A note on the sign-changing solutions to elliptic problems with critical Sobolev and Hardy terms

Let be a bounded domain such that . We obtain existence of sign-changing solutions for the Dirichlet problem on Ω,u=0 on ∂Ω for suitable positive numbers μ and λ.     

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.     

Exact multiplicity and ordering properties of positive solutions of a p-Laplacian Dirichlet problem and their applications

We study the exact multiplicity and ordering properties of positive solutions of the p-Laplacian Dirichlet problem

     

Oscillation theorems for differential equations with a nonlinear damping term

In this paper, we are concerned with a class of nonlinear second-order differential equations with a nonlinear damping term. Passage to more general class of equations allows us to remove a restrictive condition usually imposed on the nonlinearity, and, as a consequence, our results apply to wider classes of nonlinear differential equations. Two illustrative examples are considered.     

Existence for nonoscillatory solutions of higher-order nonlinear neutral difference equations

In this paper, we consider the following forced higher-order nonlinear neutral difference equation

     

Viscosity solutions of fully nonlinear parabolic systems

In this paper, we discuss the viscosity solutions of the weakly coupled systems of fully nonlinear second-order degenerate parabolic equations and their Cauchy-Dirichlet problem. We prove the existence, uniqueness and continuity of viscosity solution by combining Perron's method with the technique of coupled solutions. The results here generalize those in Proc. London Math. Soc. 63 (1991) 212-240 and Comm. Partial Differential Equations 16 (1991) 1095-1128.     

Existence results for some fourth-order nonlinear elliptic problems of local superlinearity and sublinearity

In this paper we study the existence of positive solutions for the problem

(0.1)     

Some noncoercive parabolic equations with lower order terms in divergence form

This paper deals with existence and regularity results for the problem $ \cases{u_t-\mathrm{div}(a(x,t,u )\nabla u)=-\mathrm{div}(u\,E) \qquad in \Omega\times (0,T),\cr u=0 \qquad on \partial \Omega\times (0,T), \cr u (0)= u_0 \qquad in \Omega ,\cr} $ under various assumptions on E and $ u_0 $. The main difculty in studying this problem is due to the presence of the term div(uE), which makes the differential operator non coercive on the "energy space" $ L^2 (0, T; H_0^1 (\Omega)) $.AMS Subject Classification: 35K10, 35K15, 35K65.     

Two Dimensional Interface Problems for Elliptic Equations

We study the structure of solutions to the interface problems for second order quasi-linear elliptic partial differential equations in two dimensional space. We prove that each weak solution can be decomposed into two parts near singular points, a finite sum of functions in the form of cr^α log^m rφ(θ) and a regular one w. The coefficients c and the C^{1,α} norm of w depend on the H¹-norm and the C^{º,α}-norm of the solution, and the equation only.