rbe given byH(u) = ¹_pkuk_1,p^p .

We consider the p-Laplacian operator−_∆p :W₀^1,p(_Ω)→W^−1,p0(_Ω)defined by h−_∆p(u),vi=

Z

Ω|∇u(x)|^p−2∇u(x)∇v(x)dxfor all u,v∈W₀^1,p(_Ω).

It is known that the functional H is continuously Fréchet differentiable on W₀^1,p(_Ω) _and H⁰ = −_∆p. The operator −_∆p is in fact the duality mapping Jϕ : W₀^1,p(_Ω) → W^−1,p0(_Ω) corresponding to the normalization function ϕ(t) = t^p−1 fort ∈ _R+, i.e. H⁰ = Jϕ, for details consult [6, Theorem 7 and Theorem 9]. In our example we have

NR =

v∈W₀^1,p(_Ω): 1

pkvk_1,p^p = R

and

X_R =

v∈W₀^1,p(_Ω): 1

pkvk_1,p^p ≤ R

.

Assume that f : Ω×_R → _R is a Carathéodory function such that f(_{x, 0}) 6= 0 for a.e.

x∈ _Ωand

|f(x,s)| ≤a(x)|s|^q−1+b(x) forx ∈_Ω, s∈_R,

where a ∈ L^∞(_Ω),b ∈ L^q−q1(_Ω)are positive functions and q ∈ (1,p^∗). Define the Nemytskii operatorN_f :W₀^1,p(_Ω)→W^−1,p0(_Ω)by

N_f (u) (x) = f(x,u(x)).

We have N_f(W₀^1,p(_Ω)) ,→ N_f(L^q(_Ω)) ⊂ L^q−q1(_Ω) = (L^q(_Ω))^∗ ,→ W^−1,p0(_Ω) _and N_f is a continuous function which maps bounded sets into bounded sets (see [9]).

Consider the following Dirichlet problem involving the p-Laplacian:

−_∆pu= f(x,u) _a.e. x ∈_{Ω and} u

∂Ω =_{0. (4.2)}

We callu∈W₀^1,p(_Ω)aweak solutionof (4.2) if for eachv∈W₀^1,p(_Ω)it holds Z

Ω|∇u(x)|^p−2∇u(x)∇v(x)dx=

Z

Ω f(x,u(x))v(x)dx. (4.3) We introduceF:W₀^1,p(_Ω)→_Rdefined by

F(u) = ¹

pkuk_1,p^p −

Z

Ωh(x,u)dx, whereh:Ω×_R→_Rish(x,t) =

Z _t

0 f(x,s)ds. We have (see [9, Theorem 7]) F⁰(u) =H⁰(u)−N_f (u).

The critical points ofFare the solutions of (4.3).

(A4) Assumptions for R: denote by C an upper bound for C_q and suppose that one of the following three assumptions is satisfied.

(1) If p>q: letR>0 be a solution of the inequality inR R^p

−1

p >C^qp^q

−p

p kak_L∞(_Ω)R^q

−1

p +Cp¹

−p p kbk

L

q q−1(_Ω). (2) If p=q: assume 1>C^pkak_L∞(_Ω) and let Rbe such that

R>







 Cp¹

−p p kbk

L

p p−1(_Ω)

1−C^pkak_L∞(_Ω)









p p−1

.

(3) If q> p: assume that 1> C^qp^q−ppkak_L∞(_Ω)+Cp^1−ppkbk

L

q

q−1(_Ω) and letR > 0 to be a solution of the inequality inR

R^p

−1

p −C^qp^q

−p

p kak_L∞(_Ω)R^q

−1

p >Cp¹

−p p kbk

L

q q−1(_Ω). Proposition 4.1. The following relation holds

F⁰(u) +µH⁰(u)6=0 for anyµ>0, u∈ NR, where R satisfies one of the three conditions mentioned in(A4).

Proof. We reason by contradiction: assume that there exist u ∈ N_R and µ > 0 such that F⁰(u) +µH⁰(u) =0, which implies

(₁+µ)hJ_ϕ(u)_,ui=hN_f(u)_,ui. (4.4) By our assumptions

hN_f(u),ui=

Z

Ω f(x,u(x))u(x)dx

≤

Z

Ωa(x)|u(x)|^q+b(x)|u(x)|dx≤ kak_L∞(_Ω)kuk^q

L^q(_Ω)+kbk

L

q q−1(_Ω)

kuk_Lq(_Ω). Using (4.1) and (4.4) we get

kuk_1,p^p ≤(1+µ)kuk_1,p^p ≤ kak_L∞(_Ω)kuk^q

L^q(_Ω)+kbk

L

q q−1(_Ω)

kuk_Lq(_Ω)

≤C^qkak_L∞(_Ω)kuk^q_1,p+Ckbk

L

q q−1(_Ω)

kuk_1,p.

Butu ∈N_R implieskuk_1,p= (pR)^1p, and we obtain pR≤C^qkak_L∞(_Ω)(pR)^qp +Ckbk

L

q q−1(_Ω)

(pR)^1p, which yields

R^p

−1

p ≤C^qp^q

−p

p kak_L∞(_Ω)R^q

−1

p +Cp¹

−p p kbk

L

q

q−1(_Ω). (4.5) The assumptions in (A4) imply that (4.5) cannot be satisfied.

Proposition 4.2. Suppose that R satisfies one of the three conditions mentioned in (A4). Then F satisfies the following Palais–Smale type compactness condition: if(un)_n is a sequence from XR such that one of the following statements hold

(a) F⁰(u_n)→₀as n→_∞;

(b) for each n∈_Nwe have H(u_n) =R,hH⁰(u_n), ¯JF⁰(u_n)i ≤0and F⁰(u_n)− hF⁰(u_n),u_ni

hH⁰(u_n)_,u_ni^H

0(u_n)→0 as n→_∞,

then(un)_nadmits a convergent subsequence.

Proof. Since the sequence(u_n)_nis bounded inW₀^1,p(_Ω)(it belongs toX_R) and since the embed- dingW₀^1,p(_Ω),→ L^q(_Ω)is compact forq∈(1,p^∗), there existu∈W₀^1,p(_Ω)and a subsequence of (u_n)_n, which we denote again by (u_n)_n, which converges weakly in W₀^1,p(_Ω) to u and strongly inL^q(_Ω)tou. Then by Hölder’s inequality we have

hN_f(un),un−ui ≤ kf(·,un)k

L

q

q−1(_Ω)kun−uk_Lq(_Ω)→0. (4.6) Case (a): F⁰(un)→0 asn→_{∞. Then,}

hH⁰(u_n),u_n−ui=hF⁰(u_n),u_n−ui+hN_f(u_n),u_n−ui →0.

The(S+)property ofH⁰ = Jϕ (see [6, Theorem 10]) implies(un)_nconverges strongly tou.

Case (b): For eachn∈_Nwe haveH(u_n) =R,hH⁰(u_n), ¯JF⁰(u_n)i ≤0 and F⁰(un)− hF⁰(un),uni

hH⁰(u_n),u_ni^H

0(un)→0 asn→_∞.

We denote

µ= _lim

n→_∞

hF⁰(u_n)_,u_ni hH⁰(un),uni ∈_R.

Therefore,

F⁰(u_n)−µH⁰(u_n)→0 asn→_∞. (4.7) Ifµ=0, the above convergence implies

F⁰(un)→0 asn→_∞.

As in Case (a) it follows that there existu∈W₀^1,p(_Ω) and a subsequence of(u_n)_n, which we denote again by(u_n)_n, which converges strongly inW₀^1,p(_Ω)_tou. Since F⁰ is continuous, we haveF⁰(u) =0, which implieshH⁰(u),ui= hN_f(u),ui. Butu∈ N_R(since(un)_nbelongs to the closed setN_R), which yields

pR=hN_f(u),ui ≤C^qkak_L∞(_Ω)(pR)^qp +Ckbk

L

q

q−1(_Ω)(pR)^1p.

This contradicts the assumption onRfrom (A4). Hence, the caseµ=0 is not possible.

Forµ6=0 we have by (4.7)

hF⁰(u_n)−µH⁰(u_n)_,u_n−ui →_{0 as}n→_∞.

(1−µ)hH⁰(u_n),u_n−ui+hN_f(u_n),u_n−ui →0 asn→_∞.

Ifµ6=1 we get by (4.6)

(₁−µ)hH⁰(u_n)_,u_n−ui →_{0 as}n→_∞.

The (S+) property of H⁰ = J_ϕ implies (u_n)_n converges strongly to u. The convergence (4.7) and the strong convergenceun →uimplies F⁰(u)−µH⁰(u) =0.

hF⁰(u), ¯JF⁰(u)i=µhH⁰(u), ¯JF⁰(u)i.

But hH⁰(u), ¯JF⁰(u)i ≤ 0 and hF⁰(u), ¯JF⁰(u)i ≥ 0, therefore µ < 0 and the relation F⁰(u)− µH⁰(u) =0 contradicts the statement of Proposition4.1. Hence, the caseµ6=1 is not possible.

Forµ=1: by (4.7) it follows that

hF⁰(un), ¯JF⁰(un)i − hH⁰(un), ¯JF⁰(un)i →0 asn→_∞.

Using (2.1) we have

hF⁰(un), ¯JF⁰(un)i= ϕ⁻¹(kF⁰(un)k)kF⁰(un)k ≥0 and by the assumptions of Case (b) we have

−hH⁰(un), ¯JF⁰(un)i ≥0

Then F⁰(u_n) → 0 asn → ∞. As in Case (a) it follows that there exist u∈W₀^1,p(_Ω) and a subsequence of (u_n)_n, which we denote again by(u_n)_n, which converges strongly inW₀^1,p(_Ω) to u. This yields u ∈ NR, because (un)_n belongs to the closed set NR. Since F⁰ and H⁰ are continuous, we have by the convergence (4.7) thatF⁰(u)−H⁰(u) =0,

hF⁰(u), ¯JF⁰(u)i=hH⁰(u), ¯JF⁰(u)i.

ButhH⁰(u), ¯JF⁰(u)i ≤0 (by the assumption of Case (b) and by the strong convergenceu_n→u) and hF⁰(u), ¯JF⁰(u)i ≥ 0, hence F⁰(u) = 0, which implies H⁰(u) = 0 and then u = 0. But 0 /∈ N_R, contradictsu∈ N_R. Hence the caseµ=1 is not possible.

We apply Theorem3.1 in order to localize the solution of (4.3).

Theorem 4.3. Suppose that R satisfies one of the three conditions mentioned in(A4). Then, equation (4.3)admits a weak solution u∈X_R, which minimizes F on X_R.

In what follows we discuss situations when the best Sobolev constantC_qadmits an upper estimate which can be computed:

Denote the first eigenvalue of thep-Laplace operator by λp(_Ω) = min

v∈W₀^1,p(_Ω)\{0}

R

Ω|∇v(x)|^pdx R

Ω|v(_x)|^pdx . Then,

kuk_L^p_p(Ω)≤ ¹

λ_p(_Ω)kuk_1,p^p for allu∈W₀^1,p(_Ω). Hence the best embedding constant of W₀^1,p(_Ω),→ L^p(_Ω) is Cp = ¹

λp(_Ω)

1/p

, while for q < p the best embedding constant of W₀^1,p(_Ω),→ L^q(_Ω) verifies (via Hölder’s inequality)

C_q≤ |_Ω|^{ppq−q 1}

λ_p(_Ω)

1/p

(here |_Ω| denotes the Lebesgue measure, i.e. the k-dimensional vol- ume, of the setΩ). In order to obtain upper bounds forCq(q≤ p) we need lower bounds for λ_p(_Ω).

By using the Faber–Krahn inequality [2, Theorem 1] it holds λp(_Ω)≥ λp(_Ω^∗),

whereΩ^∗ is thek-dimensional ball centered at the origin having the same volume asΩ. So it has the radiusr = ^√1

π |_Ω|_Γ(₂^k +₁)^1/k_.

By [10] we have for the ballΩ^∗ = Br⊂_R^k of radiusrthe inequality λp(Br)≥

k rp

p

.

Then the best Sobolev constant has the following upper estimate, which can be computed:

C_p≤ ^p k√

π

|_Ω|_Γ k

2+₁ ¹_k

, and for 1<q< p

Cq≤ ^p k√ π

|_Ω|^k(ppq−q)+1_Γ k

2+1 ¹_k

.

Fork =1 andΩ= (0,T)⊂_Rthe value of the first eigenvalue is known (see [7])

λ_p(_Ω) = (p−₁) ^2π T psin(^π_p)

!p

, hence

C_p = ^{T psin}(^π_p) 2π(p−1)^1p .

For the casek=1 andΩ= (0,T)the sharp Poincaré inequality is known (see [20], p. 357):

for each p>1,q>1 andu∈W₀^1,p(0,T)it holds

kuk_Lq(_Ω) ≤Cqkuk_1,p, where the embedding constant is given by

Cq= ^T

1 q+¹

p0

2B(¹_q,_p¹0) ^p

0¹_q

q^p10 p⁰+q¹_p−¹_q

,

p⁰ = _p−^p₁ andBis the Beta function.

4.2 Example 2

For 1< p<_∞we define the following subspace of radially symmetric functions ofW^1,p(_R^k) W_r^1,p(_R^k) =u∈W^1,p(_R^k)_:u(x) =u(x⁰)∀ x,x⁰ ∈ _R^k_,|x|=|x⁰| _,

endowed with the norm induced fromW^1,p(_R^k) kuk^p =

Z

R^k

|∇u(z)|^p+|u(z)|^pdz.

The spaceW^1,p(_R^k) is a separable, reflexive and uniformly convex Banach space [1, 3.6 The- orem, p. 61]; moreover, it is also uniformly smooth (which is proved by using Clarkson’s inequalities [1, 2.38 Theorem, p. 44] and [4, Definition 2.4., p. 13]).

In the context of Section 2 and Section 3 we consider X = W_r^1,p(_R^k) endowed with the above normk · k, Xis a closed subspace ofW^1,p(_R^k). Hence it is also uniformly smooth and by [4, Theorem 2.10] it follows that its dualX^∗ is uniformly convex.

LetJϕ :X→ X^∗ be the duality mapping corresponding to the weight functionϕ(t) =t^p−1, t ∈ _R+ where p ∈ (1,+_∞) (see [3, Proposition 2.2.4]). It is well known that the duality mapping Jϕ satisfies the following conditions:

kJ_ϕuk= ϕ(kuk) and hJ_ϕu,ui=kJ_ϕukkuk for all u∈X.

Moreover, the functional H : X →_R defined by H(u) = ¹_pkuk^p is convex and Fréchet differ- entiable with H⁰ = Jϕ. We take ¯J = J⁻_ϕ¹.

It is known [11, Théorème II. 1] that the embedding W_r^1,p(_Rⁿ) ,→ L^q(_Rⁿ)is compact for q∈ (p,p^∗)(wherek ≥2, p^∗ = _k^kp_−p if p <kand p^∗ =_{∞, if} p ≥k) there existsC_q>0 (the best embedding constant) such that

kuk_Lq(_R^k) ≤C_qkuk for eachu∈W_r^1,p(_R^k). (4.8) Let f :R×_R^k →_Rbe a Carathéodory function such that f(x, 0)6=0 for a.e.x∈_Rwhich satisfies

|f(x,s)| ≤a(x)|s|^q−1+b(x) for(x,s)∈_R^k×_R,

where a ∈ L^∞(_R^k),b ∈ L^q−q1(_R^k)are positive functions and q ∈ (p,p^∗)and f(x,·) = f(x⁰,·) for all x,x⁰ ∈ _Rⁿ,|x|= |x⁰|(f is radially symmetric in the first variable).

Consider the following problem involving the p-Laplacian:

−_∆pu+|u|^p−2u= f(x,u) a.e.x ∈_R^k. (4.9) We call u∈W^1,p(_R^k)aweak solutionof (4.9) if for eachv ∈W^1,p(_R^k)it holds

Z

Ω|∇u(x)|^p−2∇u(x)∇v(x) +|u(x)|^p−2u(x)v(x)dx=

Z

Ω f(x,u(x))v(x)dx. (4.10) Define the Nemytskii operatorN_f :X→ X^∗ byN_f (u) (x) = f(x,u(x))andF: X→_Rby

F(u) = ¹

pkuk_1,p^p −

Z

Rⁿh(x,u)dx, where h:Ω×_R→_Rish(x,t) =Rt

0 f(x,s)ds. We have F⁰(u) = H⁰(u)−N_f(u).

Let G= O(_R^k)be the set of all rotations onR^k. Observe that the elements of GleaveR^k invariant, i.e.g(_R^k) =_R^k for allg∈G. Ginduces an isometric linear action overW^1,p(_R^k)by

(gu)(z) =u(g⁻¹z)_, g ∈G, u∈W^1,p(_R^k)_{, a.e.} z∈_R^k_.

A functionφdefined onW^1,p(_R^k)is said to beG-invariant if

φ(gu) =φ(u) for allg ∈G, u∈W^1,p(_R^k). In factW_r^1,p(_R^k)is the fixed point set ofW^1,p(_R^k)underGand the norm

kuk= Z

R^k(|∇u(z)|^p+|u(z)|^p)dz ¹_p

isG-invariant onW^1,p(_R^k).

Observe that by the assumption on f and the above remark, the functionalFisG-invariant and then by the principle of symmetric criticality [12] every critical point ofFis also a solution of (4.10).

Consider

X_R=

v∈W_r^1,p(_R^k): 1

pkvk^p ≤R

. Reasoning as in Section4.1 one has the following result.

Theorem 4.4. Suppose that R satisfies one of the three conditions mentioned in(A4). Then, F admits a critical point u∈XR, which minimizes F on XR. Moreover, this critical point is also a weak solution of (4.10).

We discuss situations when the Sobolev constant Cq admits an upper estimate which can be computed: by [20] we have that for 1< p<kand p^∗= _k^kp_−p it holds for allu ∈W^1,p(_R^k)

kuk_Lp∗

(_R^k) ≤C_R Z

R^k|∇u(x)|^pdx ¹_p

, where

C_R= √¹ πk^1p

p−1 k−p

1−¹_p Γ(₁+₂^k)_Γ(k) Γ(^k_p)_Γ(1+k− ^k_p)

!¹_k . Obviously this implies

kuk_Lp∗

(_R^k)≤C_Rkukfor eachu∈W^1,p(_R^k).

For any q ∈ (p,p^∗) there exists θ ∈ (0, 1) such that q = θp+ (1−θ)p^∗, then by Hölder’s inequality

kuk^q

L^q(_R^k)≤ kuk^θp

L^p(_R^k)kuk^(1−θ)p∗

L^p∗(_R^k) ≤C^kq

1 p−¹_q R kuk^q,

for eachu ∈ W^1,p(_R^k). Then the Sobolev constant has the following upper estimate, which can be computed:

Cq≤C^k

1 p−¹_q

R forq∈

p, kp k−p

and 1< p<k.

Acknowledgement

The authors would like to express their special thanks to Prof. Cs. Varga from Babes_,-Bolyai University in Cluj-Napoca for his constructive discussions.

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