Marta García-Huidobro

Solvability for a nonlinear three-point boundary value problemwith p-Laplacian-like operator at resonance

Let φ be an odd increasing homeomorphism from R onto R which satisfies φ(0)= 0 and let f : [a,b]×R×R → R be a function satisfying Caratheodory conditions. Separated two-point and periodic boundary value problems containing the nonlinear operator (φ(u′))′, or its more particular form, the so-called p-Laplace operator, have received a lot of attention lately (cf. [6, 7, 8, 14, 15] and the references therein). On the other hand, three-point (or m-point) boundary value problems for the case when (φ(u′))′ = u′′, that is, the linear operator, have been considered by many authors (cf. [3, 9, 10, 12, 13]). The purpose of this paper is to study the following three-point boundary value problem which contains the nonlinear operator (φ(u′))′, ( φ(u′) )′ = f (t,u,u′), u′(a)= 0, u(η)= u(b), (1.1)

ON THE UNIQUENESS OF SIGN CHANGING BOUND STATE SOLUTIONS OF A SEMILINEAR EQUATION

Abstract We establish the uniqueness of the higher radial bound state solutions of (P) Δ u + f ( u ) = 0 , x ∈ R n . We assume that the nonlinearity f ∈ C ( − ∞ , ∞ ) is an odd function satisfying some convexity and growth conditions, and has one zero at b > 0 , is nonpositive and not-identically 0 in ( 0 , b ) , positive in [ b , ∞ ) , and is differentiable in ( 0 , ∞ ) .

On a pseudo Fuccik spectrum for strongly nonlinear second-order ODEs and an existence result

Abstract We study here the existence of solutions to the nonlinear Dirichlet problem (P): ( φ ( u ′))′ + f ( t , u ) = q ( t ), a.e. for t ∈ [ a, b ], u ( a ) = u ( b ) = 0, where φ is an increasing odd homeomorphism of R , f :[a,b] × R → R satisfies the Caratheodory assumptions and q ∈ L 1 ([a, b], R ) . With a view towards defining a pseudo Fucik spectrum (PFS) we study, using a time-mapping approach, the eigenvalue-like problem {( φ ( u ′))′ + Aφ ( u + ) − Bφ ( u − ) = 0, u ( a ) = 0 = u ( b ), where A > 0, B > 0. We show here that this PFS, which we denote by l , consists of a set of curves in ( R + ) 2 resembling the classical Fucik spectrum, i.e., when φ ( s ) = s . Our main existence result, which deals with nonresonance with respect to the PFS, can be roughly stated as follows: if for s > 0 sufficiently large and for almost every t ∈ [ a , b ], the pair ( f ( t , − s )/ φ (− s ), f ( t , s )/ φ ( s )) lies in a compact rectangle contained in an open component of ( R + ) 2 \ L which intersects the diagonal, then problem ( P ) has at least a solution .

MOUNTAIN PASS TYPE SOLUTIONS FOR QUASILINEAR ELLIPTIC INCLUSIONS

We establish the existence of weak solutions to the inclusion problem where Ω is a bounded domain in ℝN, , and ψ ∊ ℝ × ℝ is a maximal monotone odd graph. Under suitable conditions on ψ, g (which reduce to subcritical and superlinear conditions in the case of powers) we obtain the existence of non-trivial solutions which are of mountain pass type in an appropriate not necessarily reflexive Orlicz Sobolev space. The proof is based on a version of the Mountain Pass Theorem for a non-smooth case.

ON THE UNIQUENESS OF POSITIVE SOLUTIONS OF A QUASILINEAR EQUATION CONTAINING A WEIGHTED p-LAPLACIAN, THE SUPERLINEAR CASE

We consider the quasilinear equation of the form where Δpu ≔ div(|∇u|p-2∇u) is the degenerate p-Laplace operator and the weight K is a positive C1 function defined in ℝ+. We deal with the case in which f ∈ C[0,∞) has one zero at u0 > 0, is non positive and not identically 0 in (0,u0), and is locally Lipschitz, positive and satisfies some superlinear growth assumption in (u0,∞). We carefully study the behavior of the solution of the corresponding initial value problem for the radial version of the quasilinear equation, as well as the behavior of its derivative with respect to the initial value. Combining, as Cortazar, Felmer and Elgueta, comparison arguments due to Coffman and Kwong, with some separation results, we show that any zero of the solutions of the initial value problem is monotone decreasing with respect to the initial value, which leads immediately the uniqueness of positive radial ground states, and the uniqueness of positive radial solutions of the Dirichlet problem on a ball.

Behavior of positive radial solutions for quasilinear elliptic equations

We establish a necessary and sufficient condition so that positive radial solutions to −div(A(|∇u|)∇u) = f(u), in BR(0) \ {0}, R > 0, having an isolated singularity at x = 0, behave like a corresponding fundamental solution. Here, A : R \ {0} → R and f : [0,∞)→ [0,∞) are continuous functions satisfying some mild growth restrictions.

Qualitative Properties and Existence of Sign Changing Solutions with Compact Support for an Equation with a p-Laplace Operator

Abstract We consider radial solutions of an elliptic equation involving the p-Laplace operator and prove by a shooting method the existence of compactly supported solutions with any prescribed number of nodes. The method is based on a change of variables in the phase plane corresponding to an asymptotic Hamiltonian system and provides qualitative properties of the solutions.

Existence of positive radial solutions for a weakly coupled system via blow up

The existence of positive solutions to certain systems of ordi- nary differential equations is studied. Particular forms of these systems are satisfied by radial solutions of associated partial differential equations.

On the Uniqueness of the Limit for an Asymptotically Autonomous Semilinear Equation on ℝ N

We consider a parabolic equation of the form where f ∈ C 1(ℝ) is such that f(0) = 0 and f′(0) <0 and h is a suitable function on ℝ N × (0, ∞). We show that, under certain assumptions on f and u, each globally defined and nonnegative bounded solution u converges to a single steady state.

Solvability of a nonlinear Neumann problem for systems arising from a burglary model

-estimates of the type introduced by Ward, see Ward (1981) in some periodic problems.