The existence problem is solved, and global pointwise estimates of solutions are obtained for quasilinear and Hessian equations of Lane-Emden type, including the following two model problems:is the k-Hessian defined as the sum of k × k principal minors of the Hessian matrix D 2 u (k = 1, 2, . . . , n); µ is a nonnegative measurable function (or measure) on Ω.The solvability of these classes of equations in the renormalized (entropy) or viscosity sense has been an open problem even for good data µ ∈ L s (Ω), s > 1. Such results are deduced from our existence criteria with the sharp exponents s = n(q−p+1) pq for the first equation, and s = n(q−k) 2kq for the second one. Furthermore, a complete characterization of removable singularities is given.Our methods are based on systematic use of Wolff's potentials, dyadic models, and nonlinear trace inequalities. We make use of recent advances in potential theory and PDE due to Kilpeläinen and Malý, Trudinger and Wang, and Labutin. This enables us to treat singular solutions, nonlocal operators, and distributed singularities, and develop the theory simultaneously for quasilinear equations and equations of Monge-Ampère type.
Abstract. We study the solvability problem for the multidimensional Riccati equation Au--IVu Iv +w, where q > 1 and w is an arbitrary nonnegative function (or measure). We also discuss connections with the classical problem of the existence of positive solutions for the SchrSdinger equation Au--wu 0 with nonnegative potential w. We establish explicit criteria for the existence of global solutions on R n in terms involving geometric (capacity) estimates or pointwise behavior of Riesz potentials, together with sharp pointwise estimates of solutions and their gradients. We also consider the corresponding nonlinear Dirichlet problem on a bounded domain, as well as more general equations of the type Lu=f (x, u, Vu)q-w where f (x, u, Vu)~a(x)lVul ql +b(~)l~l q~, and L is a uniformly elliptic operator.
Abstract. We give necessary and sufficient conditions for the existence of weak solutions to the model equationin the case 0 < q < p − 1, where σ ≥ 0 is an arbitrary locally integrable function, or measure, and ∆pu = div(∇u|∇u| p−2 ) is the p-Laplacian. Sharp global pointwise estimates and regularity properties of solutions are obtained as well. As a consequence, we characterize the solvability of the equationwhere b > 0. These results are new even in the classical case p = 2. Our approach is based on the use of special nonlinear potentials of Wolff type adapted for "sublinear" problems, and related integral inequalities. It allows us to treat simultaneously several problems of this type, such as equations with general quasilinear operators div A(x, ∇u), fractional Laplacians (−∆) α , or fully nonlinear k-Hessian operators.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.