The analysis of some PDE s and related problems
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1 HABILITATION THESIS Specialization: Mathematics Mihai Mihăilescu The analysis of some PDE s and related problems
2 i Abstract The main goal of the present thesis is to investigate the existence, non-existence, multiplicity and other qualitative properties of solutions for some partial differential equations of elliptic type. We also plan to consider situations where the differential operators involved in such equations possess some degeneracies or singularities which lead to partial differential equations that are not completely elliptic in the whole domain where the problems will be analyzed. In a related context some difference equations will be studied. Finally, we will be concerned with the study of the asymptotic behavior of some power-law functionals. This thesis is divided into six chapters: The analysis of some eigenvalue problems; PDE s involving variable exponent growth conditions; PDE s in Orlicz-Sobolev spaces; The analysis of some difference equations; The asymptotic behavior of some inhomogeneous functionals; Final comments and further directions of research. A short description of this thesis is presented in what follows. In the first tree chapters of the thesis we will study equations associated with some elliptic partial differential operators. In a very general framework the model equations that will be considered here have one of the forms div(φ(x, u)) = f(x, u) (0.) or N xi (φ i (x, xi u)) = f(x, u), (0.2) i= where in the left-hand side we consider elliptic differential operators that can be linear or nonlinear, homogeneous or nonhomogeneous, while in the right-hand side f is a given function. Specifically, we will consider partial differential equations of type (0.) where, for example, φ(x, u) = u (in this case the left-hand side becomes the Laplace operator), φ(x, u) = u p 2 u, with p (, ), a given real number (in this case the left-hand side becomes the p-laplace operator), φ(x, u) = u p(x) 2 u, with p(x) > a continuous function (in this case the left-hand side becomes the so-called p(x)-laplace operator), φ(x, u) = ( u p(x) 2 + u q(x) 2 ) u, with p(x), q(x) > continuous functions, p(x) q(x), or φ(x, u) = a( u ) u, with a being an odd, increasing homeomorphism from R onto R. Partial differential equations involving more general differential operators, including anisotropic versions of the previous one, such as u i x i (φ i (x, xi u(x))) (leading to equations of type (0.2)), degenerate/singular p(x)-laplace type operators, or Grushin type operators with variable exponents will also be considered. In addition to its intrinsic interest from an analytical point of view, the research from these chapters has a strong interdisciplinary character. Nonlinear and nonhomogeneous differential operators have proved to be extremely important in the modeling of different phenomena coming from various areas of Science and Engineering. In particular, the study of partial differential equations involving variable exponent growth conditions, of the type considered here, has been strongly motivated during the last
3 ii decade by its applications to elasticity, electrorheological fluids, image processing, and mathematical biology. The fourth chapter is devoted to the study of some difference equations. This chapter comprises three sections. The first two sections are concerned with the study of some eigenvalue problems while the third section analyzes the existence of homoclinic solutions for a class of difference equations involving variable exponents. Chapter 5 of the thesis discusses the asymptotic behavior of some power-law functionals involving variable exponents. The chapter comprises three sections. In the first section some Γ-convergence results for power-law functionals with variable exponents are obtained. The main motivation comes from the study of (first-failure) dielectric breakdown. Some connections with the generalization of the -Laplace equation to the variable exponent setting are also explored. The second section studies, via Γ-convergence, the asymptotic behavior of several classes of power-law functionals acting on fields belonging to variable exponent Lebesgue spaces and which are subject to constant rank differential constraints. Applications of the Γ-convergence results to the derivation and analysis of several models related to polycrystal plasticity arising as limiting cases of more flexible power-law models are also discussed. In the third section of this chapter the asymptotic behavior of several classes of powerlaw functionals involving variable exponents p n ( ) is analyzed via Mosco convergence. As an application of these results we obtain a model for the growth of a sandpile in which the allowed slope of the sand explicitly depends on the position in the sample. The last chapter of the thesis presents some ideas regarding the study of some open problems starting from the research presented in the first five chapters. Further plans regarding the evolution of the professional and scientific career of the candidate will also be presented both from the point of view of research and teaching. January 202 Acknowledgments I wish to thank the many collaborators that have interacted with me after completing my Ph.D. at the University of Craiova. I express my gratitude to all my co-authors and it is my great pleasure to acknowledge them in what follows: Marian Bocea, Maria-Magdalena Boureanu, Nicuşor Costea, Alexandru Kristály, Gherghe Moroşanu, Constantin P. Niculescu, Mayte Pérez-Llanos, Cristina Popovici, Patrizia Pucci, Vicenţiu Rădulescu, Dušan Repovš, Julio D. Rossi, Ionel Rovenţa, Denisa Stancu-Dumitru, Stepan Tersian, Csaba Varga. My appreciation equally goes to my good friend Liviu Ignat for many stimulating discussions over the years. I will always be grateful to my sister Dana for her permanent help, advice and comments regarding the language style in my projects.
4 iii Rezumat Scopul principal al prezentei teze este acela de a investiga existenţa, non-existenţa, multiplicitatea şi alte proprietăţi calitative ale soluţiilor unor ecuaţii cu derivate parţiale de tip eliptic. De asemenea vom considera şi situaţii în care operatorii diferenţiali implicaţi în ecuaţii posedă degenerări sau singularităţi care conduc la ecuaţii cu derivate parţiale care nu sunt eliptice în întreg domeniul unde sunt analizate. Într-un context apropiat câteva ecuaţii cu diferenţe finite vor fi studiate. În final, ne vom ocupa de studiul comportamentului asimptotic al unor funcţionale. Teza de faţă cuprinde şase capitole: Analiza unor probleme de valori proprii; Ecuaţii cu derivate parţiale ce implică prezenţa unor condiţii de creştere date de exponenţi variabili; Ecuaţii cu derivate parţiale studiate în spaţii Orlicz-Sobolev; Analiza unor ecuaţii cu diferenţe finite; Comportamentul asimptotic al unor funcţionale neomogene; Comentarii finale şi direcţii de cercetare viitoare. O descriere succintă a tezei este dată în cele ce urmează. În primele 3 capitole ale tezei vom studia ecuaţii în care sunt prezenţi diferiţi operatori diferenţiali. Într-un context foarte general ecuaţiile model considerate aici au una din formele div(φ(x, u)) = f(x, u) (0.3) sau N xi (φ i (x, xi u)) = f(x, u), (0.4) i= unde în membrul stâng considerăm un operator diferenţial eliptic care poate fi liniar sau neliniar, omogen sau neomogen, în timp ce în membrul drept f este o funcţie dată. Mai exact, vom considera ecuaţii cu derivate parţiale de tipul (0.3) unde, de exemplu, φ(x, u) = u (în acest caz în membrul stâng este prezent operatorul lui Laplace) φ(x, u) = u p 2 u, cu p (, ), un număr real dat (în acest caz membrul stâng devine operatorul p-laplace), φ(x, u) = u p(x) 2 u, cu p(x) > o funcţie continuă (în acest caz membrul stâng devine operatorul p(x)- Laplace), φ(x, u) = ( u p(x) 2 + u q(x) 2 ) u, cu p(x), q(x) > funcţii continue, p(x) q(x), sau φ(x, u) = a( u ) u, cu a fiind un homeomorfism impar, crescător şi surjectiv de la R în R. Ecuaţiile cu derivate parţiale implicând prezenţa unor operatori diferenţiali mai generali, incluzând versiunile anisotropice ale celor de mai sus, ca de exemplu u i x i (φ i (x, xi u(x))) (ce conduce la ecuaţii de tipul (0.4)), operatori de tipul p(x)-laplace degeneraţi şi/sau singulari, sau operatori de tip Grushin cu exponenţi variabili vor fi de asemenea consideraţi. Dincolo de interesul intrinsec dat de formularea analitică, cercetarea din aceste prime trei capitole are un puternic caracter interdisciplinar. Operatori diferenţiali neliniari şi neomogeni s-au dovedit a fi extrem de importanţi în modelarea unor fenomene din diferite arii ale ştiinţei şi tehnicii. În particular, studiul ecuaţiilor cu derivate parţiale ce implică prezenţa unor exponenţi variabili, de tipul celor consideraţi aici, a fost puternic motivată în ultimul timp de aplicaţiile în elasticitate, teoria fluidelor electrorheologice, procesarea de imagini şi biologia matematică.
5 iv Capitolul patru include studiul unor ecuaţii cu diferenţe finite. Acest capitol cuprinde trei secţiuni. Primele două studiază câteva probleme de valori proprii în timp ce a treia analizează existenţa soluţiilor homoclinice pentru o clasă de ecuaţii cu diferenţe finite ce implică prezenţa unor exponenţi variabili. Capitolul 5 al acestei teze discută comportamentul asimptotic al unor funcţionale ce implică prezenţa unor exponenţi variabili. Capitolui cuprinde trei secţiuni. În prima secţiune sunt obţinute câteva rezultate de Γ-convergenţă pentru funcţionale ce implică prezenţa unor exponenţi variabili. Motivaţia principală a acestui studiu vine din cercetările în domeniul modelării materialelor compozite dielectrice care se comportă ca izolator electric (vizând pierderea calităţii de a fi izolatori electrici a unor materiale). Câteva conexiuni cu generalizări ale ecuaţiilor implicând prezenţa -Laplacianului în contextul exponenţilor variabili sunt de asemenea prezentate. A doua secţiune studiază, via Γ-convergenţă, comportamentul asimptotic al unor funcţionale care acţionează pe câmpuri ce aparţin unor spaţii Lebesgue cu exponent variabil. Aplicaţii ale rezultatelor de Γ-convergenţă la analiza unor modele legate de plasticitatea policristalelor ce apar drept cazuri limită ale unor modele mai flexibile sunt de asemenea discutate. În cea de a treia secţiune a acestui capitol investigăm comportamentul asimptotic al unor clase de funcţionale implicând prezenţa unor exponenţi variabili p n ( ) via Mosco convergenţă. Ca o aplicaţie a rezultatelor teoretice obţinem un model pentru studiul evoluţiei (creşterii) grămezilor (dunelor) de nisip în care panta grămezii depinde explicit de poziţia în eşantion. Ultimul capitol al tezei prezintă câteva idei cu privire la studiul unor probleme deschise formulate plecând de la cercetările din această teză. Sunt prezentate de asemenea câteva planuri privind evoluţia profesională şi ştiinţifică a candidatului, atât din punct de vedere al cercetării cât şi al activităţilor de predare. Ianuarie 202
6 v Publications of the Thesis Chapter. M. Mihăilescu and V. Rădulescu, Sublinear eigenvalue problems associated to the Laplace operator revisited, Israel J. Math. 8 (20), M. Mihăilescu and G. Moroşanu, Eigenvalues of the Laplace operator with nonlinear boundary conditions, Taiwanese Journal of Mathematics 5 (20), M. Mihăilescu, An eigenvalue problem possessing a continuous family of eigenvalues plus an isolated eigenvalue, Communications on Pure and Applied Analysis 0 (20), M. Mihăilescu and D. Repovš, An eigenvalue problem involving a degenerate and singular elliptic operator, Bull. Belg. Math. Soc., 8 (20), Chapter 2. M. Mihăilescu, V. Rădulescu and D. Stancu-Dumitru, On a Caffarelli-Kohn-Nirenberg type inequality in bounded domains involving variable exponent growth conditions and applications to PDE s, Complex Variables-Elliptic Equations 56 (20), M. Mihăilescu and D. Stancu-Dumitru, On an eigenvalue problem involving the p(x)-laplace operator plus a non-local term, Differential Equations & Applications (2009), M. Mihăilescu and D. Repovš, On a PDE involving the A p( ) -Laplace operator, Nonlinear Analysis 75 (202), M. Mihăilescu and V. Rădulescu, Continuous spectrum for a class of nonhomogeneous differential operators, Manuscripta Mathematica 25 (2008) M. Mihăilescu and V. Rădulescu, Concentration phenomena in nonlinear eigenvalue problems with variable exponents and sign-changing potential, Journal d Analyse Mathématique (200), M. Mihăilescu and V. Rădulescu, Spectrum consisting in an unbounded interval for a class of nonhomogeneous differential operators, Bulletin of the London Mathematical Society 40 (2008), M. Mihăilescu, P. Pucci and V. Rădulescu, Eigenvalue problems for anisotropic quasilinear elliptic equations with variable exponent, J. Math. Anal. Appl. 340 (2008), M. Mihăilescu, P. Pucci and V. Rădulescu, Nonhomogeneous boundary value problems in anisotropic Sobolev spaces, C. R. Acad. Sci. Paris, Ser. I 345 (2007) M. Mihăilescu and G. Moroşanu, On an eigenvalue problem for an anisotropic elliptic equation involving variable exponents, Glasgow Mathematical Journal 52 (200), M. Mihăilescu and G. Moroşanu, Existence and multiplicity of solutions for an anisotropic elliptic problem involving variable exponent growth conditions, Applicable Analysis 89 (2) (200),
7 vi. M. Mihăilescu, G. Moroşanu and D. Stancu-Dumitru, Equations involving a variable exponent Grushin-type operator, Nonlinearity 24 (20), M. Mihăilescu and C. Varga, Multiplicity results for some elliptic problems with nonlinear boundary conditions involving variable exponents, Computers & Mathematics with Applications 62 (20), Chapter 3. A. Kristály, M. Mihăilescu and V. Rădulescu, Two nontrivial solutions for a non-homogeneous Neumann problem: an Orlicz-Sobolev setting, Proceedings of the Royal Society of Edinburgh: Section A (Mathematics) 39A (2009), M. Mihăilescu and D. Repovš, Multiple solutions for a nonlinear and non-homogeneous problem in Orlicz-Sobolev spaces, Applied Mathematics and Computation 27 (20), M. Mihăilescu, V. Rădulescu and D. Repovš, On a non-homogeneous eigenvalue problem involving a potential: an Orlicz-Sobolev space setting, J. Math. Pures Appliquées (Journal de Liouville) 93 (200), M. Mihăilescu and V. Rădulescu, Eigenvalue problems associated to nonhomogeneous differential operators in Orlicz-Sobolev spaces, Analysis and Applications 6 (2008), No., M. Mihăilescu and V. Rădulescu, A continuous spectrum for nonhomogeneous differential operators in Orlicz-Sobolev spaces, Mathematica Scandinavica 04 (2009), M. Mihăilescu, G. Moroşanu and V. Rădulescu, Eigenvalue problems for anisotropic elliptic equations: an Orlicz-Sobolev space setting, Nonlinear Analysis 73 (200), M. Mihăilescu, G. Moroşanu and V. Rădulescu, Eigenvalue problems in anisotropic Orlicz-Sobolev spaces, C. R. Acad. Sci. Paris, Ser., I 347 (2009), M. Bocea and M. Mihăilescu, A Caffarelli-Kohn-Nirenberg inequality in Orlicz-Sobolev spaces and applications, Applicable Analysis, in press. (DOI:0.080/ ) 9. M. Mihăilescu and V. Rădulescu, Neumann problems associated to nonhomogeneous differential operators in Orlicz-Sobolev spaces, Annales de l Institut Fourier 58 (6) (2008), Chapter 4. M. Mihăilescu, V. Rădulescu and S. Tersian, Eigenvalue Problems for Anisotropic Discrete Boundary Value Problems, Journal of Difference Equations and Applications 5 (2009), A. Kristály, M. Mihăilescu, V. Rădulescu and S. Tersian, Spectral estimates for a nonhomogeneous difference problem, Communications in Contemporary Mathematics 2 (200), M. Mihăilescu, V. Rădulescu and S. Tersian, Homoclinic solutions of difference equations with variable exponents, Topological Methods in Nonlinear Analysis 38 (20), Chapter 5. M. Bocea and M. Mihăilescu, Γ-convergence of power-law functionals with variable exponents, Nonlinear Analysis 73 (200), 0-2.
8 vii 2. M. Bocea, M. Mihăilescu and C. Popovici, On the asymptotic behavior of variable exponent power-law functionals and applications, Ricerche di Matematica 59 (200), M. Bocea, M. Mihăilescu, M. Pérez-Llanos and J. D. Rossi, Models for growth of heterogeneous sandpiles via Mosco convergence, Asymptotic Analysis, in press. (DOI /ASY )
9 Contents The analysis of some eigenvalue problems. Sublinear eigenvalue problems associated to the Laplace operator revisited Eigenvalues of the Laplace operator with nonlinear boundary conditions An eigenvalue problem possessing a continuous family of eigenvalues plus an isolated eigenvalue An eigenvalue problem involving a degenerate and singular elliptic operator PDE s involving variable exponent growth conditions 2. An overview on variable exponent Lebesgue-Sobolev spaces Eigenvalue problem p(x) u = λ u p(x) 2 u and related problems An equation with variable exponents possessing singularities An eigenvalue problem involving the p(x)-laplace operator and a non-local term On a PDE involving the A p( ) -Laplace operator Continuous spectrum for a class of nonhomogeneous differential operators Concentration phenomena in nonlinear eigenvalue problems with variable exponents and sign-changing potential Spectrum in an unbounded interval for a class of nonhomogeneous differential operators Eigenvalue problems for anisotropic quasilinear elliptic equations with variable exponent Equations involving a variable exponent Grushin-type operator Multiplicity results for some elliptic problems with nonlinear boundary conditions involving variable exponents The -case The 0-case PDE s in Orlicz-Sobolev spaces An overview of Orlicz-Sobolev spaces Two nontrivial solutions for a non-homogeneous Neumann problem: an Orlicz-Sobolev setting Multiple solutions for a nonlinear and non-homogeneous problem in Orlicz-Sobolev spaces 43 viii
10 ix 3.4 On a non-homogeneous eigenvalue problem involving a potential: an Orlicz-Sobolev space setting A continuous spectrum for nonhomogeneous differential operators in Orlicz-Sobolev spaces Eigenvalue problems for anisotropic elliptic equations: an Orlicz-Sobolev space setting A singular PDE in Orlicz-Sobolev spaces Neumann problems associated to nonhomogeneous differential operators in Orlicz-Sobolev spaces The analysis of some difference equations 6 4. Eigenvalue problems for anisotropic discrete boundary value problems Spectral estimates for a nonhomogeneous difference problem Homoclinic solutions of difference equations with variable exponents The asymptotic behavior of some inhomogeneous functionals Γ-convergence of power-law functionals with variable exponents On the asymptotic behavior of variable exponent power-law functionals and applications Introduction Preliminaries Main results Dielectric breakdown and electrical resistivity Plane stress and antiplane shear polycrystal plasticity Models for growth of heterogeneous sandpiles via Mosco convergence Introduction Main results A model for sandpiles An example Final comments and further directions of research Open problems Some comments on the eigenvalue problem p(x) u = λ u p(x) 2 u On the infimum of the Rayleigh quotient corresponding to the Schwarz rearrangement of a variable exponent A maximum-type principle Further plans of the candidate regarding research and teaching activities
11 Chapter The analysis of some eigenvalue problems In this chapter we study four eigenvalue problems. The first three equations are analyzed on bounded domains and involve the presence of the Laplace operator. We will assume that the first eigenvalue problem involves the Dirichlet homogeneous boundary condition, the second problem involves a nonlinear Robin boundary condition while the third problem involves the Neumann homogeneous boundary condition. The last eigenvalue problem involves a degenerate and singular elliptic operator and is analyzed on the whole space R N.. Sublinear eigenvalue problems associated to the Laplace operator revisited Eigenvalue problems involving the Laplace operator on bounded domains lead to a discrete or a continuous set of eigenvalues. In this section we highlight the case of an eigenvalue problem involving the Laplace operator which possesses, on the one hand, a continuous family of eigenvalues and, on the other hand, at least one more eigenvalue which is isolated in the set of eigenvalues of that problem. Throughout this section we assume that R N is a bounded domain with smooth boundary. We are concerned with the study of the following eigenvalue problem u = λf(x, u), in u = 0, on, where f : R R is a given function and λ R is a real number. We say that λ is an eigenvalue of problem (.) if there exists u H0 () \ {0} such that for any v H 0 (), u v dx λ f(x, u)v dx = 0. (.)
12 . PDE s involving the Laplace operator 2 Moreover, if λ is an eigenvalue of problem (.) then u H0 () \ {0} given in the above definition is called the eigenfunction corresponding to the eigenvalue λ. We are interested in finding positive eigenvalues for problems of type (.). The study of eigenvalue problems involving the Laplace operator guides our mind back to a basic result in the elementary theory of partial differential equations which asserts that the problem below (which represents a particular case of problem (.), obtained when f(x, u) = u) u = λu, in u = 0, on, possesses an unbounded sequence of eigenvalues 0 < λ < λ 2... λ n... This celebrated result goes back to the Riesz-Fredholm theory of self-adjoint and compact operators on Hilbert spaces. In what concerns λ, the lowest eigenvalue of problem (.2), we remember that it can be characterized from a variational point of view as the minimum of the Rayleigh quotient, that is, u 2 dx λ = inf. (.3) u H0 ()\{0} u 2 dx Moreover, it is known that λ is simple, that is, all the associated eigenfunctions are merely multiples of each other (see, e.g. Gilbarg and Trudinger [80]). Furthermore, the corresponding eigenfunctions of λ never change signs in. Going further, another type of eigenvalue problems involving the Laplace operator (obtained in the case when we take in (.), f(x, u) = u p 2 u) is given by the nonlinear model equation u = λ u p 2 u, in u = 0, on, where p (, 2 ) \ {2} is a given real number and 2 denotes the critical Sobolev exponent, that is, 2N if N 3 2 = N 2 + if N {, 2}. Using a mountain-pass argument if p > 2 or the fact that the energy functional associated to problem (.4) has a nontrivial (global) minimum point for any positive λ if p < 2, it can be proved that each λ > 0 is an eigenvalue of problem (.4). Thus, in the case of problem (.4) the set of eigenvalues consists of a continuous family, namely the interval (0, ). Motivated by the above results on problems (.2) and (.4) which show that the eigenvalue problems involving the Laplace operator can lead to a discrete spectrum (see the case of problem (.2)) or a continuous spectrum (see the case of problem (.4) ) we consider it important to supplement the above (.2) (.4)
13 . PDE s involving the Laplace operator 3 situations by studying a new eigenvalue problem involving the Laplace operator which possesses, on the one hand, a continuous family of eigenvalues and, on the other hand, at least one more eigenvalue which is isolated in the set of eigenvalues of that problem. We study problem (.) in the case when h(x, t), if t 0 f(x, t) = t, if t < 0, where h : [0, ) R is a Carathéodory function satisfying the following hypotheses (H) there exists a positive constant C (0, ) such that h(x, t) Ct for any t 0 and a.e. x ; (H2) there exists t 0 > 0 such that H(x, t 0 ) := t 0 0 h(x, s) ds > 0, for a.e. x ; h(x,t) (H3) lim t t = 0, uniformly in x. Examples. We point out certain examples of functions h which satisfy hypotheses (H)-(H3):. h(x, t) = sin (t/2), for any t 0 and any x ; 2. h(x, t) = k log( + t), for any t 0 and any x, where k (0, ) is a constant; 3. h(x, t) = g(x)(t q(x) t p(x) ), for any t 0 and any x, where p, q : (, 2) are continuous functions satisfying max p < min q, and g L () satisfies 0 < inf g sup g <. The main result of this section establishes a striking property of eigenvalue problem (.), provided that f is defined as in (.5) and satisfies the above assumptions. More precisely, we prove that the first eigenvalue of the Laplace operator in H0 () is an isolated eigenvalue of (.) and, moreover, any λ sufficiently large is an eigenvalue, while the interval (0, λ ) does not contain any eigenvalue. This shows that problem (.) has both isolated eigenvalues and a continuous spectrum in a neighbourhood of +. Theorem.. ([6, Theorem ]) Assume that f is given by relation (.5) and conditions (H), (H2) and (H3) are fulfilled. (.5) Then λ defined in (.3) is an isolated eigenvalue of problem (.) and the corresponding set of eigenvectors is a cone. Moreover, any λ (0, λ ) is not an eigenvalue of problem (.) but there exists µ > λ such that any λ (µ, ) is an eigenvalue of problem (.). We notice that similar results as those given by Theorem. can be formulated for equations of type (.) if we replace the Laplace operator u by the p-laplace operator, that is p u := div( u p 2 u), with < p <. Certainly, in that case hypotheses (H)-(H3) should be modified according to the new situation. This statement is supported by the fact that the first eigenvalue of the p-laplace operator on bounded domains satisfies similar properties as the one obtained in the case of the Laplace operator (see, e.g., [9]).
14 . PDE s involving the Laplace operator 4.2 Eigenvalues of the Laplace operator with nonlinear boundary conditions An eigenvalue problem on a bounded domain for the Laplacian with a nonlinear Robin-like boundary condition is investigated in this section. We prove the existence, isolation and simplicity of the first two eigenvalues. Assume R N is a bounded domain with smooth boundary. We consider the following eigenvalue problem u = λu in, u (.6) ν = αu + on. where λ R, u/ ν denotes the outward normal derivative of u and u + (x) = max{u(x), 0} for a.e. x. The natural space for nonlinear eigenvalue problems of the type (.6) is the Sobolev space H (). Recall that if u H () then u +, u H () and 0, if [u 0] u + = u, if [u > 0], 0, if [u 0] u = u, if [u < 0], (see, e.g. [80, Theorem 7.6]), where u ± (x) = max{±u(x), 0} for a.e. x. We will say that λ R is an eigenvalue of problem (.6) if there exists u H () \ {0} such that u φ dx + α u + φ dσ(x) = λ uφ dx, (.7) for any φ H (). Such a function u will be called an eigenfunction corresponding to the eigenvalue λ. In fact, u is more regular. Indeed, it is known (see [8, Proposition 2.9, p. 63]) that A = with D(A) = {u H 2 (); u/ ν β(u) a.a. x } is a maximal (cyclically) monotone operator in L 2 (), and moreover there exist some constants C, C 2 > 0 such that v H 2 () C v v L 2 () + C 2, v D(A). Therefore, if u is an eigenfunction of problem (.6) corresponding to some λ, then it is easy to see that the (unique) solution of equation v + Av = f, where f = ( + λ)u, is v = u, thus u H 2 (), and Note that u satisfies problem (.6) in a classical sense. Define λ = u H 2 () C + λ u L 2 () + C 2. (.8) inf v H ()\{0}, v dx 0 v 2 dx + α v 2 dx The main result of this section is given by the following theorem. v+ 2 dσ(x). (.9)
15 . PDE s involving the Laplace operator 5 Theorem.2. ([05, Theorem ]) Numbers λ 0 = 0 and λ (defined by relation (.9)) represent the first two eigenvalues of problem (.6), provided that α > 0 is small. They are isolated in the set of eigenvalues of problem (.6). Moreover, the sets of eigenfunctions corresponding to λ 0 and λ are positive cones (more precisely, one-dimensional half-spaces) in H (). Since problem (.6) has a nonlinear boundary condition, the study of the existence of other eigenvalues (different from λ 0 and λ ) is more difficult than in the case of problems involving linear boundary conditions. Methods which are usually used fail in this case. In this context, we just notice that we cannot apply the Ljusternik-Schnirelman theory in this case, since the Euler-Lagrange energetic functional associated with problem (.6) is not even, a crucial condition required by the application of the quoted method. However, in the one-dimensional case the existence of infinitely many eigenvalues can be easily stated. Note that problem (.6) with = (0, ) becomes u (t) = λu(t) for t (0, ), u (0) = αu + (0), u () = αu + (). (.0) On the other hand, it is known (see, e.g., [89, p. 0]) that the one-dimensional Neumann problem u (t) = λu(t) for t (0, ), (.) u (0) = u () = 0, has the eigenvalues µ k = k 2 π 2, k = 0,,..., with the corresponding eigenfunctions u k (t) = cos(kπt). Simple computations show that for each k Z +, µ 2k is an eigenvalue of problem (.0) with the corresponding eigenfunction u 2k. Finally, let us point out that all the discussion on problem (.6) presented above can be extended (by using similar arguments) to the nonlinear eigenvalue problem p u = λ u p 2 u in, u p 2 u ν = αup + on, where p (, N) is a real number and p = div( p 2 ) stands for the p-laplace operator..3 An eigenvalue problem possessing a continuous family of eigenvalues plus an isolated eigenvalue In this section we analyze an eigenvalue problem, involving a homogeneous Neumann boundary condition, in a smooth bounded domain. We show that the problem possesses, on the one hand, a continuous family of eigenvalues and, on the other hand, exactly one more eigenvalue which is isolated in the set of eigenvalues of the problem.
16 . PDE s involving the Laplace operator 6 Assume R N (N 2) is a bounded domain with smooth boundary. Denote by ν the outward unit normal to. A classical result in the theory of eigenvalue problems assures that problem u = λu in, (.2) u ν = 0 on, possesses a sequence of non-negative eigenvalues (going to + ) and a sequence of corresponding eigenfunctions which define a Hilbert basis in L 2 () (see, e.g. [89, Theorem.2.8]). Moreover, it is known that the first eigenvalue of problem (.2) is λ = 0 and it is isolated and simple (see, e.g. [79, Proposition 4.2.]). Furthermore, the second eigenvalue is characterized from a variational point of view in the following way λ N := inf u W,2 ()\{0}, u dx=0 u 2 dx. u 2 dx Assume that p > 2 is a given real number and consider the eigenvalue problem p u = λu in, (.3) u ν = 0 on, where p u := div( u p 2 u) stands for the p-laplace operator. Using a variational technique based on the fact that the energy functional associated to this problem has a nontrivial minimum for any positive λ it is easy to show that the set of eigenvalues of problem (.3) is exactly the interval [0, ). In other words, the set of eigenvalues in this case is a continuous family. In this section we consider it is important to point out a new situation which can occur in the study of eigenvalue problems for elliptic operators involving homogeneous Neumann boundary conditions. More exactly, we analyze the following eigenvalue problem p u u = λu in, (.4) u ν = 0 on, where λ R and p > 2 is a real number. We will show that this problem possesses, on the one hand, a continuous family of eigenvalues and, on the other hand, exactly one more eigenvalue, which is isolated in the set of eigenvalues of problem (.4). Since p > 2 (and consequently W,p () W,2 ()) it is natural to analyze equation (.4) in the Sobolev space W,p (). Consequently, we will say that λ R is an eigenvalue of problem (.4) if there exists u λ W,p () \ {0} such that ( u λ p 2 + ) u λ φ dx λ u λ φ dx = 0, (.5) for all φ W,p (). Such a function u λ will be called an eigenfunction corresponding to eigenvalue λ. The first main result of this section is given by the following theorem.
17 . PDE s involving the Laplace operator 7 Theorem.3. ([02, Theorem.]) For each p > 2 define u p dx + u 2 dx p λ (p) := inf u W,p ()\{0}, 2. (.6) u dx=0 u 2 dx 2 Then λ (p) > 0 and for each p > 2 fixed, the set of eigenvalues of problem (.4) is given by {0} (λ (p), ). We point out the fact that a similar result with the one of Theorem.3 was obtained in [] for a problem of type (.4) with a homogeneous Dirichlet boundary condition instead of the homogeneous Neumann boundary condition considered in this section. However, in [] only the existence of a continuous family of eigenvalues was established. Thus, the result of Theorem.3 here is more interesting in this new case. Furthermore, Theorem.3 here describes completely the set of eigenvalues of problem (.4) while the result in [] does not describe the entire set of eigenvalues of the problem studied there. The existence of a continuous family of eigenvalues for problem (.4) is a direct consequence of the fact that we deal with a non-homogeneous eigenvalue problem while the presence of the isolated eigenvalue λ 0 = 0 is a consequence of the boundary condition considered in relation to problem (.4). On the other hand, we notice that the proofs in this new situation ask for a different framework than the one used in [] since we deal with another type of boundary condition. Finally, we recall that results concerning a continuous family of eigenvalues plus one more isolated point were also obtained for a different eigenvalue problem involving a homogeneous Dirichlet boundary condition in [6]. Next, we define function λ : [2, ) [λ N, ) where λ (p) is given by expression (.6) from Theorem.3 if p 2 and λ (2) = 2λ N, where λn stands for the second eigenvalue of problem (.2). Our second main result presents certain properties of function λ defined above. Theorem.4. ([02, Theorem.2]) a) Function λ : (2, ) [λ N, ) is non-decreasing. b) For each p (2, ) we have lim λ (s) λ (p) lim λ (s). s p s p c) Function λ : [2, ) [λ N, ) is bounded from above. d) If λ N 2 then there exists p 0 2 such that λ (p 0 ) = p 0. Remark.. We note that hypotheses λ N 2 can occur. For instance if is the ball of radius and centered in the origin in R N then λ N π2 > 2 (see, e.g. [89, Chapter 7, p.0] or L. Payne and H. Weinberger [36]).
18 . PDE s involving the Laplace operator 8 Remark.2. By Theorems.3 and.4 we deduce that there exists p > 2 for which the set of eigenvalues of problem (.4) is given by {0} (p, )..4 An eigenvalue problem involving a degenerate and singular elliptic operator In this section we study an eigenvalue problem involving a degenerate and singular elliptic operator on the whole space R N. We prove the existence of an unbounded and increasing sequence of eigenvalues. Our study generalizes to the case of degenerate and singular operators a result of A. Szulkin and M. Willem [54]. The goal of this section is to study the eigenvalue problem div( x α u(x)) = λg(x)u(x), x R N, (.7) where N 3, α (0, 2), λ > 0 and g : R N R is a function that can change sign on R N satisfying the following basic assumption (G) g L loc (RN ), g + = g + g 2 0, g L N 2 α (R N ) and lim x y x y 2 α g 2 (x) = 0, for all y R N and lim x x 2 α g 2 (x) = 0. Remark. Note that there exists functions h : R N R such that h L N 2 α (R N ) but h satisfies lim x y x y 2 α h(x) = 0, for all y R N and lim x x 2 α h(x) = 0. Indeed, simple computations show that we can take h(x) = x α 2 [log(2 + x 2 α )] (α 2)/N, if x 0 and h(0) =. In the case when α = 0 problem (.7) becomes u(x) = λg(x)u(x), x R N. (.8) For this problem A. Szulkin & M. Willem proved in [54] the existence of an unbounded and increasing sequence of eigenvalues. Motivated by this result on problem (.8) we consider in this section the natural generalization of problem (.8) given by problem (.7), obtained in the case of the presence of the degenerate and singular potential x α in the divergence operator. differential operator div( x α u(x)) This potential leads to a which is degenerate and singular in the sense that lim x 0 x α = 0 and lim x x α =, provided that α (0, 2). Consequently, we will analyze equation (.7) in the case when the operator div( x α u(x)) is not strictly elliptic in the sense pointed out in D. Gilbarg & N. S. Trudinger [80] (see, page 3 in [80] for the definition of strictly elliptic operators). It follows that some of the techniques that can be applied in solving equations involving strictly elliptic operators fail in this new context. For
19 . PDE s involving the Laplace operator 9 instance some concentration phenomena may occur in the degenerate and singular case which lead to a lack of compactness. This is in keeping, on the one hand, with the action of the non-compact group of dilations in R N and, on the other hand, with the fact that we are looking for entire solutions for problem (.7), that means solutions defined on the whole space. Regarding the real-world applications of problems of type (.7) we remember that degenerate differential operators like the one which appears in (.7) are used in the study of many physical phenomena related to equilibrium of anisotropic continuous media (see [49]). In an appropriate context we also note that problems of type (.7) come also from considerations of standing waves in anisotropic Schrödinger equations. A powerful tool that can be useful when we deal with equations of type (.7) is the Caffarelli- Kohn-Nirenberg inequality. More exactly, in 984, L. Caffarelli, R. Kohn & L. Nirenberg proved in [36] (see also [38] and [39]), in the context of some more general inequalities, the following result: given p (, N), for all u C 0 (RN ), there exists a positive constant C a,b such that where ( ) p/q x bq u q dx C a,b x ap u p dx, (.9) R N R N < a < N p, a b a +, q = p Np N p( + a b). The constant C a,b in inequality (.9) is never achieved (see the paper of F. Catrina and Z.-Q. Wang [40] for detailes). Note that the Caffarelli-Kohn-Nirenberg inequality (.9) reduces to the classical Sobolev inequality (if a = b = 0) and to the Hardy inequality (if a = 0 and b = ). Furthermore, its utility is even more important since it implies some Sobolev and Hardy type inequalities in the context of degenerate differential operators. More exactly, in the case when N 3, α (0, 2), p = q = 2, a = α/2 and b = (2 α)/2 then inequality (.9) reads RN u 2 dx C α x 2 α 2, 2 α 2 Inequality (.20) is a Hardy type inequality. The constant C α 2, 2 α 2 R N x α u 2 dx, u C 0 (R N ). (.20) can be chosen ( 2 N 2+α) 2 (see, M. Willem [63, Théorème 20.7]). On the other hand, taking N 3, α (0, 2), p = 2, q = 2N N 2+α, a = α/2, b = 0 in (.9) we find that there exists a positive constant C α := C α,0 > 0 such that the 2 following Sobolev type inequality holds true where 2 α = 2N N 2+α ( ) 2/2 α u 2 α dx Cα x α u 2 dx, u C0 (R N ), (.2) R N R N plays the role of the critical Sobolev exponent in the classical Sobolev inequality.
20 . PDE s involving the Laplace operator 0 Turning back to equation (.7) and taking into account the above discussion we notice that the natural functional space where we can analyze equation (.7) is the closure of C 0 (RN ) under the norm u 2 α = x α u 2 dx. R N Let us denote this space by Wα,2 (R N ). It is easy to see that Wα,2 (R N ) is a Hilbert space with respect to the scalar product u, v α = x α u v dx, R N for all u, v Wα,2 (R N ). Furthermore, according to [40] we have Wα,2 (R N ) = C0 (RN \ {0}) α. On the other hand, we also point out that by construction inequalities (.20) and (.2) hold for all u W,2 α (R N ). We say that λ > 0 is an eigenvalue of problem (.7) if there exists u λ Wα,2 (R N ) \ {0} such that x α u λ φ dx = λ g(x)u λ φ dx, R N R N for all φ Wα,2 (R N ). For each eigenvalue λ > 0 we will call u λ in the above definition an eigenvector corresponding to λ. The main result of this section is given by the following theorem: Theorem.5. ([23, Theorem ]) Assume that condition (G) is fulfilled. Then problem (.7) has an unbounded, increasing sequence of positive eigenvalues.
21 Chapter 2 PDE s involving variable exponent growth conditions The study of PDE s involving variable exponents becomes more and more attractive in the last decades since differential operators involving variable exponent growth conditions can serve in describing nonhomogeneous phenomena which can occur in different branches of science. In this context we remember that the first two models where such kind of operators were considered comes from fluid mechanics, more exactly from the study of electrorheological fluids (Acerbi & Mingione [, 2], Diening [52], Halsey [86], Ružička [48], Rajagopal & Ružička [42], Ružička [49]) and from the study of elastic mechanics (Zhikov [66]). After this pioneering models many other applications of differential operators with variable exponents appeared in a large range of fields, such as image restoration (Chen et al. [4]), mathematical biology (Fragnelli [75]), the study of dielectric breakdown, electrical resistivity, and polycrystal plasticity (Bocea & Mihăilescu [2], Bocea et al. [24]) or in the study of some models for growth of heterogeneous sandpiles (Bocea et al. [23]). On the other hand, important results in the context of potential theory have been obtained by the Research group on variable exponent Lebesgue and Sobolev spaces from Finland (see pharjule/varsob/index.shtml). In particular, we note that a survey paper regarding the most important results in the field of partial differential equations involving variable exponent growth conditions can be found in the section Publications on the web-site quoted above, more exactly (Harjulehto et al. [88]). 2. An overview on variable exponent Lebesgue-Sobolev spaces In this section we provide a brief review of the basic properties of the variable exponent Lebesgue- Sobolev spaces. For more details we refer to the books by Musielak [30] and Diening et al. [54] and the papers by Edmunds et al. [56, 57, 58], Kovacik & Rákosník [93], and Samko & Vakulov [52]. In the following, let R N be an open set and denote by the N-dimensional Lebesgue measure of the set. For any Lipschitz continuous function p : (, ) we denote p = ess inf x p(x) and p + = ess sup x p(x).
22 2. PDE s involving variable exponent growth conditions 2 Usually it is assumed that p + < +, since this condition is known to imply many desirable features for the associated variable exponent Lebesgue space L p( ) (). This function space is defined by { } L p( ) () = u; u is a measurable real-valued function such that u(x) p(x) dx <. On this space we define a norm, the so-called Luxemburg norm, by the formula { } u p( ) = inf µ > 0; u(x) p(x) µ dx. The variable exponent Lebesgue space is a special case of an Orlicz-Musielak space. For constant functions p, L p( ) () reduces to the classical Lebesgue space L p (), endowed with the standard norm ( /p u L p () := u(x) dx) p. We recall that variable exponent Lebesgue spaces are separable and reflexive Banach spaces. If 0 < < and p, p 2 are variable exponents such that p (x) p 2 (x) everywhere in then there exists the continuous embedding L p 2( ) () L p ( ) (). We denote by L p ( ) () the conjugate space of L p( ) (), where /p(x) + /p (x) =. For any u L p( ) () and v L p ( ) () the Hölder type inequality ( uv dx p + ) u p( ) v p ( ) (2.) holds true. Moreover, if p, p 2, p 3 : (, ) are three Lipschitz continuous functions such that /p (x) + /p 2 (x) + /p 3 (x) = then for any u L p ( ) (), v L p 2( ) () and w L p 3( ) () the following inequality holds (see [64, Proposition 2.5]) ( uvw dx p + p 2 p + ) p 3 u p ( ) v p2 ( ) w p3 ( ). (2.2) An important role in manipulating the generalized Lebesgue Sobolev spaces is played by the modular of the L p( ) () space, which is the mapping ρ p( ) : L p( ) () R defined by ρ p( ) (u) = u(x) p(x) dx. Lebesgue Sobolev spaces with p + = + have been investigated in [56, 93]. denote = {x ; p(x) = + } and define the modular by setting ρ p( ) (u) = u(x) p(x) dx + ess sup x u(x). \ If (u n ), u L p( ) () then the following relations hold true In such a case we u p( ) > u p p( ) ρ p( )(u) u p+ p( ), (2.3)
23 2. PDE s involving variable exponent growth conditions 3 Next, we define the variable exponent Sobolev space u p( ) < u p+ p( ) ρ p( )(u) u p p( ), (2.4) u n u p(x) 0 ρ p( ) (u n u) 0. (2.5) W,p( ) () = {u L p( ) () : u L p( ) ()}. On W,p( ) () we may consider one of the following equivalent norms or u = inf { µ > 0; u p( ) = u p( ) + u p( ) ( u(x) µ p(x) + u(x) p(x)) } dx, µ where, in the definition of u p( ), u p( ) stands for the Luxemburg norm of u. We also define W,p( ) 0 () as the closure of C0 () in W,p( ) (). Assuming p >, then the function spaces W,p( ) () and W,p( ) 0 () are separable and reflexive Banach spaces. Set ( ϱ p( ) (u) = u(x) p(x) + u(x) p(x)) dx. For all (u n ), u W,p( ) 0 () the following relations hold u > u p ϱ p( ) (u) u p+, (2.6) u < u p+ ϱ p( ) (u) u p, (2.7) u n u 0 ϱ p( ) (u n u) 0. (2.8) We remember some embedding results regarding variable exponent Lebesgue Sobolev spaces. If p, q : (, ) are Lipschitz continuous and p + < N and p(x) q(x) p (x) for any x where p (x) = Np(x)/(N p(x)), then there exists a continuous embedding W,p( ) 0 () L q( ) (). Furthermore, assuming that 0 is a bounded subset of, then the embedding W,p( ) 0 ( 0 ) L q( ) ( 0 ) is continuous and compact, provided that q(x) < p (x) for any x, where p (x) = Np(x)/(N p(x)) if p(x) < N and p (x) = if p(x) N. Furthermore, in this last case on the Sobolev space W,p( ) 0 ( 0 ) we can consider the equivalent norm u 0 = u p( ). Finally, we consider the case when R N is open and bounded. In this case we introduce a natural generalization of the variable exponent Sobolev space W,p( ) 0 () that will enable us to study with sufficient accuracy problems involving anisotropic variable exponent operators. For this purpose, let us denote by p : R N the vectorial function p = (p,..., p N ), where p i : (, ) are
24 2. PDE s involving variable exponent growth conditions 4 continuous functions for each i {,..., N}. We define W, p ( ) 0 (), the anisotropic variable exponent Sobolev space, as the closure of C0 () with respect to the norm u p ( ) = N xi u pi ( ). i= As it was pointed out in [0], W, p ( ) 0 () is a reflexive Banach space. We also point out that in the case when p i : (, ) are constant functions for any i {,.., N} the resulting anisotropic Sobolev space is denoted by W, p 0 (), where p is the constant vector (p,..., p N ). The theory of such spaces was developed in [74, 62, 43, 44, 6, 33]. On the other hand, in order to facilitate the manipulation of space W, p ( ) 0 () we introduce P +, P R N as and P + +, P +, P R+ as P + = (p +,..., p+ N ), P = (p,..., p N ), P + + = max{p+,..., p+ N }, P + = max{p,..., p N }, P = min{p,..., p N }. Here we always assume that and define P R + and P, R + by N p i= i >, (2.9) P = N N i= /p i, P, = max{p +, P }. We recall that if s : (, ) is continuous and satisfies < s(x) < P, for all x, then the embedding W, p ( ) 0 () L s( ) () is compact (see [0, Theorem ] or [09]). 2.2 Eigenvalue problem p(x) u = λ u p(x) 2 u and related problems Eigenvalue problems involving variable exponent growth conditions represent a starting point in analyzing more complicated equations. A first contribution in this sense is the paper of Fan et al. [66] where the following eigenvalue problem has been considered p(x) u = λ u p(x) 2 u in, where R N u = 0 on, (2.0) is a bounded domain with smooth boundary, p : (, ) is a continuous function, p(x) u := div( u p(x) 2 u) stands for the p(x)-laplace operator and λ is a real number. The result in [66] establishes the existence of infinitely many eigenvalues for problem (2.0) by using
25 2. PDE s involving variable exponent growth conditions 5 an argument based on the Ljusternik-Schnirelmann critical point theory. Denoting by Λ the set of all nonnegative eigenvalues, the authors showed that sup Λ = + and they pointed out that only under special conditions, which are somehow connected with a kind of monotony of function p(x), we have inf Λ > 0 (this is in contrast with the case when p(x) is a constant; then, we always have inf Λ > 0). We notice that the above discussion is in keeping with the fact that considering, the Rayleigh quotient associated with problem (2.0), that is µ := inf u C 0 ()\{0} u p(x) dx, u p(x) dx we often have µ = 0 for general p(x). A simple example in that sense is illustrated by Fan & Zhao in [67], pages More exactly, letting = ( 2, 2) R and defining p(x) = 3 if 0 x, and p(x) = 4 x if x 2 simple computations yield µ = 0. Actually, the following result was proved by Fan et al. (see [66, Theorem 3.]): Theorem 2.. If there exists an open set U and a point x 0 U such that either p(x 0 ) < p(x) for all x U or p(x 0 ) < p(x) for all x U then µ = 0. On the other hand, a necessary and sufficient condition such that µ > 0 has not yet been obtained excepting the case when N = (in that case, the infimum is positive if and only if p(x) is a monotone function, see [66, Theorem 3.2]). However, the authors of [66] pointed out that in the case N > a sufficient condition to have µ > 0 is to exist a vector l R N \ {0} such that, for any x, function f(t) = p(x + tl) is monotone, for t I x := {s; x + sl } (see [66, Theorem 3.3]). Assuming p is of class C the monotony of function f reads as follows: either p(x + tl) l 0, for all t I x, x, or p(x + tl) l 0, for all t I x, x. We can supplement the above results in a sense that will be described below An equation with variable exponents possessing singularities Assume R N (N 2) is an open, bounded and smooth set. For each x, x = (x,..., x N ) and i {,..., N} we denote m i = inf x x i M i = sup x i. x For each i {,..., N} let a i : [m i, M i ] R be functions of class C. Particularly, functions a i are allowed to vanish. Let a : R N be defined by a (x) = (a (x ),..., a N (x N )).
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