Multiple solutions for a problem with resonance involving the p-Laplacian

In this paper we will investigate the existence of multiple solutions for the problem (P)                                                −Δpu+g(x,u)=λ1h(x)|u|p−2u,     in     Ω,    u∈H01,p(Ω) where Δpu=div(|∇u|p−2∇u) is the p-Laplacian operator, Ω⫅ℝN is a bounded domain with smooth boundary, h and g are bounded functions, N≥1 and 1<p<∞. Using the Mountain Pass Theorem and the Ekeland Variational Principle, we will show the existence of at least three solutions for (P)

Non-autonomous perturbations for a class of quasilinear elliptic equations on R

This paper is concerned with the existence of two positive solutions for a class of quasilinear elliptic equations on R involving the p-Laplacian, with a non-autonomous perturbation. The first solution is obtained as a local minimum in a neighborhood of 0 and the second one by a mountain-pass argument. The special features of the problem here is the “complex” structure of the linear part which, in particular, oblige to work into the space W 1,p (R). Then one faces problems in the convergence of the sequences of derivatives un → u

Subcritical perturbations of a singular quasilinear elliptic equation involving the critical Hardy–Sobolev exponent

In this work we improve some known results for a singular operator and also for a wide class of lower-order terms by proving a multiplicity result. The proof is made by applying the generalized mountain-pass theorem due to Ambrosetti and Rabinowitz. To do this, we show that the minimax levels are in a convenient range by combining a special class of approximating functions, due to Gazzola and Ruf, with the concentrating functions of the best Sobolev constant