MULTIPLE SOLUTIONS FOR A CLASS OF NONLINEAR EQUATIONS VIA THE MOUNTAIN PASS THEOREM

Abstract

The aim of this paper is to study the existence of the multiple solutions
for the abstract equation Jp u=Nf u, where Jp is the duality mapping on a real reflexive and smooth Banach space X, corresponding to the gauge function φ(t)=tp-1, 1<p<∞.
It is assumed that X is compactly imbedded in Lq(Ω), where Ω is a bounded domain in RN,N≤2, 1<q<p*, p* being the Sobolev conjugate exponent, Nf:Lq(Ω)Lq'(Ω), 1/q+1/q'=1, being the Nemytskii operator generated by a Carathéodory function f:ΩxR  R which satisfies some appropriate conditions.
In order to prove the existence of the multiple solutions we use a multiple variant of the the Mountain Pass theorem.

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