No abstract

Note: Please see pdf for full abstract with equations. We consider the prescribed scalar curvature problem on SN ΔSN v −N(N − 2)/2 v + ​​K̃ (y)v N+2/N−2 = 0 on SN, v > 0 in SN, under the assumptions that the scalar curvature ​K̃ is rotationally symmetric, and has a positive local maximum point between the poles. We prove the existence of infinitely many non-radial positive solutions, whose energy can be made arbitrarily large. These solutions are invariant under some non-trivial sub-group of O(3) obtained doubling the equatorial. We use the finite dimensional Lyapunov-Schmidt reduction method.

In this paper, we are concerned with the following elliptic equationUnder some conditions on Q(x), Cao and Zhong in Nonlin. Anal. TMA (Vol 29, 1997, 461-483) proved that there exists a single-peak solution for small ε if N ≥ 4 and s ∈ (1, 2 * − 1). And they proposed in Remark 1.7 of their paper that "it is interesting to know the existence of single-peak solutions for small ε and s = 1". Also it was addressed in Remark 1.8 of their paper that "the question of solutions concentrated at several points at the same time is still open".Here we give some confirmative answers to the above two questions. Furthermore, we prove the local uniqueness of the multi-peak solutions. And our results show that the concentration of the solutions to above problem is delicate whether s = 1 or s > 1.

If B < 0, then for every ǫ small enough, we construct a family of entire solutions wǫ(z, t) ∈ C 2 in the cylindrical coordinates (z, t) ∈ R 2 × R for this system via the approach introduced by J. Dávila, M. del Pino, M. Medina and R. Rodiac in arXiv:1901.02807. These solutions are 2π-periodic in t and have multiple interacting vortex helices. The main results are the extensions of the phenomena of interacting helical vortex filaments for the classical (single) Ginzburg-Landau equation in R 3 which has been studied in arXiv:1901.02807. Our results negatively answer the Gibbons conjecture [10] for the Allen-Cahn equation in Ginzburg-Landau system version, which is an extension of the question originally proposed by H. Brezis.

<p style='text-indent:20px;'>In this paper, we are concerned with the following elliptic equation</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \begin{cases} -\Delta u = Q(x)u^{2^*-1 }+\varepsilon u^{s}, \; &amp;{\;{\rm{in}}\;\; \Omega},\\ \ u&gt;0, \; &amp;{\;{\rm{in}}\;\; \Omega}, \\ \ u = 0, &amp;{\;{\rm{on}}\;\; \partial \Omega}, \end{cases} \end{equation*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ N\geq 3 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ s\in [1, 2^*-1) $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M3">\begin{document}$ 2^* = \frac{2N}{N-2} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ \varepsilon&gt;0 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M5">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> is a smooth bounded domain in <inline-formula><tex-math id="M6">\begin{document}$ \mathbb{R}^N $\end{document}</tex-math></inline-formula>. Under some conditions on <inline-formula><tex-math id="M7">\begin{document}$ Q(x) $\end{document}</tex-math></inline-formula>, Cao and Zhong in Nonlin. Anal. TMA (Vol 29, 1997,461–483) proved that there exists a single-peak solution for small <inline-formula><tex-math id="M8">\begin{document}$ \varepsilon $\end{document}</tex-math></inline-formula> if <inline-formula><tex-math id="M9">\begin{document}$ N\geq 4 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M10">\begin{document}$ s\in (1, 2^*-1) $\end{document}</tex-math></inline-formula>. And they proposed in Remark 1.7 of their paper that</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ \begin{array}{c} ''it\; is\; interesting\; to\; know\; the \;existence\; of\; single-peak \;solutions \; for \;small \;\varepsilon\;\\ and \;s = 1''.\end{array} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>Also it was addressed in Remark 1.8 of their paper that</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE3"> \begin{document}$ \begin{array}{c} ''the\; question\; of \;solutions \;concentrated \;at \;several\; points \;at \;the \;same \;time \;is \\ still \;open''. \end{array}$\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>Here we give some confirmative answers to the above two questions. Furthermore, we prove the local uniqueness of the multi-peak solutions. And our results show that the concentration of the solutions to above problem is delicate whether <inline-formula><tex-math id="M11">\begin{document}$ s = 1 $\end{document}</tex-math></inline-formula> or <inline-formula><tex-math id="M12">\begin{document}$ s&gt;1 $\end{document}</tex-math></inline-formula>.</p>