Groundstates of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics

We consider a semilinear elliptic problem \[ - \Delta u + u = (I_\alpha \ast \abs{u}^p) \abs{u}^{p - 2} u\quad\text{in \(\R^N\),} \] where \(I_\alpha\) is a Riesz potential and \(p>1\). This family of equations includes the Choquard or nonlinear Schr\"odinger--Newton equation. For an optimal range of parameters we prove the existence of a positive groundstate solution of the equation. We also establish regularity and positivity of the groundstates and prove that all positive groundstates are radially symmetric and monotone decaying about some point. Finally, we derive the decay asymptotics at infinity of the groundstates.

[1]  Lin Zhao,et al.  Classification of Positive Solitary Solutions of the Nonlinear Choquard Equation , 2010 .

[2]  G. Burton.  Sobolev Spaces , 2013 .

[3]  Juncheng Wei,et al.  Strongly interacting bumps for the Schrödinger–Newton equations , 2009 .

[4]  M. Willem.  Functional analysis. Fundamentals and applications , 2013 .

[5]  S. Secchi,et al.  Semi-classical limit for Schrödinger equations with magnetic field and Hartree-type nonlinearities , 2009, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[6]  Jean Van Schaftingen,et al.  Symmetry of solutions of semilinear elliptic problems , 2008 .

[7]  Marc Vuffray,et al.  Stationary solutions of the Schrödinger-Newton model---an ODE approach , 2007, Differential and Integral Equations.

[8]  Elliott H. Lieb.  Existence and Uniqueness of the Minimizing Solution of Choquard's Nonlinear Equation , 1977 .

[9]  P. Lions.  The concentration-compactness principle in the Calculus of Variations , 1984 .

[10]  M. Weinstein.  Nonlinear Schrödinger equations and sharp interpolation estimates , 1983 .

[11]  Jean Van Schaftingen,et al.  Existence of groundstates for a class of nonlinear Choquard equations , 2012, 1212.2027.

[12]  Elliott H. Lieb,et al.  A Relation Between Pointwise Convergence of Functions and Convergence of Functionals , 1983 .

[13]  P. Lions,et al.  The Choquard equation and related questions , 1980 .

[14]  Irene M. Moroz,et al.  An analytical approach to the Schrödinger-Newton equations , 1999 .

[15]  Alexander Yu. Solynin,et al.  An approach to symmetrization via polarization , 1999 .

[16]  J. K. Hunter,et al.  Measure Theory , 2007 .

[17]  P. Lions.  The concentration-compactness principle in the calculus of variations. The locally compact case, part 1 , 1984 .

[18]  Tobias Weth,et al.  Partial symmetry of least energy nodal solutions to some variational problems , 2005 .

[19]  P. Bassanini,et al.  Elliptic Partial Differential Equations of Second Order , 1997 .

[20]  Jean Van Schaftingen.  Explicit approximation of the symmetric rearrangement by polarizations , 2009, 0902.0637.

[21]  Jean Van Schaftingen,et al.  Set transformations, symmetrizations and isoperimetric inequalities , 2004 .

[22]  S. I. Pekar,et al.  Untersuchungen über die Elektronentheorie der Kristalle , 1954 .

[23]  Wenxiong Chen,et al.  Classification of solutions for an integral equation , 2006 .

[24]  Otared Kavian,et al.  Introduction à la théorie des points critiques : et applications aux problèmes elliptiques , 1993 .

[25]  S. Secchi,et al.  Multiple solutions to a magnetic nonlinear Choquard equation , 2011, 1109.1386.

[26]  Shmuel Agmon,et al.  Bounds on exponential decay of eigenfunctions of Schrödinger operators , 1985 .

[27]  M. Willem.  Minimax Theorems , 1997 .

[28]  G. P. Menzala,et al.  On regular solutions of a nonlinear equation of Choquard's type , 1980, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[29]  Irene M. Moroz,et al.  Spherically symmetric solutions of the Schrodinger-Newton equations , 1998 .

[30]  A. Quaas,et al.  Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian , 2012, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[31]  M. Riesz.  L'intégrale de Riemann-Liouville et le problème de Cauchy , 1949 .

[32]  B. Gidas,et al.  Symmetry of positive solutions of nonlinear elliptic equations in R , 1981 .

[33]  Jean Van Schaftingen,et al.  Nonexistence and optimal decay of supersolutions to Choquard equations in exterior domains , 2012, 1203.3154.