Sign-changing solutions for coupled nonlinear Schrödinger equations with critical growth

详细信息    查看全文

• 作者：Jiaquan Liu^a ; Xiangqing Liu^b ; Zhi-Qiang Wang^c ; ^d ; ^{wang@math.usu.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：35A15 ; 35J50 ; 58J70
• 刊名：Journal of Differential Equations
• 出版年：2016
• 出版时间：15 December 2016
• 年：2016
• 卷：261
• 期：12
• 页码：7194-7236
• 全文大小：519 K

文摘

We consider the following nonlinear Schrödinger system with critical growth 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">R^N${\mathbb{R}}^N$, 856&_mathId=si3.gif&_user=111111111&_pii=S0022039616302856&_rdoc=1&_issn=00220396&md5=acdea9ddc5f3d5279347319163c2296f">856-si3.gif">$2^{⁎}=\frac{2N}{N-2}$, 856&_mathId=si4.gif&_user=111111111&_pii=S0022039616302856&_rdoc=1&_issn=00220396&md5=0f9904707274ba5ac8483f76ba0bf5b8" title="Click to view the MathML source">0<λ_j<λ₁(Ω)$0<{\lambda }_j<{\lambda }_1(\mathrm{\Omega })$, 856&_mathId=si29.gif&_user=111111111&_pii=S0022039616302856&_rdoc=1&_issn=00220396&md5=eb7439f1eb1a831090e8708f75e9c16a" title="Click to view the MathML source">j=1,⋯,k$j=1,\cdots ,k$, 856&_mathId=si6.gif&_user=111111111&_pii=S0022039616302856&_rdoc=1&_issn=00220396&md5=907e522fa90d453809de8c092d66d949" title="Click to view the MathML source">λ₁(Ω)${\lambda }_1(\mathrm{\Omega })$ 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">β_jj>0${\beta }_{jj}>0$, 856&_mathId=si29.gif&_user=111111111&_pii=S0022039616302856&_rdoc=1&_issn=00220396&md5=eb7439f1eb1a831090e8708f75e9c16a" title="Click to view the MathML source">j=1,⋯,k$j=1,\cdots ,k$, 856&_mathId=si30.gif&_user=111111111&_pii=S0022039616302856&_rdoc=1&_issn=00220396&md5=0162d52880e4eeb988ca68279c1b2e2d" title="Click to view the MathML source">β_ij=β_ji≤0${\beta }_{ij}={\beta }_{ji}\le 0$, 856&_mathId=si18.gif&_user=111111111&_pii=S0022039616302856&_rdoc=1&_issn=00220396&md5=8657306304783b549036d9ba9d30826c" title="Click to view the MathML source">i≠j$i\ne j$, 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$i,j=1,\cdots ,k$. 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$N\ge 7$, by approximations of systems with subcritical growth and by the concentration analysis on approximating solutions.