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Positive solutions for a quasilinear elliptic problem involving sublinear and superlinear terms. (English) Zbl 1345.35049

The authors of the present paper study a quasilinear elliptic problem involving a \(p\)-Laplacian operator (\(1<p<N,\) where \(N\in\mathbb N\) is the space dimension) and three different nonlinearities and some other parameters. The problem is set either in the whole space (with the corresponding vanishing asymptotic property at infinity) or in bounded smooth domains (with the corresponding homogeneous Dirichlet condition) and they are looking for positive solutions.
On one hand, for some values of one of the parameters the authors show the existence of a \(C^1\) (weak) solution of the problem. The method of the proof relies on the lower-upper solutions method, combined with a technique of monotone regularization of the nonlinearities. On the other hand, for some other values of this parameter, using some consequences of Picone’s identity, they show the nonexistence of the solutions.
The considered problems are general enough, in the sense that for different choices of the parameters the authors recover some well-known results from the literature. Also, their approach allows them to consider nonlinearities with combined effects of concave and convex terms, besides allowing the presence of singularities. The question of uniqueness of the solutions in the first case is not treated.

MSC:

35J92 Quasilinear elliptic equations with \(p\)-Laplacian
35J67 Boundary values of solutions to elliptic equations and elliptic systems
35B09 Positive solutions to PDEs
35B08 Entire solutions to PDEs
35J75 Singular elliptic equations
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References:

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