In this work we consider the problemswhere L is a nonlocal differential operator and Ω is a bounded domain in R N , with Lipschitz boundary. The main goal of this work is to study existence, uniqueness and summability of the solution u with respect to the summability of the datum f . In the process we establish an L p -theory, for p 1, associated to these problems and we prove some useful inequalities for the applications.2010 Mathematics Subject Classification. 45K05, 47G20, 35R09, 35D30, 35D35.
We study both existence and nonexistence of nonnegative solutions for nonlinear elliptic problems with Singular lower order terms that have natural growth with respect to the gradient, whose model is {-Delta u + vertical bar del u vertical bar(2)/u(gamma) = f in Omega. u = 0 on partial derivative Omega. where Omega is an open bounded subset of R, gamma > 0 and f is a function which is strictly positive on every compactly contained subset of Omega. As a consequence of our main results, we prove that the condition gamma < 2 is necessary and sufficient for the existence of solutions in H(0)(1) (Omega) for every sufficiently regular f as above. (C) 2009 Elsevier Inc. All rights reserved
We consider the quasilinear degenerate elliptic equation lambda u - Delta(p)u + H(x, Du) = 0 in Omega where (p) is the p-Laplace operator, p>2, 0 and is a smooth open bounded subset of (N) (N2). Under suitable structure conditions on the function H, we prove local and global gradient bounds for the solutions. We apply these estimates to study the solvability of the Dirichlet problem, and the existence, uniqueness and asymptotic behavior of maximal solutions blowing up at the boundary. The ergodic limit for those maximal solutions is also studied and the existence and uniqueness of a so-called additive eigenvalue is proved in this context
In this paper we deal with a nonlinear elliptic problem, whose model is { -Delta u = B vertical bar u vertical bar 2/u + f in Omega, u = 0 on partial derivative Omega, where B > 0 and f >= 0 belongs to a Lebesgue space. We prove the existence of positive solutions in suitable Sobolev spaces (depending on f and B). (C) 2010 Elsevier Inc. All rights reserved
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