Multiple positive solutions for quasilinear elliptic problems with sign-changing nonlinearities (original) (raw)
Multiple positive solutions for a class of nonlinear elliptic equations
Journal of Mathematical Analysis and Applications, 2004
In 2009 Loc and Schmitt established a result on sufficient conditions for multiplicity of solutions of a class of nonlinear eignvalue problems for the p-Laplace operator under Dirichlet boundary conditions, extending an earlier result of 1981 by Peter Hess for the Laplacian. Results on necessary conditions for existence were also established. In the present paper the authors extend the main results by Loc and Schmitt to the Φ-Laplacian. To overcome the difficulties with this much more general operator it was necessary to employ regularity results by Lieberman, a strong maximum principle by Pucci and Serrin and a general result on lower and upper solutions by Le [10].
On positive solutions for ppp-Laplacian systems with sign-changing nonlinearities
Hokkaido Mathematical Journal, 2010
We consider the existence and multiplicity of positive solutions to the quasilinear system (−∆ p i u i = µ i a i (x)f i (u 1 ,. .. , u n) in Ω, i = 1,. .. , n, u i = 0 on ∂Ω, where Ω is a bounded domain in R N with a smooth boundary ∂Ω, ∆ p i u i = div(|∇u i | p i −2 ∇u i), p i > 1, µ i are positive parameters, and f i are allowed to change sign.
arXiv (Cornell University), 2013
In 2009 Loc and Schmitt established a result on sufficient conditions for multiplicity of solutions of a class of nonlinear eignvalue problems for the p-Laplace operator under Dirichlet boundary conditions, extending an earlier result of 1981 by Peter Hess for the Laplacian. Results on necessary conditions for existence were also established. In the present paper the authors extend the main results by Loc and Schmitt to the Φ-Laplacian. To overcome the difficulties with this much more general operator it was necessary to employ regularity results by Lieberman, a strong maximum principle by Pucci and Serrin and a general result on lower and upper solutions by Le [10].
Positive solutions for some generalized $ p $–Laplacian type problems
Discrete & Continuous Dynamical Systems - S, 2020
In this paper, we prove the existence of nontrivial weak bounded solutions of the nonlinear elliptic problem −div(a(x, u, ∇u)) + At(x, u, ∇u) = f (x, u) in Ω, u ≥ 0 in Ω, u = 0 on ∂Ω, where Ω ⊂ R N is an open bounded domain, N ≥ 3, and A(x, t, ξ), f (x, t) are given functions, with At = ∂A ∂t , a = ∇ ξ A. To this aim, we use variational arguments which are adapted to our setting and exploit a weak version of the Cerami-Palais-Smale condition. Furthermore, if A(x, t, ξ) grows fast enough with respect to t, then the nonlinear term related to f (x, t) may have also a supercritical growth.
Multiple Solutions for a Class of Quasilinear Elliptic Problems
Proceedings of the Edinburgh Mathematical Society, 2003
We deal with a class of ppp-Laplacian Dirichlet boundary-value problems where the combined effects of ‘sublinear’ and ‘superlinear’ growths allow us to establish the existence of at least two positive solutions.AMS 2000 Mathematics subject classification: Primary 35J25; 35J60. Secondary 35J65; 35J70
Multiple positive solutions for classes of elliptic systems with combined nonlinear effects
Differential and Integral Equations
Here Δ is the Laplacian operator, λ is a positive parameter, Ω is a bounded domain in R N with smooth boundary and f, g belong to a class of positive functions that have a combined sublinear effect at ∞. Our results also easily extend to the corresponding p-Laplacian systems. We prove our results by the method of sub and super solutions.
EXISTENCE OF SIGN-CHANGING SOLUTIONS FOR THE NONLINEAR p-LAPLACIAN BOUNDARY VALUE PROBLEM
Analysis and Applications, 2013
We study the nonlinear one-dimensional p-Laplacian equation [Formula: see text] where p > 1, y(p-1) = |y|p-1 sgn y = |y|p-2y, with linear separated boundary conditions. We give sufficient conditions for the existence of solutions with prescribed nodal properties concerning the behavior of f(s)/s(p-1) when s is at infinity and zero, respectively. These results are more general and complementary than previous known ones for the case when p = 2 and q is nonnegative.
Positive solutions for the p -Laplacian with dependence on the gradient
Nonlinearity, 2012
We prove a result of existence of positive solutions of the Dirichlet problem for −∆pu = w(x)f (u, ∇u) in a bounded domain Ω ⊂ R N , where ∆p is the p-Laplacian and w is a weight function. As in previous results by the authors, and in contrast with the hypotheses usually made, no asymptotic behavior is assumed on f , but simple geometric assumptions on a neighborhood of the first eigenvalue of the p-Laplacian operator. We start by solving the problem in a radial domain by applying the Schauder Fixed Point Theorem and this result is used to construct an ordered pair of sub-and super-solution, also valid for nonlinearities which are superlinear both at the origin and at +∞. We apply our method to the Dirichlet problem −∆pu = λu(x) q−1 (1+|∇u(x)| p ) in Ω and give examples of superlinear nonlinearities which are also handled by our method. * The authors were supported in part by FAPEMIG and CNPq-Brazil.