On a parabolic equation in MEMS with fringing field

We consider the asymptotic behavior of the solutions to the equation ${u_{t}-u_{xx} = \lambda(1 + {\delta}u_{x}^{2})(1 - u)^{-2}}$ , which comes from Micro-Electromechanical Systems (MEMS) devices modeling. It is shown that when the fringing field exists (i.e., δ?> 0), there is a critical value λ _δ ^* > 0 such that if 0 < λ < λ _δ ^* , the equation has a global solution for some initial data; while for λ > λ _δ ^* , all solutions to the equation will quench at finite time. When the quenching happens, u has only finitely many quenching points for particular initial data. A one-side estimate is deduced for the quenching rate of u.     

Existence and concentration of positive solutions for semilinear Schrödinger–Poisson systems in {\mathbb{R}^{3}}

In this paper, we study the existence and concentration of positive ground state solutions for the semilinear Schrödinger–Poisson system $$\left\{\begin{array}{ll}-\varepsilon^{2}\Delta u + a(x)u + \lambda\phi(x)u = b(x)f(u), & x \in \mathbb{R}^{3},\\-\varepsilon^{2}\Delta\phi = u^{2}, \ u \in H^{1}(\mathbb{R}^{3}), &x \in \mathbb{R}^{3},\end{array}\right.$$ where ε > 0 is a small parameter and λ ≠ 0 is a real parameter, f is a continuous superlinear and subcritical nonlinearity. Suppose that a(x) has at least one minimum and b(x) has at least one maximum. We first prove the existence of least energy solution (u _ε , φ _ε ) for λ ≠ 0 and ε > 0 sufficiently small. Then we show that u _ε converges to the least energy solution of the associated limit problem and concentrates to some set. At the same time, some properties for the least energy solution are also considered. Finally, we obtain some sufficient conditions for the nonexistence of positive ground state solutions.     

Asymptotic analysis and estimates of blow‐up time for the radial symmetric semilinear heat equation in the open‐spectrum case

We estimate the blow‐up time for the reaction diffusion equation u_t=Δu+ λf(u), for the radial symmetric case, where f is a positive, increasing and convex function growing fast enough at infinity. Here λ>λ^*, where λ^* is the ‘extremal’ (critical) value for λ, such that there exists an ‘extremal’ weak but not a classical steady‐state solution at λ=λ^* with ∥w(?, λ)∥_∞→∞ as 0<λ→λ^*?. Estimates of the blow‐up time are obtained by using comparison methods. Also an asymptotic analysis is applied when f(s)=e^s, for λ?λ^*?1, regarding the form of the solution during blow‐up and an asymptotic estimate of blow‐up time is obtained. Finally, some numerical results are also presented. Copyright © 2007 John Wiley & Sons, Ltd.     

A Hard-Core Model on a Cayley Tree: An Example of a Loss Network

The paper is about a nearest-neighbor hard-core model, with fugacity λ>0, on a homogeneous Cayley tree of order k(with k+1 neighbors). This model arises as as a simple example of a loss network with a nearest-neighbor exclusion. We focus on Gibbs measures for the hard core model, in particular on ‘splitting’ Gibbs measures generating a Markov chain along each path on the tree. In this model, ?λ>0 and k≥1, there exists a unique translation-invariant splitting Gibbs measure μ^*. Define λ_c=1/(k?1)×(k/(k?1)) ^k . Then: (i) for λ≤λ_c, the Gibbs measure is unique (and coincides with the above measure μ^*), (ii) for λ>λ_c, in addition to μ^*, there exist two distinct translation-periodic measures, μ₊and μ_?, taken to each other by the unit space shift. Measures μ₊and μ_?are extreme ?λ>λ_c. We also construct a continuum of distinct, extreme, non-translational-invariant, splitting Gibbs measures. For $\lambda >1/(\sqrt k - 1) \times (\sqrt k /\sqrt k - 1))^k $ , measure μ^*is not extreme (this result can be improved). Finally, we consider a model with two fugacities, λ_eand λ_o, for even and odd sites. We discuss open problems and state several related conjectures.     

An optimal constant for the existence of least energy solutions of a coupled Schrödinger system

We study the following coupled Schrödinger system which has appeared as several models from mathematical physics: $$\left\{\begin{array}{ll}-\Delta u + \lambda_1 u = \mu_1 u^3 + \beta uv^2, \quad x \in \mathbb{R}^N,\\-\Delta v + \lambda_2 v = \mu_2 v^3 + \beta vu^2, \quad x \in \mathbb{R}^N,\\u \geq 0, v \geq 0 \,\,{\rm in}\mathbb{R}^N, \quad u, v \in H^1(\mathbb{R}^N).\end{array}\right.$$ Here, N = 2, 3, and λ ₁, λ ₂, μ ₁, μ ₂ are all positive constants. In [Ambrosetti and Colorado in C R Acad Sci Paris Ser I 342:453–438, 2006], Ambrosetti and Colorado showed that, there exists β ₀ > 0 such that this system has a nontrivial positive radially symmetric solution for any ${\beta \in (0, \beta_0)}$ . Later in [Ikoma and Tanaka in Calc Var 40:449–480, 2011], Ikoma and Tanaka showed that solutions obtained by Ambrosetti and Colorado are indeed least energy solutions for any ${\beta \in (0, {\rm min}\{\beta_0, \sqrt{\mu_1\mu_2}\})}$ . Here, in case λ ₁ = λ ₂ and μ ₁ ≠ μ ₂, we prove the uniqueness of the positive solutions for min{μ ₁, μ ₂} ? β > 0 sufficiently small. In case λ ₁ ≠ λ ₂ and (λ ₂ ? λ ₁)(μ ₂ ? μ ₁) ≤ 0, we prove that ${\beta_0 < \sqrt{\mu_1\mu_2}}$ and β ₀ is optimal, in the sense that this system has no nontrivial least energy solutions for ${\beta \in (\beta_0, \sqrt{\mu_1\mu_2})}$ . Moreover, there exists δ > 0 such that this system has no nontrivial nonnegative solutions for any ${\beta \in ({\rm min}\{\mu_1, \mu_2\} - \delta,\, \max\{\mu_1, \mu_2\} + \delta)}$ . This answers an open question of [Sirakov in Commun Math Phys 271:199–221, 2007] partially, and improves a result of [Sirakov in Commun Math Phys 271:199–221, 2007]. The asymptotic behavior of the least energy solutions is also studied as ${\beta \nearrow \beta_0}$ .     

On the existence of a continuous branch of nodal solutions of elliptic equations with convex-concave nonlinearities

We study the existence of nodal solutions of a parametrized family of Dirichlet boundary value problems for elliptic equations with convex-concave nonlinearities. In the main result, we prove the existence of nodal solutions u _λ for λ ∈ (?∞, λ*₀). The critical value λ*₀ >0 is found by a spectral analysis procedure according to Pokhozhaev’s fibering method. We show that the obtained solutions form a continuous branch (in the sense of level lines of the energy functional) with respect to the parameter λ. Moreover, we prove the existence of an interval \(( - \infty ,\tilde \lambda )\) , where \(\tilde \lambda > 0\) , on which this branch consists of solutions with exactly two nodal domains.     

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].     

Eigenvalues and the One-Dimensional p-Laplacian

We consider the boundary value problem (?_p(u′))′ + λF(t, u) = 0, with p > 1, t ∈ (0, 1), u(0) = u(1) = 0, and with λ > 0. The value of λ is chosen so that the boundary value problem has a positive solution. In addition, we derive an explicit interval for λ such that, for any λ in this interval, the existence of a positive solution to the boundary value problem is guaranteed. In addition, the existence of two positive solutions for λ in an appropriate interval is also discussed.     

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‖=+∞.     

Perturbations with several independent parameters

In this paper we discuss the problem of determining a T-periodic solution $\text{x}{}^1\text{(·, λ)}$ of the differential equation x = A(t)x + f(t, x, λ) + b(t), where the perturbation parameter λ is a vector in a parameter-space R^k. The customary approach assumes that λ = λ(?), ??R. One then establishes the existence of an ?₀ > 0 such that the differential equation has a T-periodic solution $\text{x}{}^1\text{(·, λ(?))}$ for all ? satisfying 0 < ? < ?₀. More specifically it is usually assumed that λ(?) has the form λ(?) = ?λ₀ where λ₀ is a fixed vector in R^k. This means that attention is confined in the perturbation procedure to examining the dependence of $\text{x}{}^1\text{(·, λ)}$ on λ as λ varies along a line segment terminating at the origin in the parameter-space R^k. The results established here generalize this previous work by allowing one to study the dependence of $\text{x}{}^1\text{(·, λ)}$ on λ as λ varies through a “conical-horn” whose vertex rests at the origin in R^k. In the process an implicit-function formula is developed which is of some interest in its own right.     

Multiplicity and uniqueness of positive solutions for nonhomogeneous semilinear elliptic equation with critical exponent

In this paper, we consider the following problem

$$\left\{ {\begin{array}{*{20}{c}}{ - \Delta u\left( x \right) + u\left( x \right) = \lambda \left( {{u^p}\left( x \right) + h\left( x \right)} \right),\;x \in {\mathbb{R}^N},} \\ {u\left( x \right) \in {H^1}\left( {{\mathbb{R}^N}} \right),\;u\left( x \right) \succ 0,\;x \in {\mathbb{R}^N},\;} \end{array}} \right.\;\left( * \right)$$

, where λ > 0 is a parameter, p = (N+2)/(N?2). We will prove that there exists a positive constant 0 < λ* < +∞ such that (*) has a minimal positive solution for λ ∈ (0, λ*), no solution for λ > λ*, a unique solution for λ = λ*. Furthermore, (*) possesses at least two positive solutions when λ ∈ (0, λ*) and 3 ≤ N ≤ 5. For N ≥ 6, under some monotonicity conditions of h we show that there exists a constant 0 < λ** < λ* such that problem (*) possesses a unique solution for λ ∈ (0, λ**).

     

Asymptotic behavior for nonlocal dispersal equations

This paper is concerned with the existence and asymptotic behavior of solutions of a nonlocal dispersal equation. By means of super-subsolution method and monotone iteration, we first study the existence and asymptotic behavior of solutions for a general nonlocal dispersal equation. Then, we apply these results to our equation and show that the nonnegative solution is unique, and the behavior of this solution depends on parameter λ in equation. For λ≤λ₁(Ω), the solution decays to zero as t→∞; while for λ>λ₁(Ω), the solution converges to the unique positive stationary solution as t→∞. In addition, we show that the solution blows up under some conditions.     

A bifurcation theorem for the p-logistic equation

We consider a p-logistic equation with an equidiffusive reaction. Using variational methods and truncation techniques, we show that there is a critical parameter value λ^∗ > 0 such that for λ > λ^∗ the problem has a unique positive smooth solution, and for λ ∈ (0, λ^∗] the problem has no positive solution.     

Doubly nonlinear parabolic equations with measure data

The existence of solutions to the initial boundary value problem for the equation $$u_{t}-{\rm div}(u^m|Du|^{p-2}Du)=\lambda|Du^q|^{l}+u^{\alpha},$$ with zero-Dirichlit boundary condition and Radon measure as initial condition is studied, where m > 0, p > 1, λ, q, l, and α in various situations.     

On the Frequency of Zeros of Solutions of Second Order Linear Differential Equations

We consider the equation \(\rm f^{\prime\prime}+{A}(z){f}=0\) with linearly independent solutions f₁,₂, where A(z) is a transcendental entire function of finite order. Conditions are given on A(z) which ensure that max{λ(f₁),λ(f₂)} = ∞, where λ(g) denotes the exponent of convergence of the zeros of g. We show as a special case of a further result that if P(z) is a non-constant, real, even polynomial with positive leading coefficient then every non-trivial solution of \(\rm f^{\prime\prime}+{e}^P{f}=0\) satisfies λ(f) = ∞. Finally we consider the particular equation \(\rm f^{\prime\prime}+({e}^Z-K){f}=0\) where K is a constant, which is of interest in that, depending on K, either every solution has λ(f) = ∞ or there exist two independent solutions f₁, f₂ each with λ(f_i) ≤ 1.     

Nonlinear eigenvalue problems with positively convex operators

We consider the equation u = λAu (λ > 0), where A is a forced isotone positively convex operator in a partially ordered normed space with a complete positive cone K. Let Λ be the set of positive λ for which the equation has a solution u?K, and let Λ₀ be the set of positive λ for which a positive solution—necessarily the minimum one—can be obtained by an iteration u_n = λAu_n?1, u₀ = 0. We show that if K is normal, and if Λ is nonempty, then Λ₀ is nonempty, and each set Λ₀, Λ is an interval with $\text{inf}\text{(Λ}{}_0\text{) =}\text{inf}\text{(Λ) = 0}\text{and sup}\text{(Λ}{}_0\text{) =}\text{sup}\text{(Λ) (= λ1,}\text{say}\text{)}$; but we may have λ1 ? Λ₀ and λ1 ? Λ. Furthermore, if A is bounded on the intersection of K with a neighborhood of 0, then Λ₀ is nonempty. Let u⁰(λ) = lim_n→∞(λA)ⁿ(0) be the minimum positive fixed point corresponding to λ ? Λ₀. Then u⁰(λ) is a continuous isotone convex function of λ on Λ₀.     

Constructive existence theorems for problems of Thomas-Fermi Type

A minimal positive solution of the Thomas-Fermi problem ? = λt^?1/2 w^3/2, w(0) = 1, w(1) = w(1) is shown to exist for each λ > 0. It is proved that all positive solutions, for a given value of λ, are strictly ordered and that the minimal positive solution w_λ is a decreasing function of λ. Upper and lower analytic bounds for w _λ are given and these bounds are shown to initiate sequences of Picard and Newton iterates which converge monotonically to w _λ. A comparative analysis of the efficiency of the iteration schemes is presented. The methods used are of a general nature and can be applied to a variety of nonlinear boundary value problems of convex type [14].     

An operator-theoretical treatment of the Maskawa–Nakajima equation in the massless abelian gluon model

The Maskawa–Nakajima equation has attracted considerable interest in elementary particle physics. From the viewpoint of operator theory, we study the Maskawa–Nakajima equation in the massless abelian gluon model. We first show that there is a nonzero solution to the Maskawa–Nakajima equation when the parameter λ   satisfies λ>2$\lambda >2$. Moreover, we show that the solution is infinitely differentiable and strictly decreasing. We thus conclude that the massless abelian gluon model generates the nonzero quark mass spontaneously and exhibits the spontaneous chiral symmetry breaking when λ>2$\lambda >2$. We next show that there is a unique solution 0 to the Maskawa–Nakajima equation when 0<λ<1$0<\lambda <1$, from which we conclude that each quark remains massless and that the model realizes the chiral symmetry when 0<λ<1$0<\lambda <1$.     

Grow-up of critical solutions for a non-local porous medium problem with Ohmic heating source

We investigate the behaviour of solution u = u(x, t; λ) at λ =  λ* for the non-local porous medium equation ${u_t = (u^n)_{xx} + {\lambda}f(u)/({\int_{-1}^1} f(u){\rm d}x)^2}We investigate the behaviour of solution u = u(x, t; λ) at λ =  λ* for the non-local porous medium equation u_t = (uⁿ)_xx + lf(u)/(ò_-1¹ f(u)dx)²{u_t = (u^n)_{xx} + {\lambda}f(u)/({\int_{-1}^1} f(u){\rm d}x)^2} with Dirichlet boundary conditions and positive initial data. The function f satisfies: f(s),−f ′ (s) > 0 for s ≥ 0 and s ^n-1 f(s) is integrable at infinity. Due to the conditions on f, there exists a critical value of parameter λ, say λ*, such that for λ > λ* the solution u = u(x, t; λ) blows up globally in finite time, while for λ ≥ λ* the corresponding steady-state problem does not have any solution. For 0 < λ < λ* there exists a unique steady-state solution w = w(x; λ) while u = u(x, t; λ) is global in time and converges to w as t → ∞. Here we show the global grow-up of critical solution u* =  u(x, t; λ*) (u* (x, t) → ∞, as t → ∞ for all x ? (-1,1){x\in(-1,1)}.     

The Brezis-Nirenberg problem near criticality in dimension 3

We consider the problem of finding positive solutions of Δu+λu+u^q=0 in a bounded, smooth domain Ω in , under zero Dirichlet boundary conditions. Here q is a number close to the critical exponent 5 and 0<λ<λ₁. We analyze the role of Green's function of Δ+λ in the presence of solutions exhibiting single and multiple bubbling behavior at one point of the domain when either q or λ are regarded as parameters. As a special case of our results, we find that if , where λ∗ is the Brezis-Nirenberg number, i.e., the smallest value of λ for which least energy solutions for q=5 exist, then this problem is solvable if q>5 and q−5 is sufficiently small.