Nguyen Phuong Các

On nontrivial solutions of a Dirichlet problem whose jumping nonlinearity crosses a multiple eigenvalue

Let Ω be a bounded domain in RN (N ⩾ 2) with smooth boundary ∂Ω. Let p(·)R → R be a continuous function such that p(0) = 0 and limt → ± ∞(p(t)t) = p± exist. We discuss the existence of nontrivial solutions of the Dirichlet problem, −Δu = p(u) in Ω, u = 0 on ∂Ω. Our hypotheses relate the position of the point (p−, p+) in the plane with the extended spectrum of −Δ. They seem to allow noticeably greater freedom for the numbers p± than is usually found in the literature in the sense that the interval with end points p± is here permitted to contain in its interior an eigenvalue of arbitrarily multiplicity of −Δ. It also appears that we significantly weaken some technical conditions on p(·) of another paper which treats the crossing of a simple eigenvalue.

On an elliptic boundary value problem at double resonance

Abstract Let Ω be a bounded domain in R n (n ⩾ 2) with smooth boundary ∂Ω. We discuss the existence and multiplicity of solutions of the boundary value problem −Δu = p(x, u) in Ω, u = 0 on ∂Ω , roughly speaking, under the assumption that λ l ⩽ lim t → ± ∞ ( p(x, t) t ) ⩽ λ l + 1 for two distinct consecutive eigenvalues λl, λl + 1 of the first boundary value problem of −Δ.

Positive solutions to semilinear problems with coefficient that changes sign

where Ω will be assumed to be a bounded domain in R with smooth boundary. We will concentrate on the case when the coefficient a(x) is allowed to change sign; this brings about an interesting mathematical problem. This is not a new problem: due to its appearance in many mathematical models in physics, several special cases of it have been studied since the middle 1800’s. For example, it was used by Lord Kelvin [T,W] and J. Lane [L,J] to study the equilibrium configuration of mass in a spherical cloud gas; this model was further studied by R.H. Fowler [F,R], leaving the Emden Fowler equation which, in a generalized form is still studied today, as seen in, for example [AP]. For a review of the work done up to 1975 in the ordinary differential equations setting, see [W]; for a review of the work done for the existence of positive solutions of semilinear elliptic equations, see [L]. Positive solutions are usually the ones of interest and because of the difficulties that this brings in using the techniques developed for nonlinear functional analysis, most of the recent work assumes nonnegativity of a(x) and f(u) in order to generate positive operators using the Green’s function of −∆ on the region where the problem is posed. In nuclear physics, a form of (1.1) was proposed and used by E. Fermi [F,E] and by L. H. Thomas [T,L], and now it is widely known as the Thomas-Fermi equation. Both the Emden-Fowler equation as well as the Thomas Fermi equation continue to be subjects of considerable interest in theoretical physics to this day, as a search on the Wide World Web will quickly convince you. In combustion theory equation (1.1) has also been used as a model, for example in [ACP], [BK],[CL], [G],[L]. Equation (1.1) has also found applications to geometry: see [N] As mentioned above, most of the literature deals with the problem under the assumption that the coefficient function a(x) be positive in its domain; we are interested in the case when this function is allowed to change sign. In particular we are interested in determining sufficient conditions on a(x) to assure the existence of positive solutions for

On the number of solutions of an elliptic boundary value problem with jumping nonlinearity

Soit Ω un domaine borne de R N (N>2) a frontiere lisse ∂Ω. On considere la determination du nombre de solutions du probleme aux valeurs limites −Δu=f(u)+h+τφ h dans Ω, u=0 sur ∂Ω quand τ varie

On nontrivial solutions of an asymptotically linear Dirichlet problem

Let Ω be a bounded domain in RN (N ⩾ 2) with smooth boundary ∂Ω. Let ƒ(·):R → R be a continuous function such that ƒ(0) = 0 and limζ → ± ∞ƒ(ζ)ζ exist. We discuss the existence of nontrivial solutions of the Dirichlet problem −Δu = ƒ(u) in Ω, u = 0 on ∂Ω. Our hypotheses seem to allow significantly greater freedom for the numbers ƒ± = limζ → ± ∞ƒ(ζ)ζ than is usually found in the literature in that the interval with end points ƒ± is here permitted to contain an eigenvalue of −Δ subject to zero Dirichlet data. Furthermore we do not require any differentiability of ƒ and we weaken other technical conditions on ƒ that are generally assumed in other papers dealing with this problem.

On some quasilinear elliptic boundary value problems with conditions at infinity

in various, possibly weighted, Sobolev spaces of real functions defined on a. In the case the function p depends on the third variable v in a special way, this problem has been considered by Benci and Fortunato [I]. They point out the necessity of considering Sobolev spaces with weights by giving an interesting example for which the BVP (I), (2) has no solution in IY’*p(0), p > 2. This means we have to accept solutions with more liberal behaviour at infinity depending on the weight function a(.) used. Benci and Fortunato prove the solvability in w’(Q, o) of the BVP (1) (2) under the assumption that it has an upper solution I,U and a lower solution q with c~ < w. It seems to us they make use in an essential way of the regularity of the solution of an elliptic inequality [ 31. Thus for their method to be applicable, apparently the coefficients of the elliptic operator A have to be reasonably smooth; the function p has to depend on the third variable q in a special way; the upper and lower solutions v, p, and the weight function CJ have to belong to C’(4). In Theorem 1, we shall establish solvability in I+“(.Q, a) under fewer smoothness requirements. Our proof is based on a result that we proved earlier [4] concerning the existence of a weak solution of the BVP (l), (2) assuming that it has a weak upper solution v/ and a weak lower solution v, with 9 < v/. Thus, in Theorem 1 we shall require, for example, only that 9, v/ belong to some Sobolev space W’(R, a). This improvement as compared to

A note on Köthe spaces

Let E be a locally compact space which can be expressed as the union of an increasing sequence of compact subsets K n ( n =1, 2, …) and let μ be a positive Radon measure on E . Ω is the space of equivalence classes of locally integrable functions on E . We denote the equivalence class of a function f by and if is an equivalence class then f denotes any function belonging to f. Provided with the topology defined by the sequence of seminorms Ω is a Frechet space. The dual of Ω is the space φ of equivalence classes of measurable, p.p. bounded functions vanishing outside a compact subset of E . For a subset Γ of Ω, the collection Λ of all ∊Ω, such that for each g ∊Γ the product fg is integrable, is called a Kothe space and Γ is said to be the denning set of Λ. The Kothe space Λ x which has Λ as a denning set is called the associated Kothe space of Λ. Λ and Λ x are put into duality by the bilinear form

The exterior Dirichlet problem for a strongly nonlinear elliptic equation

Let Ω be the exterior of a bounded domain in RN (N ⩾ 2). For a possibly nonlinear elliptic operator A in divergence form and a differentialbe function p: R→R with p(0) = 0, p′(t)⩾0∀t∈R we discuss the solvability of the boundary value problem Au+p(u)=ƒ−Σi=1N∂ƒi∂xi inΩ, u=0 on ∂Ω under various restrictive conditions on p(·) which, however, would still allow more or less liberal exponential growths. The right-hand side of the equation belongs to some subspaces of the dual of W01,2(Ω).

Solutions with Asymptotic Conditions of a Nonlinear Boundary Value Problem

Publisher Summary This chapter presents the solutions with asymptotic conditions of a nonlinear boundary value problem (BVP), where A is a nonlinear elliptic differential operator in divergence form of Leray–Lions type. The chapter focuses on the linear operators A and the solution obtained is local in nature. The chapter presents the method of upper and lower solutions that is conceptually simple and particularly useful in proving the existence of a solution for noncoercive and possibly strongly nonlinear BVPs.