Abstract
We consider the following nonlinear problem in R N (0.1) { − Δ u + V ( | y | ) u = u N + 2 N − 2 , u > 0 , in R N ; u ∈ H 1 ( R N ) , where V ( r ) is a bounded non-negative function, N ⩾ 5 . We show that if r 2 V ( r ) has a local maximum point, or local minimum point r 0 > 0 with V ( r 0 ) > 0 , then (0.1) has infinitely many non-radial solutions, whose energy can be made arbitrarily large. As an application, we show that the solution set of the following problem − Δ u = λ u + u N + 2 N − 2 , u > 0 on S N has unbounded energy, as long as λ < − N ( N − 2 ) 4 , N ⩾ 5 .
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