Abstract
We study nonhomogeneous elliptic problems considering a general linear elliptic operator with singular critical potentials and nonlinearities depending on multiplier operators that can be derivatives (even fractional) and singular integral operators. The general elliptic operator can contain derivatives of high-order and fractional type like polyharmonic operators and fractional Laplacian. We obtain results about existence and qualitative properties in a space whose norm is based on the Fourier transform. Our approach is of non-variational type and consists in a contraction argument in a critical space for the studied elliptic PDEs. Examples of applications are given.
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A. Ambrosetti, A. Malchiodi, Nonlinear Analysis and Semilinear Elliptic Problems. Cambridge University Press, 2007.
Alarcón S., García-Melián J., Quaas A.: Keller-Osserman type conditions for some elliptic problems with gradient terms. J. Differential Equations 255(2), 886–914 (2012)
Benci V., Fortunato D.: Solitary waves of the nonlinear Klein–Gordon equation coupled with Maxwell equations. Rev. Math. Phys. 14, 409–420 (2002)
Brézis H., Nirenberg L.: Louis Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Comm. Pure Appl. Math. 36(4), 437–477 (1983)
Brézis H., Nirenberg L.: Characterizations of the ranges of some nonlinear operators and applications to boundary value problems. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 5(2), 225–326 (1978)
Biler P., Cannone M., Guerra I. A., Karch G.: Global regular and singular solutions for a model of gravitating particles. Math. Ann. 330(4), 693–708 (2004)
Cabré X., Tan J.: Positive solutions of nonlinear problems involving the square root of the Laplacian. Adv. Math., 224(5), 2052–2093 (2010)
Caffarelli L., Jin T., Sire Y., Xiong J.: Local analysis of solutions of fractional semilinear elliptic equations with isolated singularities. Arch. Ration. Mech. Anal. 213(1), 245–268 (2014)
Cannone M., Karch G.: Smooth or singular solutions to the Navier-Stokes system. J. Differential Equations 197(2), 247–274 (2004)
Carrillo J. A., Ferreira L. C. F.: Self-similar solutions and large time asymptotics for the dissipative quasi-geostrophic equation. Monatsh. Math. 151(2), 111–142 (2007)
Castro A., Córdoba D.: Global existence, singularities and ill-posedness for a nonlocal flux. Adv. Math. 219(6), 1916–1936 (2008)
Chae D., Constantin P., Wu J.: Dissipative models generalizing the 2D Navier–Stokes and the surface quasi-geostrophic equations. Indiana Univ. Math. J. 61(5), 1997–2018 (2012)
Chen H., Felmer P., Quaas A.: Large solutions to elliptic equations involving fractional Laplacian. Ann. Inst. H. Poincar Anal. Non Linaire 32(6), 1199–1228 (2015)
Chen H., Véron L.: Semilinear fractional elliptic equations involving measures. J. Differential Equations 257(5), 1457–1486 (2014)
Chen H., Véron L.: Semilinear fractional elliptic equations with gradient nonlinearity involving measures. J. Funct. Anal. 266(8), 5467–5492 (2014)
Colin M., Ohta M.: Stability of solitary waves for derivative nonlinear Schrödinger equation. Ann. Inst. H. Poincaré Anal. Non Linéaire 23(5), 753–764 (2006)
Córdoba A., Córdoba D., Fontelos M. A.: Formation of singularities for a transport equation with nonlocal velocity. Ann. of Math. 162(3), 1377–1389 (2005)
Z. Du, Singular solutions of semilinear elliptic equations with fractional Laplacian in entire space, Commun. Contemp. Math. 18 (1) (2016), 1550024, 8 pp.
Fall M. M., Felli V.: Unique continuation property and local asymptotics of solutions to fractional elliptic equations. Comm. Partial Differential Equations 39(2), 354–397 (2014)
Felli V., Marchini E. M., Terracini S.: On Schrödinger operators with multisingular inverse-square anisotropic potentials. Indiana Univ. Math. J. 58, 617–676 (2009)
Ferreira L. C. F., Montenegro M.: A Fourier approach for nonlinear equations with singular data. Israel Journal of Mathematics 193, 83–107 (2013)
Ferreira L. C. F., Mesquita C. A. A. S.: Existence and symmetries for elliptic equations with multipolar potentials and polyharmonic operators. Indiana University Mathematics Journal 62, 1955–1982 (2013)
L. C. F. Ferreira, C. A. A. S. Mesquita, An approach without using Hardy inequality for the linear heat equation with singular potential, Commun. Contemp. Math. 17 (5) (2015), 1550041, 16 pp.
Fila M., Quittner P.: Radial positive solutions for a semilinear elliptic equation with a gradient term. Adv. Math. Sci. Appl. 2(1), 39–45 (1993)
Huang S., Li W.-T., Tian Q., Mu C.: Large solution to nonlinear elliptic equation with nonlinear gradient terms. J. Differential Equations 251, 3297–3328 (2011)
Karch G., Schonbek M. E.: On zero mass solutions of viscous conservation law. Communications in Partial Differential Equations 27(9-10), 2071–2100 (2002)
E. M. Landesman, A. C. Lazer, Nonlinear perturbations of linear elliptic boundary value problems at resonance, J. Math. Mech. 19 (1969/1970), 609–623.
Lasry J. M., Lions P. L.: Nonlinear elliptic equations with singular boundary conditions and stochastic control with state constraints. I. The model problem. Math. Ann. 283, 583–630 (1989)
Lions P.-L.: Solutions of Hartree–Fock equations for Coulomb systems. Comm. Math. Phys. 109, 33–97 (1984)
E. Lieb, M. Loss, Analysis, 2nd ed., Graduate Studies in Mathematics 14, American Mathematical Society, Providence, RI, 2001.
Lieb E. H., Simon B.: The Thomas–Fermi theory of atoms, molecules and solids. Adv. Math. 23, 22–116 (1977)
Miao C., Yuan B.: Solutions to some nonlinear parabolic equations in pseudomeasure spaces. Math. Nachr. 280(1-2), 171–186 (2007)
Nagle K., Pothoven K., Singkofer K.: Nonlinear elliptic equations at resonance where the nonlinearity depends essentially on the derivatives. J. Differential Equations 38(2), 210–225 (1980)
Ohta M.: Instability of solitary waves for nonlinear Schr¨odinger equations of derivative type. SUT J. Math. 50(2), 399–415 (2014)
Pohozaev S. I.: The Dirichlet problem for the equation \({\Delta u = u^2}\). Dokl. Akad. Nauk SSSR 134, 769–772 (1960)
A. Porretta, S. Segura de León, Nonlinear elliptic equations having a gradient term with natural growth, J. Math. Pures Appl. 85 (2006), 465–492.
P. H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, CBMS Regional Conference Series in Mathematics 65, American Mathematical Society, Providence, RI, 1986.
Ruiz D.: The Schrödinger–Poisson equation under the effect of a nonlinear local term. J. Funct. Anal. 237, 655–674 (2006)
Souplet P., Tayachi S., Weissler F. B.: Exact Self-Similar Blow-up of Solutions of a Semilinear Parabolic Equation with a Nonlinear Gradient Term. Indiana Univ. Math. J. 45, 655–682 (1996)
Tan J.: The Brezis-Nirenberg type problem involving the square root of the Laplacian. Calc. Var. Partial Differential Equations, 42(1-2), 21–41 (2011)
R. Temam, Navier-Stokes equations. Theory and numerical analysis, 3nd ed., Studies in Mathematics and its Applications 2, North-Holland Publishing Co., Amsterdam, 1984.
M. Willem, Minimax theorems, Progress in Nonlinear Differential Equations and their Applications 24, New York, Birkhauser, 1996.
Xue H., Shao X.: Existence of positive entire solutions of a semilinear elliptic problem with a gradient term. Nonlinear Anal. 71(7-8), 3113–3118 (2009)
Zhang Z.: Boundary blow-up elliptic problems with nonlinear gradient terms and singular weights. Proceedings of the Royal Society of Edinburgh 138A, 1403–1424 (2008)
Zhang Z.: The asymptotic behaviour of solutions with boundary blow-up for semilinear elliptic equations with nonlinear gradient terms. Nonlinear Analysis 62, 1137–1148 (2005)
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L.C.F. Ferreira was supported by FAPESP-SP and CNPQ, Brazil. (corresponding author).
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Ferreira, L.C., Castañeda-Centurión, N.F. A Fourier Analysis Approach to Elliptic Equations with Critical Potentials and Nonlinear Derivative Terms. Milan J. Math. 85, 187–213 (2017). https://doi.org/10.1007/s00032-017-0269-6
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DOI: https://doi.org/10.1007/s00032-017-0269-6