Multiple solutions for weighted nonlinear elliptic system involving critical exponents

In this paper,by using the idea of category,we investigate how the shape of the graph of h(x)affects the number of positive solutions to the following weighted nonlinear elliptic system:-div(|x|-2au)-μu|x|2(a+1)=αα+βh(x)|u|α-2|v|βu|x|b2*(a,b)+λK1(x)|u|q-2u,in,-div(|x|-2av)-μv|x|2(a+1)=βα+βh(x)|u|α|v|β-2v|x|b2*(a,b)+σK2(x)|v|q-2v,in,u=v=0,on,where 0∈is a smooth bounded domain in RN(N 3),λ,σ0 are parameters,0μμa(N-2-2a2)2;h(x),K1(x)and K2(x)are positive continuous functions in,1 q2,α,β1 andα+β=2*(a,b)(2*(a,b)2N N-2(1+a-b),is critical Sobolev-Hardy exponent).We prove that the system has at least k nontrivial nonnegative solutions when the pair of the parameters(λ,σ)belongs to a certain subset of R2.     

Existence of positive solutions to nonlinear elliptic equations involving critical Sobolev exponents

In this paper we extend the results of Brezis and Nirenberg in [4] to the problem $$\left\{ \begin{gathered} Lu = - D_i (a_{ij} (x)D_j u) = b(x)u^p + f(x,u) in\Omega , \hfill \\ p = (n + 2)/(n - 2) \hfill \\ u > 0 in\Omega , u = 0 \partial \Omega , \hfill \\ \end{gathered} \right.$$ whereL is a uniformly elliptic operator,b(x)≥0,f(x,u) is a lower order perturbation ofu ^p at infinity. The existence of solutions to (A) is strongly dependent on the behaviour ofa _ij (x), b(x) andf(x, u). For example, for any bounded smooth domain Ω, we have \(a_{ij} \left( x \right) \in C\left( {\bar \Omega } \right)\) such thatLu=u ^p possesses a positive solution inH ₀ ¹ (Ω). We also prove the existence of nonradial solutions to the problem ?Δu=f(|x|, u) in Ω,u>0 in Ωu=0 on ?Ω, Ω=B(0,1). for a class off(r, u).     

Existence of multiple positive solutions to singular elliptic systems involving critical exponents

This paper is devoted to investigate multiple positive solutions to a singular elliptic system where the nonlinearity involves a combination of concave and convex terms. By exploiting the effect of the coefficient of the critical nonlinearity and a variational method, we establish the main result which is based on the argument of the compactness.     

Existence and multiplicity of positive solutions to elliptic systems involving critical exponents

In this paper, a singular elliptic system involving multiple critical exponents and the Caffarelli-Kohn-Nirenberg inequality is investigated. By using the extremals of the best Hardy-Sobolev constants, the existence and multiplicity of positive solutions to the system are established.     

Multiple positive solutions for semilinear elliptic equations with critical weighted Hardy-Sobolev exponents

The existence and multiplicity of positive solutions are obtained for a class of semilinear elliptic equations with critical weighted Hardy-Sobolev exponents and the concave-convex nonlinearity by variational methods and some analysis techniques.     

Nontrivial solutions to semilinear elliptic problems involving two critical Hardy-Sobolev exponents

In this paper, we investigate a semilinear elliptic equation, which involves doubly critical Hardy-Sobolev exponents and a Hardy-type term. By means of the Linking Theorem and delicate energy estimates, the existence of nontrivial solutions to the problem is established.     

Multiplicity of solutions for a class of quasilinear elliptic equation involving the critical Sobolev and Hardy exponents

In this paper, by using concentration-compactness principle and a new version of the symmetric mountain-pass lemma due to Kajikiya (J Funct Anal 225:352–370, 2005), infinitely many small solutions are obtained for a class of quasilinear elliptic equation with singular potential $$- \Delta_p u - \mu \frac{|u|^{p-2}u}{|x|^p} =\frac{|u|^{p^\ast(s)-2}u}{|x|^s} + \lambda f(x, u),\quad u\in H_0^{1,p}(\Omega).$$      

Infinitely many solutions to elliptic systems involving critical exponents and Hardy potential

In this paper, we consider the following elliptic systems involving critical Sobolev growth and Hardy potential: where N ≥ 3,λ₁,λ₂ ∈ [0,Λ_N), is the best Hardy constant. is the critical Sobolev exponent. a₁,a₂, b are positive parameters, α,β > 0 and 1 < α + β : = q < 2 < 2*. with . By means of the concentration‐compactness principle and Kajikiya's new version of symmetric mountain pass lemma, we obtain infinitely many solutions that tend to zero for suitable positive parameters a₁,a₂,b and λ₁,λ₂. Copyright © 2012 John Wiley & Sons, Ltd.     

MULTIPLICITY OF POSITIVE SOLUTIONS FOR AN ELLIPTIC SYSTEM

By considering the properties of f(t,u,v)/u v, g(t,u,v)/u v, we show the multiplicity of at least two positive solutions of the elliptic system.     

Existence and asymptotic properties of solutions to elliptic systems involving multiple critical exponents

This paper is concerned with a singular elliptic system, which involves the Caffarelli-Kohn-Nirenberg inequality and multiple critical exponents. By analytic technics and variational methods, the extremals of the corresponding bet Hardy-Sobolev constant are found, the existence of positive solutions to the system is established and the asymptotic properties of solutions at the singular point are proved.     

Multiplicity of positive solutions of nonlinear elliptic equations with critical Sobolev exponent in some contractible domains

In this paper we prove that, for every positive integer k, there exists a contractible bounded domain in ^N with N3, where the problem (*) (see Introduction) has at least k solutions.     

Multiplicity of positive solutions to a p-Laplacian equation involving critical nonlinearity

In this paper we deal with multiplicity of positive solutions to the p-Laplacian equation of the type