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 existence and multiplicity of positive solutions for the following problemwhere λ is a positive parameter, Ω is a bounded and smooth domain in R N , p ∈ (1, N), f (x, t) behaves, for instance, like o(|t| p−1 ) near 0 and +∞, and satisfies some further properties. In particular, our assumptions allow us to consider both positive and sign changing nonlinearitites f , the latter describing logistic as well as reaction-diffusion processes. By using sub-and supersolutions and variational arguments, we prove that there exists a positive constant λ such that the above problem has at least two positive solutions for λ > λ, at least one positive solution for λ = λ and no solution for λ < λ. An important rôle plays the fact that local minimizers of certain functionals in the C 1 -topology are also minimizers in W 1,p 0 (Ω). We give a short new proof of this known result.
In this paper we study the geometry of certain functionals associated to quasilinear elliptic boundary value problems with a degenerate nonlocal term of Kirchhoff type.Due to the degeneration of the nonlocal term it is not possible to directly use classical results such as uniform a-priori estimates and "Sobolev versus Hölder local minimizers" type of results. We prove that results similar to these hold true or not, depending on how degenerate the problem is.We apply our findings in order to show existence and multiplicity of solutions for the associated quasilinear equations, considering several different interactions between the nonlocal term and the nonlinearity.Mathematical Subject Classification MSC2010: 35J20 (35J70)
We study the existence of a positive solution of a p-superlinear equation involving the p-Laplacian operator. The main difficulty here is that the nonlinearity considered does not necessarily verify the well-known Ambrosetti–Rabinowitz condition. As an application, by performing an adequate change of variables we obtain an existence result of a quasilinear equation depending on the gradient.
Abstract. We study the existence and multiplicity of positive solutions to p-Laplace equations where the nonlinear term depends on a p-power of the gradient. For this purpose we combine Picone's identity, blow-up arguments, the strong maximum principle and Liouville-type theorems to obtain a priori estimates.
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