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摘要
In this paper, we consider the following critical equation: @@@ -Delta u + V(y)u = K(y)u (N + 2/N - 2), u > 0, u is an element of H-1 (R-N), @@@ where (y', y '') is an element of R-2 x RN-2, V(vertical bar y'vertical bar, y '') and K(vertical bar y'vertical bar, y '') are two nonnegative and bounded functions. Using a finite-dimensional reduction argument and local Pohozaev type of identities, we show that if N >= 5, K(r, y '') has a stable critical point (r(0), y(0)'') with r(0) > 0, K(r(0), y(0)'') > 0 and B-1 := V(r(0), y(0)'') integral(RN) U-0,1(2) dy - Delta K (r(0), y(0)'')/2* N integral(RN) vertical bar y vertical bar 2U(0,1)(2)* dy > 0, then the above equation has infinitely many positive solutions, where U-0,U- 1 is the unique positive solution of -Delta u = u(N + 2/N -2) with u(0) = max(y is an element of RN) u(y). Combining the results of [S. Peng, C. Wang and S. Wei, Constructing solutions for the prescribed scalar curvature problem via local Pohozaev identities, to appear in J. Differential Equations; S. Peng, C. Wang and S. Yan, Construction of solutions via local Pohozaev identities, J. Funct. Anal. 274 (2018) 2606-26331, it implies that the role of stable critical points of K(r, y '') in constructing bump solutions is more important than that of V(r, y '') and that V(r(0), y(0)'') can influence the sign of Delta K(r(0), y(0)''), i.e. Delta K(r(0), y(0)'') can be nonnegative, different from that in [S. Peng, C. Wang and S. Wei, Constructing solutions for the prescribed scalar curvature problem via local Pohozaev identities, to appear in J. Differential Equations]. The concentration points of the solutions locate near the stable critical points of K (r, y ''), which include the case of a saddle point.
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