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December 2015 Positive Solutions for a Quasilinear Elliptic Problem Involving Sublinear and Superlinear Terms
Manuela C. REZENDE, Carlos Alberto SANTOS
Tokyo J. Math. 38(2): 381-407 (December 2015). DOI: 10.3836/tjm/1452806047

Abstract

We deal with the existence and non-existence of positive solutions for the problem \begin{eqnarray*} \displaystyle \left \{ \begin{array}{@{\,}c} -\Delta_p u +m(x)u^{p-1} = a(x)f(u) + \lambda b(x)g(u)~~ \text{in}~~ \mathbf{R}^N\,,\\ \\ \displaystyle u > 0~~ \text{in}~~\mathbf{R}^N\,,~~~ u(x)\rightarrow 0 \ \text{when} \ |x|\rightarrow\infty\,, \end{array} \right. \end{eqnarray*} where $\Delta_p$ is the $p$-Laplacian operator, $1<p<N$, $\lambda>0$ is a real parameter, $ f, g: (0, \infty)\rightarrow (0,\infty)$ and $m, a, b: \mathbf{R}^N\rightarrow[0,\infty )$; $ a, b\neq 0$ are continuous functions. In this work we consider, for example, nonlinearities with combined effects of concave and convex terms, besides allowing the presence of singularities. For existence of solutions, we exploit the lower and upper solutions method, combined with a technique of monotone-regularization on the nonlinearities $f$ and $g$ and for non-existence we use a consequence of Picone identity.

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Manuela C. REZENDE. Carlos Alberto SANTOS. "Positive Solutions for a Quasilinear Elliptic Problem Involving Sublinear and Superlinear Terms." Tokyo J. Math. 38 (2) 381 - 407, December 2015. https://doi.org/10.3836/tjm/1452806047

Information

Published: December 2015
First available in Project Euclid: 14 January 2016

zbMATH: 1345.35049
MathSciNet: MR3448864
Digital Object Identifier: 10.3836/tjm/1452806047

Subjects:
Primary: 35J62
Secondary: 35B08 , 35B09 , 35B25 , 35B40 , 35J67 , 35J75 , 35J92

Rights: Copyright © 2015 Publication Committee for the Tokyo Journal of Mathematics

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Vol.38 • No. 2 • December 2015
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