On an eigenvalue problem involving variable exponent growth conditions

We study the problem in Ω, u=0 on ∂Ω, where Ω is a bounded domain in R^N, is a continuous function and λ and ε are two positive constants. We prove that for any ε>0 each λ∈(0,λ₁) is an eigenvalue of the above problem, where λ₁ is the principal eigenvalue of the Laplace operator on Ω. Moreover, for each eigenvalue λ∈(0,λ₁) it corresponds a unique eigenfunction. The proofs will be based on the Banach fixed point theorem combined with adequate variational techniques.     

Existence and regularity of positive solutions of a degenerate elliptic problem

Existence and regularity of positive solutions of a degenerate elliptic Dirichlet problem of the form in Ω, on , where Ω is a bounded smooth domain in , , are obtained via new embeddings of some weighted Sobolev spaces with singular weights and . It is seen that and admit many singular points in Ω. The main embedding results in this paper provide some generalizations of the well‐known Caffarelli–Kohn–Nirenberg inequality.     

Existence and nonexistence of least energy solutions of the Neumann problem for a semilinear elliptic equation with critical Sobolev exponent and a critical lower-order perturbation

Let Ω be a smooth bounded domain in , with N?5, a>0, α?0 and . We show that the exponent plays a critical role regarding the existence of least energy (or ground state) solutions of the Neumann problem

     

Existence and concentration of positive solutions for a class of gradient systems

Some gradient systems with two competing potential functions are considered. Bound states (solutions with finite energy) are proved to exist and to concentrate at a point in the limit. The proof relies on variational methods, where the existence and concentration of positive solutions are related to a suitable ground energy function.     

Existence of positive solutions for a semilinear elliptic problem with critical Sobolev and Hardy terms

Let , let and let be a bounded domain with a smooth boundary . Our purpose in this paper is to consider the existence of solutions of the problem:

where

     

A Jacobi matrix inverse eigenvalue problem with mixed data

In this paper, the inverse eigenvalue problem of reconstructing a Jacobi matrix from part of its eigenvalues and its leading principal submatrix is considered. The necessary and sufficient conditions for the existence and uniqueness of the solution are derived. Furthermore, a numerical algorithm and some numerical examples are given.     

Spectral estimates of least energy solutions to the Brezis-Nirenberg problem with a variable coefficient

Let uε be a least energy solution to the Brezis-Nirenberg problem:

     

Optimal solvability for a nonlocal problem at critical growth

We provide optimal solvability conditions for a nonlocal minimization problem at critical growth involving an external potential function a. Furthermore, we get an existence and uniqueness result for a related nonlocal equation.     

Existence and multiple solutions for a variable exponent system

In this paper, we study the following variable exponent system

     

Least energy solutions of a critical Neumann problem with a weight

We investigate the effect of the coefficient of the critical nonlinearity for the Neumann problem on the existence of least energy solutions. As a by-product we establish a Sobolev inequality with interior norm. Received: 26 April 2000 / Accepted: 25 February 2001 / Published online: 5 September 2002     

Nonhomogeneous eigenvalue problem with indefinite weight

This paper, following the theory of partial differential equations on the Orlicz–Sobolev spaces, is mainly concerned with the nonhomogeneous eigenvalue problem involving variable growth conditions and a sign-changing potential. The results show that the spectrum of such problems contains a continuous family of eigenvalues.     

Existence and multiplicity of positive solutions of a nonlinear eigenvalue problem with indefinite weight function

This paper is concerned with the existence, multiplicity and stability of positive solutions of an indefinite weight boundary value problem

where aC[0,1] changes sign. The proof of our main result is based upon bifurcation techniques.     

Neumann problems with resonance in the first eigenvalue

The aim of this work is to present results of existence of solutions for a class of superlinear asymmetric elliptic systems with resonance in the first eigenvalue. The asymmetry that we consider has linear behavior on and superlinear on . To obtain these results we apply topological degree theory.     

Existence and multiplicity of solutions for a general p(x)-Laplacian Neumann problem

In this paper, we verify that a general p(x)$p\left(x\right)$-Laplacian Neumann problem has at least three weak solutions, which generalizes the corresponding result of the reference [Mihai Mihilescu, Existence and multiplicity of solutions for a Neumann problem involving the p(x)$p\left(x\right)$-Laplace operator, Nonlinear Anal. 67 (2007) 1419–1425].     

Existence of bounded solutions for nonlinear elliptic equations in unbounded domains

In this paper we study the existence of bounded weak solutions for some nonlinear Dirichlet problems in unbounded domains. The principal part of the operator behaves like the p-laplacian operator, and the lower order terms, which depend on the solution u and its gradient u, have a power growth of order p–1 with respect to these variables, while they are bounded in the x variable. The source term belongs to a Lebesgue space with a prescribed asymptotic behaviour at infinity.     

Existence of nonnegative solutions for a class of semilinear elliptic systems with indefinite weight

We prove the existence of nonnegative solutions to the system