On the Connection Between Two Quasilinear Elliptic Problems With Source Terms of Order 0 or 1

Abstract

We establish a precise connection between two elliptic quasilinear problems with Dirichlet data in a bounded domain of R N . The first one, of the forminvolves a source gradient term with natural growth, where β is nonnegative, λ > 0, f (x) ≧ 0, and α is a nonnegative measure. The second one, of the formpresents a source term of order 0, where g is nondecreasing, and µ is a nonnegative measure. Here β and g can present an asymptote. The correlation gives new results of existence, nonexistence, regularity and multiplicity of the solutions for the two problems, without or with measures. New informations on the extremal solutions are given when g is superlinear.

Nonlinear elliptic equations and intrinsic potentials of Wolff type

“…Equations of the type (1.1) and (1.5) have been extensively studied, mostly in bounded domains Ω ⊂ R n , with σ ∈ L r (Ω) for some r > 1, in [Kra64,BO86,BoOr96,ABL10,ABV10,AGP11]. Various existence and uniqueness results for solutions in certain Sobolev spaces, and further references, can be found there.…”

Local and Global Behaviour of Solutions to Nonlinear Equations with Natural Growth Terms

“…In Remark 1.3 below we consider a simple condition on σ which ensures the existence of a solution u for any measure ω with corresponding pointwise bound. This condition in particular covers the work of [2] cited above.…”

“…2) In our previous work [25], we studied an important special case of (1.1), namely, the fundamental solution:…”

“…In recent papers [2] and [19] it is pointed out that the existence problem for (1.1) is non-trivial for general measure right-hand side ω, even under the assumption that σ ∈ L q ( ) for q > n/ p. In these papers, the existence problem for (1.1) for measure data ω is solved under the assumption that σ ∈ L q ( ) for q > n/ p with small norm. If one avoids the phenomenon of interaction between σ and ω, then a simple analysis (see Remark 6.1 of [2]) shows that this L q class of potentials σ is optimal on the Lebesgue scale in order to solve the equation (1.1) for all finite measures ω.…”

“…If one avoids the phenomenon of interaction between σ and ω, then a simple analysis (see Remark 6.1 of [2]) shows that this L q class of potentials σ is optimal on the Lebesgue scale in order to solve the equation (1.1) for all finite measures ω.…”