On Hamiltonian elliptic systems with exponential growth in dimension two. Yony Raúl Santaria Leuyacc

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1 On Hamiltonian elliptic systems with exponential growth in dimension two Yony Raúl Santaria Leuyacc

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3 SERVIÇO DE PÓS-GRADUAÇÃO DO ICMC-USP Data de Depósito: Assinatura: Yony Raúl Santaria Leuyacc On Hamiltonian elliptic systems with exponential growth in dimension two Doctoral dissertation submitted to the Instituto de Ciências Matemáticas e de Computação ICMC- USP, in partial fulfillment of the requirements for the degree of the Doctorate Program in Mathematics. FINAL VERSION Concentration Area: Mathematics Advisor: Prof. Dr. Sérgio Henrique Monari Soares USP São Carlos June 17

4 Ficha catalográfica elaborada pela Biblioteca Prof. Achille Bassi e Seção Técnica de Informática, ICMC/USP, com os dados fornecidos peloa) autora) L65o Leuyacc, Yony Raúl Santaria On Hamiltonian elliptic systems with exponential growth in dimension two / Yony Raúl Santaria Leuyacc ; orientador Sérgio Henrique Monari Soares. -- São Carlos, p. Tese Doutorado - Programa de Pós-Graduação em Matemática) -- Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, Hamiltonian systems.. Exponential growth. 3. Variational methods. 4. Trudinger-Moser inequality. 5. Lorentz-Sobolev spaces. I., Sérgio Henrique Monari Soares, orient. II. Título.

5 Yony Raúl Santaria Leuyacc Sistemas elípticos hamiltonianos com crescimento exponencial em dimensão dois Tese apresentada ao Instituto de Ciências Matemáticas e de Computação ICMC-USP, como parte dos requisitos para obtenção do título de Doutor em Ciências Matemática. VERSÃO REVISADA Área de Concentração: Matemática Orientador: Prof. Dr. Sérgio Henrique Monari Soares USP São Carlos Junho de 17

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7 To my dear family.

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9 ACKNOWLEDGEMENTS I want to start by expressing my sincerest gratitude to my advisor Sérgio Monari, for all his outsting supervision, valuable advice great guidance. I feel very fortunate to have worked with an advisor who was so involved with my research. I also have to thank the members of my PhD committee, Professors Raquel Lehrer, Jefferson Abrantes, Ederson Moreira dos Santos for their helpful feedback suggestions in general. I must express my very profound appreciation to all the people who provided me support continuous encouragement. I wish to thank all the Brazilians for their generosity giving me such a comfortable place to stay. I have had a wonderful time in this lovely country. I would also like to show gratitude to my friends in Philippines. I would like to thank CAPES, for the financial support.

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11 Take what you need, do what you should, you will get what you want. Gottfried Leibniz)

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13 RESUMO LEUYACC, R. Y. S. Sistemas elípticos hamiltonianos com crescimento exponencial em dimensão dois p. Doctoral dissertation Doctorate Cidate Program in Mathematics) Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, São Carlos SP, 17. Neste trabalho estudamos a existência de soluções fracas não triviais para sistemas hamiltonianos do tipo elíptico, em dimensão dois, envolvendo uma função potencial e não linearidades tendo crescimento exponencial máximo com respeito a uma curva hipérbole) crítica. Consideramos quatro casos diferentes. Primeiramente estudamos sistemas de equações em domínios limitados com potencial nulo. No segundo caso, consideramos sistemas de equações em domínio ilimitado, sendo a função potencial limitada inferiormente por alguma constante positiva e satisfazendo algumas de integrabilidade, enquanto as não linearidades contêm funções-peso tendo uma singularidade na origem. A classe seguinte envolve potenciais coercivos e não linearidades com funções peso que podem ter singularidade na origem ou decaimento no infinito. O quarto caso é dedicado ao estudo de sistemas em que o potencial pode ser ilimitado ou decair a zero no infinito. Para estabelecer a existência de soluções, utilizamos métodos variacionais combinados com desigualdades do tipo Trudinger-Moser em espaços de Lorentz-Sobolev e a técnica de aproximação em dimensão finita Palavras-chave: Sistemas hamiltonianos, Crescimento exponencial, Métodos variacionais, Desigualdade de Trudinger-Moser, Espaços de Lorentz-Sobolev.

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15 ABSTRACT LEUYACC, R. Y. S. On Hamiltonian elliptic systems with exponential growth in dimension two p. Doctoral dissertation Doctorate Cidate Program in Mathematics) Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, São Carlos SP, 17. In this work we study the existence of nontrivial weak solutions for some Hamiltonian elliptic systems in dimension two, involving a potential function nonlinearities which possess maximal growth with respect to a critical curve hyperbola). We consider four different cases. First, we study Hamiltonian systems in bounded domains with potential function identically zero. The second case deals with systems of equations on the whole space, the potential function is bounded from below for some positive constant satisfies some integrability conditions, while the nonlinearities involve weight functions containing a singulatity at the origin. In the third case, we consider systems with coercivity potential functions nonlinearities with weight functions which may have singularity at the origin or decay at infinity. In the last case, we study Hamiltonian systems, where the potential can be unbounded or can vanish at infinity. To establish the existence of solutions, we use variational methods combined with Trudinger-Moser type inequalities for Lorentz-Sobolev spaces a finite-dimensional approximation. Keywords: Hamiltonian systems, Exponential growth, Variational methods, Trudinger-Moser inequality, Lorentz-Sobolev spaces.

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17 LIST OF SYMBOLS A B A is a subset of B A B A is a proper subset of B. X\A The the complement of a set A X χ E The the characteristic function of the set E. R The set of real numbers. R + {x R : x }. N The set of natural numbers. R N The Euclidean N-space. e i The vector,...,,1,,...,) with 1 in the i-th entry elsewhere. [x] The integer part of the real number x. B R x) The ball of radius R centered at x in R N. x x x n when x = x 1,...,x n ) R N. S N The unit sphere {x R N : x = 1}. ω N 1 The surface area of the unit sphere S N 1. The Lebesgue measure of the set R N X, µ) Measure space M X,R) The collection of all extended real-valued µ-measurable functions on X M X,R) Class of functions in M X,R) that are finite µ-almost everywhere in X L p X, µ) The Lebesque space over the measuare X, µ) L p R N ) The space L p R N, ). L p loc RN ) The space of functios that lie in L p K) for any compact set K in R N. supp f Support of a function f. f The decreasing rearrangement of a function f. f n f The sequence f n increases monotonically to a function f. f n f The sequence f n decreases monotonically to a function f.

18 f = Og) Means f x) M gx) for some M for x near x. f = og) Means f x) gx) 1 as x x. f n = o n 1) Means f n as n +. α indicates the size α α N of a multi-index α = α 1,...,α N ). m i f The m-th partial derivative of f x 1,...,x N ) with respect to x i. α f α 1 1 α N N f. C k The space of functions f with α f continuous for all α k. C The space of continuous functions with compact support C The space of smooth functions k 1 C k. C The space of smooth functions with compact support.

19 CONTENTS 1 INTRODUCTION LORENTZ AND LORENTZ-SOBOLEV SPACES Distribution functions decreasing rearrangement Lorentz spaces Lorentz-Sobolev spaces Lorentz-Sobolev spaces in R The tilde-map HAMILTONIAN SYSTEM WITH CRITICAL EXPONENTIAL GROWTH IN A BOUNDED DOMAIN Introduction Variational setting The geometry of the linking theorem Finite-dimensional approximation Proof of Theorem SINGULAR HAMILTONIAN SYSTEM WITH CRITICAL EXPO- NENTIAL GROWTH IN R Introduction main results Preliminary results The concentrating hole functions Variational setting On Palais-Smale sequences Theorem The geometry of the Linking theorem Approximation finite dimensional Estimate of the minimax level Proof of Theorem Theorem The geometry of the Linking theorem Finite-dimensional approximation Proof of Theorem

20 5 HAMILTONIAN SYSTEMS WITH CRITICAL EXPONENTIAL GROWTH AND COERCIVE POTENTIALS Introduction main result Preliminaries A Trudinger-Moser type inequality Variational setting On Palais-Smale sequences Linking geometry Approximation finite dimensional Estimate of the minimax level Proof of Theorem HAMILTONIAN SYSTEMS WITH POTENTIALS WHICH CAN VANISH AT INFINITY Introduction Preliminaries The auxiliary functional The geometry of the linking theorem Estimates Finite-dimensional approximation Proof of Theorem BIBLIOGRAPHY

21 19 CHAPTER 1 INTRODUCTION In recent years, many authors have considered the existence of nontrivial solutions for Hamiltonian systems of the form { u +V x)u = H v x,u,v), x, v +V x)v = H u x,u,v), x, 1.1) where is a smooth domain in R N, N Hx,u,v) is a nonlinear function. Hamiltonian systems have been widely use in applied sciences, mainly in the mathematical study of sting wave solutions in models in population dynamics Murray 1993)), in nonlinear optics Bulgan et al. 4), Christodoulides et al. 1)) in the study of Bose-Einstein condensates Chang et al. 4)). In dimension N 3, the simplest example of 1.1) is H v x,u,v) = gv), H u x,u,v) = f u), gv) v p f u) u q ). Even for this case, relevant open questions still persist see Bonheure, Santos Tavares 14)). In order to suppose that this systems is in variational form, that is 1.1) is the Euler-Lagrange equation of some functional defined on a suitable product of Sobolev type spaces, the couple p,q) lies on or below the critical hyperbola see Figueiredo Felmer 1994), Hulshof Vorst 1993), Mitidieri 1993)): 1 p q + 1 N N. 1.) In dimension N = one sees that the critical hyperbola is not defined. More precisely, Let R N be a domain of finite measure. The classical Sobolev space embeddings say that W 1, ) L q ) for all 1 q N/N ). In the limiting case N = we have q = +, but easy examples show that W 1, ) L ), in particular from that any polynomial growth for f g is admitted. Thus, one is lead to ask if there is another kind of maximal growth in this situation. The answer was obtained independently by Pohozaev 1964) Trudinger 1967), it states that e αu L 1 ) for all u H 1 ) α >. Furthermore, Moser 197/71) showed

22 Chapter 1. Introduction that there exists a positive constant C = Cα,) such that C, α 4π, sup e αu dx u H 1) = +, α > 4π. u 1 1.3) Estimate 1.3) from now on will referred to as Trudinger-Moser inequality, similar results wwere obtained for = R see Cao 199), Ruf 5)). A singular type extension of inequality 1.3) for bounded domains was given by Adimurthi Seep 7) its version in the whole space R N was obtained by Adimurthi Yang 1). They showed that there exists a positive constant C = Cα, β, N) such that 1 sup u H 1 R N ) R N x β e α u N/N 1) u + u 1 where α N = Nω 1/N N )N/N 1). N k= α k u kn/n 1) ) C, α 1 β/n)α N dx k! = +, α > 1 β/n)α N, 1.4) In dimension two, inequalities 1.3) 1.4) show that, if the setting space of the system 1.1) is given by H 1) H1 ) the maximal growth of the functions f g can be consider such as gv) e v f u) e u. An important point is the fact that Trudinger-Moser type inequalities can be sharpened using Lorentz-Sobolev spaces. First, we recall the Lorentz spaces: for a measurable function u : R, u denote its decreasing rearrangement. Then, u belongs to the Lorentz space L p,q ) p,q > 1) if u p,q = + [ u t)t 1/p] q dt ) 1/q < +. t These spaces represent an extension of the Lebesgue spaces, in particular when p = q we have L p,p ) = L p ). Using these spaces we can define the Lorentz-Sobolev spaces, roughly speaking we say that u belongs to the Lorentz-Sobolev space W 1 Lp,q ) if u its weak derivatives belongs to L p,q ). Using Lorentz-Sobolev spaces, Brézis Wainger 198) showed : If be a bounded domain in R s > 1, then, e u s 1 s belongs to L 1 ) for all u W 1L,s ). Furthermore, Alvino, Ferone Trombetti 1996) obtained the following refinement of 1.3), there exists a positive constant C = C,s,α) such that sup e α u s u W 1L,s ) u,s 1 s 1 dx C, α 4π) s/s 1), = +, α > 4π) s/s 1). 1.5) As it was showed in Ruf 6), if the setting space of the system 1.1) is given by the product space W 1 L,q ) W 1 L,p ) the maximal growth of the nonlinearities can be

23 1 considered like f u) e u p gv) e v q with p,q > 1 satisfying 1 p + 1 = ) q Trudinger-Moser inequalities in the case = R were studied by Cassani Tarsi 9) with some natural modifications. Recently Lu Tang 16) obtained the following result which represents an extension of 1.4) in Lorentz-Sobolev spaces: Let 1 < s < +, β < N. Then, there exists a positive constant C = CN,s,β) such that Φα u s/s 1) ) C, α 1 β/n)α N,s, sup W 1 L N,s R N ) R N x β dx = +, α > 1 β/n)α N,s, u s N,s + u s N,s 1 where Φt) = e t k k= t k k!, k = [ s 1)N ] s α N,s = Nω 1/N N )s/s 1). 1.7) In dimension two the last inequality allows us toconsider the nonlinearities of the system 1.1) such as gx,v) e v p / x a f x,u) e u q / x b with a,b [,) p,q) belonging to 1.6). Finally, we illustrate the content of each chapter of this thesis. In Chapter, we show important properties which will be used in the chapters 3,4 5. We start introducing some basic concepts about distribution decreasing rearrangement of a function in order to define Lorentz spaces, which represent a generalization of L p -spaces. Furthermore, with the help of these spaces we can construct Lorentz-Sobolev spaces as generalization of Sobolev spaces. Finally, following Figueiredo, Ó Ruf 5), Ruf 8) we define an application called tilde-map which is very useful in the variational formulation of the systems which will be presented in the next chapters. In Chapter 3, we study the existence of nontrivial weak solution to the following Hamiltonian elliptic system u = gv), in, v = f u), in, u = v =, on, 1.8) where is a smooth bounded domain in R the nonlinearities f g possess maximal growth which allows us to treat the system 1.8) variationally in the cartesian product of Lorentz- Sobolev spaces. In Ruf 8) it was shown the existence of nontrivial solution of the system 1.8) in the case where f u) e u p gv) e v q where p, q > such that 1/ p +1/ q > 1. In this case, we can obtain p,q) belongs to the hyperbola 1.6) such that f s) gs) lim = lim =, for all α >. 1.9) s e α s p s e α s q

24 Chapter 1. Introduction The existence of solutions for the system 1.8) when f u) e u p gv) e v p has been solved for the case p = q = in Figueiredo, Ó Ruf 4). Our main result in this chapter is to prove the existence of nontrivial weak solutions for the general case, that is p,q) satisfies 1.6). Motivated by the above results, we call the curve 1.6) as exponential critical hyperbola in analogy to 1.) in the sense that for p,q) belongs to this hyperbola gives the maximal growth range the solutions is proved when p,q) lies on or below to 1.6). Therefore, from this results we have naturally associated notions of criticality subcriticality, namely: Given p > 1, we say that a function f has p-subcritical exponential growth, if f satisfies condition 1.9), whereas a function f has p-critical exponential growth, if there exists α > such that f s) lim = s e α s p, α > α, +, α < α. In order to study the existence of solutions of the system 1.8) we are going to impose the following conditions: A 1 ) f g are continuous functions, with f s) = gs) = os) near the origin. A ) There exist constants µ >, ν > s > such that < µfs) s f s), < νgs) sgs), for all s > s. where Fs) = s f t) dt Gs) = s gt) dt. A 3 ) There exist α > p > 1, such that A 4 ) There exists β >, such that where q = p p 1. f s) lim = s e α s p gs) lim = s e β s q A 5 ) There exist constants θ > C θ > such that, α > α, +, α < α., β > β, +, β < β. Fs) C θ s θ Gs) C θ s θ, for all s R, where C θ > { 5 δ θ R θ, π µ R = α 1/p β 1/q max µ, ν } ν δ θ is a positive constant which will be explicit later on.

25 3 Now we state the main result of chapter 3. Theorem 1.1. Suppose A 1 ) A 5 ) hold. Then, the system 1.8) possesses a nontrivial weak solution. Note that the above theorem permits to work with p,q) lying in the exponential critical hyperbola thanks to assumptions A 3 ) A 4 ). Consequently, this result completes the study made in Figueiredo, Ó Ruf 4) which corresponds to the diagonal case p = q =. We point out the condition A 5 ) will be crucial in our proof, this condition is of type as considered in many works see Cao 199) the references therein). We remark that from the choose of C θ we do not need the following usual assumption: A ) There exist positive constants M s such that < Fs) M f s) < Gs) M gs), for all s > s. which is used to get some convergence results. Since the system 1.8) is a special case of a Hamiltonian system, some difficulties appear; for example, the associated functional is strongly indefinite, that is, its leading part is respectively coercive anti-coercive on infinite-dimensional subspaces of the energy space. To overcome these difficulties, we will use a finite-dimensional approximation combine with the Linking theorem. In Chapter 4, we study the following singular Hamiltonian system: u +V x)u = gv) x a, x R, v +V x)v = f u) x b, x R, 1.1) where a,b [,) the functions f g possess critical exponential growth. This system is motivated by inequality 1.7). In order to have properties like embedding theorems we consider that V is a continuous potential verifying the following conditions: V 1 ) There exists a positive constant V such that V x) V for all x R. V ) There exist constants p > q = p/p 1) such that 1 V 1/q L,p R ) 1 V 1/p L,q R ). System 1.1) was studied by Souza 1) in the case where p = q = its solution was found in H 1 R ) H 1 R ), for this case the author use the respective assumption instead of

26 4 Chapter 1. Introduction V ), that is 1/V L 1 R ) similar conditions on the functions f g as A ) A 4 ) given above. Moreover, it is considered a following condition: there exist θ > a positive constant C θ sufficiently large such that f t) C θ t θ 1 gt) C θ t θ 1, for all t. 1.11) Cassani Tarsi 15) proved the existence of nontrivial solutions of the system 1.1) in the case where a = b =. The authors have assumed V 1 ) V ) on V A ) A 4 ) on the nonlinearities. Furthermore, in order to estimate the minimax level it was considered the following conditions: lim t f t + t)e α t p = lim t + tgt)e β t q = + α 1/p β 1/q. 1.1) Motivated by these results, we will prove the existence of nontrivial weak solution of 1.1) in two different ways, that means, in addition to A ) A 4 ) we will adapt the conditions A 5 ) 1.1) we use each one independently in the proofs. More precisely, we describe the following additional conditions on the functions f g. A 6 ) The following limits holds s f s) sgs) lim s + e α = + lim s p s + e β = +. s q A 7 ) For a,b given by 1.1), p,q given by V ), α β given by A 3 ) A 4 ) respectively, it satisfies α ) 1/p β ) 1/q. 1 b/ 1 a/ A 8 ) Let a,b [,) given by 1.1). Then, there exist θ > a positive constant C θ,a,b such that where Fs) C θ,a,b s θ Gs) C θ,a,b s θ, for all s R, C θ,a,b > δ θ,a,b R θ, R = 4π1 b/)1/p 1 a/) 1/q α 1/p β 1/p whereas the constant δ θ,a,b will be explicit later on. { µ max µ, ν }, ν The following theorems contains our main results in Chapter 4. Theorem 1.. Suppose that V satisfies V 1 ) V ) f g satisfy A ) A 4 ) A 6 ) A 7 ). Then, the system 1.1) possesses a nontrivial weak solution.

27 5 Theorem 1.3. Suppose that V satisfies V 1 ) V ) f g satisfy A 1 ) A 4 ) A 8 ). Then, the system 1.1) possesses a nontrivial weak solution. We remark that the class of functions which satisfy the hypotheses of the above theorems are different. The conclusion of Theorems extends the result given in Cassani Tarsi 15) in the sense that we add the singularities x a x b on the nonlinearities considered in that paper. Moreover, our result complements the study made in Souza 1) in the sense that, in this work, we study the class of Hamiltonian systems where the nonlinearities possess maximal growth with respect to the exponential critical hyperbola. Our proof of Theorems is based on variational methods a finite dimensional approximation. In Chapter 5, we discuss the existence of nontrivial solutions for the Hamiltonian system { u +V x)u = Q x)gv), x R, v +V x)v = Q 1 x) f u), x R 1.13), where V,Q 1,Q are continuous functions the nonlinearities f g possess critical exponential growth with p,q) lying on the exponential critical hyperbola. On the potential V we assume the following condition: V ) V C R,R), V x) V > for all x R, there exists a such that V x) lim inf x x a >. Assumption V ) implies that, if a > the potential V is coercive. On the functions Q i for i = 1,, we consider: Q i ) Q i C R \{},R), Q i x) > for x there exist d i < a/max{p,q} 1) 1 b i > such that Q i x) Q i x) < lim x x b < + lim sup i x x d < +. i The existence of solutions of system 1.13) was studied in Cassani Tarsi 15) for the case Q 1 x) = Q x) 1. The case Q 1 x) = x a Q x) = x b with a,b [,) was treated in Souza 1) for the diagonal case p = q = also considered in Chapter 4 when p,q) belongs to the exponential critical hyperbola. In this section we treat a more general class of nonlinearities studied in previously mentioned papers. We also mention that the systems studied in Cassani Tarsi 15), Souza 1) also in Chapter 4 the potential V satisfy some integrability conditions. In our case, we consider coercive potentials which represent a different class of potential from the mentioned works.

28 6 Chapter 1. Introduction On assumption V ) for s = p or s = q, we consider the following weighted Lorentz- Sobolev space W 1 L,s V R ) which is defined to be the closure of compactly supported smooth functions, with respect to the quasinorm u W 1 L,s V R ) := u s,s + V 1/s u s ) 1/s.,s For any λ 1 i = 1, we also define L λ R,Q i ) := {u : Q ix) u λ dx < + }, R endowed with the norm u L λ R,Q i ) := R Q ix) u λ dx) 1/λ. In these spaces we obtain the next result which will be proved later. Proposition 1.4. Assume V ) Q i ) for i = 1, let s = q or s = p. Then, the following embeddings are compact W 1 L,s V R ) L λ R,Q i ), for all λ min{p,q}. Concerning the functions f g we suppose the following assumptions: B 1 ) f,g C R), f s) = os η 1) gs) = os η ), as s, where η 1 = max{1/q 1),min{p,q}} η = max{1/p 1),min{p,q}}. B ) There exist constants µ > ν > such that < µfs) s f s), < νgs) sgs), for all s, where Fs) = s f t) dt Gs) = s gt) dt. B 3 ) There exist positive constants M s such that < Fs) M f s) < Gs) M gs), for all s > s. B 4 ) There exists α > such that B 5 ) There exists β > such that f s) lim = s e α s p gs) lim = s e β s q +, α < α, α > α. +, β < β, β > β.

29 7 B 6 ) The following limits holds s f s) sgs) lim s + e α = + lim s p s + e β = +. s q B 7 ) For b i given by Q i ), i = 1, α, β given by B 4 ) B 5 ) respectively, it satisfies or α min{1,1 + b b 1 ) α min{1,1 + b 1 } } ) 1/p > β min{1,1 + b } ) 1/p < β min{1,1 + b 1 + b ) ) 1/q }) 1/q. Theorem 1.5. Suppose that V satisfies V ), Q i satisfy Q i ) for i = 1, the nonlinearities f g satisfy B 1 ) B 7 ). Then, the system 1.13) possesses a nontrivial weak solution. In Costa 1994) was studied the existence of solutions for gradient elliptic systems involving coercive potentials in dimension N 3 where the growth of the nonlinearities were of polinomial type. In our case we study a Hamiltonian elliptic system in dimension two, the potential is coercive which is of the class different considered in the systems studied in Cassani Tarsi 15), Souza Ó 16), Souza 1). Moreover, due to the fact of the weights Q i allow us to complement the results with more general class of nonlinearities. We recall that under the hypothesis V 1 ) V ) considered in Cassani Tarsi 15) Chapter 4 or V 1 ) 1/V L 1 R ) assumed in Souza 1)) implies that the space W 1 L,s V R ) or H 1 V R ) = {u H 1 R ) : R V x) u dx < + }) is compactly embedded in L λ R ) for any λ 1. In view of Proposition 1.4 in order to overcome some difficulties due to lack of embeddings, we compensate with condition B 1 ) which will be used to show control the boundedness of Palais-Smale sequences. Observe also that B 1 ) implies the usual assumption, that is, f s) = gs) = os), as s. In our argument to prove the existence results, it was crucial a Trudinger-Moser inequality some embeddings type properties in weighted Lorentz-Sobolev spaces W 1 L,s V R ). In the proof we used a linking theorem finite dimensional approximation as in the proofs of Theorems In Chapter 6, we establish the existence of the following Hamiltonian system { u +V x)u = gv), x R, v +V x)v = f u), x R, 1.14) where the functions f g possess critical exponential growth V is a continuous potential. First, in the systems 1.1) 1.13) considered in the last chapters, the condition V 1 ) says that V is bounded below for a some positive constant V ) gives some conditions of integrability or coercivity.

30 8 Chapter 1. Introduction In Albuquerque, Ó Medeiros 16), the authors proved that the system { u +V x )u = Q x )gv), x R, v +V x )v = Q x ) f u), x R, 1.15) has a nontrivial solution under the potential V the weight function Q being radially symmetric satisfying the following assumptions: V ) V C,+ ), V r) > there exists a > such that V r) lim inf r + r a >. Q) Q C,+ ), Qr) > there exists b < a )/ b > such that lim inf r + Qr) r b < + lim sup r + Qr) r b < +. In Souza Ó 16), the authors established the existence of nontrivial solutions for Hamiltonian systems of the form { u +V x)u = gx,v), x R, v +V x)v = f x,u), x R, 1.16) when the potential V is neither bounded away from zero, nor bounded from above. The nonlinear terms f x, s) gx, s) are superlinear at infinity have exponential subcritical or critical growth for the case p = q =. Among other things, it is assumed that potential V satisfies the following assumptions lim ν sr \B R ) = +, for some s [,+ ), 1.17) R + or for any r > any sequence x k ) R, which goes to infinity lim ν sb r x k )) = +, for some s [,+ ), 1.18) k + where ν s is defined by, if R is an open set s, ν s ) = +. ν s ) = inf u H 1 )\ u +V x)u ) dx u s dx ) /s Motivated by the above mentioned results we are interested in studying the system 1.14) for the exponential critical case p = q =, when the potential V can be bounded or can vanish at infinity. More precisely, we assume: V 1 ) V C R,R) is a radially symmetric positive function.

31 9 V ) There exist constants < a <, b a R > 1 such that L a x a V x) L b x b for all x R, where L a L b are positive constants depending on a,b R. V 3 ) V x) = 1 if x 1 V x) 1 if 1 < x < R. Under these conditions on V, we set for 1 < p < + L p V,rad R ) := {u : R R : u is measurable, radial we consider the following Sobolev space R V x) u p dx < + } H 1 V,rad R ) = {u L V,rad R ) : u L R )}, these spaces were considered by Su, Wang Willem 7a), Su, Wang Willem 7b). Concerning the functions f g, we suppose the following assumptions: H 1 ) f,g C R) f s) = gs) = for all s. Setting b = b + a)/ a) where a b are given by V ), consider H ) There exist constants µ > b ν > b such that < µfs) s f s), < νgs) sgs), for all s >, where Fs) = s f t) dt Gs) = s gt) dt. H 3 ) There exist constants s 1 > M > such that < Fs) M f s) < Gs) Mgs), for all s > s 1. Setting µ ν given by H ) a given by V ), we suppose: H 4 ) There exists θ 4a/ a) such that f s) = Os µ 1+θ ) gs) = Os ν 1+θ ) as s +. H 5 ) There exists α > such that f s) lim s e =, α > α, αs +, α < α, gs) lim s e = αs, α > α, +, α < α. H 6 ) For α > given by H 5 ), we have t f t) lim inf t + e α t > 4e liminf α t + tgt) e α t > 4e α.

32 3 Chapter 1. Introduction The following theorem contains our main result in Chapter 5. Theorem 1.6. Suppose that V satisfies V 1 ) V 3 ) f g satisfy H 1 ) H 6 ). Then, there exists L = L f,g, µ,ν,α,θ,a,b,r ) > such that system 6.1) possesses a nontrivial weak solution u,v) H 1 V,rad R ) H 1 V,rad R ) provided that L a L, namely u,v) H 1 V,rad R ) H 1 V,rad R ) satisfies ) u ψ +V x)uψ + v φ +V x)vφ dx = f u)φ + gv)ψ) dx, R R for all φ,ψ) H 1 V,rad R ) H 1 V,rad R ). Our theorem may be seen as complement of the above mentioned results. We recall that condition V ) allows V x) as x. The condition V ) in the system 1.1) its relationed works considered in Chapter 4 requires that V be large at infinity. We also note when Q 1 in condition Q), this implies that V x) + as x. Thus, although the class of Hamiltonian systems considered in Albuquerque, Ó Medeiros 16) is very general, the main result in that paper can not be applied to the model case V x) = L/ x a, for x sufficiently large, considered here. In the recent paper Souza Ó 16), a fairly general result was proved on system 1.16), but under the hypotheses 1.17) 1.18), which implies that V is large at infinity, as we can verified with the following example: taking u C R ) such that u x,y) = 1 if x,y) 3 u x,y) = if x,y) 3 4. Setting u k ) C R ) defined by u k x,y) = u x k,y k) for k N. Thus, for every k N for all s [, ), we have u k H 1 )\{} supp u k B 1 k,k), 1.19) If V is bounded near infinity there exist k > C > 1 such that u k s dx 1 for all B 1 k,k). 1.) V x) C for all x k. 1.1) For given R > let k 1 > max{r,k } + 1 using 1.19), 1.) 1.1) we have that ν s R \B R ) R \B R uk1 +V x)u k 1 ) dx R \B R u k1 s dx ) /s uk1 +Cu ) R k 1 dx \B R uk1 +Cu ) k 1 dx C u H 1 B 1), B 1 k 1,k 1 )

33 31 which contradicts 1.17). Note also that by 1.19), 1.) 1.1), for every k > k + 1, we get ν s B 1 k,k)) B 1 k,k) uk +V x)u k) dx B 1 k,k) u k s dx ) C u /s H 1 B 1 ), which contradicts 1.18). Observe that, the associated functional with 1.14) is strongly indefinite the space HV 1R ) presents some phenomenons such as lack of compactness. In order to prove the existence, we combine a truncation argument with a finite-dimensional approximation Linking theorem. The truncation argument employed here is an adaptation of the reasoning used in Alves Souto 1) to study the existence of positive solutions to a scalar equation. We point out that this chapter is contained in the accepted paper Leuyacc Soares 17).

34

35 33 CHAPTER LORENTZ AND LORENTZ-SOBOLEV SPACES In this chapter we introduce prove some properties which will be important in the development of this thesis..1 Distribution functions decreasing rearrangement Let X = X,Σ, µ) be a σ finite measure space, denote by M X,R) the collection of all extended real-valued µ-measurable functions on X M X,R) the class of functions in M X,R) that are finite µ-almost everywhere in X. As usual, any two functions coinciding almost everywhere in X will be identified. Moreover, natural vector space operations are well defined on M X,R). Definition.1. The distribution function µ φ of a function φ M X,R) is defined by µ φ t) := µ{x X : φx) > t}, for t. The distribution function satisfies the following properties see Hunt 1966), Bennett Sharpley 1988)). Proposition.. Let φ,ψ M X,R). Then, the distribution function µ φ is nonnegative, nonincreasing continuous from the right on [,+ ). Furthermore, i) If φx) ψx) µ-almost everywhere in X, then µ φ t) µ ψ t), for all t. t ) ii) µ λφ t) = µ φ, for all t λ. λ iii) µ φ+ψ t 1 +t ) µ φ t 1 ) + µ ψ t ), for all t 1,t. iv) µ φψ t 1 t ) µ φ t 1 )µ φ t ), for all t 1,t.

36 34 Chapter. Lorentz Lorentz-Sobolev spaces v) Let φ n ) be a sequence in M X,R) such that φx) liminf n φ n x), µ-a.e in X. Then, µ φ t) liminf n µ φn t) a.e in R +. In particular, if φ n φ µ-a.e in X. Then, µ φ µ φn a.e in R +. Using distribution functions we can consider the following spaces: Definition.3. Weak L p -spaces) If f M X,R), let We define the weak-l p as follows: [ f ] p = [ f ] p,x = supt [ µ f t) ] 1/p. t> Weak L p X) := { f : f M X,R), [ f ] p < + }. More details about these spaces can be found on Adams Fournier 3). Definition.4. The decreasing rearrangement of φ M X,R) is defined by φ s) := inf{t : µ φ t) s}, for s. The decreasing rearrangement satisfies the following properties see Hunt 1966), Bennett Sharpley 1988)). Proposition.5. Let φ,ψ M X,R). Then, the distribution function φ is nonnegative, nonincreasing continuous from the right on [,+ ). Furthermore, i) If µ φ t) µ ψ t), for all t, then φ s) ψ s), for all s. ii) λφ) = λ φ, for all λ R. iii) φ + ψ) s 1 + s ) φ s 1 ) + ψ s ), for all s 1,s. iv) φψ) s 1 s ) φ s 1 )φ s ), for all s 1,s. v) Let φ n ) a sequence in M X,R) such that φ liminf n φ n, µ-a.e in X. Then, φ liminf n φn a.e in R +. In particular, if φ n φ µ-a.e in X. Then, φn φ a.e in R +. In the following example, we compute the distribution decreasing rearrangement of a simple function. Example.6. Let φ M X,R) be a simple function, that is, φ is a linear combination of characteristic functions, in particular, we can write φ = n a j χ E j j=1

37 .1. Distribution functions decreasing rearrangement 35 where a 1 > a > a n > E j = {x X : φ x) = a j }. Indeed, since φx) a 1 for all x X, for each t a 1, we have µ φ t) = µ{x X : φx) > a 1 } = µ ) =. Let a t < a 1. Thus, µ φ t) = µ{x X : φx) > t} = µ{x X : φx) = a 1 } = µe 1 ). In general, if a j+1 t < a j for j = 1,,n a n+1 = ), we have Thus, µ φ t) = µ{x X : φx) > t} = µ{x X : φx) = a 1,a,...,a j } = If s < m 1, we have µ φ t) = n m j χ [a j+1,a j ), where m j = j=1 φ s) = inf{t : j i=1 µe i ). n m j χ [a j+1,a j )t) s} = a 1. j=1 j i=1 µe i ). If m 1 s < m, we have φ s) = inf{t : n m j χ [a j+1,a j )t) s} = a. j=1 In general, if m j 1 s < m j for j = 1,...,n m = ), we have φ s) = inf{t : n m j χ [a j+1,a j )t) s} = a j. j=1 Thus, φ s) = n a j χ [m j 1,m j ), where m j = j=1 j i=1 µe i ). See the following figures for a specific example:

38 36 Chapter. Lorentz Lorentz-Sobolev spaces Figure 1 A simple function φ. a) φ = χ [ 3, 1) + χ [ 1,1) + 4χ [1,) + χ [,3]. Source: Elaborated by the author. Figure Distribution function decreasing rearrangement of φ. a) µ φ = 6χ [,1) + 3χ [1,) + χ [,4]. b) φ = 4χ [,1) + χ [1,3) + χ [3,6). Source: Elaborated by the author. Example.7. Let r > φ : R R defined by 1 ) r. φx) = 1 + π x Then, φ s) = 1, for all s. 1 + s) r

39 .1. Distribution functions decreasing rearrangement 37 Indeed, since φx) 1 for all x R, for each t 1 we have If < t < 1, we have µ φ t) = {x R : φx) > t} = =. µ φ t) = { x R : φx) > t } { = x R 1 : 1 + π x > t1/r} { = x R : x < = 1 1. t1/r 1 1 π t 1/r 1)} On the other h, let s fixed t such that µ φ t) s. Thus, there are two possibilities: t 1 or that is, Thus, we conclude that 1 1 s, t1/r s) r t. φ s) = inf{t : µ φ t) s} = s) r. Example.8. Let f x) = 1 e x defined on R. Then, for each t 1 we have µ f t) = for each t < 1 we have { µ f t) = {x R : 1 e x > t} = x R : x > ln 1 ) } = +. 1 t Therefore, f s) = 1 for all s. In particular, we conclude that, if f M X,R) not necessarily µ f is an almost everywhere finite-valued function. Let φ be a simple function as given by Example.6. Then, + φ s) ds = A more general result is given by the following Lemma: n n a j m j m j 1 ) = a j µe j ) = φx) dµx)..1) j=1 j=1 X Lemma.9. Let φ M X,R) G : [,+ ) [,+ ) be a nondecreasing function such that G φ ) L 1 X) G) =. Then, Gφ ) L 1 [,+ )) + Gφ s)) ds = G φx) ) dµx). X

40 38 Chapter. Lorentz Lorentz-Sobolev spaces Proof. Let φ be a simple function, using notation given by Example.6 the fact that G) = we have ) n ) n G φx) ) = Ga j )χ E j x) = Ga j )χ [m j 1,m j )s) = Gφ s))..) j=1 j=1 Thus, from.).1) we obtain + Gφ s)) ds = + G φs) ) ) ds = X G φx) ) dµx)..3) In the general case, there exists a increasing sequence φ n ) of simple functions converging almost everywhere to φ. By Proposition.5-v), φ n converges monotonically to φ almost everywhere. Consequently, the sequences G φ n ),Gφ n ) converges monotonically to G φ ) Gφ ) respectively. Moreover, G φ n ) Gφ n ) are simple functions. By.3) Monotone converge theorem, we have + Gφ ) ds = lim n + Gφn ) ds = lim G φ n ) dµx) = G φ ) dµx). n X X Reducing to simple functions taking limit we can obtain the following result: Lemma.1. Hardy-Littlewood inequality) See Hunt 1966).) Let φ,ψ M X,R). Then, + φx)ψx) dµx) φ s)ψ s) ds.. Lorentz spaces X In this section we present Lorentz spaces which were introduced by Lorentz 195). For simplicity we consider throughout this section the following measure space X, µ) =, m) where is a measurable subset in R N with N 1 m is the Lebesgue measure. Definition.11. Let 1 < p < +, 1 q +. The Lorentz space L p,q ) is the collection of functions φ M,R) such that φ p,q < + where + [φ t)t 1/p ] q dt ) 1/q, t if 1 q < +, φ p,q = sup t> t 1/p φ t), if q = +. In particular, two functions in L p,q ) are identified if they are equal almost everywhere in..4) For a mensurable function f = f 1,, f N ) : R N, we say that f L p,q ) if only if f L p,q ) we set f p,q := f p,q. Therefore, f L p,q ) if only if f i L p,q ) for 1 i N.

41 .. Lorentz spaces 39 Proposition.1. The map given by.4) is a quasinorm L p,q ) is a vector space. Proof. Let 1 < p < + 1 q < +. i) It is clear that φ p,q for all φ L p,q ) φ p,q = if only if φ =. ii) Let λ R, by Proposition.5-ii) we have λφ) t) = λ φ. Then, λφ p,q = + [λφ) t)t 1/p ] q dt t iii) Let φ,ψ L p,q ) using Proposition.5-iii), we have Hence, φ + ψ q p,q = + + [ φ + ψ) t)t 1/p] q dt t [φ t ) + ψ t ) ) t 1/p] q dt t ) 1/q = λ φ p,q. = q + [ φ p s) + ψ s) ) s 1/p] q ds s q + p +q 1 = q p +q 1 φ q p,q + ψ q p,q). [φ s)s 1/p] q + [ ψ s)s 1/p] q ) ds s φ + ψ p,q 1 p +1 1 q φ p,q + ψ p,q ), for all φ,ψ L p,q ). The properties i)-iii) are also true for the case when 1 < p < + q = +. Thus, p,q represents a quasinorm. Moreover, if φ,ψ L p,q ) λ R, using ii) iii), we have φ + ψ λψ belong to L p,q ), that is, L p,q ) is a vector space. The following result says that p,q is a norm for some cases. Proposition.13. See Bennett Sharpley 1988).) The map p,q is a norm if only if 1 q p. Now, we build a topology T in Lorentz spaces. For every x L p,q ) every r >, we consider the following open ball: we set the collection of balls B r x) = {y L p,q ) : y x p,q < r} B := {B r x) : x L p,q ),r > }. A subset U in L p,q ) is said to be open in L p,q ) U T ) if only if U = i I ) Bi1 B i B ini, where Bik B I is an index set.

42 4 Chapter. Lorentz Lorentz-Sobolev spaces Consequently, L p,q ) turns out a topological vector space. Note that, each ball B r x) is an open set. Thus, we say that, the sequence φ n ) L p,q ) converges to φ L p,q ), in the topology T if only if φ n φ p,q. In the following we define a metric d such that L p,q ),d ) is a metric space. Definition.14. Let 1 < p < +, 1 q +, R N φ M,R), the maximal function is defined by + φ t) := 1 t φ s) ds, for all t >. Definition.15. Let 1 < p < +, 1 q +, we define + [φ t)t 1/p ] q dt ) 1/q, t if 1 q < +, φ p,q = sup t> t 1/p φ t), if q = +. Proposition.16. See Adams Fournier 3).) Let 1 < p < + 1 q +. Then, the functional p,q represents a norm on L p,q ). Moreover, L p,q ) endowed with this norm is a Banach space Setting the metric φ p,q φ p,q p p 1 φ p,q, for all φ L p,q )..5) d : L p,q ) L p,q ) R + φ,ψ), φ ψ p,q. Let T the topology induced by the metric d. Using.5), we have the topologies T T defined on L p,q ) are equals. Remark.17. i) The Lorentz space L,q ) with 1 < q < + is not interesting, since the only function in this space is given by the zero function. ii) Using Lemma.9 with Gs) = s p, p > 1, we have + ) 1/p φ p,p = [φ t)] p dt = This implies, L p,p ) = L p ). Thus, Lorentz spaces are intermediate between L p -spaces. iii) For 1 < p < +, we have φx) p dx) 1/p = φ p. Thus, φ p, = sup t> t 1/p φ t) = supt[µ φ t)] 1/p = [φ] p. t> L p, ) = weak-l p ).

43 .. Lorentz spaces 41 iv) Given a function φ defined on, we denote by φ its extension outside, that is, φx), x φx) =, x R N \. In particular, if is a bounded set, we have φ) φ t), t t) =, t >. Thus, φ L p,q R N ) = + [ φ) t)t 1/p] p dt t ) 1/p = + [ φ t)t 1/p] p dt ) 1/p = φ L t p,q ). Lemma.18. Hölder s inequality in Lorentz spaces) Let 1 < p,q < + p, q denoted the conjugate exponents defined by p = p/p 1) q = q/q 1). If f L p,q ) q L p,q ). Then, f g L 1 ) f x)gx) dx f p,q g p,q. Proof. Using Lemma.1 we have f x)gx) dx t 1/q t 1/q + f t)g t) dt = + f t)t 1/p g t)t 1/p t 1/q t 1/q dt..6) By classical Hölder s inequality with 1/q + 1/q = 1 we obtain + f t)t 1/p g t)t 1/p + ds f t)t 1/p) q dt ) 1/q + g ) t)t 1/p q dt t t Joining.6).7) the claim follows. ) 1/q dt.7) Lemma.19. Generalized Hölder s inequality in Lorentz spaces) Let the following constants 1 < p, p 1, p,q,q 1,q < + such that 1 p = 1 p p 1 q = 1 q q. If f L p 1,q 1 ) g L p,q ). Then, f g L p,q ) f g p,q 1/p f p1,q 1 g p,q. Proof. Using Proposition.5-iv) Hölder s inequality, we have + [ f g) t)t 1/p] q dt + [ f t ) g t ) t 1/p] q dt t t + [ q/p f t)g t)t 1/p] q dt t + [ f q/p t)t 1/p 1 g t)t 1/p ] q dt Then, the claim follows. q/p + t 1/q 1 [ f t)t 1/p 1 ] q 1 dt t t 1/q ) q/q1 + [ g t)t 1/p ] q dt ) q/q. t

44 4 Chapter. Lorentz Lorentz-Sobolev spaces Remark.. i) The claim in Lemma.18 is still valid if we consider 1 + as conjugated exponents. ii) The claim in Lemma.19 is still valid if we consider q = q 1 q = + or q = q q 1 = +. Lemma.1. See Hunt 1966).) Let 1 q 1 q + p > 1. Then, the following embedding is continuous L p,q 1 ) L p,q ). Lemma.. Let < +, 1 < p 1 < p < + 1 q 1 q +. Then, the following embedding is continuous L p,q ) L p 1,q 1 ). Proof. Let f L p 1,q 1 ) taking p 3 q 3 such that 1 p 1 = 1 p + 1 p 3 1 q 1 = 1 q + 1 q 3. Using Lemma.18 we have f p1,q 1 1/p 1 f p,q 1 p3,q 3 = 1/p 1 1/q 3 f p,q. the embedding follows. Lemma.3. Let < +, 1 < p < + 1 q +. Then, the following embeddings are continuous. L p,q ) L p δ ), for all < δ p 1 Proof. If 1 q p, by Lemma.1, we have for all < δ p 1 L p,q ) L p,p ) = L p ) L p δ ). If q > p, by Lemma. we have for all < δ p 1 L p,q ) L p δ,p δ ) = L p δ ). Then, for all < δ p 1 L p,q ) L p δ ). Lemma.4. Let < +, 1 < p < + 1 q +. Then, the following embeddings L p+δ ) L p,q ), for all δ > are continuous.

45 .. Lorentz spaces 43 Proof. If 1 p q, by Lemma.1 we have L p+δ ) L p ) = L p,p ) L p,q ), for all δ >..8) If p > q, by Lemma. we have L p+δ ) = L p+δ,p+δ ) L p,q ), for all δ >..9) Joining.8).9) the lemma follows. Proposition.5. See Hunt 1966).) Let 1 < p < +, 1 q < +. Then, the set of simple functions are dense in L p,q ). Proposition.6. Let 1 < p < +, 1 q < + a open subset in R N. Then, C c ) is dense in L p,q ). Proof. Let f L p,q ) ε >, by Lemma.5 there exists a simple function s defined on with compact support such that f s L p,q ) < ε 4..1) Let K = supp f ) consider = x K B1 x) ). Thus, is an open bounded set such that K. From Lemma.4 the space L p+1 ) continuously embedded in L p,q ), denoting by S p > its best embedding constant. Since s L p+1 ) using the density of Cc ) in L p+1 ), there exists g Cc ) Cc ) such that s g L p+1 ) < ε 4S p..11) Note that, s g L p,q ) = s g L p,q ).1) Thus, combining.1),.11).1) we obtain f g L p,q ) f s L p,q ) + s g L p,q ) < ε + s g L p,q ) < ε + S p s g L p+1 ) < ε. Thus, C c ) is dense in L p,q ). Proposition.7. See Hunt 1966).) Let an open subset in R N. Then, the following results holds: i) Let 1 < p < +. Then, the dual space of L p,1 ) is given by L p, ) where 1/p+1/p = 1.

46 44 Chapter. Lorentz Lorentz-Sobolev spaces ii) Let 1 < p < + 1 < q < +. Then, the dual space of L p,q ) is given by L p,q ) where 1/p + 1/p = 1 1/q + 1/q = 1. Moreover, these spaces are reflexive. Proposition.8. See Halperin 1954).) Let an open subset in R N, 1 < p < + 1 < q < +. Then, the Lorentz space L p,q ) is a uniformly convex space. Lemma.9. Let φ L p,q R N ). Then, for every ε > there exists R > such that φ φ χ BR p,q < ε. where χ BR is the characteristic function of B R. Proof. Let ε >, since φ L p,q R N ), we have {x R N : φx) > ε} < +. Observe that {x B R : φx) > ε} {x R N : φx) > ε} as R +. Thus, for each δ > there exists R = Rε,δ) > such that I R < δ where I R = {x R N \B R : φx) > ε}. Setting φ R = φ χ R, then, φ φ R )x) ε, for all x R N \I R. Thus, Therefore, Then, {x R N : φ φ R )x) > ε} < δ. µ φ φr )ε) < δ. φ φ R ) δ) = inf{s : µ φ φr )s) < δ} ε. Using the fact that φ φ R ) is nonincreasing we have Thus, for each n N, there exists φ n such that φ φ R ) t) ε, for all t δ. φ φ n ) t) 1 n, for all t 1 n, where φ n = φ χ Rn. Consequently, there exists a sequence φ n ) such that φ n x) φx), almost everywhere in R N.13)

47 .. Lorentz spaces 45 φ φ n ) t), almost everywhere in R +..14) Using.13) Lemma.5, we obtain φ φ n ) t) φ + φ n ) t) φ t ) + φ t ) n φ t ),.15) almost everywhere in R +. Now, by.14),.15) Dominated convergence theorem, we have φ φ n q p,q = Thus, there exists R > such that + [ t 1/p φ φ n ) t) ] q dt t φ φ χ R p,q < ε.. Proposition.3. Let 1 < p < + 1 q +. If u n ) is a sequence in L p,q ) u L p,q ) such that u n u in L p,q ). Then, i) u n u in measure. ii) There exists a subsequence u nk ) such that Proof. u nk x) ux), almost everywhere in. i) By Lemma.1 we have L p,q ) L p, ) continuously. Thus, u n u in L p, ). Consequently, for given ε > there exists n 1 such that By Remark.17-iii), we have u n u p, < ε p+1)/p, for all n n. supt[µ un u)t)] 1/p < ε p+1)/p, for all n n. t> In particular, taking t = ε in last inequality we obtain µ un u)ε) < ε, for all n n. That means, {x : u n x) ux) > ε} < ε, for all n n. which proves the assertion.

48 46 Chapter. Lorentz Lorentz-Sobolev spaces ii) It is a consequence of i). Proposition.31. Let f n ) be a sequence of functions in L p,q ) satisfying i) f 1 f f n f n+1, almost everywhere in. ii) sup f n p,q < +. n N Then, f n converges pointwise on to a measurable function f, that is finite almost everywhere, furthermore f n f in L p,q ). Proof. Let E be a measurable set in with measure zero such that, for any x \E the sequence f n x)) is nondecreasing. Then, we can define lim f nx) = sup f n x), x \E. n f x) = n 1 +, x E. Now, we show that f is finite almost everywhere in. Since L p,q ) is continuous embedding in L p,, by assumption ii) there exists C > such that Taking t = m N in last inequality, we have supt[µ fn t)] 1/p = f n p, C, for all n 1. t> Setting for each m,n 1 the following measurable sets µ fn m) Cp, for all n,m 1..16) mp F m,n = {x \E : f n x) > m}, F m = {x \E : f x) > m}. F = {x \E : f x) = + }. Fixing m 1, if x F m we have sup n 1 f n x) = f x) > m. Then, there exists n 1 such that f n x) > m that is x F m,n. Therefore, F m F m,n. n=1

49 .. Lorentz spaces 47 Moreover, from assumption i) we have F m,n F m,n+1 for all n 1. Then, From.16) F m n=1 F m,n = lim n F m,n..17) F n,m = {x \E : f n x) > m} {x : f n x) > m} = µ fn m) Cp m p. Combining the last inequality with.17), we get F m Cp, for all m 1..18) mp On the other h, we have F m+1 F m for all m 1 the measure of F 1 is finite. Using.18), we obtain F = n=1 F m = lim F C p m lim m m m p =. Thus, f is finite in \{E F} with E F =. Since f n ) is nondecreasing almost everywhere in f n f almost everywhere in, by Propositions..5, we have f n ) is nondecreasing almost everywhere in R + f n f almost everywhere in R +..19) By the Monotone convergence theorem, + [t 1/p f t)] 1/q dt t + = lim [t 1/p f 1/q dt n n t)] t = lim f n q n p,q sup f m q p,q < +. m N Thus, f L p,q ). Moreover, by Propositions..5, we have f n t) f t) almost everywhere in R +. Combining the last inequality,.19), the fact that f L p,q ) Dominated convergence theorem, we get + lim [t 1/p f n t) f t) ) 1/q dt ] n t = + lim [t1/p f n n t) f t) ) ] 1/q dt t =. Thus, f n f in L p,q ).

50 48 Chapter. Lorentz Lorentz-Sobolev spaces.3 Lorentz-Sobolev spaces Definition.3. Let be an open domain in R N, assume that 1 < p < +, 1 < q + define W 1 Lp,q ) the closure of the set {u C ) : u p,q + u p,q < + } with respect to the quasinorm u 1,p,q) := u q p,q + u q p,q) 1/q.) where u = D 1 u,,d N u) D i is the weak derivative with respect to x i for 1 i N. The space W 1 Lp,q ) can also be equipped with the norm u 1,p,q) := [ u p,q ) q + u p,q ) q ] 1/q..1) Proposition.33. Let an open domain in R N, assume that 1 < p,q < +. Then, W 1 Lp,q ) endowed with the norm defined by.1) is uniformly convex Banach space hence a reflexive space). Proof. Let u n ) be a Cauchy sequence in W 1 Lp,q ). Then, u n ), D i u n ) for 1 i n are Cauchy sequences in L p,q ). Since L p,q ), p,q) is a Banach space, there exist u, vi in L p,q ) such that u n u in L p,q ) D i u n v i in L p,q ). Let φ C ). By generalized Hölder s inequality in Lorentz spaces, we have u n u)d i φ dx u n u p,q D i φ p,q which implies u n D i φ dx Thus, for all φ C D i u n v i )φ dx D i u n v i p,q φ p,q ud i φ dx ), we have ud i φ dx = lim n D i u n )φ dx u n D i φ dx = lim D i u n )φ dx n = v i φ dx. v i φ dx, for all φ C ). That is, D i u = v i for 1 i N in the weak sense. Therefore, u W 1 Lp,q ) u n u 1,p,q). Now, consider the following isometry J : W 1 Lp,q ) L p,q ) L p,q ) N u, u, u).