Quasilinear elliptic equations with Hardy terms and ... - Springer Link

Chen Boundary Value Problems (2015) 2015:171 DOI 10.1186/s13661-015-0440-3

RESEARCH

Open Access

Quasilinear elliptic equations with Hardy terms and Hardy-Sobolev critical exponents: nontrivial solutions Guanwei Chen* *

Correspondence: [email protected] School of Mathematical Sciences, University of Jinan, Jinan, Shandong 250022, P.R. China

Abstract In this paper, we obtain one positive solution and two nontrivial solutions of a quasilinear elliptic equation with p-Laplacian, Hardy term and Hardy-Sobolev critical exponent by using variational methods and some analysis techniques. In particular, our results extend some existing ones. MSC: 35B33; 35J60 Keywords: quasilinear elliptic equations; Hardy terms; Hardy-Sobolev critical exponents; (PS)c condition; p-Laplacian

1 Introduction and main results We shall study the following quasilinear elliptic equation: p– p∗ (s)– –p u – μ |u||x|p u = |u| |x|s u + f (x, u), x ∈ \ {}, u = , x ∈ ∂,

(.)

where p u = div(|∇u|p– ∇u) denotes the p-Laplacian differential operator, is an open bounded domain in RN (N ≥ ) with smooth boundary ∂ and  ∈ ,  ≤ μ < μ := )p ,  ≤ s < p,  < p < N , p∗ (s) = p(N–s) is the Hardy-Sobolev critical exponent and p∗ = ( N–p p N–p Np p∗ () = N–p is the Sobolev critical exponent. The conditions of f will be given later. ,p

On the Sobolev space W (), we set u :=

p |u|p |∇u|p – μ p dx , |x|

(.)

,p

which is well defined on W () by the Hardy inequality []. It is comparable with the ,p standard Sobolev norm of W (), but it is not a norm since the triangle inequality or subadditivity may fail, which has been clarified in []. The following minimization problem will be useful in what follows: Aμ,s := Aμ,s () :=

inf

,p

u∈W ()\{}

(

up ∗

|u|p (s) |x|s

p

dx) p∗ (s)

,

(.)

which is the best Hardy-Sobolev constant. © 2015 Chen. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Chen Boundary Value Problems (2015) 2015:171

Page 2 of 11

For some special f and p = , some authors ([–], s = ) ([, –], s = ) have studied the existence of solutions for (.). If p = , then (.) becomes

–u – μ |x|u = u = ,

∗

|u| (s)– u + f (x, u), |x|s

x ∈ \ {},

(.)

x ∈ ∂.

Ding and Tang [] obtained the existence and multiplicity of solutions for (.) if  ≤ μ < ) ,  ≤ s <  and f satisfies some suitable conditions. Kang [] considered another ( N–  q– special case of (.) with f (x, u) = λ |u||x|t u ; for details, we refer the readers to see Remark .. u Let F(x, u) :=  f (x, s) ds, x ∈ , u ∈ R. In order to state our results, we make the following assumptions: (A ) f ∈ C( × R+ , R), limu→+ fu(x,u) p– =  and limu→∞ (A ) There exists a constant ρ with ρ > p such that  < ρ F(x, u) ≤ f (x, u)u,

f (x,u) ∗ up –

=  uniformly for x ∈ .

∀x ∈ , ∀u ∈ R+ \ {}.

f (x,u) f (x,u) (A ) f ∈ C( × R, R), limu→ |u| p– =  and lim|u|→∞ |u|p∗ – =  uniformly for x ∈ . (A ) There exists a constant ρ with ρ > p such that

 < ρ F(x, u) ≤ f (x, u)u,

∀x ∈ , ∀u ∈ R \ {}.

Let b(μ) be one of zeroes of the function g(t) = (p – )t p – (N – p)t p– + μ, where t ≥  and  ≤ μ < μ . Now, our main results read as follows. Theorem . Suppose that N ≥ ,  ≤ μ < μ ,  ≤ s < p,  < p < N . If (A ), (A ) and N p[N – p – b(μ)p] ρ > max p, , b(μ) N –p

(.)

hold, then problem (.) has at least one positive solution. Theorem . Suppose that N ≥ ,  ≤ μ < μ ,  ≤ s < p,  < p < N . If (A ), (A ) and (.) hold, then problem (.) has at least two distinct nontrivial solutions. Remark . We extend the special case p =  in [] to a more general situation  < p < N . The author [] obtained one positive solution for a special case of (.) with f (x, u) = q– λ |u||x|t u , where λ > ,  ≤ t < p, q < q < p∗ (t) and N – t p[N – t – b(μ)p – p] q = max p, , . b(μ) N –p Note that the function f in [] has to be a homogeneous function, but in the present paper it is not the case. Besides, we also obtain multiple solutions of (.) (see our Theorem .). Remark . We prove Theorems . and . by critical point theory. Due to the lack of ∗ ,p ,p compactness of the embeddings in W () → Lp (), W () → Lp (, |x|–p dx) and ∗ ,p W () → Lp (s) (, |x|–s dx), we cannot use the standard variational argument directly.


Page 3 of 11

The corresponding energy functional fails to satisfy the classical Palais-Smale ((PS) for ,p short) condition in W (). However, a local (PS) condition can be established in a suitable range. Then the existence result is obtained via constructing a minimax level within this range and the mountain pass lemma due to Ambrosetti and Rabinowitz (see also []). Notations For the functional I : X → R (X is a Banach space), we say that I satisfies the classical Palais-Smale ((PS) for short) condition if every sequence {un } in X such that I(un ) is bounded in X and I (un ) →  as n → ∞ contains a convergent subsequence. We say that I satisfies (PS)c condition (a local Palais-Smale condition) if every sequence {un } such that I(un ) → c and I (un ) →  as n → ∞ (c ∈ R) contains a convergent subsequence. The rest of this paper is organized as follows. In Section , we establish some preliminary lemmas, which are useful in the proofs of our main results. In Section , we give detailed proofs of our main results.

2 Preliminaries In what follows, we let · p denote the norm in Lp (). It is obvious that the values of f (x, u) for u <  are irrelevant in Theorem ., and we may define f (x, u) ≡ 

if u ≤ , ∀x ∈ .

We shall firstly consider the existence of nontrivial solutions for the following problem:

p–

–p u – μ |u||x|p u = u = ,

x ∈ ∂.

∗

(u+ )p (s)– |x|s

+ f (x, u),

x ∈ \ {},

(.)

The energy functional corresponding to (.) is given by I(u) =

∗  |u|p (u+ )p (s) |∇u|p – μ p dx – ∗ dx |x| p (s) |x|s ,p – F(x, u) dx, u ∈ W ().

 p

,p

By the Hardy and Hardy-Sobolev inequalities (see [, ]) and (A ), I ∈ C  (W (), R). Now it is well known that there exists a one-to-one correspondence between the weak ,p solutions of problem (.) and the critical points of I on W (). More precisely, we say ,p ,p that u ∈ W () is a weak solution of problem (.) if for any v ∈ W (), there holds

I (u), v =

|u|p– uv p– |∇u| ∇u∇v – μ dx |x|p ∗ (u+ )p (s)– v dx – f (x, u)v dx = . – |x|s

Next, we shall give some lemmas which are needed in proving our main results. Lemma . ([]) If fn → f a.e. in and fn p ≤ C < ∞ for all n and some  < p < ∞, then lim fn pp – fn – f pp = f pp .

n→∞


Page 4 of 11

Lemma . ([]) Suppose  < p < N ,  ≤ s < p and  ≤ μ < μ . Then the limiting problem ⎧ ∗ up– up (s)– ⎪ ⎨ –p u – μ |x|p = |x|s u >  in RN \ {}, ⎪ ⎩ u ∈ D,p (RN )

in RN \ {}, (.)

has radially symmetric ground states ε (x) := ε– V

N–p p

p,μ U

N–p x |x| p,μ = ε– p U , ε ε

∀ε > ,

and satisfies N–s ε (x)|p∗ (s) (x)|p |V |V p–s ∇ V ε (x)p – μ ε dx = dx = Aμ,s , p s |x| |x| RN RN

p,μ (|x|) is the unique radial solution of (.) satisfying p,μ (x) = U where U p,μ () = U

(N – s)(μ – μ) N –p

 p∗ (s)–p

.

p,μ has the following properties: Moreover, U p,μ (r) = c > , lim ra(μ) U

r→

p,μ (r) = c > , lim rb(μ) U

r→+∞

p,μ lim ra(μ)+ U (r) = c a(μ) ≥ ,

r→

p,μ lim rb(μ)+ U (r) = c b(μ) > ,

r→+∞

where c and c are positive constants depending on p and N ; a(μ) and b(μ) are zeroes of < the function g(t) = (p – )t p – (N – p)t p– + μ (t ≥ ,  ≤ μ < μ ) satisfying  ≤ a(μ) < N–p p N–p b(μ) < p– . N–s

p–s p–s Aμ,s ), then I satisfies (PS)c Lemma . Assume that (A ) and (A ) hold. If c ∈ (, p(N–s) condition.

,p

Proof Suppose that {un } is a (PS)c sequence of I in W (), that is, I (un ) → ,

I(un ) → c,

n → ∞.

By (A ), we have c +  + o()un

 I (un ), un θ ∗ (u+n )p (s)      p F(x, un ) – f (x, un )un dx – un + – dx – = p θ θ p∗ (s) |x|s θ   ≥ – un p , p θ ≥ I(un ) –


Page 5 of 11

where θ = min{ρ , p∗ (s)}. Hence we conclude that {un } is a bounded sequence in W (). ,p So there exists u ∈ W () such that (going if necessary to a subsequence) ,p

un → u in Lγ (),  < γ < p∗

,p

un u

in W (),

un → u

a.e. in as n → ∞.

and

p∗

By the continuity of embedding, we have un p∗ ≤ C < ∞. Going if necessary to a subsequence, from [] one can get that ⎧ ⎪ ⎪ ∇un → ∇u a.e. in , ⎨ p– u

p |un |p– un |u| weakly in Lp (), p = p– , p– p– |x| |x| ⎪ ∗ (s)– ∗ (s)– p p ⎪ ,p un u |u| ⎩ |un | v dx → v dx, ∀v ∈ W ()

|x|s

|x|s

(.)



as n → ∞. By (A ), for any ε > , there exists a(ε) >  such that f (x, t)t ≤ a(ε) +  ε|t|p∗ C

for (x, t) ∈ × (, +∞). p∗

ε ε Set δ := a(ε) > . Let E ⊂ with meas(E) < δ = a(ε) , it follows from the fact un p∗ ≤ C that ∗ f (x, un )un dx ≤ a(ε) dx + ε |un |p < ε →  as meas(E) → . C E



E

E

It follows from the fact that f (x, un )un → f (x, u)u as n → ∞ a.e. in and Vitali’s theorem that f (x, un )un dx → f (x, u)u dx as n → ∞.

Similarly, we can also get

F(x, un ) dx →

F(x, u) dx

as n → ∞.

Let vn = un – u. By the definition of · , we get up = ∇upp – μu/xpp ,

p vn p = ∇un – ∇upp – μ(un – u)/xp ,

it follows from un → u a.e. in , ∇un → ∇u a.e. in (see (.)) and Lemma . that lim un p – vn p = up .

n→∞

It follows from I (un ) → , [] that

f (x, un )un dx

→

f (x, u)u dx

 = lim I (un ), un n→∞

∗ (u+n )p (s) = lim un p – dx – f (x, u )u dx n n n→∞ |x|s

and the Brezis-Lieb lemma


Page 6 of 11

∗ ∗ (u+n )p (s) – (v+n )p (s) = lim un p – vn p – dx n→∞ |x|s ∗ (v+n )p (s) dx – f (x, un )un dx + vn p – |x|s ∗ ∗ (u+ )p (s) (v+n )p (s) p = up – dx – f (x, u)u dx + lim – dx . v n n→∞ |x|s |x|s

(.)

From (.) and I (un ) → , we get

 = lim I (un ), u = up –

n→∞

∗

(u+ )p (s) dx – |x|s

f (x, u)u dx.

(.)

By I(un ) → c, F(x, un ) dx → F(x, u) dx, limn→∞ (un p – vn p ) = up and the BrezisLieb lemma, we have c = lim I(un ) n→∞

∗    (v+n )p (s) vn p + up – ∗ dx n→∞ p p p (s) |x|s ∗  (u+ )p (s) dx – F(x, un ) dx – ∗ p (s) |x|s ∗  (v+n )p (s)  p = I(u) + lim vn – ∗ dx . n→∞ p p (s) |x|s

= lim

That is, I(u) + lim

n→∞

  vn p – ∗ p p (s)

∗ (v+n )p (s) dx = c. |x|s

(.)

Obviously, (.) and (.) imply ∗ (v+n )p (s) dx = . lim vn p – n→∞ |x|s We claim that vn p →  as n → ∞. Otherwise, there exists a subsequence (still denoted by vn ) such that lim vn p = k,

lim

n→∞

n→∞

∗

(v+n )p (s) dx = k, |x|s

k > .

(.)

By the definition of Aμ,s in (.), we have vn p ≥ Aμ,s

∗

(v+n )p (s) dx |x|s

p p∗ (s)

, p

∀n ∈ N. N–s p–s

It follows from (.) that k ≥ Aμ,s k p∗ (s) , so we have k ≥ Aμ,s , which together with (.) N–s

and c
p in (A )

≥ . So we get a contradiction. Therefore, we can obtain vn p →  as n → ∞.

From the discussion above, I satisfies (PS)c condition.

In the following, we shall give some estimates for the extremal functions. Define a function ϕ ∈ C∞ () such that ϕ(x) =  for |x| ≤ R, ϕ(x) =  for |x| ≥ R,  ≤ ϕ(x) ≤ , where ε (x), ε > , where V ε (x) see the definition in Lemma .. Then BR () ⊂ . Set vε (x) = ϕ(x)V we can get the following results by the method used in []: N–s p–s vε p = Aμ,s + O εb(μ)p+p–N , ∗ N–s |vε |p (s) ∗ p–s dx = Aμ,s + O εb(μ)p (s)+s–N s |x|

(.) (.)

and ⎧ N N N

r(b(μ)+– p ) ⎪ ≤ |vε |r dx ≤ Cεr(b(μ)+– p ) , r < b(μ) , ⎪ ⎨C ε N N N+r(– p )

N+r(– p ) r | ln ε| ≤ |vε | dx ≤ Cε | ln ε|, r = Cε ⎪ ⎪ ⎩ N+r(– Np ) N+r(– N ) N r p , Cε ≤ |vε | dx ≤ Cε r > b(μ) .

N , b(μ)

(.)

Lemma . Suppose that  ≤ s < p and  ≤ μ < μ . If (A ), (A ) and (.) hold, then there ,p exists u ∈ W () with u ≡  such that sup I(tu ) < t≥

N–s p–s p–s Aμ,s . p(N – s)

Proof We consider the functions ∗ ∗ |vε |p (s) tp t p (s) vε p – ∗ dx – F(x, tvε ) dx, p p (s) |x|s ∗ ∗ tp t p (s) |vε |p (s) g(t) = vε p – ∗ dx. p p (s) |x|s g(t) = I(tvε ) =


Page 8 of 11

Since limt→∞ g(t) = –∞, g() =  and g(t) >  for t small enough, supt≥ g(t) is attained for some tε > . Therefore, we have =g

(tε ) = tεp–

∗  |vε |p (s) p p∗ (s)–p dx – p– f (x, tε vε )vε dx , vε – tε |x|s tε

hence ∗ (s)–p

∗

vε p = tεp

|vε |p (s)  dx + p– s |x| tε

∗ (s)–p

∗

f (x, tε vε )vε dx ≥ tεp

|vε |p (s) dx. |x|s

It follows from (.) and (.) that tε ≤ C for ε small enough. From the above inequality, we obtain ∗ ∗ p (s)–p |vε |p (s) p dx := tε . tε ≤ vε s |x| By (A ), we can easily get f (x, t) ≤ εt p∗ – + d(ε)t p–

for some d(ε) > .

Hence, together with tε ≤ C, we can get vε

p

∗ ≤ tεp (s)–p

∗

|vε |p (s) dx + εC |x|s

p∗

|vε | dx + d(ε)

|vε |p dx.

By (.)-(.), when ε is small enough, we conclude that ∗ (s)–p

tεp

 ≥ . 

(.)

On the other hand, the function g(t) attains its maximum at tε and is increasing in the interval [, tε ], together with (.), (.) and (.) and F(x, t) ≥ C |t|ρ which is directly got from (A ), we deduce g(tε ) ≤ g tε –

F(x, tε vε ) dx

p(N–s) p–s = vε p–s p(N – s)

=

∗

|vε |p (s) dx |x|s

– N–p p–s

–

F(x, tε vε ) dx

N–s N–s – N–p p – s N–s ∗ p–s p–s p–s Aμ,s + O εb(μ)p+p–N p–s Aμ,s + O εb(μ)p (s)+s–N p(N – s) – F(x, tε vε ) dx

N–s p–s ∗ p–s + O εb(μ)p+p–N + O εb(μ)p (s)+s–N – = Aμ,s p(N – s)

≤

F(x, tε vε ) dx

N–s p–s ∗ p–s Aμ,s + O εb(μ)p+p–N + O εb(μ)p (s)+s–N – C p(N – s)

|vε |ρ dx.


Page 9 of 11

Furthermore, from (.) and (.), we get

N

|vε |ρ dx ≥ C εN+ρ(– p ) .

Note that b(μ) >

N–p p

implies

p[N – p – b(μ)p] p[N – s – p∗ (s)b(μ)] > . N –p N –p By (.), we have ρ >

p[N–p–b(μ)p] , N–p

which implies

N b(μ)p + p – N > N + ρ  – p and N . b(μ)p∗ (s) + s – N > N + ρ  – p Therefore, by choosing ε small enough, we have sup I(tvε ) = g(tε ) < t≥


Hence the proof of this lemma is then completed by taking u = vε .

3 Proofs of our main results Proof of Theorem . From the Sobolev and Hardy-Sobolev inequalities, we can easily get upp

≤ Cu ,

p∗ up∗

p

p∗

≤ Cu ,

∗

|u|p (s) ∗ dx ≤ Cup (s) , |x|s

(.)

,p ∀u ∈ W ().

The condition (A ) implies that for any ε >  there exist δ > δ >  such that f (x, u) < εup– ,

∀u ∈ (, δ )

f (x, u) < up∗ – ,

∀u > δ .

and

Therefore, there exists a constant Cε >  such that f (x, u) ≤ εup– + Cε up∗ – ,

∀u ∈ R+ , ∀x ∈ .

Then one gets F(x, u) ≤  ε|u|p + C |u|p∗ , p

∀u ∈ R+ , ∀x ∈ .

(.)


Page 10 of 11

By (.) and (.), we have I(u) = ≥

  up – ∗ p p (s)

∗

(u+ )p (s) dx – |x|s

F(x, u) dx

C C  ∗ ∗ up – ∗ u+ p (s) – εup – CC up p p (s) p

for ε >  small enough. So there exists β >  such that I(u) ≥ β

,p for all u ∈ ∂Br = u ∈ W (), u = r , r >  small enough. ,p

By Lemma ., there exists u ∈ W () with u ≡  such that N–s p–s p–s Aμ,s . p(N – s)

sup I(tu ) < t≥

It follows from the nonnegativity of F(x, t) that   ∗ I(tu ) = t p u p – ∗ t p (s) p p (s)   ∗ ≤ t p u p – ∗ t p (s) p p (s)

∗

(u+ )p (s) dx – |x|s

(u+ )p (s) dx, |x|s

F(x, tu ) dx

∗

therefore, limt→+∞ I(tu ) → –∞. Hence, we can choose t >  such that t u > r

and

I(t u ) ≤ . ,p

By virtue of the mountain pass lemma in [], there is a sequence {un } ⊂ W () satisfying I(un ) → c ≥ β

and I (un ) → ,

where c = inf max I h(t) h∈ t∈[,]

and ,p = h ∈ C [, ], W () | h() = , h() = t u . Note that  < β ≤ c = inf max I h(t) ≤ max I(tt u ) ≤ sup I(tu ) < h∈ t∈[,]

t∈[,]

t≥


By Lemma ., we can assume that un → u in W (). From the continuity of I , we know that u is a weak solution of problem (.). Then I (u), u– = , where u– = min{u, }. Thus u ≥ . Therefore u is a nonnegative solution of (.). By the strong maximum principle, u is a positive solution of problem (.), so Theorem . holds. ,p


Page 11 of 11

Proof of Theorem . By Theorem ., problem (.) has a positive solution u . Set g(x, t) = –f (x, –t) for t ∈ R. It follows from Theorem . that the equation ∗

–p u – μ

|u|p– u |u+ |p (s)– = u + g(x, u) |x|p |x|s

has at least one positive solution v. Let u = –v, then u is a solution of ∗

–p u – μ

|u|p– u |u+ |p (s)– = u + f (x, u). |x|p |x|s

It is obvious that u = , u =  and u = u . So problem (.) has at least two nontrivial solutions. Therefore, Theorem . holds. Competing interests The author declares that he has no competing interests. Acknowledgements The author thanks the referees and the editors for their helpful comments and suggestions. Research was supported by the National Natural Science Foundation of China (No. 11401011). Received: 27 May 2015 Accepted: 12 September 2015 References 1. Ghoussoub, N, Yuan, C: Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents. Trans. Am. Math. Soc. 352, 5703-5743 (2000) 2. Filippucci, R, Pucci, P, Robert, F: On a p-Laplace equation with multiple critical nonlinearities. J. Math. Pures Appl. 91, 156-177 (2009) 3. Ambrosetti, A, Brezis, H, Cerami, G: Combined effects of concave and convex nonlinearities in some elliptic problems. J. Funct. Anal. 122, 519-543 (1994) 4. Abdellaoui, B, Peral, I: Some results for semilinear elliptic equations with critical potential. Proc. R. Soc. Edinb., Sect. A 132(1), 1-24 (2002) 5. Cao, DM, Han, PG: Solutions for semilinear elliptic equations with critical exponents and Hardy potential. J. Differ. Equ. 205(2), 521-537 (2004) 6. Ferrero, A, Gazzola, F: Existence of solutions for singular critical growth semi-linear elliptic equations. J. Differ. Equ. 177(2), 494-522 (2001) 7. Garcia Azorero, JP, Peral Alonso, I: Hardy inequalities and some critical elliptic and parabolic problems. J. Differ. Equ. 144(2), 441-476 (1998) 8. Terracini, S: On positive entire solutions to a class of equations with a singular coefficient and critical exponent. Adv. Differ. Equ. 1(2), 241-264 (1996) 9. Ghoussoub, N, Kang, XS: Hardy-Sobolev critical elliptic equations with boundary singularities. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 21(6), 767-793 (2004) 10. Kang, DS, Peng, SJ: Positive solutions for singular critical elliptic problems. Appl. Math. Lett. 17(4), 411-416 (2004) 11. Kang, DS, Peng, SJ: Solutions for semi-linear elliptic problems with critical Sobolev-Hardy exponents and Hardy potential. Appl. Math. Lett. 18(10), 1094-1100 (2005) 12. Shen, ZF, Yang, MB: Nontrivial solutions for Hardy-Sobolev critical elliptic equations. Acta Math. Sin. 48(5), 999-1010 (2005) (in Chinese) 13. Ding, L, Tang, CL: Existence and multiplicity of solutions for semilinear elliptic equations with Hardy terms and Hardy-Sobolev critical exponents. Appl. Math. Lett. 20, 1175-1183 (2007) 14. Kang, DS: On the quasilinear elliptic problems with critical Sobolev-Hardy exponents and Hardy terms. Nonlinear Anal. 68, 1973-1985 (2008) 15. Rabinowitz, PH: Minimax Methods in Critical Point Theory with Applications to Differential Equations. CBMS Reg. Conf. Series Math., vol. 65. Am. Math. Soc., Providence (1986) 16. Caffarelli, L, Kohn, R, Nirenberg, L: First order interpolation inequality with weights. Compos. Math. 53, 259-275 (1984) 17. Brezis, H, Lieb, E: A relation between pointwise convergence of functions and convergence of functionals. Proc. Am. Math. Soc. 88, 486-490 (1983) 18. Kang, D, Huang, Y, Liu, S: Asymptotic estimates on the extremal functions of a quasilinear elliptic problem. J. South Cent. Univ. Natl. 27(3), 91-95 (2008) 19. Struwe, M: Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems. Springer, Berlin (1996)