Regularity for solutions of nonlinear second order evolution equations

This paper deals with the existence of solutions for the class of nonlinear second order evolution equations. The regularity and a variation of solutions of the given equations are also given. As particular cases of our general formulation, some results for Volterra integrodifferential equations of the hyperbolic type are given.     

Periodic solutions for second order differential equations with discontinuous restoring forces

We deal with the existence of periodic solutions for problems with a jump discontinuity. We use an approximation procedure and the method of the lower and upper solutions.     

Periodic solutions of singular second order equations at resonance

In this paper, we study the existence of positive periodic solutions for singular second order equations x" + n²/4x+h(x) = p(t), where h has a singularity at the origin and n is a positive integer. We give an explicit condition to ensure the existence of positive periodic solutions when h is an unbounded perturbation at infinity by using qualitative analysis and topological degree theory.     

Nodal solutions for fourth order elliptic equations with critical exponent on compact manifolds

Using a variational method, we prove the existence of nodal solutions to prescribed scalar Q- curvature type equations on compact Riemannian manifolds with boundary. These equations are fourth-order elliptic equations with critical Sobolev growth.     

Dirichlet problem for properly elliptic equations of fourth order in the exterior of an ellipse

The present paper considers the Dirichlet problem for properly elliptic equations of fourth order in the exterior of an ellipse. No restrictions on the multiplicities of the roots of the characteristic polynomial are assumed.     

On positive solutions for a fourth order asymptotically linear elliptic equation under Navier boundary conditions

A result on existence of positive solution for a fourth order nonlinear elliptic equation under Navier boundary conditions is established. The nonlinear term involved is asymptotically linear both at the origin and at infinity. We exploit topological degree theory and global bifurcation.     

Oscillation criteria for second order forced ordinary differential equations with mixed nonlinearities

We present new oscillation criteria for the second order forced ordinary differential equation with mixed nonlinearities:

     

Positive solutions of fourth order Emden-Fowler type differential equations in the framework of regular variation

The fourth order nonlinear differential equations

(A)     

Maximum principles and bounds for a class of fourth order nonlinear elliptic equations

We deduce maximum principles for a class of fourth order nonlinear elliptic equations by using auxiliary functions containing the square of the second gradient of the solution of such equations. A priori bounds on various quantities of interest are obtained.     

Sun-Wong type theorems for second order damped elliptic equations

The oscillation criteria of Sun and Wong [Y.G. Sun, J.S.W. Wong, Oscillation criteria for second order forced ordinary differential equations with mixed nonlinearities, J. Math. Anal. Appl. 334 (2007) 549-560] are extended to the forced second order damped elliptic differential equation with mixed nonlinearities

     

Nonlinearity and nontrivial solutions of fourth order semilinear elliptic equations

We consider the existence of nontrivial solutions of a fourth order semilinear elliptic boundary value problem with Dirichlet boundary condition, Δ²u+cΔu=b₁[(u+1)⁺−1]+b₂u⁺ in Ω, where Ω is a bounded open set in R^N with smooth boundary ∂Ω. The variation of linking theorem is useful to investigate them. We investigate them in six regions of (b₁,b₂) when λ₁<c<λ₂.     

Nondegeneracy and uniqueness of positive solutions for Robin problem of second order ordinary differential equations and its applications

This article studies positive solutions of Robin problem for semi-linear second order ordinary differential equations. Nondegeneracy and uniqueness results are proven for homogeneous differential equations. Necessary and sufficient conditions for the existence of one or two positive solutions for inhomogeneous differential equations or differential equations with concave-convex nonlinearities are obtained by making use of the nondegeneracy and uniqueness results for positive solutions of homogeneous differential equations.     

On the comparison principle for unbounded solutions of elliptic equations with first order terms

We prove a comparison principle for unbounded weak sub/super solutions of the equation

$\lambda u?\text{div}(A(x)Du)=H(x,Du)\text{ in }\mathrm{\Omega }$

where $A(x)$ is a bounded coercive matrix with measurable ingredients, $\lambda \ge 0$ and $\xi ?H(x,\xi )$ has a super linear growth and is convex at infinity. We improve earlier results where the convexity of $H(x,?)$ was required to hold globally.     

Two homoclinic solutions for a nonperiodic fourth order differential equation with a perturbation

In this paper, we study multiple homoclinic solutions for a class of fourth order differential equations with a perturbation. By establishing a compactness lemma and using variational methods, the existence result of two homoclinic solutions is obtained under some suitable assumptions, but not requiring the periodicity condition. Some recent results are improved and extended.     

Stationary distributions of second order stochastic evolution equations with memory in Hilbert spaces

In this paper, we consider stationarity of a class of second-order stochastic evolution equations with memory, driven by Wiener processes or Lévy jump processes, in Hilbert spaces. The strategy is to formulate by reduction some first-order systems in connection with the stochastic equations under investigation. We develop asymptotic behavior of dissipative second-order equations and then apply them to time delay systems through Gearhart–Prüss–Greiner’s theorem. The stationary distribution of the system under consideration is the projection on the first coordinate of the corresponding stationary results of a lift-up stochastic system without delay on some product Hilbert space. Last, two examples of stochastic damped wave equations with memory are presented to illustrate our theory.     

Homoclinic solutions for fourth order differential equations with superlinear nonlinearities

In this paper we investigate the existence of homoclinic solutions for a class of fourth order differential equations with superlinear nonlinearities. Under some superlinear conditions weaker than the well-known (AR) condition, by using the variant fountain theorem, we establish one new criterion to guarantee the existence of infinitely many homoclinic solutions.