Geometrically distinct solutions of nonlinear elliptic systems with periodic potentials

Abstract

In this paper, we study the following nonlinear elliptic systems: -Δu1+V1(x)u1=∂u1F(x,u)x∈RN,-Δu2+V2(x)u2=∂u2F(x,u)x∈RN,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} {\\left\\{ \\begin{array}{ll} -\\Delta u_1+V_1(x)u_1=\\partial _{u_1}F(x,u)&{}\\quad x\\in {\\mathbb {R}}^N,\\\\ -\\Delta u_2+V_2(x)u_2=\\partial _{u_2}F(x,u)&{}\\quad x\\in {\\mathbb {R}}^N, \\end{array}\\right. } \\end{aligned}$$\\end{document}where u=(u_1,u_2):{mathbb {R}}^Nrightarrow {mathbb {R}}^2, F and V_i are periodic in x_1,ldots ,x_N and 0notin sigma (-,Delta +V_i) for i=1,2, where sigma (-,Delta +V_i) stands for the spectrum of the Schrödinger operator -,Delta +V_i. Under some suitable assumptions on F and V_i, we obtain the existence of infinitely many geometrically distinct solutions. The result presented in this paper generalizes the result in Szulkin and Weth (J Funct Anal 257(12):3802–3822, 2009).