Existence results for degenerate elliptic equations with critical cone Sobolev exponents

In this paper, we study the existence result for degenerate elliptic equations with singular potential and critical cone sobolev exponents on singular manifolds. With the help of the variational method and the theory of genus, we obtain several results under different conditions.     

CRITICAL EXPONENTS AND CRITICAL DIMENSIONS FOR NONLINEAR ELLIPTIC PROBLEMS WITH SINGULAR COEFFICIENTS

Let B1 ■ RNbe a unit ball centered at the origin. The main purpose of this paper is to discuss the critical dimension phenomenon for radial solutions of the following quasilinear elliptic problem involving critical Sobolev exponent and singular coefficients:-div(|▽u|p-2▽u) = |x|s|u|p*(s)-2u + λ|x|t|u|p-2u, x ∈ B1,u|■B1= 0,where t, s -p, 2 ≤ p N, p*(s) =(N+s)p N-pand λ is a real parameter. We show particularly that the above problem exists infinitely many radial solutions if the space dimension N p(p- 1)t + p(p2- p + 1) and λ∈(0, λ1,t), where λ1,t is the first eigenvalue of-△p with the Dirichlet boundary condition. Meanwhile, the nonexistence of sign-changing radial solutions is proved if the space dimension N ≤(ps+p) min{1,p+t p+s}+p2p-(p-1) min{1,p+t p+s}and λ 0 is small.     

Existence of sign-changing solutions for semilinear singular elliptic equations with critical Sobolev exponents

We consider the semilinear elliptic problem in Ω, u=0 on ∂Ω, where 0∈Ω is a smooth bounded domain in RN, N?4, , is the critical Sobolev exponent, f(x,⋅) has subcritical growth at infinity, K(x)>0 is continuous. We prove the existence of sign-changing solutions under different assumptions when Ω is a usual domain and a symmetric domain, respectively.     

Nodal and positive solutions for singular semilinear elliptic equations with critical exponents in symmetric domains

In this paper, we consider the semilinear elliptic problem in Ω, u=0 on ∂Ω, where Ω is a smooth bounded domain in RN, N?4, , is the critical Sobolev exponent, K(x) is a continuous function. When Ω and K(x) are invariant under a group of orthogonal transformations, we prove the existence of nodal and positive solutions for 0<λ<λ₁, where λ₁ is the first Dirichlet eigenvalue of on Ω.     

Existence of ground-state solutions for p-Choquard equations with singular potential and doubly critical exponents

We are concerned with the following Choquard equation: $$-{\mathrm{\Delta }}_{p}u+\frac{A}{{|x|}^{\theta }}{|u|}^{p-2}u=\left(I_{\alpha }\ast F(u)\right)f(u),x\in {\mathbb{R}}^N,$$ where $p\in (1,N)$, $\alpha \in (0,N)$, $\theta \in [0,p)\cup \left(p,\frac{(N-1)p}{p-1}\right)$, $A>0$, ${\mathrm{\Delta }}_p$ is the p-Laplacian, $I_{\alpha }$ is the Riesz potential, and F is the primitive of f which is of critical growth due to the Hardy–Littlewood–Sobolev inequality. Under different range of θ and almost necessary conditions on the nonlinearity f in the spirit of Berestycki–Lions-type conditions, we divide this paper into three parts. By applying the refined Sobolev inequality with Morrey norm and the generalized version of the Lions-type theorem, some existence results are established. It is worth noting that our method is not involving the concentration–compactness principle.     

Anisotropic elliptic equations with VMO coefficients

This paper presents the study of maximal regularity properties for anisotropic differential-operator equations with VMO (vanishing mean oscillation) coefficients. We prove that the corresponding differential operator is separable and is a generator of analytic semigroup in vector-valued L^p spaces. Moreover, discreetness of spectrum and completeness of root elements of this operator is obtained.     

Infinitely many solutions to elliptic systems with critical exponents and Hardy potentials

In this paper,a system of elliptic equations is investigated,which involves Hardy potential and multiple critical Sobolev exponents.By a global compactness argument of variational method and a fine analysis on the Palais-Smale sequences created from related approximation problems,the existence of infinitely many solutions to the system is established.     

Existence of nodal solution for semi-linear elliptic equations with critical sobolev exponent on singular manifold

In this article, we prove that semi-linear elliptic equations with critical cone Sobolev exponents possess a nodal solution.     

On symmetric solutions of a singular elliptic equation with critical Sobolev-Hardy exponent

This paper is concerned with the existence of the nontrivial solutions of the following problem:

     

Multiple symmetric results for singular quasilinear elliptic systems with critical homogeneous nonlinearity

This paper deals with the existence and multiplicity of symmetric solutions for a class of singular quasilinear elliptic systems with critical homogeneous nonlinearity in a bounded symmetric domain. Applying variational methods and the symmetric criticality principle of Palais, we establish several existence and multiplicity results of G‐symmetric solutions under some appropriate assumptions on the weighted functions and the parameters. Copyright © 2016 John Wiley & Sons, Ltd.     

Critical singular problems via concentration-compactness lemma

In this work we consider existence and multiplicity results of nontrivial solutions for a class of quasilinear degenerate elliptic equations in RN of the form

(P)     

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.     

Linking solutions for p-Laplace equations with nonlinearity at critical growth

Under a suitable condition on n and p, the quasilinear equation at critical growth −Δpu=λ|u|^p−2u+|u|^p∗−2u is shown to admit a nontrivial weak solution for any λ?λ₁. Nonstandard linking structures, for the associated functional, are recognized.     

Characterization of the shape stability for nonlinear elliptic problems

We characterize all geometric perturbations of an open set, for which the solution of a nonlinear elliptic PDE of p-Laplacian type with Dirichlet boundary condition is stable in the L^∞-norm. The necessary and sufficient conditions are jointly expressed by a geometric property associated to the γp-convergence.If the dimension N of the space satisfies N−1<p?N and if the number of the connected components of the complements of the moving domains are uniformly bounded, a simple characterization of the uniform convergence can be derived in a purely geometric frame, in terms of the Hausdorff complementary convergence. Several examples are presented.     

Solutions for quasilinear elliptic problems with critical Sobolev-Hardy exponents

Let N?3, 2<p<N, 0?s<p and . Via the variational methods and analytic technique, we prove the existence of nontrivial solution to the singular quasilinear problem , for N?p² and suitable functions f(u).     

Positive solutions for higher order singular p-Laplacian boundary value problems

This paper investigates 2m-th (m ≥ 2) order singular p-Laplacian boundary value problems, and obtains the necessary and sufficient conditions for existence of positive solutions for sublinear 2m-th order singular p-Laplacian BVPs on closed interval.