Multi-bump solutions of -Δn = K(x)u (n+2)/(n-2) on lattices in ℝ n

By Yanyan Li, Jun-cheng Wei, Hao Xu. Journal für die Reine und Angewandte Mathematik, 2018

64 citations recorded by the source corpus.

<jats:p>We consider the following semilinear elliptic equation with critical exponent: Δ <jats:italic>u</jats:italic> <jats:inline-formula id="j_crelle-2015-0090_ineq_9999"> <jats:alternatives> <jats:inline-graphic xlink:href="graphic/j_crelle-2015-0090_eq_0552.png" /> <jats:tex-math>=</jats:tex-math> </jats:alternatives> </jats:inline-formula> <jats:italic>K</jats:italic>(<jats:italic>x</jats:italic>)<jats:inline-formula id="j_crelle-2015-0090_ineq_9998"> <jats:alternatives> <jats:inline-graphic xlink:href="graphic/j_crelle-2015-0090_eq_0576.png" /> <jats:tex-math>u^{(n+2)/(n-2)}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, <jats:italic>u</jats:italic> <jats:inline-formula id="j_crelle-2015-0090_ineq_9997"> <jats:alternatives> <jats:inline-graphic xlink:href="graphic/j_crelle-2015-0090_eq_0553.png" /> <jats:tex-math>></jats:tex-math> </jats:alternatives> </jats:inline-formula> 0 in <jats:inline-formula id="j_crelle-2015-0090_ineq_9996"> <jats:alternatives> <jats:inline-graphic xlink:href="graphic/j_crelle-2015-0090_eq_0567.png" /> <jats:tex-math>\mathbb{R}^{n}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, where <jats:inline-formula id="j_crelle-2015-0090_ineq_9995"> <jats:alternatives> <jats:inline-graphic xlink:href="graphic/j_crelle-2015-0090_eq_0989.png" /> <jats:tex-math>{n\geq 3}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, <jats:inline-formula id="j_crelle-2015-0090_ineq_9994"> <jats:alternatives> <jats:inline-graphic xlink:href="graphic/j_crelle-2015-0090_eq_0698.png" /> <jats:tex-math>{K>0}</jats:tex-math> </jats:alternatives> </jats:inline-formula> is periodic in (<jats:inline-formula id="j_crelle-2015-0090_ineq_9993"> <jats:alternatives> <jats:inline-graphic xlink:href="graphic/j_crelle-2015-0090_eq_0578.png" /> <jats:tex-math>x_{1}</jats:tex-math> </jats:alternatives> </jats:inline-formula>,…,<jats:inline-formula id="j_crelle-2015-0090_ineq_9992"> <jats:alternatives> <jats:inline-graphic xlink:href="graphic/j_crelle-2015-0090_eq_0579.png" /> <jats:tex-math>x_{k}</jats:tex-math> </jats:alternatives> </jats:inline-formula>) with 1 <jats:inline-formula id="j_crelle-2015-0090_ineq_9991"> <jats:alternatives> <jats:inline-graphic xlink:href="graphic/j_crelle-2015-0090_eq_0566.png" /> <jats:tex-math>\leq</jats:tex-math> </jats:alternatives> </jats:inline-formula> <jats:italic>k</jats:italic> <jats:inline-formula id="j_crelle-2015-0090_ineq_9990"> <jats:alternatives> <jats:inline-graphic xlink:href="graphic/j_crelle-2015-0090_eq_0551.png" /> <jats:tex-math><</jats:tex-math> </jats:alternatives> </jats:inline-formula> (<jats:italic>n</jats:italic>-2)/2. Under some natural conditions on <jats:italic>K</jats:italic> near a critical point, we prove the existence of multi-bump solutions where the centers of bumps can be placed in some lattices in <jats:inline-formula id="j_crelle-2015-0090_ineq_9989"> <jats:alternatives> <jats:inline-graphic xlink:href="graphic/j_crelle-2015-0090_eq_0847.png" /> <jats:tex-math>{\mathbb{R}^{k}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, including infinite lattices. We also show that for <jats:italic>k</jats:italic> <jats:inline-formula id="j_crelle-2015-0090_ineq_9988"> <jats:alternatives> <jats:inline-graphic xlink:href="graphic/j_crelle-2015-0090_eq_0565.png" /> <jats:tex-math>\geq</jats:tex-math> </jats:alternatives> </jats:inline-formula> (<jats:italic>n</jats:italic>-2)/2, no such solutions exist.</jats:p>

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/4f07ae8e51c859b4ff9c47f41ce16db2da47876e