Regularizing effect of absorption terms in singular problems

Abstract

A. We prove existence of solutions to problems whose model iswhere Ω is an open bounded subset of R N (N ≥ 2), ∆ is the -laplacian operator for 1 ≤ < N, > 0, γ ≥ 0 and is a nonnegative function in L (Ω) for some ≥ 1. In particular we analyze the regularizing effect produced by the absorption term in order to infer the existence of finite energy solutions in case γ ≤ 1. We also study uniqueness of these solutions as well as examples which show the optimality of the results. Finally, we find local W 1 -solutions in case γ > 1.

The impact of a lower order term in a Dirichlet problem with a singular nonlinearity

“…Remark 4.3. We observe that the uniqueness of the solutions cannot be inferred as in [14], since we are not dealing with finite energy solutions. We recall that even in the linear case the distributional solutions are not unique, as J. Serrin showed in [15].…”

“…In [11] a nonlinear version of the above problem was studied, considering an operator as the p−laplacian instead of −div(M (x)∇u) (see also Remark 4.2). In [14] the author added a lower order term growing as |u| r−1 u, and studied existence, regularity and uniqueness of solutions to problem (1). Although the right hand side is singular at u, the lower order term in the left hand side has a regularizing effect.…”

The Dirichlet problem for possibly singular elliptic equations with degenerate coercivity

“…For our scope, we also limit the presentation to the case of solutions with zero Sobolev trace, which (in general) are the ones found in case γ 1 ≤ 1 and which is mainly the content of the present paper. Firstly we highlight that distributional solutions with W 1,p -finite energy are always unique; this is shown in various papers and it relies on an extension argument of the set of test functions and on a classical comparison technique (see for instance [9,33]). On the other hand, explicit examples show that the solutions may have infinite energy if the datum is merely integrable.…”

Nonlocal Lazer-McKenna type problem perturbed by the Hardy's potential and its parabolic equivalence

“…where Ω is a bounded domain of R N , N ≥ 2, γ > 0 and µ a general Radon measure in Ω. Also, see the papers [27,28,29,30,31,32,33,34,35,36,37,38,39,40,41] for more related problems. These types of problems have been extensively studied for their relations with some physical phenomena in the theory of pseudoplastic fluids, [42].…”