On a Positive Solution for (p,q)-Laplace Equation with Indefinite Weight
Keywords:
(p,q)-Laplacian, nonlinear eigenvalue problems, indefinite weight, mountain pass theorem, global minimizerAbstract
This paper provides existence and non-existence results for a positive solution of the quasilinear elliptic equation
−∆pu−µ∆qu=λ(mp(x)|u|p−2u+µmq(x)|u|q−2u) in Ω driven by the nonhomogeneous operator (p,q)-Laplacian under Dirichlet boundary condition, with µ > 0 and 1 < q < p < ∞. We show that in the case where µ > 0 the results are completely different from those for the usual eigenvalue problem for the p-Laplacian, which is retrieved when µ = 0. For instance, we prove that when µ > 0 there exists an interval of eigenvalues. Existence of positive solutions is obtained in resonant cases, too. A non-existence result is also given.
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