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Register a new userA nonlinear degenerate elliptic problem with W^{1,1}_0 solutions
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We study a nonlinear equation with an elliptic operator having degenerate coercivity. We prove the existence of a unique W^{1,1}_0 distributional solution under suitable summability assumptions on the source in Lebesgue spaces. Moreover, we prove that our problem has no solution if the source is a Radon measure concentrated on a set of zero harmonic capacity.
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We study a degenerate elliptic equation, proving the existence of a W^{1,1}_0 distributional solution.
We study a degenerate elliptic equation, proving existence results of distributional solutions in some borderline cases.
We study an integral non coercive functional defined on H^1_0, proving the existence of a minimum in W^{1,1}_0.
The authors of this paper deal with the existence and regularities of weak solutions to the homogenous $hbox{Dirichlet}$ boundary value problem for the equation $-hbox{div}(| abla u|^{p-2} abla u)+|u|^{p-2}u=frac{f(x)}{u^{alpha}}$. The authors apply the method of regularization and $hbox{Leray-Schauder}$ fixed point theorem as well as a necessary compactness argument to prove the existence of solutions and then obtain some maximum norm estimates by constructing three suitable iterative sequences. Furthermore, we find that the critical exponent of $m$ in $|f|_{L^{m}(Omega)}$. That is, when $m$ lies in different intervals, the solutions of the problem mentioned belongs to different $hbox{Sobolev}$ spaces. Besides, we prove that the solution of this problem is not in $W^{1,p}_{0}(Omega)$ when $alpha>2$, while the solution of this problem is in $W^{1,p}_{0}(Omega)$ when $1<alpha<2$.
We consider the Dirichlet problem for a class of elliptic and parabolic equations in the upper-half space $mathbb{R}^d_+$, where the coefficients are the product of $x_d^alpha, alpha in (-infty, 1),$ and a bounded uniformly elliptic matrix of coefficients. Thus, the coefficients are singular or degenerate near the boundary ${x_d =0}$ and they may not locally integrable. The novelty of the work is that we find proper weights under which the existence, uniqueness, and regularity of solutions in Sobolev spaces are established. These results appear to be the first of their kind and are new even if the coefficients are constant. They are also readily extended to systems of equations.
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