Pointwise bounds for positive supersolutions of nonlinear elliptic problems involving the p-Laplacian
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We derive a priori bounds for positive supersolutions of -∆pu = ρ(x)ƒ(u), where p > 1 and ∆p is the p-Laplace operator, in a smooth bounded domain of ℝN with zero Dirichlet boundary conditions. We apply our results to the nonlinear elliptic eigenvalue problem -∆pu = λƒ(u), with Dirichlet boundary condition, where ƒ is a nondecreasing continuous differentiable function on such that ƒ(0) > 0, ƒ(t)1/(p-1) is superlinear at infinity, and give sharp upper and lower bounds for the extremal parameter λ*p. In particular, we consider the nonlinearities ƒ(u) = eu and ƒ(u) = (1 + u)m (m > p - 1) and give explicit estimates on λ*p. As a by-product of our results, we obtain a lower bound for the principal eigenvalue of the p-Laplacian that improves obtained results in the recent literature for some range of p and N.
CitationAghajani, A., & Mosleh Tehrani, A. (2017). Pointwise bounds for positive supersolutions of nonlinear elliptic problems involving the p-Laplacian. Electronic Journal of Differential Equations, 2017(46), pp. 1-14.
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