zbMATH — the first resource for mathematics

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.

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
Full Text: DOI
[1] S. Ahmad - A. C. Lazer - J. L. Paul, Elementary critical point theory and pertur- bations of elliptic boundary value problems at resonance, Indiana Math. J. 25 (1976), 933-944. · Zbl 0351.35036
[2] A. Ambrosetti - P. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), 349-381. · Zbl 0273.49063
[3] C. Bereanu - P. Jebelean - J. Mawhin, Radial solutions for Neumann problems involving mean curvature operators in Euclidean and Minkowsi spaces, Math. Nachr. 283 (2010), 379-391. · Zbl 1185.35113
[4] C. Bereanu - P. Jebelean - J. Mawhin, Radial solutions for Neumann problems involving mean extrinsic curvature and periodic nonlinearities, submitted. · Zbl 1262.35088
[5] H. Brezis - J. Mawhin, Periodic solutions of the forced relativistic pendulum, Di\?er- ential Integral Equations 23 (2010), 801-810.
[6] H. Brezis - J. Mawhin, Periodic solutions of Lagrangian systems of relativistic oscil- lators, Comm. Appl. Anal., to appear.
[7] G. Dinca\check - P. Jebelean - J. Mawhin, Variational and topological methods for Dirichlet problems with p-Laplacian, Portug. Math. (N.S.) 58 (2001), 339-378. · Zbl 0991.35023
[8] A. Hammerstein, Nichtlineare Integralgleichungen nebst Anwendungen, Acta Math. 54 (1930), 117-176. · JFM 56.0343.03
[9] P. Jebelean, Variational methods for ordinary p-Laplacian systems with potential boundary conditions, Adv. Di\?erential Equations 14 (2008), 273-322. · Zbl 1177.34021
[10] J. Mawhin, Proble‘mes de Dirichlet variationnels non lineáires, Seḿin. Math. Sup. No. 104, Presses Univ. Montreál, Montreál, 1987.
[11] J. Mawhin, Semi-coercive monotone variational problems, Bull. Cl. Sci. Acad. Roy. Belgique (5) 73 (1987), 118-130. · Zbl 0647.49007
[12] J. Mawhin, Periodic solutions of second order nonlinear di\?erence systems with f-Laplacian: a variational approach, to appear. 111
[13] A. Szulkin, Minimax principles for lower semicontinuous functions and applications to nonlinear boundary value problems, Ann. Inst. H. PoincareÁnal. Non Lineáire 3 (1986), 77-109. · Zbl 0612.58011
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.