Abstract
Abstract In this paper, we study the existence of viscosity solutions to the Gelfand problem for the 1-homogeneous p-Laplacian in a bounded domain {\Omega\subset\mathbb{R}^{N}} , that is, we deal with -\frac{1}{p-1}\lvert\nabla u\rvert^{2-p}\operatorname{div}(\lvert\nabla u% \rvert^{p-2}\nabla u)=\lambda e^{u} in Ω with {u=0} on {\partial\Omega} . For this problem we show that, for {p\in[2,\infty]} , there exists a positive critical value {\lambda^{*}=\lambda^{*}(\Omega,N,p)} such that the following holds: • If {\lambda<\lambda^{*}} , the problem admits a minimal positive solution {w_{\lambda}} . • If {\lambda>\lambda^{*}} , the problem admits no solution. Moreover, the branch of minimal solutions {\{w_{\lambda}\}} is increasing with λ. In addition, using degree theory, for fixed p we show that there exists an unbounded continuum of solutions that emanates from the trivial solution {u=0} with {\lambda=0} , and for a small fixed λ we also obtain a continuum of solutions with {p\in[2,\infty]} .
Highlights
This paper deals with the Gelfand problem corresponding to the 1-homogeneous p-Laplacian,{ −∆Np u = λeu in Ω, u = 0 on ∂Ω,(Pλ,p) where Ω ⊂ RN is a regular bounded domain, p ∈ [2, ∞] and the operator ∆Np is the 1-homogeneous p-Laplacian defined, for p < ∞, by ∆ N p u :=|∇u|2−p 1 div(|∇u|p−2∇u) α∆u +β∆∞u, with α = 1/(p − 1) and β = (p − 2)/(p − 1), and for p = ∞, ∆∞u ≡
Using degree theory, for fixed p we show that there exists an unbounded continuum of solutions that emanates from the trivial solution u = 0 with λ = 0, and for a small fixed λ we obtain a continuum of solutions with p ∈ [2, ∞]
Since the normalized infinity Laplacian is not well defined at the points where |∇u(x)| = 0, we have to use the semicontinuous envelopes of the operator ξ ξ (ξ, X) → |ξ| ⋅ (X |ξ| ), ξ ∈ RN, X ∈ SN, in order to define viscosity solutions for problem (2.1)
Summary
This paper deals with the Gelfand problem corresponding to the 1-homogeneous p-Laplacian,. We remark the lack of variational structure and differentiability of this operator, in contrast to what happens with the classical p-Laplacian This fact implies that the theory concerning “stable solutions” can not be applied to our problem. One of our main tools for the proof of this result is a comparison principle (that we prove here) adapted to the particular structure of the 1-homogeneous p-Laplacian (see Theorem 3.3). This result generalizes previous ones in [3, 20].
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