Noncoercive resonant (<i>p</i>,2)-equations with concave terms

Abstract

Abstract We consider a nonlinear Dirichlet problem driven by the sum of a p -Laplace and a Laplacian (a <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mo stretchy="false">(</m:mo><m:mi>p</m:mi><m:mo>,</m:mo><m:mn>2</m:mn><m:mo stretchy="false">)</m:mo></m:mrow></m:math> {(p,2)} -equation). The reaction exhibits the competing effects of a parametric concave term plus a Caratheodory perturbation which is resonant with respect to the principle eigenvalue of the Dirichlet p -Laplacian. Using variational methods together with truncation and comparison techniques and Morse theory (critical groups), we show that for all small values of the parameter, the problem has as least six nontrivial smooth solutions all with sign information (two positive, two negative and two nodal (sign changing)).