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May 13, 2019 at 1:08 comment added Math604 for $p<0$ you might want to look at some papers or books by Marius Ghergu (i believe he has done a bunch of this stuff)
May 13, 2019 at 1:06 comment added Math604 for $ 0<p<1$ consider the energy that one would normally use to apply the Mountain Pass Theorem to obtain a solution when $p>1$ (so i mean not the contrained minimization approach). You can just minimize this energy to obtain a nonzero solution (when $p>1$ this minimization would give you $- \infty$)
May 12, 2019 at 17:08 history asked Hheepp CC BY-SA 4.0