On a conformally invariant elliptic equation on \(R^ n\). (English) Zbl 0608.35017

Ding, Weiyue

Commun. Math. Phys. 107, 331-335 (1986).

It is proved here that there exists a sequence \(u_ k\) of solutions of \[ \Delta u+| u|^{4/(m-2)}u=0\quad in\quad R^ m,\quad m\geq 3, \] with finite energy, namely, \(\int_{R^ m} | \nabla u_ k|^ 2 dx<\infty\), for which the energy tends to \(\infty\) as \(k\to \infty\). The proof is accomplished by reducing the problem to an equivalent problem on the Euclidean n-sphere and then employing standard variational techniques.

Reviewer: P.Schaefer

MSC:

35J60	Nonlinear elliptic equations
35B05	Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35J20	Variational methods for second-order elliptic equations

Keywords:

finite energy; variational techniques

References:

[1]	Ambrosetti, A., Rabinowitz, P.H.: Dual variational methods in critical point theory and applications. J. Funct. Anal.14, 349-381 (1973) · Zbl 0273.49063
[2]	Aubin, T.: Nonlinear analysis on manifolds, Monge-Ampere equations. Berlin, Heidelberg, New York: Springer 1982 · Zbl 0512.53044
[3]	Ding, W.-Y., Ni, W.-M.: On the elliptic equation ?u+Ku (n+2)/(n ? 2)=0 and related topics. Duke Math. J.52, 485-506 (1985) · Zbl 0592.35048
[4]	Gidas, B., Ni, W.-M., Nirenberg, L.: Symmetry and related properties via the maximum principle. Commun. Math. Phys.68, 209-243 (1979) · Zbl 0425.35020
[5]	Palais, R.: The principle of symmetric criticality. Commun. Math. Phys.69, 19-30 (1979) · Zbl 0417.58007