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Variational methods for nonlinear perturbations of singular \(\varphi \)-Laplacians. (English) Zbl 1219.35062
Summary: Motivated by the existence of radial solutions to the Neumann problem involving the mean extrinsic curvature operator in Minkowski space
\[ \text{div}\left( \frac{\nabla v}{\sqrt{1-|\nabla v|^2}}\right)= g(|x|,v) \quad \text{in }{\mathcal A}, \qquad \frac{\partial v}{\partial \nu}=0 \quad \text{on } \partial{\mathcal A}, \]
where \(0\leq R_1<R_2\), \({\mathcal A}=\{x\in \mathbb R^N: R_1\leq |x|\leq R_2\}\) and \(g:[R_1,R_2]\times\mathbb R\to\mathbb R\) is continuous, we study the more general problem
\[ \big[r^{N-1} \varphi(u')\big]'= r^{N-1}g(r,u), \quad u'(R_1)=0=u'(R_2), \]
where \(\varphi:=\Phi':(-a,a) \to\mathbb R\) is an increasing homeomorphism with \(\varphi(0)=0\) and the continuous function \(\Phi:[-a,a] \to\mathbb R\) is of class \(C^1\) on \((-a,a)\). The associated functional in the space of continuous functions over \([R_1,R_2]\) is the sum of a convex lower semicontinuous functional and of a functional of class \(C^1\). Using the critical point theory of Szulkin, we obtain various existence and multiplicity results for several classes of nonlinearities. We also discuss the case of the periodic problem.

MSC:
35J20 Variational methods for second-order elliptic equations
35J60 Nonlinear elliptic equations
35J93 Quasilinear elliptic equations with mean curvature operator
35J87 Unilateral problems for nonlinear elliptic equations and variational inequalities with nonlinear elliptic operators
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
49J40 Variational inequalities
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