THE EXISTENCE OF INFINITELY MANY SOLUTIONS FOR p-LAPLACIAN TYPE EQUATIONS ON R WITH LINKING GEOMETRY
نویسندگان
چکیده
In this paper, we study the existence of infinitely many solutions to the following quasilinear equation of p-Laplacian type in R (0.1) −△pu+ |u|p−2u = λV (x)|u|p−2u+ g(x, u), u ∈ W (R ) with sign-changing radially symmetric potential V (x), where 1 < p < N, λ ∈ R and △pu = div(|Du|p−2Du) is the p-Laplacian operator, g(x, u) ∈ C(R×R,R) is subcritical and p-superlinear at 0 as well as at infinity. We prove that under certain assumptions on the potential V and the nonlinearity g, for any λ ∈ R, the problem (0.1) has infinitely many solutions by using a fountain theorem over cones under Cerami condition. A minimax approach, allowing an estimate of the corresponding critical level, is used. New linking structures, associated to certain variational eigenvalues of −△pu + |u|p−2u = λV (x)|u|p−2u are recognized, even in absence of any direct sum decomposition of W (R ) related to the eigenvalue itself. Our main result can be viewed as an extension to a recent result of Degiovanni and Lancelotti in [10] concerning the existence of nontrivial solutions for the quasilinear elliptic problem: (0.2) { −△pu = λV (x)|u|p−2u+ g(x, u), in Ω, u = 0, on ∂Ω, where Ω ⊂ R is a bounded open domain.
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