Existence of Weak Solutions to a Class of Singular Elliptic Equations

Nonlinear singular problems with indefinite potential term

“…Hypotheses H (a)(i), (ii), (iii) are dictated by the nonlinear regularity theory of Lieberman [10] and the nonlinear maximum principle of Pucci and Serrin [21]. Hypothesis H (a)(iv) serves the needs of our problem, but in fact, it is a mild condition and it is satisfied in all cases of interest (see the examples below).…”

“…Proposition 3.2. If hypotheses H(a), H(ξ), H(ϑ) hold, then for all small enough λ > 0 problem (10) λ has a smallest positive solution…”

“…It is also related to the work of Papageorgiou & Smyrlis [17] and Papageorgiou & Winkert [19], where the differential operator is the p-Laplacian, ξ ≡ 0 and no concave terms are allowed. Singular p-Laplacian equations with no potential term and reactions of special form were considered by Chu, Gao & Sun [2], Giacomoni, Schindler & Takač [5], Li & Gao [10], Mohammed [12], Perera & Zhang [20], and Papageorgiou, Rȃdulescu & Repovš [14].…”

“…(from the nonlinear maximum principle, seePucci & Serrin [21, pp. 111,120]).Let Ŝλ denote the set of positive solutions of problem(10) λ . We have just seen that∅ = Ŝλ ⊆ int C + for λ ∈ (0, λ 0 ].Moreover, from Papageorgiou, Rȃdulescu & Repovš [16, Proposition 18], we know that Ŝλ is downward directed (that is, if u 1 , u 2 , ∈ Ŝλ , then we can find u ∈ Ŝλ such that u u 1 , u u 2 ).…”

Degenerate elliptic problem with a singular nonlinearity

“…Theorem 5. Let f be in L m (Ω) with m > N/p, 0 ≤ θ ≤ 1 and let u n be solution of (16). Then the norm of u n in L ∞ (Ω).…”