### Nontrivial solutions for nonvariational quasilinear Neumann problems

#### Abstract

We consider a nonlinear nonvariational Neumann problem with a nonsmooth

potential. Using the spectrum of the assymptotic (as $\vert x\vert

\rightarrow\infty)$ differential operator and degree theoretic techniques

based on the degree map of certain multivalued perturbations of

(S)$_{+}$-operators, we establish the existence of at least one

nontrivial smooth solution.

potential. Using the spectrum of the assymptotic (as $\vert x\vert

\rightarrow\infty)$ differential operator and degree theoretic techniques

based on the degree map of certain multivalued perturbations of

(S)$_{+}$-operators, we establish the existence of at least one

nontrivial smooth solution.

#### Keywords

Nonvariational problem; asymptotic differential operator; degree theory; (S)$_{+}$-operator; nonsmooth potential; generalized subdifferential

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