Positive solutions of the p-Laplacian involving a superlinear nonlinearity with zeros

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Abstract

Using 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 Δpu=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 p-Laplacian operator, where the nonlinearity involved is superlinear, nonnegative, and has positive zeros.

MSC

35J60
35B40
35B45
35B50

Keywords

Multiplicity of positive solutions
p-Laplacian
Liouville-type theorems
Asymptotic behavior
Variational methods
Comparison principle

Cited by (0)

1

Partially supported by FONDECYT No 11080203 and Convenio de desempeño UTA–MECESUP 2.

2

The author was partially supported by Fapesp and CNPq/Brazil.

3

Partially supported by FONDECYT grant 1080430.

4

The author was partially supported by FONDECYT grant 1080430.