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 \(1\).


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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