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.

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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)