MULTIPLICITY RESULT FOR CRITICAL P–LAPLACIAN SYSTEMS WITH SINGULAR POTENTIAL

The aim of this paper is to prove the existence of multiple positive solutions to the following critical p–Laplacian system with singular potential [Formula: see text] where Ω is a starshaped bounded domain of ℝN with respect to the origin, p* and q* denote the critical Sobolev exponents and the parameters λ, α, β and γ satisfy some mild conditions. Our system corresponds to a perturbed critical problem which has no positive solutions (λ = 0). We show that the form of the associated energy functional has a local property and satisfies the requirements of the Mountain–Pass geometry; then the Ekeland variational principle and the concept of Palais–Smale sequences will be useful.

NONTRIVIAL SOLUTIONS OF p-SUPERLINEAR p-LAPLACIAN PROBLEMS VIA A COHOMOLOGICAL LOCAL SPLITTING

We consider a quasilinear equation, involving the p-Laplace operator, with a p-superlinear nonlinearity. We prove the existence of a nontrivial solution, also when there is no mountain pass geometry, without imposing a global sign condition. Techniques of Morse theory are employed.