Positive Radial Solutions of Some Nonlinear Partial Differential Equations (original) (raw)

EXISTENCE AND ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO NONLINEAR RADIAL p-LAPLACIAN EQUATIONS

2015

This article concerns the existence, uniqueness and boundary behavior of positive solutions to the nonlinear problem 1 A (AΦp(u)) + a 1 (x)u α 1 + a 2 (x)u α 2 = 0, in (0, 1), lim x→0 AΦp(u)(x) = 0, u(1) = 0, where p > 1, α 1 , α 2 ∈ (1 − p, p − 1), Φp(t) = t|t| p−2 , t ∈ R, A is a positive differentiable function and a 1 , a 2 are two positive measurable functions in (0, 1) satisfying some assumptions related to Karamata regular variation theory.

Existence of radial solutions for an asymptotically linear p-Laplacian problem

Journal of Mathematical Analysis and Applications, 2008

We prove that an asymptotically linear Dirichlet problem which involves the p-Laplacian operator has multiple radial solutions when the nonlinearity has a positive zero and the range of the 'p-derivative' of the nonlinearity includes at least the first j radial eigenvalues of the p-Laplacian operator. The main tools that we use are a uniqueness result for the p-Laplacian operator and bifurcation theory.

A Lower/Upper Solutions Result for Generalised Radial p-Laplacian Boundary Value Problems

Mediterranean Journal of Mathematics

We provide existence results to some planar nonlinear boundary value problems, in the presence of lower and upper solutions. Our results apply to a class of systems generalising radial elliptic equations driven by the p-Laplace operator, and to some problems involving the Laplace–Beltrami operator on the sphere. After extending the definition of lower and upper solutions to the planar system, we prove our results by a shooting method involving a careful analysis of the solutions in the phase plane.

Uniqueness of positive radial solutions for n-Laplacian Dirichlet problems

Proceedings, 2000

We prove a theorem on the uniqueness of positive radial solutions to a Dirichlet problem of the n-Laplacian in a¯nite ball of R n. Our proofs use only elementary analysis based on an identity due to Erbe and Tang. The result can be applied to a large class of nonlinearities, including some polynomials and functions with exponential growth; in particular, the one recently studied by Adimurthi.

Nonnegative solutions of the radial Laplacian with nonlinearity that changes sign

Proceedings of the American Mathematical Society, 1995

We find a solution to the radial Laplacian equation y + N − 1 x y ′ + λ a ( x ) f ( y ) = 0 , y ′ ( 0 ) = y ( 1 ) = 0 y + \frac {{N - 1}}{x}y’ + \lambda a(x)f(y) = 0,y’ (0) = y(1) = 0 when a may change sign and is "sufficiently positive". The function f is qualitatively like e y {e^y} , and we conclude solutions for 0 ≤ λ ≤ λ 0 0 \leq \lambda \leq {\lambda _0} .

Radial solutions of Dirichlet problems with concave–convex nonlinearities

Nonlinear Analysis: Theory, Methods & Applications, 2011

(AAM) is copyrighted and published by Elsevier. It is posted here by agreement between Elsevier and the University of Turin. Changes resulting from the publishing process-such as editing, corrections, structural formatting, and other quality control mechanismsmay not be reflected in this version of the text. The definitive version of the text was subsequently published in [Nonlinear Analysis: theory, methods and applications,