Contributions To Nonlinear Elliptic Equations And Systems: A Tribute To Djairo Guedes De Figueiredo On The Occasion Of His 80th Birthday (progress In Nonlinear Differential Equations And Their Applications 86)

This volume of contributions pays tribute to the life and work of Djairo Guedes de Figueiredo on the occasion of his 80th birthday. The articles it contains were born out of the ICMC Summer Meeting on Differential Equations – 2014 Chapter, also dedicated to de Figueiredo and held at the Universidade de São Paulo at São Carlos, Brazil from February 3-7, 2014. The contributing authors represent a group of international experts in the field and discuss recent trends and new directions in nonlinear elliptic partial differential equations and systems. Djairo Guedes de Figueiredo has had a very active scientific career, publishing 29 monographs and over one hundred research articles. His influence on Brazilian mathematics has made him one of the pillars of the subject in that country. He had a major impact on the development of analysis, especially in its application to nonlinear elliptic partial differential equations and systems throughout the entire world. The articles collected here pay tribute to him and his legacy and are intended for graduate students and researchers in mathematics and related areas who are interested in nonlinear elliptic partial differential equations and systems. Preface 8 Contents 10 Contributors 12 Nonexistence of positive classical solutions for the nonlinear Schrödinger equation with unbounded or decaying weights 16 1 Introduction and main result 16 2 Proof of Theorem 1.3 19 References 21 Hylomorphic solitons for the generalized KdV equation 23 1 Introduction 23 2 Solitary waves and solitons 24 2.1 Solitary waves 25 2.2 Orbitally stable states and solitons 26 2.3 Hylomorphic solitons 27 3 Hylomorphic solitons for the nonlinear Schrödinger equation 28 4 Hylomorphic solitons for the generalized KdV equation 30 References 34 A consequence of Djairo's Lectures on the Ekelandvariational principle 36 1 Introduction and statement of the result 36 2 Proof of the main result 38 3 The impact of a lower order term 42 References 43 Waveguide solutions for a nonlinear Schrödinger equation with mixed dispersion 44 1 Introduction 44 2 Functional framework 48 3 Existence of minimizers 50 4 Sign and symmetry 52 5 The effect of a small fourth order dissipation 54 6 Sign-changing radial minimizer 62 7 Comments 65 References 66 Fibers and global geometry of functions 67 1 Introduction 67 2 The first examples in infinite dimension 69 2.1 Homeomorphisms: Dolph and Hammerstein 69 2.2 Breaking the barrier: the Ambrosetti-Prodi theorem 69 3 Fibers and height functions 71 4 Adapted coordinates and a plan 72 5 Obtaining fibers in other contexts 75 5.1 Podolak's approach 75 5.2 Transplanting fibers 75 5.3 Fibers and Numerics 76 6 Asymptotics of F on fibers and vertical lines 77 6.1 F along fibers 78 6.2 Asymptotic geometry of fibers 78 6.3 Comparing F on fibers and on vertical lines 79 6.4 Fibers and the properness of F 80 7 Singularities 80 7.1 Morin theory in adapted coordinates 81 7.2 Critical points of the height function 82 8 Some examples 82 8.1 The non-autonomous case 82 8.2 Schrödinger operators on Rn 83 8.3 Perturbations of compact operators 84 8.4 Folds as perturbations of non-self-adjoint operators 85 References 85 Existence of infinitely many continua of radial singular polytropes with gain-loss function 88 1 Introduction 88 2 Existence of a two parameter family of singular solutions 89 3 Proof of Theorems 1.1 and 1.2 97 References 97 Branches of positive solutions for subcritical elliptic equations 98 1 Introduction 98 2 On the a-priori bounds for any regular domain 100 References 107 Multiple solutions to anisotropic critical and supercritical problems in symmetric domains 109 1 Introduction and statement of results 109 1.1 The critical problem 110 1.2 The supercritical case 111 2 Proofs of the main results 113 2.1 The proof of Theorems 1.1 and 1.2 113 2.2 The proof of Theorems 1.3 and 1.4 116 3 Representation of G-invariant Palais-Smale sequences 117 References 129 Compactness and non-compactness for Yamabe-type problems 131 1 Introduction 131 2 Compactness and noncompactness 132 3 Manifolds with boundary 135 4 Stability question and systems 137 5 Higher order Yamabe problems 138 References 139 Asymptotics of ground states for fractional Hénon systems 142 1 Introduction and main results 142 2 Preliminaries 148 3 Proof of Theorem 1.3 155 4 Proof of Theorem 1.4 159 References 168 Positive solutions for certain classes of fourth-order ordinary elliptic systems 171 1 Introduction 171 2 The associate functional differential equations 173 3 Locally superlinear nonlinearities 173 4 A fixed-point technique 174 5 Proof of Theorem 3.1 177 6 Proof of Theorem 1.1 179 References 179 Remarks on the behavior of the best Sobolev constants 180 1 Introduction 180 2 Preliminaries 184 3 Differentiability 184 4 Hölder regularity 189 References 191 Bubbling solutions to an anisotropic Hénon equation 193 1 Introduction 193 2 Preliminaries 195 3 The finite-dimensional reduction 199 3.1 The linear theory 199 3.2 The one-dimensional reduction 209 3.3 The one-dimensional problem 212 References 220 The sub-supersolution method for Kirchhoff systems: applications 222 1 Introduction 222 2 The sub-super method for non-local systems 223 3 The sub-supersolution for Kirchhoff systems 225 4 Applications 227 4.1 Non-local Lotka–Volterra models 227 4.2 Kirchhoff systems 230 References 231 Nodal structure of the solution of a cooperative elliptic system 233 1 Introduction 233 2 Preliminaries 235 3 Main Result 236 4 Proofs 239 References 246 Nonquadraticity condition on superlinear problems 248 1 Introduction 248 2 Proof of the main theorem 251 References 257 Radial solutions of a Neumann problem coming from aburglary model 259 1 The problem 259 2 The homotopy 262 3 The a priori estimates 262 4 The associated linear system 266 5 The fixed point reduction 268 6 The existence theorem and applications 269 References 270 Modeling suspension bridges through the von Kármán quasilinear plate equations 271 1 Introduction and motivations: nonlinear behavior of suspension bridges 271 2 Functional framework and the quasilinear equations 274 2.1 Elastic energies of a plate 274 2.2 The Euler-Lagrange equation 277 2.3 Boundary conditions 279 2.4 The quasilinear von Kármán equations modeling suspension bridges 280 3 Main results 282 4 Preliminaries: some useful operators and functionals 285 5 Proof of Theorem 3.1 289 6 Proof of Theorem 3.2 292 7 Proof of Theorem 3.3 296 References 298 Nonlinear Klein-Gordon-Maxwell systems with Neumann boundary conditions on a Riemannian manifold with boundary 300 1 Introduction 300 2 Preliminary results 302 2.1 The Lyapunov Schmidt reduction 308 3 Reduction to finite dimensional space 309 4 The reduced functional 313 4.1 Sketch of the proof of Theorem 1.2 319 5 Appendix: Technical lemmas 319 References 323 Asymptotic analysis for radial sign-changing solutions of the Brezis-Nirenberg problem in low dimensions 325 1 Introduction 325 2 Some preliminary results 328 3 Energy and asymptotic analysis of the positive part 332 4 Asymptotic analysis of the negative part in dimension N=6 335 5 Asymptotic analysis of the negative part in dimension N=3,4,5 341 References 342 Critical and noncritical regions on the critical hyperbola 344 1 Introduction 344 2 Compactness and proof of Theorem 1.2 350 3 On the ground state solutions of (S) 354 4 More on compactness 356 5 Multiplicity of solutions: proof of Theorem 1.3 357 6 Appendix: Proof of Lemma 2.6 362 References 368 A nonlinear Steklov problem arising in corrosion modeling 370 1 Introduction 370 2 Global existence of the bifurcation solutions 372 3 Analysis of the first branch 375 Blow-up analysis 379 4 Numerical simulations 381 References 385 Remarks on the p-Laplacian on thin domains 387 1 Introduction 387 2 Dirichlet boundary condition 389 2.1 The thin domain problem 389 2.2 Nonlinearities 392 3 Neumann boundary condition 393 3.1 Convergence 396 3.2 Nonlinearities 398 4 A specific example 399 References 400 Spatial trajectories and convergence to traveling fronts for bistable reaction-diffusion equations 402 1 Introduction 402 2 Preliminaries 406 2.1 Phase space and traveling fronts 406 2.2 Properties of Ω(u) 410 2.3 Zero number 411 3 Proofs of Theorems 1.1, 1.3, and Corollary 1.2 412 References 420 Regularity estimates for fully non linear elliptic equations which are asymptotically convex 421 1 Introduction 421 2 Preliminaries 424 3 C1,α0- estimates 426 4 Proof of Corollary 1.3 429 5 Proof of Theorem 1.4 429 References 433 Front Matter....Pages i-xiv Nonexistence of positive classical solutions for the nonlinear Schrödinger equation with unbounded or decaying weights....Pages 1-7 Hylomorphic solitons for the generalized KdV equation....Pages 9-21 A consequence of Djairo’s Lectures on the Ekeland variational principle....Pages 23-30 Waveguide solutions for a nonlinear Schrödinger equation with mixed dispersion....Pages 31-53 Fibers and global geometry of functions....Pages 55-75 Existence of infinitely many continua of radial singular polytropes with gain-loss function....Pages 77-86 Branches of positive solutions for subcritical elliptic equations....Pages 87-97 Multiple solutions to anisotropic critical and supercritical problems in symmetric domains....Pages 99-120 Compactness and non-compactness for Yamabe-type problems....Pages 121-131 Asymptotics of ground states for fractional Hénon systems....Pages 133-161 Positive solutions for certain classes of fourth-order ordinary elliptic systems....Pages 163-171 Remarks on the behavior of the best Sobolev constants....Pages 173-185 Bubbling solutions to an anisotropic Hénon equation....Pages 187-215 The sub-supersolution method for Kirchhoff systems: applications....Pages 217-227 Nodal structure of the solution of a cooperative elliptic system....Pages 229-243 Nonquadraticity condition on superlinear problems....Pages 245-255 Radial solutions of a Neumann problem coming from a burglary model....Pages 257-268 Modeling suspension bridges through the von Kármán quasilinear plate equations....Pages 269-297 Nonlinear Klein-Gordon-Maxwell systems with Neumann boundary conditions on a Riemannian manifold with boundary....Pages 299-323 Asymptotic analysis for radial sign-changing solutions of the Brezis-Nirenberg problem in low dimensions....Pages 325-343 Critical and noncritical regions on the critical hyperbola....Pages 345-370 A nonlinear Steklov problem arising in corrosion modeling....Pages 371-387 Remarks on the p-Laplacian on thin domains....Pages 389-403 Spatial trajectories and convergence to traveling fronts for bistable reaction-diffusion equations....Pages 405-423 Regularity estimates for fully non linear elliptic equations which are asymptotically convex....Pages 425-438 This volume of contributions pays tribute to the life and work of Djairo Guedes de Figueiredo on the occasion of his 80th birthday. The articles it contains were born out of the ICMC Summer Meeting on Differential Equations ĺl 2014 Chapter, also dedicated to de Figueiredo and held at the Universidade de S©Đo Paulo at S©Đo Carlos, Brazil from February 3-7, 2014. The contributing authors represent a group of international experts in the field and discuss recent trends and new directions in nonlinear elliptic partial differential equations and systems. Djairo Guedes de Figueiredo has had a very active scientific career, publishing 29 monographs and over one hundred research articles. His influence on Brazilian mathematics has made him one of the pillars of the subject in that country. He had a major impact on the development of analysis, especially in its application to nonlinear elliptic partial differential equations and systems throughout the entire world. The articles collected here pay tribute to him and his legacy, and are intended for graduate students and researchers in mathematics and related areas who are interested in nonlinear elliptic partial differential equations and systems