Existence of solutions for quasilinear Dirichlet problems with gradient terms

Roberta Filippucci, ,Chiara Lini

doi:10.3934/dcdss.2019019

Abstract

In this paper we prove an existence theorem for positive solutions of a nonlinear Dirichlet problem involving the p-Laplacian operator on a smooth bounded domain when a nonlinearity depending on the gradient is considered. Our main theorem extends a previous result by Ruiz in [ 19 ], in which a slight modification of the celebrated blowup technique due to Gidas and Spruck, [ 11 ] and [ 12 ] is introduced.

Highlights

Motivated by [19], in this paper we prove an existence theorem for positive weak solutions of a nonlinear Dirichlet problem involving the pLaplacian operator, namely∆pu + f (x, u, ∇u) = 0, x∈Ω (1)u(x) = 0, x ∈ ∂Ω where Ω ⊂ RN, N > 1, is a bounded smooth domain, ∆pu = div |∇u|p−2∇u is the p-Laplacian operator with 1 < p < N and f : Ω × R+0 × RN → R is a nonnegative continuous function such that (F )uδ − M us|η|θ ≤ f (x, u, η) ≤ c0uδ + M us|η|θ, for all (x, u, η) ∈ Ω × + R0 RN, where c0 andM are positive constants, with c0
In this paper we prove an existence theorem for positive solutions of a nonlinear Dirichlet problem involving the p-Laplacian operator on a smooth bounded domain when a nonlinearity depending on the gradient is considered
The novelty we introduce, respect to [19], is to consider a nonlinearity f involving an explicit dependence on the solution u in the gradient term

Summary

Introduction

Motivated by [19], in this paper we prove an existence theorem for positive weak solutions of a nonlinear Dirichlet problem involving the pLaplacian operator, namely. U(x) = 0, x ∈ ∂Ω where Ω ⊂ RN , N > 1, is a bounded smooth domain, ∆pu = div |∇u|p−2∇u is the p-Laplacian operator with 1 < p < N and f : Ω × R+0 × RN → R is a nonnegative continuous function such that (F ). The novelty we introduce, respect to [19], is to consider a nonlinearity f involving an explicit dependence on the solution u in the gradient term. We point out that the presence of a factor depending both on a power of u and of |∇u| makes the analysis fairly delicate. Ruiz in [19], considers the subcase of (F ) when s = 0. Blowup method, a priori estimates. ∗ Corresponding author: Roberta Filippucci

ROBERTA FILIPPUCCI AND CHIARA LINI

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