Optimization problems for weighted Sobolev constants

In this paper, we study a variational problem under a constraint on the mass. Using a penalty method we prove the existence of an optimal shape. It will be shown that the minimizers are Hölder continuous and that for a large class they are even Lipschitz continuous. Necessary conditions in form of a variational inequality in the interior of the optimal domain and a condition on the free boundary are derived.     

Sign changing solutions of semilinear elliptic problems in exterior domains

We prove the existence of a sign changing solution to the semilinear elliptic problem , in an exterior domain Ω having finite symmetries.     

Uniqueness of solutions for some nonlinear Dirichlet problems

We consider here a class of nonlinear Dirichlet problems, in a bounded domain , of the form

Positive solutions of semilinear biharmonic equations with critical Sobolev exponents

In this paper we study the critical growth biharmonic problem with a parameter λ and establish uniform lower bounds for Λ, which is the supremum of the set of λ, related to the existence of positive solutions of the biharmonic problem.     

Effect of topology on the multiplicity of solutions for some semilinear elliptic systems with critical Sobolev exponent

In this paper, we are concerned with the effect of the domain topology on the multiplicity of solutions for some semilinear elliptic systems involving critical Sobolev exponent. We show that if the interaction term is sufficiently small, then the number of solutions of the system is estimated from below by 2 catΩ. The proof of this fact requires analysis of the structure of the Nehari manifold associated with the system. This research was partially supported by the Grant-in-Aid for Young Scientists (B), The Ministry of Education, Culture, Sports, Science and Technology, Japan.     

Multiplicity of solutions for a noncooperative elliptic system with critical Sobolev exponent

In this paper, we study a system of elliptic equations by applying the Limit Index Theory. Under some assumptions on nonlinear part, we can obtain the existence of multiple solutions for the equations. The research is supported by NNSF of China (10471024) and Fujian Provincial Natural Science Foundation of China (A0410015).     

On nontrivial solutions of critical polyharmonic elliptic systems

In the present work, we consider elliptic systems involving polyharmonic operators and critical exponents. We discuss the existence and nonexistence of nontrivial solutions to these systems. Our theorems improve and/or extend the ones established by Bartsch and Guo [T. Bartsch, Y. Guo, Existence and nonexistence results for critical growth polyharmonic elliptic systems, J. Differential Equations 220 (2006) 531-543] in both aspects of spectral interaction and regularity of lower order perturbations.     

Four nontrivial solutions for subcritical exponential equations

We show that a semilinear Dirichlet problem in bounded domains of in presence of subcritical exponential nonlinearities has four nontrivial solutions near resonance. Research supported by the Italian National Project Metodi Variazionali ed Equazioni Differenziali Non Lineari.     

Fourth order equations of critical Sobolev growth. Energy function and solutions of bounded energy in the conformally flat case

Given (M,g) a smooth compact Riemannian manifold of dimension n ≥ 5, we consider equations like

Local regularity of weak solutions of semilinear parabolic systems with critical growth

We show that, under so called controllable growth conditions, any weak solution in the energy class of the semilinear parabolic system

Symmetry breaking for a class of semilinear elliptic problems and the bifurcation diagram for a 1-dimensional problem

We study the behaviour of the positive solutions to the Dirichlet problem IR ⁿ in the unit ball in IR ^R wherep<(N+2)/(N−2) ifN≥3 and λ varies over IR. For a special class of functionsg viz.,g(x)=u ₀ ^p (x) whereu ₀ is the unique positive solution at λ=0, we prove that for certain λ’s nonradial solutions bifurcate from radially symmetric positive solutions. WhenN=1, we obtain the complete bifurcation diagram for the positive solution curve.     

Brake orbits type solutions to some class of semilinear elliptic equations

We consider a class of semilinear elliptic equations of the form

On the existence of solutions to a special variational problem

In this paper we establish an existence and regularity result for solutions to the problem

Finite element approximation of elliptic control problems with constraints on the gradient

We consider an elliptic optimal control problem with control constraints and pointwise bounds on the gradient of the state. We present a tailored finite element approximation to this optimal control problem, where the cost functional is approximated by a sequence of functionals which are obtained by discretizing the state equation with the help of the lowest order Raviart–Thomas mixed finite element. Pointwise bounds on the gradient variable are enforced in the elements of the triangulation. Controls are not discretized. Error bounds for control and state are obtained in two and three space dimensions. A numerical example confirms our analytical findings.     

Solutions to nonlinear Neumann problems with an inverse square potential

Let Ω be an open bounded domain in with smooth boundary . We are concerned with the critical Neumann problem

Infinitely many solutions of some nonlinear variational equations

The aim of this paper is investigating the existence of one or more critical points of a family of functionals which generalizes the model problem

Existence of a pair of positive solutions for a class of nonlinear eigenvalue problems

In this paper, we consider a nonlinear elliptic equation driven by the p-Laplacian and with a parameter λ > 0. Using a combination of variational and degree theoretic methods, we show that there exists λ^* > 0 such that, if λ > λ^*, then the problem has two positive smooth solutions. Our result extends earlier ones by Rabinowitz (semilinear equations) and Guo (nonlinear equations).      

Existence and nonexistence results for critical growth polyharmonic elliptic systems

In this work, we consider semilinear elliptic systems for the polyharmonic operator having a critical growth nonlinearity. We establish conditions for existence and nonexistence of nontrivial solutions to these systems.