INFINITELY MANY SIGN-CHANGING SOLUTIONS FOR THE BRZIS-NIRENBERG PROBLEM INVOLVING HARDY POTENTIAL

In this article, we give a new proof on the existence of infinitely many sign- changing solutions for the following Brezis-Nirenberg problem with critical exponent and a Hardy potential -△u-μ（u/｜x｜^2）=λu＋｜u｜^2＊-2u inΩ, u=0 on eΩ,where Ω is a smooth open bounded domain of R^N which contains the origin, 2＊=2N/n-2 is the critical Sobolev exponent. More precisely, under the assumptions that N ≥ 7, μ ∈ [0, μ- 4）, and μ=（N-2）^2/4, we show that the problem admits infinitely many sign-changing solutions for each fixed λ 〉 0. Our proof is based on a combination of invariant sets method and Lj usternik-Schnirelman theory.

SIGN-CHANGING SOLUTIONS FOR p-BIHARMONIC EQUATIONS WITH HARDY POTENTIAL IN R^N

In this article, by using the method of invariant sets of descending flow, we obtain the existence of sign-changing solutions of p-biharmonic equations with Hardy potential in RN.

Multiple Solutions for an Elliptic Equation with Hardy Potential and Critical Nonlinearity on Compact Riemannian Manifolds

We prove the existence of multiple solutions of an elliptic equation with critical Sobolev growth and critical Hardy potential on compact Riemannian manifolds.

MULTIPLICITY OF POSITIVE SOLUTIONS FOR A NONLOCAL ELLIPTIC PROBLEM INVOLVING CRITICAL SOBOLEV-HARDY EXPONENTS AND CONCAVE-CONVEX NONLINEARITIES

In this article,we study the following critical problem involving the fractional Laplacian:{(−Δ)^α/2u−γu/|x|^α=λ|u|^q−2/|x|^s+|u|^2^∗α^(t)−2u/|x|^t in Ω,u=0 in R^N∖Ω,whereΩ⊂R^N(N>α)is a bounded smooth domain containing the origin,α∈(0,2),0≤s,t<α,1≤q<2,λ>0,2α^*(t)=2(N-t)/N-αis the fractional critical Sobolev-Hardy exponent,0≤γ<γH,and γH is the sharp constant of the Sobolev-Hardy inequality.We deal with the existence of multiple solutions for the above problem by means of variational methods and analytic techniques.

BOUND STATES FOR A CLASS OF QUASILINEAR SCALAR FIELD EQUATIONS WITH POTENTIALS VANISHING AT INFINITY

We study the existence and non-existence of bound states （i.e., solutions in W1,P（RN）） for a class of quasilinear scalar field equations of the for -△pu＋V（x）｜u｜p-2u=a（x）｜u｜q-2u,x∈RN,1〈P〈N,mwhen the potentials V（.）≥ 0 and a（.） decay to zero at infinity.

Multiplicity of Solutions for Nonlinear Biharmonic Equation Involving Critical Parameter and Critical Exponent

By using variational method, the multiplicity of solutions for nonlinear biharmonic equation involving critical parameter and critical exponent are established.

THE ASYMPTOTIC BEHAVIOR AND SYMMETRY OF POSITIVE SOLUTIONS TO p-LAPLACIAN EQUATIONS IN A HALF-SPACE

We study a nonlinear equation in the half-space with a Hardy potential,specifically,−Δ_(p)u=λu^(p−1)x_(1)^(p)−x_(1)^(θ)f(u)in T,where Δp stands for the p-Laplacian operator defined by Δ_(p)u=div(∣Δu∣^(p−2)Δu),p>1,θ>−p,and T is a half-space{x_(1)>0}.When λ>Θ(where Θ is the Hardy constant),we show that under suitable conditions on f andθ,the equation has a unique positive solution.Moreover,the exact behavior of the unique positive solution as x_(1)→0^(+),and the symmetric property of the positive solution are obtained.

Elliptic systems involving multiple critical nonlinearities and symmetric multi-polar potentials

In this paper,a system of elliptic equations is investigated,which involves multiple critical Sobolev exponents and symmetric multi-polar potentials.By variational methods and analytic techniques,the relevant best constants are studied and the existence of(Zk×SO(N.2))2-invariant solutions to the system is established.

SOLUTION TO p(x)-LAPLACIAN PROBLEM WITH HARDY TERM

We study the existence of solutions to a class of p(x)-Laplacian equations involving singular Hardy terms for nonlinear terms of the type f(x, t) = ±(-λ|t|m(x)-2t +|t|q(x)-2t). First we show the existence of infinitely many weak solutions for anyλ 】 0, and next prove that if λ is large enough then there exists a nontrivial weak solution. Our approach relies on direct variational methods and the theory of variable exponent Lebesgue-Sobolev spaces.

Existence of Nontrivial Solutions for p-Laplacian-Like Equations

In this paper, we study the p-Laplacian-Like equations involving Hardy potential or involving critical exponent and prove the existence of one or infinitely many nontrivial solutions. The results of the equations discussed can be applied to a variety of different fields in applied mechanics.