Existence and Nonexistence Results for Classes of Singular Elliptic Problem
نویسندگان
چکیده
and Applied Analysis 3 2. Proof of Theorems Consider the more general semilinear elliptic problem −Δu f x, u , in Ω, u > 0, in Ω, u 0, on ∂Ω, 2.1 where the function f x, s is locally Hölder continuous in Ω × 0,∞ and continuously differentiable with respect to the variable s. A function u is called to be a subsolution of problem 2.1 if u ∈ C2 Ω ∩ C Ω , and −Δu ≤ fx, u, in Ω, u > 0, in Ω, u 0, on ∂Ω. 2.2 A function u is called to be a supersolution of problem 2.1 if u ∈ C2 Ω ∩ C Ω , and −Δu ≥ f x, u , in Ω, u > 0, in Ω, u 0, on ∂Ω. 2.3 According to Lemma 3 in the study of Cui 15 , we can easily have the following basic existence of classical solution to problem 2.1 . Lemma 2.1. Let f ∈ C loc Ω × 0,∞ be continuously differentiable with respect to the variable s. Suppose that problem 2.1 has a supersolution u and a subsolution u such that u x ≤ u x , in Ω, 2.4 then problem 2.1 has at least one solution u ∈ C2 α Ω ∩ C Ω satisfying u x ≤ u x ≤ u x , in Ω. 2.5 Let λ1 be the first eigenvalue of the eigenvalue problem −Δu λu, in Ω, u 0, on ∂Ω, 2.6 and φ1 > 0 in Ω the corresponding eigenfunction. Then φ1 ∈ C2 α Ω . Moreover one has the following lemma. 4 Abstract and Applied Analysis Lemma 2.2 see 10 . One has ∫
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