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Existence of five nonzero solutions with exact sign for a $p$-Laplacian equation

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  • We consider nonlinear elliptic problems driven by the $p$-Laplacian with a nonsmooth potential depending on a parameter $\lambda\ >\ 0$. The main result guarantees the existence of two positive, two negative and a nodal (sign-changing) solution for the studied problem whenever $\lambda\ >\ 0$ belongs to a small interval (0, λ*) and $p$ ≥ 2. We do not impose any symmetry hypothesis on the nonlinear potential. The constant-sign solutions are obtained by using variational techniques based on nonsmooth critical point theory (minimization argument, Mountain Pass theorem, and a Brézis-Nirenberg type result for C1-minimizers), while the nodal solution is constructed by an upper-lower solutions argument combined with the Zorn lemma and a nonsmooth second deformation theorem.
    Mathematics Subject Classification: Primary: 35J20, 35J60, 58J70.


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