NontriviaL solutions of some nonlinear elliptic problems

This paper is concerned with a class of semilinear elliptic Dirichlet problems approximating degenerate equations. The aim is to prove the existence of at least 4^k?1 nontrivial solutions when the degeneration set consists of k distinct connected components     

Optimal summability of solutions to nonlinear elliptic problems

The strongest possible norms in a priori estimates for solutions to a class of second-order nonlinear elliptic boundary value problems are exhibited.     

Positive solutions of nonlinear elliptic boundary value problems

This paper considers the problem of finding positive vector-valued solutions U of the nonlinear elliptic boundary value problem L(U) + f(x, U) = 0 on a bounded region Ω, $\text{U +}\text{?U}\text{?v}\text{= 0}$ on ?Ω. The operator L is uniformly elliptic and in divergence form, and f is, roughly speaking, superlinear; by the positivity of U is meant the positivity of each component of U on Ω. Under certain growth conditions on f and some further technical assumptions, the existence of a positive solution is proved, an a priori bound on all positive solutions is obtained, and a certain fixed point index is proved equal to ? 1. As an example, information about fixed point indices is used to allow perturbations of the form ?h(x, U, DU). In the final section, an essentially best possible theorem is given for Ω a ball and for radially symmetric solutions of the Laplacian with Dirichlet boundary conditions.     

Sign-changing solutions for some fourth-order nonlinear elliptic problems

In this paper, we consider the existence and multiplicity of sign-changing solutions for some fourth-order nonlinear elliptic problems and some existence and multiple are obtained. The weak solutions are sought by means of sign-changing critical theorems.     

Multibump solutions for a class of nonlinear elliptic problems

The paper is concerned with a class of semilinear elliptic Dirichlet problems approximating degenerate equations. By using variational methods, it is proved that, if the degeneration set consists of connected components, then there exist at least multibump positive solutions. Received March 25, 1996 / Accepted May 26, 1997     

Existence of solutions for some fourth-order nonlinear elliptic problems

In this paper, we consider the existence of positive, negative and sign-changing solutions for some fourth order semilinear elliptic boundary value problems. We present new results on invariant sets of the gradient flows of the corresponding variational functionals. The structure of the invariant sets will be built into minimax procedures to construct the sign-changing solutions.     

Sturm type theorems for solutions of elliptic nonlinear problems

In this paper a variety of nonexistence results are obtained for positive solutions to the m-Laplacean and related equations by means of generalized Picone Identity arguments. These procedures allow the treatment of nonradial problems and extend several recent theorems.     

Asymptotic expansion of solutions to nonlinear elliptic eigenvalue problems

We consider the nonlinear eigenvalue problem

where is an appropriately smooth bounded domain and 0$"> is a parameter. It is known that if , then the corresponding solution is almost flat and almost equal to inside . We establish an asymptotic expansion of when , which is explicitly represented by .

     

Multiple solutions of nonlinear elliptic equations for oscillation problems

We discussed oscillating equations with Neumann boundary value in [Nonlinear Anal. 54 (2003) 431-443] and [J. Math. Anal. Appl. 298 (2004) 14-32] and prove the existence of infinitely many nonconstant solutions. However, it seems difficult to find infinitely many disjoint order intervals for oscillating equations with Dirichlet boundary value. To get rid of this difficulty, in this paper, we build up a mountain pass theorem in half-order intervals and use it to study oscillating problems with Dirichlet boundary value in which we only have the existence of super-solutions (or sub-solutions) and obtain new results on the exactly infinitely many solutions.     

Detection of arbitrarily many solutions for perturbed elliptic problems involving oscillatory terms

We propose a direct approach for detecting arbitrarily many solutions for perturbed elliptic problems involving oscillatory terms. Although the method works in various frameworks, we illustrate it on the problem

(P_ε)     

Positive solutions of a class of nonlinear elliptic eigenvalue problems

This paper is mainly concerned with the natural order relationship between positive solutions of the elliptic eigenvalue Dirichlet problem: in and u=0 on . Under suitable conditions, we prove that there are 2m-1 positive solutions satisfying . It seems that standard arguments do not provide such a result. Several authors, including P. Hess, proved the existence of equal number of positive solutions without such a relationship between them. We also prove that in Hess's result as well as in ours some sufficient condition is also necessary if the domain possesses a particular shape. At last, as an illustrative example, we study the diagram of positive solutions when with and d being both parameters. Received: 13 July 2000; in final form: 22 August 2001 / Published online: 1 February 2002     

Multiple solutions for strongly resonant nonlinear elliptic problems with discontinuities

We examine a nonlinear strongly resonant elliptic problem driven by the -Laplacian and with a discontinuous nonlinearity. We assume that the discontinuity points are countable and at them the nonlinearity has an upward jump discontinuity. We show that the problem has at least two nontrivial solutions without using a multivalued interpretation of the problem as it is often the case in the literature. Our approach is variational based on the nonsmooth critical point theory for locally Lipschitz functions.