In this paper we investigate the existence of positive solutions for nonlinear elliptic problems driven by the $p$-Laplacian with a nonsmooth potential (hemivariational inequality). Under asymptotic conditions that make the Euler functional indefinite and incorporate in our framework the asymptotically linear problems, using a variational approach based on nonsmooth critical point theory, we obtain positive smooth solutions. Our analysis also leads naturally to multiplicity results.

Keywords:

$p$-Laplacian, locally Lipschitz potential, nonsmooth critical point theory, principal eigenvalue, positive solutions, nonsmooth Mountain Pass Theorem $p$-Laplacian, locally Lipschitz potential, nonsmooth critical point theory, principal eigenvalue, positive solutions, nonsmooth Mountain Pass Theorem

MSC Classifications:

35J20, 35J60, 35J85 show english descriptions Variational methods for second-order elliptic equations
Nonlinear elliptic equations
Unilateral problems and variational inequalities for elliptic PDE (See also 35R35, 49J40) 35J20 - Variational methods for second-order elliptic equations
35J60 - Nonlinear elliptic equations
35J85 - Unilateral problems and variational inequalities for elliptic PDE (See also 35R35, 49J40)

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