Elliptic equations with nonlinear absorption depending on the solution\n and its gradient

2014/09/25 by Moshe Marcus, Marcus, Moshe, Phuoc‐Tai Nguyen +1
Mathematics · Computer Science · #Nonlinear Partial Differential Equations #Advanced Mathematical Modeling in Engineering #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.1409.7191

Abstract

We study positive solutions of equation (E1) -\Δ u + u^p|\∇ u|^q= 0\n(0\≤ p, 0\≤ q\≤ 2, p+q>1) and (E2) -\Δ u + u^p + |\∇ u|^q\n=0 (p>1, 1<q\≤ 2) in a smooth bounded domain \Ω \⊂\n\ℝ^N. We obtain a sharp condition on p and q under which, for\nevery positive, finite Borel measure \μ on \∂ \Ω, there exists a\nsolution such that u=\μ on \∂ \Ω. Furthermore, if the condition\nmentioned above fails then any isolated point singularity on \∂ \Ω\nis removable, namely there is no positive solution that vanishes on \∂\n\Ω everywhere except at one point. With respect to (E2) we also prove\nuniqueness and discuss solutions that blow-up on a compact subset of \∂\n\Ω. In both cases we obtain a classification of positive solutions with an\nisolated boundary singularity. Finally, in Appendix A a uniqueness result for a\nclass of quasilinear equations is provided. This class includes (E1) when p=0\nbut not the general case.\n

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