Existence and Multiplicity Results for the p-Laplacian with a p-Gradient Term

Abstract

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.

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

“…Now, we work with τ > 0 and we show that a second solution exists: we will apply a topological degree argument, adapting a result obtained, for p = 2, by de Figueiredo and Lions in [4], see also [10] for the general case.…”

Existence of positive solutions for the p-Laplacian with p-gradient term

“…However, since the power of growth of the nonlinearities for problem (P ) with respect to u and ∇u may be critical or supercritical (here we call p * and p the critical exponents corresponding to u and ∇u, respectively), we cannot use the argument directly to the original problem (P ). To overcome this difficulty, we use the same change of variables applied in [19] to transform the problem (P ) into an equivalent one which satisfies the requirement of blow-up argument. The most important two tools used in blow-up argument are global C 1,α estimates and Liouville type results, see Sections 2 and 3.…”

“…When f (x, u, ξ) = λu q , by using variational method, several sufficient conditions for the existence of positive solutions for the problem (1.1) can be implied in [17] for the case that p = 2, and in [18] for the case that p = 2. While the case that g ≡ 1 and f = f (x, u) was considered in [19] with f satisfying some power like growth with respect to u.…”

Nonlinear nonhomogeneous Robin problems with dependence on the gradient

“…We mention the works of de Figueiredo et al [4], Girardi and Matzeu [5] for semilinear equations driven by the Dirichlet Laplacian; the works of Faraci et al [6], Huy et al [7], Iturriaga et al [8] and Ruiz [9] for nonlinear equations driven by the Dirichlet p-Laplacian; and the works of Averna et al [10], Faria et al [11] and Tanaka [12] for equations driven by the Dirichlet (p, q)-Laplacian. Finally, we mention also the recent work of Gasiński and Papageorgiou [13] for Neumann problems driven by a differential operator of the form div(a(u)Du).…”