Multiple nonnegative solutions for an elliptic boundary value problem involving combined nonlinearities

THREE NONTRIVIAL NONNEGATIVE SOLUTIONS FOR SOME CRITICAL p-LAPLACIAN SYSTEMS WITH LOWER-ORDER NEGATIVE PERTURBATIONS

“…In recent years, more and more attention have been paid to the existence and multiplicity of nonnegative or positive solutions for the elliptic problems involving concave terms and critical Sobolev exponent. Results relating to these problems can be found in [1], [2], [4,5,12,13], [7,8,9], [11,14,15,16,17,18,19,20,21], and the references therein. By the results of the above papers we know that the number of nontrivial solutions for problem (1) is affected by the concave-convex nonlinearities.…”

“…Applying the strong maximum principle, it is easy to obtain the positive solutions for problem (1) when µ = 0. However, if the concave terms of problem (1) are negative or local negative in Ω as |z| near origin, then the strong maximum principle can not be applied (see [4] and [21]). When F , G, H depends only on the first variable, problem (1) reduces to the following Dirichlet problem (2) −△ p u = u p * −1 + λu q−1 − µu r−1 , in Ω, u = 0, on ∂Ω, where 1 < r < q < p < p * .…”

“…When F , G, H depends only on the first variable, problem (1) reduces to the following Dirichlet problem (2) −△ p u = u p * −1 + λu q−1 − µu r−1 , in Ω, u = 0, on ∂Ω, where 1 < r < q < p < p * . Anello in [4] proved that problem (2) has at least two nontrivial nonnegative solutions for λ > 0 and µ > 0 small enough by truncation techniques and variational methods. Anello also considered the subcritical growth case and obtain three nontrivial nonnegative solutions of the related problems (see Theorem 1 in [4]).…”

“…Anello in [4] proved that problem (2) has at least two nontrivial nonnegative solutions for λ > 0 and µ > 0 small enough by truncation techniques and variational methods. Anello also considered the subcritical growth case and obtain three nontrivial nonnegative solutions of the related problems (see Theorem 1 in [4]). The purpose of this paper is to apply the ideas of Theorem 1 in [4] to the critical growth case to obtain more than two solutions.…”

“…Anello also considered the subcritical growth case and obtain three nontrivial nonnegative solutions of the related problems (see Theorem 1 in [4]). The purpose of this paper is to apply the ideas of Theorem 1 in [4] to the critical growth case to obtain more than two solutions.…”