where Ω is a bounded smooth domain in 856&_mathId=si2.gif&_user=111111111&_pii=S0022039616302856&_rdoc=1&_issn=00220396&md5=2c282fa48350786cd1eed3088848bf19" title="Click to view the MathML source">RN, 856&_mathId=si3.gif&_user=111111111&_pii=S0022039616302856&_rdoc=1&_issn=00220396&md5=acdea9ddc5f3d5279347319163c2296f">856-si3.gif">, 856&_mathId=si4.gif&_user=111111111&_pii=S0022039616302856&_rdoc=1&_issn=00220396&md5=0f9904707274ba5ac8483f76ba0bf5b8" title="Click to view the MathML source">0<λj<λ1(Ω), 856&_mathId=si29.gif&_user=111111111&_pii=S0022039616302856&_rdoc=1&_issn=00220396&md5=eb7439f1eb1a831090e8708f75e9c16a" title="Click to view the MathML source">j=1,⋯,k, 856&_mathId=si6.gif&_user=111111111&_pii=S0022039616302856&_rdoc=1&_issn=00220396&md5=907e522fa90d453809de8c092d66d949" title="Click to view the MathML source">λ1(Ω) is the first eigenvalue of −Δ with zero Dirichlet boundary condition. We consider the repulsive case, namely 856&_mathId=si7.gif&_user=111111111&_pii=S0022039616302856&_rdoc=1&_issn=00220396&md5=046554a6fbc15ba958a3eb5648dcae81" title="Click to view the MathML source">βjj>0, 856&_mathId=si29.gif&_user=111111111&_pii=S0022039616302856&_rdoc=1&_issn=00220396&md5=eb7439f1eb1a831090e8708f75e9c16a" title="Click to view the MathML source">j=1,⋯,k, 856&_mathId=si30.gif&_user=111111111&_pii=S0022039616302856&_rdoc=1&_issn=00220396&md5=0162d52880e4eeb988ca68279c1b2e2d" title="Click to view the MathML source">βij=βji≤0, 856&_mathId=si18.gif&_user=111111111&_pii=S0022039616302856&_rdoc=1&_issn=00220396&md5=8657306304783b549036d9ba9d30826c" title="Click to view the MathML source">i≠j, 856&_mathId=si10.gif&_user=111111111&_pii=S0022039616302856&_rdoc=1&_issn=00220396&md5=89c2401714d6ab0eeaa140b38a299549" title="Click to view the MathML source">i,j=1,⋯,k. The existence of infinitely many sign-changing solutions as bound states is proved, provided 856&_mathId=si11.gif&_user=111111111&_pii=S0022039616302856&_rdoc=1&_issn=00220396&md5=ba0b12a71fe7db047fd008dbf25f5494" title="Click to view the MathML source">N≥7, by approximations of systems with subcritical growth and by the concentration analysis on approximating solutions.