On a class of parametric $(p,2)$-equations

Abstract

We consider parametric equations driven by the sum of a $p$-Laplacian and a Laplace operator (the so-called $(p,2)$-equations). We study the existence and multiplicity of solutions when the parameter $\lambda >0$ is near the principal eigenvalue ${\hat{\lambda }}_1(p)>0$ of $(-{\Delta }_p,W_0^{1,p}(\Omega ))$. We prove multiplicity results with precise sign information when the near resonance occurs from above and from below of ${\hat{\lambda }}_1(p)>0$.

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