Semilinear elliptic problems near resonance with a nonprincipal eigenvalue

We consider the Dirichlet problem for the equation −Δu=λu±f(x,u)+h(x) in a bounded domain, where f has a sublinear growth and h∈L². We find suitable conditions on f and h in order to have at least two solutions for λ near to an eigenvalue of −Δ. A typical example to which our results apply is when f(x,u) behaves at infinity like a(x)|u|^q−2u, with M>a(x)>δ>0, and 1<q<2.     

Existence and asymptotic behavior for elliptic equations with singular anisotropic potentials

We study the equation Δu+u|u|^p−1+V(x)u+f(x)=0 in Rn, where n?3 and p>n/(n−2). The forcing term f and the potential V can be singular at zero, change sign and decay polynomially at infinity. We can consider anisotropic potentials of form h(x)|x|⁻² where h is not purely angular. We obtain solutions u which blow up at the origin and do not belong to any Lebesgue space Lr. Also, u is positive and radial, in case f and V are. Asymptotic stability properties of solutions, their behavior near the singularity, and decay are addressed.     

Boundary layer solutions to problems with infinite-dimensional singular and regular perturbations

We prove existence, local uniqueness and asymptotic estimates for boundary layer solutions to singularly perturbed equations of the type (ε²(x)u^′′(x))=f(x,u(x))+g(x,u(x),ε(x)u^′(x)), 0<x<1, with Dirichlet and Neumann boundary conditions. Here the functions ε and g are small and, hence, regarded as singular and regular functional perturbation parameters. The main tool of the proofs is a generalization (to Banach space bundles) of an implicit function theorem of R. Magnus.     

Positive solutions and eigenvalue intervals of a nonlinear singular fourth-order boundary value problem

We consider the classical nonlinear fourth-order two-point boundary value problem . In this problem, the nonlinear term h(t)f(t, u(t), u′(t), u″(t)) contains the first and second derivatives of the unknown function, and the function h(t)f(t, x, y, z) may be singular at t = 0, t = 1 and at x = 0, y = 0, z = 0. By introducing suitable height functions and applying the fixed point theorem on the cone, we establish several local existence theorems on positive solutions and obtain the corresponding eigenvalue intervals.     

Solitary waves for a class of quasilinear Schrödinger equations in dimension two

In this paper we prove the existence and concentration behavior of positive ground state solutions for quasilinear Schrödinger equations of the form ?ε ²Δu + V(z)u ? ε ² [Δ(u ²)]u = h(u) in the whole two-dimension space where ε is a small positive parameter and V is a continuous potential uniformly positive. The main feature of this paper is that the nonlinear term h(u) is allowed to enjoy the critical exponential growth with respect to the Trudinger–Moser inequality and also the presence of the second order nonhomogeneous term [Δ(u ²)]u which prevents us to work in a classical Sobolev space. Using a version of the Trudinger–Moser inequality, a penalization technique and mountain-pass arguments in a nonstandard Orlicz space we establish the existence of solutions that concentrate near a local minimum of V.     

On singular perturbations of superlinear elliptic systems

We study the existence, multiplicity and shape of positive solutions of the system −ε²Δu+V(x)u=K(x)g(v), −ε²Δv+V(x)v=H(x)f(u) in RN, as ε→0. The functions f and g are power-like nonlinearities with superlinear and subcritical growth at infinity, and V, H, K are positive and locally Hölder continuous.     

The singular limit of the Allen-Cahn equation and the FitzHugh-Nagumo system

We consider an Allen-Cahn type equation of the form ut=Δu+ε⁻²fε(x,t,u), where ε is a small parameter and fε(x,t,u)=f(u)−εgε(x,t,u) a bistable nonlinearity associated with a double-well potential whose well-depths can be slightly unbalanced. Given a rather general initial data u₀ that is independent of ε, we perform a rigorous analysis of both the generation and the motion of interface. More precisely we show that the solution develops a steep transition layer within the time scale of order ε²|lnε|, and that the layer obeys the law of motion that coincides with the formal asymptotic limit within an error margin of order ε. This is an optimal estimate that has not been known before for solutions with general initial data, even in the case where gε≡0.Next we consider systems of reaction-diffusion equations of the form

     

Stability of stationary solutions of a semilinear parabolic partial differential equation

We consider a parabolic partial differential equation u_t = u_xx + f(u) on a compact interval of spatial variable x with Dirichlet boundary conditions. The stability of stationary solutions of this system is studied by the use of Liapunov's second method. We obtain necessary and sufficient conditions for the stability, asymptotic stability, neutral stability, instability, and conditional stability. These conditions are closely connected with the conditions for the existence of the stationary solutions.     

Existence and stability of solutions with periodically moving weak internal layers

We consider the periodic parabolic differential equation under the assumption that ε is a small positive parameter and that the degenerate equation f(u,x,t,0)=0 has two intersecting solutions. We derive conditions such that there exists an asymptotically stable solution up(x,t,ε) which is T-periodic in t, satisfies no-flux boundary conditions and tends to the stable composed root of the degenerate equation as ε→0.     

Existence and nonuniqueness of solutions to a functional-differential equation

We examine the functional-differential equation Δu(x) — div(u(H(x))f (x)) = 0 on a torus which is a generalization of the stationary Fokker-Planck equation. Under sufficiently general assumptions on the vector field f and the map H, we prove the existence of a nontrivial solution. In some cases the subspace of solutions is established to be multidimensional.     

Nontrivial solutions of Schrödinger equations with indefinite nonlinearities

We prove the existence of nontrivial solutions for the Schrödinger equation −Δu+V(x)u=aγ(x)f(u) in RN, where f is superlinear and subcritical at zero and infinity respectively, V is periodic and a(x) changes sign.     

Spike solutions of a nonlinear Schrödinger equation with degenerate potential

We study the existence of positive solutions to the elliptic equation ε²Δu(x,y)−V(y)u(x,y)+f(u(x,y))=0 for (x,y) in an unbounded domain subject to the boundary condition u=0 whenever is nonempty. Our potential V depends only on the y variable and is a bounded or unbounded domain which may coincide with . The positive parameter ε is tending to zero and our solutions u_ε concentrate along minimum points of the unbounded manifold of critical points of V.     

Necessary and sufficient conditions for the existence and ɛ-uniqueness of bounded solutions of the equation x′ = f(x) − h(t)

We introduce the notion of ?-unique bounded solution to the nonlinear differential equation x′ = f(x) ? h(t), where f: ? → ? is a continuous function and h(t) is an arbitrary continuous function bounded on ?. We derive necessary and sufficient conditions for the existence and ?-uniqueness of bounded solutions to this equation.     

Uniqueness results and asymptotic behavior of solutions with boundary blow-up for logistic-type porous media equations

Under the proper structure conditions on the nonlinear term f(u) and weight function b(x), the paper shows the uniqueness and asymptotic behavior near the boundary of boundary blow-up solutions to the porous media equations of logistic type ?Δu = a(x)u ^1/m ? b(x)f(u) with m > 1.     

Stability of a simple Levi–Civitá functional equation on non-unital commutative semigroups

In this paper, we study the Hyers–Ulam stability of a simple Levi–Civitá functional equation f(x+y)=f(x)h(y)+f(y) and its pexiderization f(x+y)= g(x) h(y)+k(y) on non-unital commutative semigroups by investigating the functional inequalities |f(x+y)?f(x)h(y)?f(y)|≤?? and |f(x+y)?g(x)h(y)?k(y)|≤??, respectively. We also study the bounded solutions of the simple Levi–Civitá functional inequality.     

Existence and asymptotic behavior of blow-up solutions to weighted quasilinear equations

Given a bounded domain Ω we consider local weak blow-up solutions to the equation Δ_pu=g(x)f(u) on Ω. The non-linearity f is a non-negative non-decreasing function and the weight g is a non-negative continuous function on Ω which is allowed to be unbounded on Ω. We show that if Δ_pw=−g(x) in the weak sense for some and f satisfies a generalized Keller-Osserman condition, then the equation Δ_pu=g(x)f(u) admits a non-negative local weak solution such that u(x)→∞ as x→∂Ω. Asymptotic boundary estimates of such blow-up solutions will also be investigated.     

Boundary behaviour for solutions of boundary blow-up problems in a borderline case

We investigate boundary blow-up solutions of the equation Δu=f(u) in a bounded domain Ω⊂RN under the condition that f(t) has a relatively slow growth as t goes to infinity. We show how the mean curvature of the boundary ∂Ω appears in the asymptotic expansion of the solution u(x) in terms of the distance of x from ∂Ω.     

Global existence and blowup solutions for quasilinear parabolic equations

The authors discuss the quasilinear parabolic equation ut=∇⋅(g(u)∇u)+h(u,∇u)+f(u) with u_|∂Ω=0, u(x,0)=?(x). If f, g and h are polynomials with proper degrees and proper coefficients, they show that the blowup property only depends on the first eigenvalue of −Δ in Ω with Dirichlet boundary condition. For a special case, they obtain a sharp result.     

On the asymptotic behavior of the solutions of a class of second order nonlinear differential equations

In this paper, we investigate properties of the solutions of a class of second-order nonlinear differential equation such as [p(t)f(x(t))x′(t)]′ + q(t)g(x′(t))e(x(t)) = r(t)c(x(t)). We prove the theorems of monotonicity, nonoscillation and continuation of the solutions of the equation, the sufficient and necessary conditions that the solutions of the equation are bounded, and the asymptotic behavior of the solutions of the equation when t → ∞ on condition that the solutions are bounded. Also we provide the asymptotic relationship between the solutions of this equation and those of the following second-order linear differential equation: [p(t)u′(t)]′ = r(t)u(t)     

On the generalized Burgers equation

The present paper is concerned with the initial boundary value problem for the generalized Burgers equation u _t + g(t, u)u _x + f(t, u) = εu _xx which arises in many applications. We formulate a condition guaranteeing the a priori estimate of max |u _x | independent of ε and t and give an example demonstrating the optimality of this condition. Based on this estimate we prove the global existence of a unique classical solution of the problem and investigate the behavior of this solution for ε → 0 and t → + ∞. The Cauchy problem for this equation is considered as well.