Polyharmonic k -Hessian equations inRN

Abstract This work is focused on the study of the nonlinear elliptic higher order equation ( − Δ ) m u = S k [ − u ] + λ f , x ∈ R N , where the k-Hessian S k [ u ] is the kth elementary symmetric polynomial of eigenvalues of the Hessian matrix of the solution and the datum f belongs to a suitable functional space. This problem is posed in R N and we prove the existence of at least one solution by means of topological fixed point methods for suitable values of m ∈ N . Questions related to the regularity of the solutions and extensions of these results to the nonlocal setting are also addressed. On the way to construct these proofs, some technical results such as a fixed point theorem and a refinement of the critical Sobolev embedding, which could be of independent interest, are introduced.

[1]  J. Dávila,et al.  Multiplicity of solutions for a fourth order equation with power-type nonlinearity , 2010 .

[2]  Yu. G. Reshetnyak.  Weak convergence of completely additive vector functions on a set , 1968 .

[3]  Y. Meyer,et al.  Compensated compactness and Hardy spaces , 1993 .

[4]  H. Brezis,et al.  Degree theory and BMO; part I: Compact manifolds without boundaries , 1995 .

[5]  E. Berchio,et al.  ftp ejde.math.txstate.edu (login: ftp) SOME REMARKS ON BIHARMONIC ELLIPTIC PROBLEMS WITH POSITIVE, INCREASING AND CONVEX NONLINEARITIES , 2022 .

[6]  S. Müller.  Higher integrability of determinants and weak convergence in L1. , 1990 .

[7]  Xu-jia Wang.  Existence of multiple solutions to the equations of Monge-Ampère type , 1992 .

[8]  S. Muêller.  Det = det. A remark on the distributional determinant , 1990 .

[9]  Luis A. Caffarelli,et al.  The Dirichlet problem for nonlinear second-order elliptic equations I , 1984 .

[10]  H. Brezis,et al.  Integrability for the Jacobian of orientation preserving mappings , 1993 .

[11]  B. Dacorogna,et al.  On the optimality of certain Sobolev exponents for the weak continuity of determinants , 1992 .

[12]  The Yamabe problem for higher order curvatures , 2005, math/0505463.

[13]  H. Grunau,et al.  Supercritical biharmonic equations with power-type nonlinearity , 2007, 0711.2202.

[14]  M. Giaquinta.  Cartesian currents in the calculus of variations , 1983 .

[15]  Tadeusz Iwaniec,et al.  Geometric Function Theory and Non-linear Analysis , 2002 .

[16]  N. Trudinger,et al.  Boundary regularity for the Monge-Ampere and affine maximal surface equations , 2005, math/0509342.

[17]  Stefan Müller,et al.  On a new class of elastic deformations not allowing for cavitation , 1994 .

[18]  Xu-jia Wang,et al.  A variational theory of the Hessian equation , 2001 .

[19]  T. Iwaniec,et al.  ${\cal H}^1$-estimates of Jacobians by subdeterminants , 2002 .

[20]  Filippo Gazzola,et al.  Polyharmonic Boundary Value Problems: Positivity Preserving and Nonlinear Higher Order Elliptic Equations in Bounded Domains , 2010 .

[21]  Amir Moradifam.  THE SINGULAR EXTREMAL SOLUTIONS OF THE BI-LAPLACIAN WITH EXPONENTIAL NONLINEARITY , 2009, 0905.1937.

[22]  N. Trudinger,et al.  Hessian Measures III , 2002 .

[23]  F. Bonsall,et al.  Lectures on some fixed point theorems of functional analysis , 1962 .

[24]  Carlos Escudero,et al.  On polyharmonic regularizations of $k-$Hessian equations: Variational methods , 2015, 1507.05435.

[25]  P. Hajłasz.  A note on weak approximation of minors , 1995 .

[26]  C. Escudero,et al.  Existence of radial solutions to biharmonic $k-$Hessian equations , 2015, 1503.06700.

[27]  H. Grunau,et al.  The Dirichlet problem for supercritical biharmonic equations with power-type nonlinearity , 2007 .

[28]  Xu Wang.  Some counterexamples to the regularity of Monge-Ampère equations , 1995 .

[29]  Xu-jia Wang.  The k-Hessian Equation , 2009 .

[30]  Timothy S. Murphy,et al.  Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals , 1993 .

[31]  Juan Dávila,et al.  Stable Solutions for the Bilaplacian with Exponential Nonlinearity , 2007, SIAM J. Math. Anal..

[32]  Michiel Hazewinkel,et al.  Cartesian currents in the calculus of variations , 2003 .

[33]  Hessian measures II , 1999, math/9909199.

[34]  Gianni Arioli,et al.  A Semilinear Fourth Order Elliptic Problem with Exponential Nonlinearity , 2005, SIAM J. Math. Anal..

[35]  Elias M. Stein,et al.  On the theory of harmonic functions of several variables , 1960 .

[36]  T. Iwaniec,et al.  Estimates of Jacobians by subdeterminants , 2002 .

[37]  C. Escudero,et al.  On radial stationary solutions to a model of non-equilibrium growth , 2013, European Journal of Applied Mathematics.

[38]  C. B. Morrey.  Multiple Integrals in the Calculus of Variations , 1966 .

[39]  James R. Munkres,et al.  Elements of algebraic topology , 1984 .

[40]  N. Trudinger,et al.  A Poincaré type inequality for Hessian integrals , 1998 .

[41]  J. Spruck,et al.  The Dirichlet problem for nonlinear second order elliptic equations, III: Functions of the eigenvalues of the Hessian , 1985 .

[42]  Existence and nonexistence results for a singular boundary value problem arising in the theory of epitaxial growth , 2013, 1309.5659.

[43]  D. Labutin.  Potential estimates for a class of fully nonlinear elliptic equations , 2002 .

[44]  Neil S. Trudinger,et al.  On the weak continuity of elliptic operators and applications to potential theory , 2002 .

[45]  N. Trudinger,et al.  Hessian measures I , 1997 .

[46]  H. Brezis,et al.  On the distributional Jacobian of maps from SN into SN in fractional Sobolev and Holder spaces , 2011 .

[47]  Neil S. Trudinger,et al.  On the Dirichlet problem for Hessian equations , 1995 .

[48]  N. Ghoussoub,et al.  The Critical Dimension for a Fourth Order Elliptic Problem with Singular Nonlinearity , 2008, 0904.2414.

[49]  L. Grafakos.  HARDY SPACE ESTIMATES FOR MULTILINEAR OPERATORS , II , 1992 .

[50]  Mark S. C. Reed,et al.  Method of Modern Mathematical Physics , 1972 .

[51]  L. Grafakos.  Hardy Space Estimates for Multilinear Operators, II , 1992 .

[52]  J. Ball.  Convexity conditions and existence theorems in nonlinear elasticity , 1976 .

[53]  S. Trudinger.  Weak solutions of hessian equations , 1997 .

[54]  Claudio Procesi,et al.  The invariant theory of n × n matrices , 1976 .

[55]  H. Brezis,et al.  The Jacobian determinant revisited , 2011 .

[56]  T. Iwaniec,et al.  On the integrability of the Jacobian under minimal hypotheses , 1992 .

[57]  N. Trudinger,et al.  The Dirichlet Problem for Degenerate Hessian Equations , 2005 .

[58]  H. Brezis,et al.  Degree theory and BMO; part II: Compact manifolds with boundaries , 1995 .

[59]  L. Caffarelli.  Interior $W^{2,p}$ estimates for solutions of the Monge-Ampère equation , 1990 .

[60]  F. Gazzola,et al.  Global existence versus blow-up results for a fourth order parabolic PDE involving the Hessian , 2015, 1503.06697.

[61]  C. Escudero,et al.  Some fourth order nonlinear elliptic problems related to epitaxial growth , 2013, 1309.5656.

[62]  S. Müller.  On the singular support of the distributional determinant , 1993 .