Existence of three nontrivial solutions for elliptic systems with critical exponents and weights

In this paper, an elliptic system with critical Sobolev exponents and weights is studied by the linking theorem on product space and the classical minimax theorem. By investigating the coefficients of the critical nonlinearities, we establish the existence of three nontrivial solutions.     

Minimizers to Rayleigh quotients of critical elliptic systems involving different Hardy-type terms

In this paper, a system of semilinear elliptic equations is investigated, which involves homogeneous critical nonlinearities and different Hardy-type terms. By variational methods, the existence of minimizers to the Rayleigh quotient and ground state solutions to the system is verified completely.     

EXISTENCE OF MULTIPLE SOLUTIONS FOR SINGULAR QUASILINEAR ELLIPTIC SYSTEM WITH CRITICAL SOBOLEV-HARDY EXPONENTS AND CONCAVE-CONVEX TERMS

The main purpose of this paper is to establish the existence of multiple solutions for singular elliptic system involving the critical Sobolev-Hardy exponents and concave-convex nonlinearities.It is shown,by means of variational methods,that under certain conditions,the system has at least two positive solutions.     

Multiplicity of positive solutions for a critical quasilinear elliptic system with concave and convex nonlinearities

In this paper, by using the Lusternik–Schnirelmann category, we obtain a multiplicity result for a quasilinear elliptic system with both concave and convex nonlinearities and critical growth terms in bounded domains.     

Existence of solution for a nonvariational elliptic system with exponential growth in dimension two

We prove existence of solution for an elliptic system on a bounded domain in dimension two. We use the Galerkin scheme in the product of Hilbert spaces. The nonlinearities may have subcritical or critical exponential growth.     

Nontrivial solutions for the fractional Schrödinger-Poisson system with subcritical or critical nonlinearities

In this paper, we are concerned with a class of fractional Schrödinger-Poisson system involving subcritical or critical nonlinearities. By using the Nehari manifold and variational methods, we obtain the existence and multiplicity of nontrivial solutions.     

An Orlicz-space approach to superlinear elliptic systems

In this paper we study superlinear elliptic systems in Hamiltonian form. Using an Orlicz-space setting, we extend the notion of critical growth to superlinear nonlinearities which do not have a polynomial growth. Existence of nontrivial solutions is proved for superlinear nonlinearities which are subcritical in this generalized sense.     

On a class of semipositone discrete boundary value problems

In this paper, we deal with a class of semipositone discrete boundary value problems via critical point theory developed by Chang, and obtain nonexistence and multiplicity results on sublinear nonlinearities and an existence result on superlinear nonlinearities, respectively.     

A note on a conjecture for the critical curve of a weakly coupled system of semilinear wave equations with scale-invariant lower order terms

In this note, two blow-up results are proved for a weakly coupled system of semilinear wave equations with distinct scale-invariant lower order terms both in the subcritical case and in the critical case when the damping and the mass terms make both equations in some sense “wave-like.” In the proof of the subcritical case, an iteration argument is used. This approach is based on a coupled system of nonlinear ordinary integral inequalities and lower bound estimates for the spatial integral of the nonlinearities. In the critical case, we employ a test function-type method that has been developed recently by Ikeda-Sobajima-Wakasa and relies strongly on a family of certain self-similar solutions of the adjoint linear equation. Therefore, as critical curve in the p−q plane of the exponents of the power nonlinearities for this weakly coupled system, we conjecture a shift of the critical curve for the corresponding weakly coupled system of semilinear wave equations.     

Existence and multiplicity of solutions for elliptic problems with indefinite discontinuous nonlinearities

In this paper we study the critical points for a locally Lipschitz functional that in some sense will be solutions of an elliptic problem with indefinite discontinuous nonlinearities. We should mention that our results were inspired by the work of Ambrosetti-Badiale [3], Arcoya-Calahorrano [5], Alama-Tarantello [1] and Chang [8]. For the problem studied in [3] we introduce indefinite nonlinearities as in [1] and [6]. To obtain the existence and multiplicity of solutions we use the critical points theory developed by Chang. Applications for Plasma Physics are considered with nonlinearities that change sign. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Multiple blow-up rates in a coupled heat system with mixed-type nonlinearities

This paper studies heat equations with inner absorptions and coupled boundary fluxes of mixed-type nonlinearities. At first, the critical exponent is obtained, and simply described via a characteristic algebraic system introduced by us. Then, as the main results of the paper, three blow-up rates are established under different dominations of nonlinearities for the one-dimensional case, and represented in another characteristic algebraic system. In particular, it is observed that unlike those in previous literature on parabolic models with absorptions, two of the multiple blow-up rates obtained here do depend on the absorption exponents. In the known works, the absorptions affect the blow-up criteria, the blow-up time, as well as the initial data required for the blow-up of solutions, all without changing the blow-up rates. To our knowledge, this is the first example of absorption-dependent blow-up rates, exploiting the significant interactions among diffusions, inner absorptions and nonlinear boundary fluxes in the coupled system. It is also proved that the blow-up of solutions in the model occurs on the boundary only.     

Modelling and optimization of parallel reservoirs having nonlinear storage curves under critical water conditions for long-term regulation

This paper deals with the problem of parallel reservoirs having nonlinear storage-elevation curves (quadratic functions) for long-term regulation under critical water conditions using the minimum norm formulation. To overcome these nonlinearities, we introduce a set of pseudo-state variables. A set of optimizing equations is obtained. The proposed method is efficient in computing time and in calculating the expected benefits of generation from the system during the critical period. Numerical results are reported for a real system in operation consisting of two rivers; each river has two reservoirs in series.This work was supported by the National Research Council of Canada, Grant No. A4146. The authors would like to acknowledge data obtained from B. C. Hydro.     

Existence Results for Quasilinear Elliptic Equations with Discontinuous Nonlinearities

The nonsmooth critical point theory is applied to prove the existence of solutions and multiple solutions of a quasilinear elliptic equation with discontinuous nonlinearities.     

Divergence symmetries of semilinear polyharmonic equations involving critical nonlinearities

It is shown that a Lie point symmetry of the semilinear polyharmonic equations involving nonlinearities of power or exponential type is a variational/divergence symmetry if and only if the equation parameters assume critical values. The corresponding conservation laws for critical polyharmonic semilinear equations are established.     

Multiplicity result for a discrete eigenvalue problem with discontinuous nonlinearities

Using a three critical points theorem for nondifferentiable functionals, we investigate a class of second order difference equation with discontinuous nonlinearities. A new multiplicity result is obtained.     

The Neumann Problem for Semilinear Elliptic Equations with Critical Sobolev Exponent

In this survey article we discuss the existence and the properties of least energy solutions of a semilinear critical Neumann problem. The main focus is on the joint effect of the shape of the graph of coefficients of the critical nonlinearities and the geometry of the boundary on the existence of solutions. Received: July 2006     

Three Solutions of a Kirchhoff Type Problem Involving Critical Growth and Near Resonance

The purpose of this work is to study a Kirchhoff type equation involving critical and singular nonlinearities. Based on variational methods, we obtain the existence of three nontrivial solutions for this problem.     

An eigenvalue Dirichlet problem involving the p-Laplacian with discontinuous nonlinearities

A multiplicity result for an eigenvalue Dirichlet problem involving the p-Laplacian with discontinuous nonlinearities is obtained. The proof is based on a three critical points theorem for nondifferentiable functionals.     

具有多重非线性的非牛顿多方渗流方程的整体存在性与非存在性

考虑了一个具有多重非线性的抛物模型中,非线性扩散项、非线性反应项和非线性边界流三种非线性机制之间的相互作用.通过构造自相似上解和自相似下解,获得了临界整体存在性曲线和临界Fujita曲线.     

Large Critical Exponents for Some Second Order Uniformly Elliptic Operators

In this paper we investigate the critical exponents of two families of Pucci's extremal operators. The notion of critical exponent that we have chosen for these fully nonlinear operators which are not variational is that of threshold between existence and nonexistence of the solutions for semilinear equations with pure power nonlinearities. Interesting new exponents appear in this context.