Minimum action solutions of some vector field equations

The system of equations studied in this paper is −Δui=gi(u) on ℝd,d≧2, withu:ℝd→ℝn andgi(u)=∂G/∂ui. Associated with this system is the action,S(u)=ε{1/2|∇u|2−G(u)}. Under appropriate conditions onG (which differ ford=2 andd≧3) it is proved that the system has a solution,u ≢0, of finite action and that this solution also minimizes the action within the class {v is a solution,v has finite action,v ≢0}.

[1]  L. Nirenberg,et al.  On elliptic partial differential equations , 1959 .

[2]  G. Stampacchia,et al.  Inverse Problem for a Curved Quantum Guide , 2012, Int. J. Math. Math. Sci..

[3]  W. Strauss,et al.  Existence of localized solutions for certain model field theories , 1981 .

[4]  Pierre-Louis Lions,et al.  Nonlinear scalar field equations, I existence of a ground state , 1983 .

[5]  Walter A. Strauss,et al.  Existence of solitary waves in higher dimensions , 1977 .

[6]  P. Lions,et al.  Existence d'états multiples dans des équations de champs scalaires non linéaires dans le cas de masse nulle , 1983 .

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

[8]  Tosis Kato,et al.  Schrödinger operators with singular potentials , 1972 .

[9]  M. Crandall,et al.  A semilinear equation in $L^1 (\mathbb {R}^N)$ , 1975 .

[10]  C. Keller.  Large-time asymptotic behavior of solutions of nonlinear wave equations perturbed from a stationary ground state , 1983 .

[11]  Elliott H. Lieb,et al.  On the lowest eigenvalue of the Laplacian for the intersection of two domains , 1983 .

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

[13]  C. Bardos,et al.  Bifurcation phenomena in mathematical physics and related topics : proceedings of the NATO Advanced Study Institute held at Cargèse, Corsica, France, June 24-July 7, 1979 , 1980 .

[14]  D. Gilbarg,et al.  Elliptic Partial Differential Equa-tions of Second Order , 1977 .

[15]  La méthode de concentration-compacité en calcul des variations , 1983 .

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

[17]  I. Affleck.  Two-dimensional disorder in the presence of a uniform magnetic field , 1983 .

[18]  P. Lions,et al.  Existence of Stationary States in Nonlinear Scalar Field Equations , 1980 .

[19]  H. Berestycki,et al.  Une methode locale pour l’existence de solutions positives de problemes semi-lineaires elliptiques dans RN , 1980 .

[20]  S. Coleman,et al.  Action minima among solutions to a class of Euclidean scalar field equations , 1978 .

[21]  E. Lieb.  Some Vector Field Equations , 1984 .