A class of quasilinear degenerate elliptic problems (original) (raw)
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An Elliptic Problem Arising from the Unsteady Transonic Small Disturbance Equation
Journal of Differential Equations, 1996
We prove a theorem on existence of a weak solution of the Dirichlet problem for a quasilinear elliptic equation with a degeneracy on one part of the boundary. The degeneracy is of a type ("Keldysh type") associated with singular behavior-blow-up of a derivativeat the boundary. We define an associated operator which is continuous: pseudo-monotone and coercive and show that a weak.solution &playing singular behavior at the boundary e~sts.
Journal of Mathematical Fluid Mechanics, 2021
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Eigenvalue problems for degenerate nonlinear elliptic equations in anisotropic media
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We study nonlinear eigenvalue problems of the type − div(a(x)∇u) = g (λ,x,u) in R N , where a(x) is a degenerate nonnegative weight. We establish the existence of solutions and we obtain information on qualitative properties as multiplicity and location of solutions. Our approach is based on the critical point theory in Sobolev weighted spaces combined with a Caffarelli-Kohn-Nirenberg-type inequality. A specific minimax method is developed without making use of Palais-Smale condition.
Boundary blow-up solutions to a class of degenerate elliptic equations
Analysis and Mathematical Physics, 2018
Let be a bounded domain in R N = R N 1 × R N 2 with N 1 , N 2 ≥ 1, and N (s) = N 1 + (1 + s)N 2 be the homogeneous dimension of R N for s ≥ 0. In this paper, we prove the existence and uniqueness of boundary blow-up solutions to the following semilinear degenerate elliptic equation G s u = |x| 2s u p + in , u(z) → +∞ as d(z) → 0, where u + = max{u, 0}, 1 < p < N (s)+2s N (s)−2 , and d(z) denotes the Grushin distance from z to the boundary of. Here G s is the Grushin operator of the form G s u = x u + |x| 2s y u, s ≥ 0. It is worth noticing that our results do not require any assumption on the smoothness of the domain , and when s = 0, we cover the previous results for the Laplace operator .
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Based on a comparison principle, we discuss the existence, uniqueness and asymptotic behaviour of various boundary blow-up solutions for a class of quasilinear elliptic equations, which are then used to obtain a rather complete understandin~ of some quasilinear elliptic problems on a bounded domain or over the entire R ~* .
Existence results for some nonlinear degenerate problems in the anisotropic spaces
Boletim da Sociedade Paranaense de Matemática
Our goal in this study is to prove the existence of solutions for the following nonlinear anisotropic degenerate elliptic problem:- \partial_{x_i} a_i(x,u,\nabla u)+ \sum_{i=1}^NH_i(x,u,\nabla u)= f- \partial_{x_i} g_i \quad \mbox{in} \ \ \Omega,where for i=1,...,Ni=1,...,Ni=1,...,N $ a_i(x,u,\nabla u)$ is allowed to degenerate with respect to the unknown u, and Hi(x,u,nablau)H_i(x,u,\nabla u)Hi(x,u,nablau) is a nonlinear term without a sign condition. Under suitable conditions on aia_iai and HiH_iHi, we prove the existence of weak solutions.
Quasi-One-Dimensional Riemann Problems and Their Role in Self-Similar Two-Dimensional Problems
Archive for Rational Mechanics and Analysis, 1998
We study two-dimensional Riemann problems with piecewise constant data. We identify a class of two-dimensional systems, including many standard equations of compressible flow, which are simplified by a transformation to similarity variables. For equations in this class, a two-dimensional Riemann problem with sectorially constant data becomes a boundary-value problem in the finite plane. For data leading to shock interactions, this problem separates into two parts: a quasi-one-dimensional problem in supersonic regions, and an equation of mixed type in subsonic regions. We prove a theorem on local existence of solutions of quasi-one-dimensional Riemann problems. For 2 × 2 systems, we generalize a theorem of Courant & Friedrichs, that any hyperbolic state adjacent to a constant state must be a simple wave. In the subsonic regions, where the governing equation is of mixed hyperbolic-elliptic type, we show that the elliptic part is degenerate at the boundary, with a nonlinear variant of a degeneracy first described by Keldysh.