Exact number of positive solutions for semipositone problems

This paper is concerned with the exact number of positive solutions for the boundary value problem ^′(|y^′|^p−2y^′)+λf(y)=0 and y(−1)=y(1)=0, where p>1 and λ>0 is a positive parameter. We consider the case in which both f(u) and g(u)=(p−1)f(u)−uf^′(u) change sign exactly once from negative to positive on (0,∞).     

Exact multiplicity results for a class of two-point boundary value problems

This paper is concerned with the exact number of positive solutions for boundary value problems ^′(|y^′|^p−2y^′)+λf(y)=0 and y(−1)=y(1)=0, where p>1 and λ>0 is a positive parameter. We consider the case in which the nonlinearity f is positive on (0,∞) and (p−1)f(u)−uf^′(u) changes sign from negative to positive.     

Multiple positive solutions to a singular boundary value problem for a superlinear Emden-Fowler equation

A multiplicity result for the singular ordinary differential equation y^″+λx⁻²yσ=0, posed in the interval (0,1), with the boundary conditions y(0)=0 and y(1)=γ, where σ>1, λ>0 and γ?0 are real parameters, is presented. Using a logarithmic transformation and an integral equation method, we show that there exists Σ^?∈(0,σ/2] such that a solution to the above problem is possible if and only if λγσ−1?Σ^?. For 0<λγσ−1<Σ^?, there are multiple positive solutions, while if γ=(λ⁻¹Σ^?)^1/(σ−1) the problem has a unique positive solution which is monotonic increasing. The asymptotic behavior of y(x) as x→⁰⁺ is also given, which allows us to establish the absence of positive solution to the singular Dirichlet elliptic problem −Δu=d⁻²(x)uσ in Ω, where Ω⊂RN, N?2, is a smooth bounded domain and d(x)=dist(x,∂Ω).     

Multiple positive solutions of nonhomogeneous elliptic systems with strongly indefinite structure and critical Sobolev exponents

In this paper, we study the existence of multiple positive solutions to some Hamiltonian elliptic systems −Δv=λu+u^p+εf(x), −Δu=μv+v^q+δg(x) in Ω;u,v>0 in Ω; u=v=0 on ∂Ω, where Ω is a bounded domain in R^N (N?3); 0?f, g∈L∞(Ω); 1/(p+1)+1/(q+1)=(N−2)/N, p,q>1; λ,μ>0. Using sub- and supersolution method and based on an adaptation of the dual variational approach, we prove the existence of at least two nontrivial positive solutions for all λ,μ∈(0,λ₁) and ε,δ∈(0,δ₀), where λ₁ is the first eigenvalue of the Laplace operator −Δ with zero Dirichlet boundary conditions and δ₀ is a positive number.     

On an eigenvalue problem involving the one-dimensional p-Laplacian

We apply general results on operator equations in ordered spaces and properties of the principal eigenvalues for weighted semi-linear equations to prove the existence of a global continua of positive solutions and eigenvalue intervals to the problem (?(x′))′+λf(t,x,x′)=0 in (0,1), x(0)=x(1)=0, where ?(x)=|x|^p−2x, p>1, λ>0.     

Uniqueness results for the one-dimensional p-Laplacian

This paper is concerned with positive solutions of the boundary value problem ^′(|y^′|^p−2y^′)+f(y)=0, y(−b)=0=y(b) where p>1, b is a positive parameter. Assume that f is continuous on (0,+∞), changes sign from nonpositive to positive, and f(y)/yp−1 is nondecreasing in the interval of f>0. The uniqueness results are proved using a time-mapping analysis.     

Existence and multiplicity of solutions for quasilinear nonhomogeneous problems: An Orlicz-Sobolev space setting

We study the boundary value problem −div(log(1+q|∇u|)|∇u|^p−2∇u)=f(u) in Ω, u=0 on ∂Ω, where Ω is a bounded domain in RN with smooth boundary. We distinguish the cases where either f(u)=−λ|u|^p−2u+|u|^r−2u or f(u)=λ|u|^p−2u−|u|^r−2u, with p, q>1, p+q<min{N,r}, and r<(Np−N+p)/(N−p). In the first case we show the existence of infinitely many weak solutions for any λ>0. In the second case we prove the existence of a nontrivial weak solution if λ is sufficiently large. Our approach relies on adequate variational methods in Orlicz-Sobolev spaces.     

Equivalence of inverse Sturm-Liouville problems with boundary conditions rationally dependent on the eigenparameter

Three inverse problems for a Sturm-Liouville boundary value problem −y″+qy=λy, y(0)cosα=y′(0)sinα and y′(1)=f(λ)y(1) are considered for rational f. It is shown that the Weyl m-function uniquely determines α, f, and q, and is in turn uniquely determined by either two spectra from different values of α or by the Prüfer angle. For this it is necessary to produce direct results, of independent interest, on asymptotics and oscillation.     

Absolute continuity and spectral concentration for slowly decaying potentials

We consider the spectral function ϱ(μ) (μ⩾0) for the Sturm-Liouville equation y″ + (λ −qq)y=0 on [0,∞) with the boundary condition y(0)=0 and where q has slow decay O(x^−a (a > 0) as x → ∞. We develop our previous methods of locating spectral concentration for q with rapid exponential decay (this Journal 81 (1997) 333–348) to deal with the new theoretical and computational complexities which arise for slow decay.     

Existence-uniqueness results for a class of singular boundary value problems-II

In this paper we establish existence-uniqueness of solution of a class of singular boundary value problem −(p(x)y^′′(x))=q(x)f(x,y) for 0<x?b and y(0)=a, α₁y(b)+β₁y^′(b)=γ₁, where p(x) satisfies (i) p(x)>0 in (0,b), (ii) p(x)∈C¹(0,r), and for some r>b, (iii) is analytic in and q(x) satisfies (i) q(x)>0 in (0,b), (ii) q(x)∈L¹(0,b) and for some r>b, (iii) is analytic in with quite general conditions on f(x,y). Region for multiple solutions have also been determined.     

On p−q=c and related three term exponential Diophantine equations with prime bases

Using a theorem on linear forms in logarithms, we show that the equation p^x−2^y=p^u−2^v has no solutions (p,x,y,u,v) with x≠u, where p is a positive prime and x,y,u, and v are positive integers, except for four specific cases, or unless p is a Wieferich prime greater than 10¹⁵. More generally, we obtain a similar result for p^x−q^y=p^u−q^v>0 where q is a positive prime, . We solve a question of Edgar showing there is at most one solution (x,y) to p^x−q^y=2^h for positive primes p and q and positive integer h. Finally, we use elementary methods to show that, with a few explicitly listed exceptions, there are at most two solutions (x,y) to |p^x±q^y|=c and at most two solutions (x,y,z) to p^x±q^y±2^z=0, for given positive primes p and q and integer c.     

Positive solutions of the nonlinear fourth-order beam equation with three parameters

This paper is concerned with the existence and nonexistence of positive solutions of the nonlinear fourth-order beam equation u(4)(t)+ηu^″(t)−ζu(t)=λf(t,u(t)), 0<t<1, u(0)=u(1)=u^″(0)=u^″(1)=0, where is continuous and ζ, η and λ are parameters. We show that there exists a such that the above boundary value problem (BVP) has at least two, one and no positive solutions for 0<λ<λ^*, λ=λ^* and λ>λ^*, respectively. Furthermore, by using the semiorder method on cones of Banach space, we establish a uniqueness criterion for positive solution of the BVP. In particular such a positive solution uλ(t) of the BVP depends continuously on the parameter λ, i.e., uλ(t) is nondecreasing in λ, limλ→⁰⁺‖uλ(t)‖=0 and limλ→+∞‖uλ(t)‖=+∞ for any t∈[0,1].     

Entire solutions of quasilinear elliptic equations

We study entire solutions of non-homogeneous quasilinear elliptic equations, with Eqs. (1) and (2) below being typical. A particular special case of interest is the following: Let u be an entire distribution solution of the equation Δpu=|u|^q−1u, where p>1. If q>p−1 then u≡0. On the other hand, if 0<q<p−1 and u(x)=o(|x|^{p/(p−q−1)}) as |x|→∞, then again u≡0. If q=p−1 then u≡0 for all solutions with at most algebraic growth at infinity.     

The behavior of solutions of second order delay differential equations

In this paper, we study the behavior of solutions of second order delay differential equation

y^″(t)=p₁y^′(t)+p₂y^′(t−τ)+q₁y(t)+q₂y(t−τ),     

A Variational Approach to Discontinuous Problems with Critical Sobolev Exponents

We employ variational techniques to study the existence and multiplicity of positive solutions of semilinear equations of the form − Δu = λh(x)H(u − a)u^q + u^2* − 1 in R^N, where λ, a > 0 are parameters, h(x) is both nonnegative and integrable on R^N, H is the Heaviside function, 2* is the critical Sobolev exponent, and 0 ≤ q < 2* − 1. We obtain existence, multiplicity and regularity of solutions by distinguishing the cases 0 ≤ q ≤ 1 and 1 < q < 2* − 1.     

Eigenvalue problems for second-order nonlinear dynamic equations on time scales

This paper is concerned with the existence and nonexistence of positive solutions of the second-order nonlinear dynamic equation uΔΔ(t)+λa(t)f(u(σ(t)))=0, t∈[0,1], satisfying either the conjugate boundary conditions u(0)=u(σ(1))=0 or the right focal boundary conditions u(0)=uΔ(σ(1))=0, where a and f are positive. We show that there exists a λ^∗>0 such that the above boundary value problem has at least two, one and no positive solutions for 0<λ<λ^∗, λ=λ^∗ and λ>λ^∗, respectively. Furthermore, by using the semiorder method on cones of the Banach space, we establish an existence and uniqueness criterion for positive solution of the problem. In particular, such a positive solution uλ(t) of the problem depends continuously on the parameter λ, i.e., uλ(t) is nondecreasing in λ, limλ→⁰⁺‖uλ‖=0 and limλ→+∞‖uλ‖=+∞.     

A complete classification of bifurcation diagrams of a p -Laplacian Dirichlet problem

We study the bifurcation diagrams of (classical) positive solutions u with |u |_∞ ∈ (0, ∞) of the p -Laplacian Dirichlet problem (φ_p (u ′(x)))′ + λf_q (u (x))) = 0, –1 ≤ x ≤ 1, u (–1) = 0 = u (1), where p > 1, φ_p (y) = |y |^{p –2} y, (φ_p (u ′))′ is the one-dimensional p -Laplacian, λ > 0 is a bifurcation parameter, and the nonlinearity f_q (u) = |1 – u |^q is defined on [0, ∞) with constant q > 0. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Some inverse results for Hill's equation

We consider Hill's equation y″+(λ−q)y=0 where q∈L¹[0,π]. We show that if l_n—the length of the n-th instability interval—is of order O(n^−(k+2)) then the real Fourier coefficients a_n^k,b_n^k of q^(k)—k-th derivative of q—are of order O(n⁻²), which implies that q^(k) is absolutely continuous almost everywhere for k=0,1,2,….     

Multi-bump solutions to Carrier's problem

We study the existence of well-known singularly perturbed BVP problem ε²y″=1−y²−2b(1−x²)y, y(−1)=y(1)=0 introduced by G.F. Carrier. In particular, we show that there exist multi-spike solutions, and the locations of interior spikes are clustered near x=0 and are separated by an amount of O(ε|lnε|), while only single spikes are allowed near the boundaries x=±1.     

Isolated singularities for weighted quasilinear elliptic equations

We classify all the possible asymptotic behavior at the origin for positive solutions of quasilinear elliptic equations of the form div(|∇u|^p−2∇u)=b(x)h(u) in Ω?{0}, where 1<p?N and Ω is an open subset of RN with 0∈Ω. Our main result provides a sharp extension of a well-known theorem of Friedman and Véron for h(u)=uq and b(x)≡1, and a recent result of the authors for p=2 and b(x)≡1. We assume that the function h is regularly varying at ∞ with index q (that is, limt→∞h(λt)/h(t)=λq for every λ>0) and the weight function b(x) behaves near the origin as a function b₀(|x|) varying regularly at zero with index θ greater than −p. This condition includes b(x)=θ|x| and some of its perturbations, for instance, b(x)=θ|x|m(−log|x|) for any m∈R. Our approach makes use of the theory of regular variation and a new perturbation method for constructing sub- and super-solutions.