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An alternate variational principle for the equation in the title has been proposed by H. A. Levine. We analyse the relation between this principle and the Rayleigh quotient that has been used previously for the variational study of this problem in      R N {R^N}      . The main result is an existence theorem for      W 1 , 2 ( R N ) {W^{1,2}}({R^N})      -solutions of the variational problem posed by Levine.

An alternate variational principle for Δ𝑢-𝑢+∣𝑢∣^{𝑟-1}𝑠𝑔𝑛𝑢=0