Positive Solutions for Asymptotically Linear Cone-Degenerate Elliptic Equations

In this paper, the authors study the asymptotically linear elliptic equation on manifold with conical singularities ??_Bu + λu = a(z)f(u), u ≥ 0 in R^N₊,where N = n + 1 ≥ 3, λ > 0, z = t,x₁,· · · ,x_n, and ?_B = (t?_t)^{2 + ?2_x1 + · · · + ?2_xn. Combining properties of cone-degenerate operator, the Pohozaev manifold and qualitative properties of the ground state solution for the limit equation, we obtain a positive solution under some suitable conditions on a and f.}     

Multiple periodic solutions for asymptotically linear Duffing equations with resonance

We investigate multiple periodic solutions of asymptotically linear Duffing equation with resonance using index theory and Morse theory and obtain a new result.     

Asymptotically linear solutions of differential equations via Lyapunov functions

We discuss the existence of solutions with oblique asymptotes to a class of second order nonlinear ordinary differential equations by means of Lyapunov functions. The approach is new in this field and allows for simpler proofs of general results regarding Emden-Fowler like equations.     

Homoclinic solutions of a class of periodic difference equations with asymptotically linear nonlinearities

By using critical point theory and periodic approximations, new sufficient conditions are obtained on the existence and nonexistence of homoclinic solutions for a class of discrete nonlinear periodic equations with asymptotically linear nonlinearities. These results partially answer an open problem proposed by Pankov (2006) [2] under rather weaker conditions and greatly improve the related results before.     

Positive solutions for asymptotically linear Schrödinger–Kirchhoff‐type equations

In this paper, we focus on the Schrödinger–Kirchhoff‐type equation (SK) where a,b > 0 are constants, may not be radially symmetric, and f(x,u) is asymptotically linear with respect to u at infinity. Under some technical assumptions on V and f, we prove that the problem (SK) has a positive solution. Copyright © 2013 John Wiley & Sons, Ltd.     

Construction of solutions via local Pohozaev identities

This paper deals with the following nonlinear elliptic equation

$?\mathrm{\Delta }u+V(|y^{\prime }|,y^{″})u=u^{\frac{N+2}{N?2}},u>0,u\in H^1({\mathbb{R}}^N),$

where $(y^{\prime },y^{″})\in {\mathbb{R}}^2\times {\mathbb{R}}^{N?2}$, $V(|y^{\prime }|,y^{″})$ is a bounded non-negative function in ${\mathbb{R}}^{+}\times {\mathbb{R}}^{N?2}$. By combining a finite reduction argument and local Pohozaev type of identities, we prove that if $N\ge 5$ and $r^{2}V(r,y^{″})$ has a stable critical point $(r_0,y_0^{″})$ with $r_0>0$ and $V(r_0,y_0^{″})>0$, then the above problem has infinitely many solutions. This paper overcomes the difficulty appearing in using the standard reduction method to locate the concentrating points of the solutions.     

Existence of nontrivial solutions of an asymptotically linear fourth-order elliptic equation

In this paper, using the mountain pass theorem, we give the existence result for nontrivial solutions for a class of asymptotically linear fourth-order elliptic equations.     

Multiple positive solutions for Kirchhoff type problems with singularity and asymptotically linear nonlinearities

In this paper, we devote ourselves to investigating a nonlocal problem involving singularity and asymptotically linear nonlinearities. By using the variational and perturbation methods, we obtain the existence of two positive solutions which improve the existing result in the literature.     

Nontrivial periodic solutions for asymptotically linear resonant difference problem

In this paper the existence of nontrivial periodic solution for second order asymptotically linear difference equation at resonance is obtained. The methods used here are based on combining the minimax methods and the Morse theory, especially the observation on the critical groups.     

Existence of sign-changing solutions for some asymptotically linear three-point boundary value problems

In this paper, by using the topological degree theory and the fixed point index theory, the existence of three kinds of solutions (i.e., sign-changing solutions, positive solutions and negative solutions) for asymptotically linear three-point boundary value problems is obtained.     

拟线性椭圆型方程组奇性解的Pohozaev恒等式与解不存在性

该文分别研究了超临界指数和自然增长条件下拟线椭圆型方程组奇性解成立Poho-zaev恒等式的一些充分条件．     

Existence of asymptotically almost periodic solutions for damped wave equations

In this paper, a class of nonlinear damped wave equations of the form αu^?(t)+u^″(t)=βAu(t)+γAu^′(t)+f(t,u(t)), t?0, satisfying αβ<γ with prescribed initial conditions are studied. Some sufficient conditions are established for the existence and uniqueness of an asymptotically almost periodic solution. These results have significance in the study of vibrations of flexible structures possessing internal material damping. Finally, an example is presented to illustrate the feasibility and effectiveness of the results.     

Biharmonic equations with asymptotically linear nonlinearities

This article considers the equation △2u = f(x, u)with boundary conditions either u|(a)Ω = (a)u/(a)n|(a)Ω = 0 or u|(a)Ω = △u|(a)Ω = 0, where f(x,t) is asymptotically linear with respect to t at infinity, and Ω is a smooth bounded domain in RN, N ＞ 4. By a variant version of Mountain Pass Theorem, it is proved that the above problems have a nontrivial solution under suitable assumptions of f(x, t).     

Multiple solutions for an asymptotically linear Duffing equation with Newmann boundary value conditions

We investigate multiple solutions of an asymptotically linear Duffing equation satisfying Newmann boundary value conditions with resonance using index theory and Morse theory and obtain some new results.     

Positive solution for asymptotically linear elliptic systems

In this paper, we show that semilinear elliptic systems of the form (1) possess at least one positive solution pair (u, v)∈H¹(?^N)×H¹(?^N), where λ and µ are nonnegative numbers, f(x, t) and g(x, t) are continuous functions on ?^N×? and asymptotically linear as t→+∞. Copyright © 2009 John Wiley & Sons, Ltd.     

Multipeak solutions for asymptotically critical elliptic equations on Riemannian manifolds

Let (M,g) be a smooth compact Riemannian manifold of dimension n≥3. We are concerned with the following asymptotically critical elliptic problem

(0.1)     

Multiple periodic orbits of asymptotically linear autonomous convex Hamiltonian systems

We investigate multiple periodic solutions of convex asymptotically linear autonomous Hamiltonian systems. Our theorem generalizes a result by Mawhin and Willem.