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

Abstract

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.

Positive solutions for the p-Laplacian involving critical and supercritical nonlinearities with zeros

“…Even in the subcritical case, our result gives new information with respect to [11] in the sense that we may substitute global hypotheses on f with (much weaker) local ones (see Remark 1.1).…”

“…In [11], two solutions for the problem (P λ ) were encountered by mainly variational techniques, the first solution being a local minimum (which could also be obtained via sub-and supersolutions), while the second one was obtained via mountain pass. It was also proved, using a combination of a Liouville-type theorem, a priori estimates and the blow-up technique, that both solutions tend pointwise to 1.…”

“…Finally, the first solution in Theorem 1.1 is just the same as in [11], while the second one is obtained as the limit of the solutions of the perturbed problem; since we need to distinguish these two solutions, it is crucial to know that one lies below 1 and the other does not: for this reason, instead of the mountain pass theorem used in [11], we obtain the second solution for the perturbed problem by a degree approach, which has the advantage to furnish the information that its maximum is greater than the supersolution available.…”

“…On the other hand, the constant function 1 is always a supersolution for (P λ ), but not for the perturbed problem: hypothesis (F 3 ) will be used to obtain a family of supersolutions, in particular a supersolution strictly below 1 and one strictly above: this will help to distinguish the two solutions when taking limit. • Hypothesis (F 3 ) is required also to obtain inequality (2.1) for the truncated nonlinearity, which is necessary for applying the Liouville-type theorem from [11] (Lemma 2.2). We also remark that, in fact, (F 3 ) avoids the formation of the so-called flat core (a solution which coincides with 1 in a whole open set, see [12,20] for instance); actually, if this phenomenon could occur, then it would become difficult to separate the two solutions and obtain the multiplicity result.…”

“…The existence of a solution whose maximum is above 1 is more delicate and usually requires some hypotheses on the growth of f at infinity. In [11], we considered the p-Laplacian operator and we allowed f to depend also on the variable x ∈ Ω, but only in the subcritical case: two positive solutions were obtained for λ above the first eigenvalue of the asymptotical problem at the origin, and it was proved that both solutions converge at least pointwise to 1 when λ → ∞.…”