Abstract. We study the existence, nonexistence and multiplicity of positive solutions for the family of problems − u = f λ (x, u), u ∈ H 1 0 ( ), where is a bounded domain in R N , N ≥ 3 and λ > 0 is a parameter. The results include the well-known nonlinearities of the Ambrosetti-Brezis-Cerami type in a more general form, namely λa(x)u q + b(x)u p , where 0 ≤ q < 1 < p ≤ 2 * − 1. The coefficient a(x) is assumed to be nonnegative but b(x) is allowed to change sign, even in the critical case. The notions of local superlinearity and local sublinearity introduced in [9] are essential in this more general framework. The techniques used in the proofs are lower and upper solutions and variational methods.
MSC: 35J60 35B40 35B45 35B50 Keywords: Multiplicity of positive solutions p-Laplacian Liouville-type theorems Asymptotic behavior Variational methods Comparison principleUsing a combination of several methods, such as variational methods, the sub and supersolutions method, comparison principles and a priori estimates, we study existence, multiplicity, and the behavior with respect to λ of positive solutions of p-Laplace equations of the form − p u = λh(x, u), where the nonlinear term has p-superlinear growth at infinity, is nonnegative, and satisfies h(x, a(x)) = 0 for a suitable positive function a. In order to manage the asymptotic behavior of the solutions we extend a result due to Redheffer and we establish a new Liouville-type theorem for the pLaplacian operator, where the nonlinearity involved is superlinear, nonnegative, and has positive zeros.
We study the existence, nonexistence and multiplicity of positive solutions for a family of problemswhere Ω is a bounded domain in R N , N > p, and λ > 0 is a parameter. The family we consider includes the well-known nonlinearities of Ambrosetti-Brezis-Cerami type in a more general form, namely λa(x)u q + b(x)u r , where 0 q < p − 1 < r p * − 1. Here the coefficient a(x) is assumed to be nonnegative but b(x) is allowed to change sign, even in the critical case. Preliminary results of independent interest include the extension to the p-Laplacian context of the Brezis-Nirenberg result on local minimization in W 1,p 0 and C 1 0 , a C 1,α estimate for equations of the form − p u = h(x, u) with h of critical growth, a strong comparison result for the p-Laplacian, and a variational approach to the method of upper-lower solutions for the p-Laplacian.
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