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Positive solutions for the \(p\)-Laplacian and bounds for its first eigenvalue. (English) Zbl 1181.35115

Summary: We prove a result of existence and localization of positive solutions of the Dirichlet problem for \(-\Delta_p u=w(x)f(u)\) in a bounded domain \(\Omega\), where \(\Delta_p\) is the \(p\)-Laplacian, \(w\) is a weight function and the nonlinearity \(f(u)\) satisfies certain local bounds. As in previous results in radially symmetric domains by two of the authors, and in contrast with the hypotheses usually made, no asymptotic behavior is assumed on \(f\). A positive solution is obtained by applying the Schauder fixed point theorem and such approach allows us to construct ordered sub- and super-solutions for the problem, thus producing an iterative method to obtain a positive solution. Our result leads not only to asymptotic conditions on the nonlinearity which provides the existence of a sequence of positive solutions for the problem with arbitrarily large sup norm. but also to an estimate of the first eigenvalue \(\lambda_p(\Omega, w)\) of the \(p\)-Laplacian operator with weight \(w\). For \(w\equiv 1\), we compare our lower bound for \(\lambda_p(\Omega, 1)\) with that obtained by means of the Cheeger constant \(h(\Omega)\). We give a characterization of this constant in terms of the solution of the torsional creep problem \(-\Delta_p\phi_p= 1\) in \(\Omega\) with Dirichlet boundary data, which offers a good approximation of the first eigenvalue of the \(p\)-Laplacian for \(p\) near \(1\).

MSC:

35J92 Quasilinear elliptic equations with \(p\)-Laplacian
35B09 Positive solutions to PDEs
35J25 Boundary value problems for second-order elliptic equations
35P15 Estimates of eigenvalues in context of PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations
35J70 Degenerate elliptic equations
52A38 Length, area, volume and convex sets (aspects of convex geometry)
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[1] Damascelli, Comparison theorems for some quasilinear degenerate elliptic operators and applications to symmetry and monotonicity resutls Henry Poincare, Inst 15 pp 493– (1998) · Zbl 0911.35009
[2] Kawohl, On a family of torsional creep problems reine, angew Math pp 410– (1990) · Zbl 0701.35015
[3] Lieberman, Bondary regularity for solutions of degenerate elliptic equations Nonlinear, Analysis 12 pp 1203– (1988)
[4] DiBenedetto, {\(\alpha\)} local regularity of weak solutions of degenerate elliptic equations Nonlinear, Analysis 7 pp 827– (1998)
[5] Anane, Etude des Valeurs Propres et de la Re sonnance Pour l Ope rateur p - Laplacien Universite Libre de Bruxelles, Doc (1987)
[6] Hernandez, Dra bek and Existence and uniqueness of positive solutions for some quasilinear elliptic problems Nonlinear, Analysis 44 pp 189– (2001)
[7] Anane, Simplicite et isolation de la premie re valeur propre du p Laplacien avec poids Paris, Acad Sci Math pp 305– (1987) · Zbl 0633.35061
[8] Kawohl, Isoperimetric estimates for the first eigenvalue of the p - Laplace operator and the Cheeger constant, Math Univ Carol pp 44– (2003) · Zbl 1105.35029
[9] Ercole, Positive solutions for the p Laplacian in annuli, Soc Edin pp 132– (2002) · Zbl 1064.35058
[10] Grieser, The first eigenvalue of the Laplacian isoperimetric constants and the Max Flow Min Cut Theorem, Arch Math pp 87– (2006) · Zbl 1105.35062
[11] Aranda, Existence and Multiplicity of positive solutions for a singular problem associated to the p Laplacian operator, Eqs pp 132– (2004) · Zbl 1129.35365
[12] Huang, A note on the asymptotic behavior of positive solutions for some elliptic equations Nonlinear, Analysis 29 pp 533– (1997)
[13] Allegretto, A Picone s identity for the p Laplacian and applica - tions Nonlinear, Analysis 32 pp 819– (1998)
[14] Azizieh, Cle ment A priori estimates and continuation methods for positive solutions of p - Laplace equations, Eqs pp 179– (2002) · Zbl 1109.35350
[15] Belloni, The p Laplacian as p in a Finsler metric no Existence resuts for quasilinear problems via ordered sub and supersolu - tions Toulouse, Eur Math Soc Sciences 8 pp 123– (2006) · Zbl 1245.35078
[16] Ercole, Existence of positive radial solutions for the n - dimensional Nonlinear, Analysis 44 pp 355– (2001) · Zbl 0992.34012
[17] Lefton, Numerical approximation of the first eigenpair of the p - Laplacian using finite elements and the penalty method no, Funct Anal Optim 18 pp 389– (1997) · Zbl 0884.65103
[18] Juutine, The - eigenvalue problem Arch, Ration Anal pp 148– (1999)
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