Existence and non-existence of solutions to elliptic equations related to the Caffarelli–Kohn–Nirenberg inequalities

We investigate elliptic equations related to the Caffarelli–Kohn–Nirenberg inequalities: and such that . For various parameters α, β and various domains Ω, we establish some existence and non-existence results of solutions in rather general, possibly degenerate or singular settings.     

On the singular elliptic systems involving multiple critical Sobolev exponents

In this paper, a singular elliptic system is investigated, which involves multiple critical Sobolev exponents and Hardy-type terms. By using variational methods and analytical techniques, the existence of positive and sign-changing solutions to the system is established.     

Existence and uniqueness of multi-bump bound states of nonlinear Schrödinger equations with electromagnetic fields

In this paper, we are concerned with the existence and uniqueness of multi-bump bound states of the nonlinear Schrödinger equations with electromagnetic potential

     

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.     

Multi-bump bound states of nonlinear Schrödinger equations with electromagnetic fields and critical frequency

In this paper, we are concerned with the existence and asymptotic behavior of standing wave solutions ψ(x,t)=e−iλEt of nonlinear Schrödinger equations with electromagnetic fields , (t,x)∈R×RN, with E being a critical frequency in the sense that infx∈RNW(x)=E. We show that if the zero set of W−E has several isolated connected components Ω₁,…,Ωk such that the interior of Ωi is not empty and ∂Ωi is smooth, then for λ>0 large there exists, for any non-empty subset J⊂{1,2,…,k}, a standing wave solution which is trapped in a neighborhood of _?j∈JΩj.     

On the quasilinear elliptic problem with a critical Hardy–Sobolev exponent and a Hardy term

In the present paper, a quasilinear elliptic problem with a critical Sobolev exponent and a Hardy-type term is considered. By means of a variational method, the existence of nontrivial solutions for the problem is obtained. The result depends crucially on the parameters p,t,s,λ$p,t,s,\lambda$ and μ$\mu$.     

Sign-changing solutions of critical growth nonlinear elliptic systems

In this paper, we are concerned with the existence of sign-changing solutions of a class of nonlinear elliptic systems with critical growth.     

Elliptic systems involving the critical exponents and potentials

In this paper, we investigate a singular elliptic system, which involves the critical Sobolev exponent and multiple Hardy-type terms. By employing variational methods, the existence of its positive solutions is established. By the Moser iteration method, some asymptotic properties of its nontrivial solutions at the singular points are verified.     

Some properties of solutions to the singular quasilinear problems

In this paper, a kind of quasilinear elliptic problem is studied, which involves the critical exponent and singular potentials. By the Caffarelli-Kohn-Nirenberg inequality and variational methods, some important properties of the positive solution to the problem are established.     

A singular Moser-Trudinger embedding and its applications

Let Ω be a bounded domain in , we prove the singular Moser-Trudinger embedding: if and only if where and . We will also study the corresponding critical exponent problem.     

Multiplicity of solutions to uniformly elliptic fully nonlinear equations with concave-convex right-hand side

We deal with existence, non-existence and multiplicity of solutions to the model problem

(P)     

An Improved Hardy's Inequality

In this paper we shall extend Hardy's inequality associated with Fourier transform to the strip n(2-p) ≤ σ < n+p(N +1) where N = [n(1/p-1)], the greatest integer not exceeding n(1/p−1).     

Soliton solutions for quasilinear Schrödinger equations: The critical exponential case

Quasilinear elliptic equations in R²${\mathbb{R}}^2$ of second order with critical exponential growth are considered. By using a change of variable, the quasilinear equations are reduced to semilinear equations, whose respective associated functionals are well defined in H¹(R²)$H^1\left({\mathbb{R}}^2\right)$ and satisfy the geometric hypotheses of the mountain pass theorem. Using this fact, we obtain a Cerami sequence converging weakly to a solution v$v$. In the proof that v$v$ is nontrivial, the main tool is the concentration–compactness principle [P.L. Lions, The concentration compactness principle in the calculus of variations. The locally compact case. Part I and II, Ann. Inst. H. Poincaré Anal. Non. Linéaire 1 (1984) 109–145, 223–283] combined with test functions connected with optimal Trudinger–Moser inequality.     

Concentrating solutions for a planar elliptic problem involving nonlinearities with large exponent

We consider the boundary value problem Δu+up=0 in a bounded, smooth domain Ω in R² with homogeneous Dirichlet boundary condition and p a large exponent. We find topological conditions on Ω which ensure the existence of a positive solution up concentrating at exactly m points as p→∞. In particular, for a nonsimply connected domain such a solution exists for any given m?1.     

Bound states of nonlinear Schrödinger equations with potentials tending to zero at infinity

In this paper, we are concerned with the existence of solutions to the N-dimensional nonlinear Schrödinger equation −ε²Δu+V(x)u=K(x)up with u(x)>0, u∈H¹(RN), N?3 and . When the potential V(x) decays at infinity faster than ⁻²(1+|x|) and K(x)?0 is permitted to be unbounded, we will show that the positive H¹(RN)-solutions exist if it is assumed that G(x) has local minimum points for small ε>0, here with denotes the ground energy function which is introduced in [X. Wang, B. Zeng, On concentration of positive bound states of nonlinear Schrödinger equations with competing potential functions, SIAM J. Math. Anal. 28 (1997) 633-655]. In addition, when the potential V(x) decays to zero at most like (1+|x|)^−α with 0<α?2, we also discuss the existence of positive H¹(RN)-solutions for unbounded K(x). Compared with some previous papers [A. Ambrosetti, A. Malchiodi, D. Ruiz, Bound states of nonlinear Schrödinger equations with potentials vanishing at infinity, J. Anal. Math. 98 (2006) 317-348; A. Ambrosetti, D. Ruiz, Radial solutions concentrating on spheres of NLS with vanishing potentials, Proc. Roy. Soc. Edinburgh Sect. A 136 (2006) 889-907; A. Ambrosetti, Z.Q. Wang, Nonlinear Schrödinger equations with vanishing and decaying potentials, Differential Integral Equations 18 (2005) 1321-1332] and so on, we remove the restrictions on the potential function V(x) which decays at infinity like (1+|x|)^−α with 0<α?2 as well as the restrictions on the boundedness of K(x)>0. Therefore, we partly answer a question posed in the reference [A. Ambrosetti, A. Malchiodi, Concentration phenomena for NLS: Recent results and new perspectives, preprint, 2006].     

Soliton solutions for quasilinear Schrödinger equations with critical growth

In this paper we establish the existence of standing wave solutions for quasilinear Schrödinger equations involving critical growth. By using a change of variables, the quasilinear equations are reduced to semilinear one, whose associated functionals are well defined in the usual Sobolev space and satisfy the geometric conditions of the mountain pass theorem. Using this fact, we obtain a Cerami sequence converging weakly to a solution v. In the proof that v is nontrivial, the main tool is the concentration-compactness principle due to P.L. Lions together with some classical arguments used by H. Brezis and L. Nirenberg (1983) in [9].