Abstract
Abstract In the present paper, we study the existence of nonnegative solutions to the Dirichlet problem ℒ p , q M u := - Δ u + u p - M | ∇ u | q = μ {{\mathcal{L}}^{{M}}_{p,q}u:=-\Delta u+u^{p}-M|\nabla u|^{q}=\mu} in a domain Ω ⊂ ℝ N {\Omega\subset\mathbb{R}^{N}} where μ is a nonnegative Radon measure, when p > 1 {p>1} , q > 1 {q>1} and M ≥ 0 {M\geq 0} . We also give conditions under which nonnegative solutions of ℒ p , q M u = 0 {{\mathcal{L}}^{{M}}_{p,q}u=0} in Ω ∖ K {\Omega\setminus K} , where K is a compact subset of Ω, can be extended as a solution of the same equation in Ω.
Highlights
Let Ω be a bounded domain of RN, N ≥ 2, and LM p,q be the operator u → LM p,qu := −∆u + |u|p−1u − M |∇u|q for all u ∈ C2(Ω)
We study under what conditions on the parameters any solution of LM p,qu = 0 in Ω \ K, (1.2)
Where μ is a nonnegative bounded Radon measure in Ω and exhibit conditions which guarantee the existence of nonnegative solutions to this problem
Summary
Let Ω be a bounded domain of RN , N ≥ 2, and LM p,q be the operator u → LM p,qu := −∆u + |u|p−1u − M |∇u|q for all u ∈ C2(Ω). Where μ is a nonnegative bounded Radon measure in Ω and exhibit conditions which guarantee the existence of nonnegative solutions to this problem. Extended the result in [13] to more general removable sets, introducing the good framework They obtained a necessary and sufficient condition expressed in terms of the Bessel capacities capR2,Np′. Both for the removability of compact subsets of Ω and the solvability of the associated Dirichlet problem with measure data. The method introduced in the proof of Theorem 1.1 combined with the result of [3] yields a more general removability result For such a task we denote by capRk,Nb the Bessel capacity relative to RN with order k > 0 and power b ∈ (1, ∞). In the above mentioned article we construct many types of local or global singular solutions using methods inherited from dynamical systems
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