Non-quadratic Conditions Research Articles

Multiplicity of Solutions for a Class of Elliptic Problem of p-Laplacian Type with a p-Gradient Term

We consider the following problem: -Δpu=c(x)|u|q-1u+μ|∇u|p+h(x) in Ω, u=0 on ∂Ω, where Ω is a bounded set in RN (N≥3) with a smooth boundary, 1<p<N, q>0, μ∈R⁎, and c and h belong to Lk(Ω) for some k>N/p. In this paper, we assume that c≩0 a.e. in Ω and h without sign condition and then we prove the existence of at least two bounded solutions under the condition that ck and hk are suitably small. For this purpose, we use the Mountain Pass theorem, on an equivalent problem to (P) with variational structure. Here, the main difficulty is that the nonlinearity term considered does not satisfy Ambrosetti and Rabinowitz condition. The key idea is to replace the former condition by the nonquadraticity condition at infinity.

Cancellation-Based Nonquadratic Controller Design for Nonlinear Systems via Takagi-Sugeno Models.

This paper is concerned with nonquadratic conditions for stabilization of continuous-time nonlinear systems via exact Takagi-Sugeno models and generalized fuzzy Lyapunov functions. The approach hereby proposed feedback to the time derivatives of the membership functions through a multi-index control law that cancels out the terms responsible of former a priori local conditions. Thus, a nonquadratic controller design in the form of linear matrix inequalities is achieved; it does not require bounds on the time derivatives nor any extra parameters. The examples included are shown to outperform former approaches.

Superlinear elliptic problems under the non-quadraticity condition at infinity

We present some sufficient conditions to obtain compactness properties for the Euler–Lagrange functional of an elliptic equation. As an application, we extend some existence and multiplicity results for superlinear problems.

Some united existence results of periodic solutions for non-quadratic second order Hamiltonian systems

In this paper, some existence theorems are obtained for periodic solutions of second order Hamiltonian systems under non-quadratic conditions by using the minimax principle. Our results unite, extend and improve those relative works in some known literature.

Existence of subharmonic solutions for non-quadratic second-order Hamiltonian systems

In this paper, some existence theorems are obtained for subharmonic solutions of second-order Hamiltonian systems with linear part under non-quadratic conditions. The approach is the minimax principle. We consider some new cases and obtain some new existence results.MSC:34C25, 58E50, 70H05.

Nonquadratic Stabilization of Continuous T–S Fuzzy Models: LMI Solution for a Local Approach

This paper is concerned with nonquadratic stabilization design problem for continuous-time nonlinear models in the Takagi-Sugeno (T-S) form obtained by sector nonlinearity approach. Most of the previous results found in the literature intended to establish global nonquadratic stabilization conditions which are hard to uphold due to the difficulty of handling time derivatives of the membership function. By changing the paradigm of global stabilization for something less restrictive, a local solution to overcome infeasible quadratic stabilization conditions is offered in this paper. It is shown that the derived local nonquadratic conditions actually lead to reasonable advantages over the existing quadratic approach, as well as some previous nonquadratic attempts. Moreover, conditions for the solvability of state feedback controller design given here are written in the form of linear matrix inequalities (LMIs) which can be efficiently solved by convex optimization techniques. Simulation examples are given to demonstrate the validity and applicability of the proposed approaches.

Subharmonic solutions for a class of non-quadratic second order Hamiltonian systems

In this paper, some existence theorems are obtained for infinitely subharmonic solutions of second order Hamiltonian systems under non-quadratic conditions. The approach is the minimax principle. Our results greatly extend and improve some known results.

Existence of solutions for the coupled systems of second and fourth order elliptic equations

In this paper, under the nonquadraticity condition, we obtain two existence theorems of nontrivial solutions for a coupled system of second and fourth order elliptic equations.

SOLUTIONS FOR A RESONANT ELLIPTIC SYSTEM WITH COUPLING IN

ABSTRACT Existence and multiplicity of solutions are established, via the Variational Method, for a class of resonant semilinear elliptic system in under a local nonquadraticity condition at infinity. The main goal is to consider systems with coupling where one of the potentials does not satisfy any coercivity condition. The existence of solution is proved under a critical growth condition on the nonlinearity.

On a double resonant problem in $\Bbb R^N$

It is established, via variational methods, the existence and multiplicity of solutions for a class of semilinear elliptic equations in ${{\mathbb R}^N}$. The condition on the potential implies that the associated linear problem possesses a sequence of positive eigenvalues. This fact is used to study double resonant problems under a local nonquadraticity condition at infinity and pointwise limits. For the existence of solution the nonlinearity may satisfy a critical growth condition.