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February 2021 Existence results for $p$-Laplace equations with nonlinearities having growth in the function and its gradient
Haydar Abdelhamid
Rocky Mountain J. Math. 51(1): 1-15 (February 2021). DOI: 10.1216/rmj.2021.51.1

Abstract

In a bounded domain ΩN, N2, we study the solvability of the boundary value problem

Δpu=H(x,u,u) inΩ,u=0 onΩ,

where Δpu=div(|u|p2u) is the p-Laplace operator with 1<p<N and the nonlinearity H satisfies |H(x,u,u)|a|u|q+b|u|r+f(x) with a,b0 and q,r>0. Assuming fLm(Ω) for a suitable exponent m>1, we prove the existence of solutions for different sets of values (q,r). To this end, we apply the classical Schauder fixed point theorem, relying on some well-known uniqueness, regularity estimates and stability results for renormalized and weak solutions. In the last section we make some examples to illustrate the applicability of our results to a large class of equations.

Citation

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Haydar Abdelhamid. "Existence results for $p$-Laplace equations with nonlinearities having growth in the function and its gradient." Rocky Mountain J. Math. 51 (1) 1 - 15, February 2021. https://doi.org/10.1216/rmj.2021.51.1

Information

Received: 11 March 2019; Revised: 6 June 2020; Accepted: 7 June 2020; Published: February 2021
First available in Project Euclid: 28 May 2021

Digital Object Identifier: 10.1216/rmj.2021.51.1

Subjects:
Primary: 35A01 , 35J60 , 35J66 , 35J92

Keywords: $p$-Laplace equations , existence , general growth

Rights: Copyright © 2021 Rocky Mountain Mathematics Consortium

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Vol.51 • No. 1 • February 2021
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