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Eigenvalues of the p -Laplacian in 𝐑 N with indefinite weight

Yin Xi Huang — 1995

Commentationes Mathematicae Universitatis Carolinae

We consider the nonlinear eigenvalue problem - div ( | u | p - 2 u ) = λ g ( x ) | u | p - 2 u in 𝐑 N with p > 1 . A condition on indefinite weight function g is given so that the problem has a sequence of eigenvalues tending to infinity with decaying eigenfunctions in W 1 , p ( 𝐑 N ) . A nonexistence result is also given for the case p N .

Positive solutions of critical quasilinear elliptic equations in R N

Paul A. BindingPavel DrábekYin Xi Huang — 1999

Mathematica Bohemica

We consider the existence of positive solutions of -pu=g(x)|u|p-2u+h(x)|u|q-2u+f(x)|u|p*-2u(1) in N , where λ , α , 1 < p < N , p * = N p / ( N - p ) , the critical Sobolev exponent, and 1 < q < p * , q p . Let λ 1 + > 0 be the principal eigenvalue of -pu=g(x)|u|p-2u    in ,        g(x)|u|p>0, (2) with u 1 + > 0 the associated eigenfunction. We prove that, if N f | u 1 + | p * < 0 , N h | u 1 + | q > 0 if 1 < q < p and N h | u 1 + | q < 0 if p < q < p * , then there exist λ * > λ 1 + and α * > 0 , such that for λ [ λ 1 + , λ * ) and α [ 0 , α * ) , (1) has at least one positive solution.

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