On existence and stability of solutions for higher order semilinear Dirichlet problems

We provide existence and stability results for semilinear Dirichlet problems with nonlinearity satisfying general growth conditions. We consider the case when both the coefficients of the differential operator and the nonlinear term depend on the numerical parameter. We show applications for the fourth order semilinear Dirichlet problem.     

On the existence and stability of higher order Dirichlet problems

We provide the existence and the stability results for higher order Dirichlet problems with convex nonlinearity which satisfies general local growth conditions. We construct a dual variational method and investigate relations between primal action functional and dual action functional.     

Three weak solutions for elliptic Dirichlet problems

An existence result of three non-zero solutions for non-autonomous elliptic Dirichlet problems, under suitable assumptions on the nonlinear term, is presented. The approach is based on a recent three critical points theorem for differentiable functionals.     

Spike-layered solutions of singularly perturbed quasilinear Dirichlet problems

In this paper we study the existence and structure of a least-energy solution for a class of singularly perturbed quasilinear Dirichlet problems. Using the moving plane method we show that this least-energy solution develops to a spike-layer solution on convex domains.     

Remarks on existence and multiplicity of solutions for a class of semilinear elliptic equations

The existence and multiplicity results are obtained for solutions of a class of the Dirichlet problem for semilinear elliptic equations by the least action principle and the minimax methods, respectively.     

一类奇异半线性椭圆型方程的Dirichlet问题

得到了一类奇异半线性椭圆型方程 Ｄｉｒｉｃｈｌｅｔ问题解的存在性．     

Positive solutions for Dirichlet problems of singular nonlinear fractional differential equations

In this paper, we investigate the existence of positive solutions for the singular fractional boundary value problem: Dαu(t)+f(t,u(t),Dμu(t))=0, u(0)=u(1)=0, where 1<α<2, 0<μ?α−1, Dα is the standard Riemann-Liouville fractional derivative, f is a positive Carathéodory function and f(t,x,y) is singular at x=0. By means of a fixed point theorem on a cone, the existence of positive solutions is obtained. The proofs are based on regularization and sequential techniques.     

Positive solutions for a Dirichlet problem

1. IntroductionSince the work of Ambrosetti and Rabinowitz[l], the problems similar to{;t2:<::l">, (l.l)have been studied extensively But it is well known that, for applying the Moulltain PassTheOrem, we atway8 assum that g(x, 8) is suPerlineax in s at indnity; moeove) a strongercondition like (AR) (see later on) is required. If these conditions are not satisfied, can wealso get solutions for problem (1.1) by a Mountain Pass Theorem? So, ill this paPer, westudy the following Dirichlet pr…     

Dirichlet problem for semilinear hyperbolic equation. Multidimensional case

We discuss the solvability of the Dirichlet problem for the semilinear equation of the vibrating string xtt(t,y)−Δx(t,y)+f(t,y,x(t,y))=0 in higher dimensions with sides length being irrational numbers and superlinear nonlinearity. To this effect we derive a new dual variational method.     

A finite element approach for finding positive solutions of semilinear elliptic Dirichlet problems

In the present article, we described the finite element method for finding positive solutions for the elliptic problems of the type ‐ Δu = λf(x)g(u) for x ε Ω, with Dirichlet boundary condition. By using Matlab, we visualize the range of λ in which this problem achieves a numerical solution, and also discussed the behavior of the branch of this solution. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

On the existence of positive solutions of semilinear elliptic equations with Dirichlet boundary conditions

Research supported in part by NSF Grants DMS 8657483 and GER 9023335     

On the number of radially symmetric solutions to Dirichlet problems with jumping nonlinearities of superlinear order

This paper is concerned with the multiplicity of radially symmetric solutions to the Dirichlet problem

on the unit ball with boundary condition on . Here is a positive function and is a function that is superlinear (but of subcritical growth) for large positive , while for large negative we have that , where is the smallest positive eigenvalue for in with on . It is shown that, given any integer , the value may be chosen so large that there are solutions with or less interior nodes. Existence of positive solutions is excluded for large enough values of .

     

On the n-dimensional Dirichlet problem for isometric maps

We exhibit explicit Lipschitz maps from Rn to Rn which have almost everywhere orthogonal gradient and are equal to zero on the boundary of a cube. We solve the problem by induction on the dimension n.     

Positive radial solutions for Dirichlet problems via a Harnack-type inequality

We deal with the existence and localization of positive radial solutions for Dirichlet problems involving $\varphi$-Laplacian operators in a ball. In particular, $p$-Laplacian and Minkowski-curvature equations are considered. Our approach relies on fixed point index techniques, which work thanks to a Harnack-type inequality in terms of a seminorm. As a consequence of the localization result, it is also derived the existence of several (even infinitely many) positive solutions.     

On the decay of the solutions of second order parabolic equations with Dirichlet conditions

We use rearrangement techniques to investigate the decay of the parabolic Dirichlet problem in a bounded domain. The coefficients of the second order term are used to introduce an isoperimetric problem. The resulting isoperimetric function together with the divergence of the first order coefficients and the value distribution of the zero order part are then used to construct a symmetric comparison equation having a slower heat‐flow than the original equation. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Multiple positive solutions of Dirichlet boundary value problems for second order impulsive differential equations

This paper is devoted to study the existence of multiple positive solutions for the second order Dirichlet boundary value problem with impulse effects. The main results here is the generalization of Liu and Li [L. Liu, F.Y. Li, Multiple positive solution of nonlinear two-point boundary value problems, J. Math. Anal. Appl. 203 (1996) 610-625] for ordinary differential equations. Existence is established via the theory of fixed point index in cones.     

On the existence of bubble-type solutions of nonlinear singular problems

Considered in this paper is a class of singular boundary value&nbsp;problem, arising&nbsp;in hydrodynamics and nonlinear field theory, when centrally bubble-type solutions are&nbsp;sought: \((p(t)u0)0 = c(t)p(t)f(u); u0(0) = 0; u(+1) = L &gt; 0\) in the half-line \([0;+1)\), where \(p(0) = 0\). We are interested in strictly increasing solutions&nbsp;of this problem in \([0;1)\) having just one zero in \((0;+1) \)and finite limit at zero, which has&nbsp;great importance in applications or pure and applied mathematics. Su&plusmn;cient conditions&nbsp;of the existence of such solutions are obtained by applying the critical point theory and&nbsp;by using shooting argument [9,10] to better analysis the properties of certain solutions&nbsp;associated with the singular di&reg;erential equation. To the authors' knowledge, for the first&nbsp;time, the above problem is dealt with when f satis&macr;es non-Lipschitz condition. Recent&nbsp;results in the literature are generalized and signi&macr;cantly improved.