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.
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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