Multiple solutions for nonlinear Neumann problems driven by a nonhomogeneous differential operator

Motreanu, D.; Papageorgiou, N.

doi:10.1090/S0002-9939-2011-10884-0

Multiple solutions for nonlinear Neumann problems driven by a nonhomogeneous differential operator
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by D. Motreanu and N. S. Papageorgiou PDF

Proc. Amer. Math. Soc. 139 (2011), 3527-3535 Request permission

Abstract:

We consider a nonlinear Neumann problem driven by a nonhomogeneous quasilinear degenerate elliptic differential operator $\operatorname {div} a(x,\nabla u)$, a special case of which is the $p$-Laplacian. The reaction term is a Carathéodory function $f(x,s)$ which exhibits subcritical growth in $s$. Using variational methods based on the mountain pass and second deformation theorems, together with truncation and minimization techniques, we show that the problem has three nontrivial smooth solutions, two of which have constant sign (one positive, the other negative). A crucial tool in our analysis is a result of independent interest which we prove here and which relates $W^{1,p}$ and $C^1$ local minimizers of a $C^1$-functional constructed with the general differential operator $\operatorname {div} a(x,\nabla u)$.

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