Ground state solutions for a modified fractional Schrödinger equation with critical exponent

In this paper, we study the existence of ground state solutions for the modified fractional Schrödinger equations $${(-\mathrm{\Delta })}^{\alpha }u+\mu u+\kappa [{(-\mathrm{\Delta })}^{\alpha }{u}^2]u=\sigma |u{|}^{p-1}u+|u{|}^{q-2}u,x\in R^N,$$ where $N\ge 2$, $\alpha \in (0,1)$, $\mu$, $\sigma$ and $\kappa$ are positive parameters, $2<p+1<q\le 2_{\alpha }^{\ast }:=\frac{2N}{N-2\alpha }$, ${(-\mathrm{\Delta })}^{\alpha }$ denotes the fractional Laplacian of order $\alpha$. For the case $2<p+1<q<2_{\alpha }^{\ast }$ and the case $q=2_{\alpha }^{\ast }$, the existence results of ground state solutions are given, respectively.     

Nonexistence of bounded energy solutions for a critical exponent problem

In this paper, we discuss positive solutions for certain weighted elliptic equations with critical Sobolev exponent in R^N. The weights depend on a positive parameter γ, which is allowed to increase to infinity. While for small values of γ solutions are completely classified, an attempt to such a classification is much more difficult for large values of the parameter. In the present work we prove the nonexistence of solutions with bounded energy as γ increases to infinity. We also prove a multiplicity result for high energy solutions.     

Multiple positive solutions for a Dirichlet problem involving critical Sobolev exponent

In this paper, we study the decomposition of the Nehari manifold via the combination of concave and convex nonlinearities. Furthermore, we use this result and the Ljusternik-Schnirelmann category to prove that the existence of multiple positive solutions for a Dirichlet problem involving critical Sobolev exponent.     

The uniqueness of symmetric single peak solutions for a Neumann problem involving critical exponent

In this paper, by a constructive method, we consider a Neumann problem involving critical Sobolev exponent and obtain a uniqueness result of the symmetric single peak solutions.     

Ground state of solutions for a class of fractional Schrödinger equations with critical Sobolev exponent and steep potential well

In this paper, we study the following fractional Schrödinger equations: (1) where (?△)^α is the fractional Laplacian operator with , 0≤s ≤2α , λ >0, κ and β are real parameter. is the critical Sobolev exponent. We prove a fractional Sobolev‐Hardy inequality and use it together with concentration compact theory to get a ground state solution. Moreover, concentration behaviors of nontrivial solutions are obtained when the coefficient of the potential function tends to infinity.     

Ground state solutions for a semilinear elliptic problem with a convection term

By a sub-supersolution method and a perturbed argument, we improve the earlier results concerning the existence of ground state solutions to a semilinear elliptic problem −Δu+p(x)q|∇u|=f(x,u), u>0, x∈RN, , where q∈(1,2], for some α∈(0,1), p(x)?0, ∀x∈RN, and f:RN×(0,∞)→[0,∞) is a locally Hölder continuous function which may be singular at zero.     

Existence and multiple solutions for a variable exponent system

In this paper, we study the following variable exponent system

     

Minimal nodal solutions of the pure critical exponent problem on a symmetric domain

We establish existence of nodal solutions to the pure critical exponent problem in u = 0 on where a bounded smooth domain which is invariant under an orthogonal involution of We extend previous results for positive solutions due to Coron, Dancer, Ding, and Passaseo to existence and multiplicity results for solutions which change sign exactly once.Received: 4 April 2003, Accepted: 26 August 2003, Published online: 24 November 2003Mathematics Subject Classification (2000): 35J65, 35J20Research partially supported by PAPIIT, UNAM, under grant IN110902-3.     

Solutions of a quasilinear elliptic problem involving a critical Sobolev exponent and multiple Hardy-type terms

In the present paper, a quasilinear elliptic problem with a critical Sobolev exponent and multiple Hardy-type terms is considered. By means of a variational method, the existence of positive solutions of the problem is obtained.     

Ground state solutions for a class of fractional Kirchhoff equations with critical growth

In this paper, we study the effect of lower order perturbations in the existence of positive solutions to the fractional Kirchhoff equation with critical growth■ where a, b 0 are constants, μ 0 is a parameter,■ , and V : R~3→ R is a continuous potential function. For suitable assumptions on V, we show the existence of a positive ground state solution, by using the methods of the Pohozaev-Nehari manifold, Jeanjean's monotonicity trick and the concentration-compactness principle due to Lions(1984).     

Ground state solutions for a Kirchhoff-type elliptic system involving critical exponential growth nonlinearities

The aim of this paper is to study the ground state solution for a Kirchhoff-type elliptic system without the Ambrosetti–Rabinowitz condition.     

Multiple solutions for a semilinear equation involving singular potential and critical exponent

Variational methods are used to prove the existence of positive and sign-changing solutions for a semilinear equation involving singular potential and critical exponent in any bounded domain.*supported in part by Tian Yuan Foundation of NNSF (A0324612)**Supported by 973 Chinese NSF and Foundation of Chinese Academy of Sciences.***Supported in part by NNSF of China.Received: September 23, 2002; revised: November 30, 2003     

Multiple solutions for a semilinear equation involving singular potential and critical exponent

Variational methods are used to prove the existence of positive and sign-changing solutions for a semilinear equation involving singular potential and critical exponent in any bounded domain.     

Infinitely many peak solutions for a biharmonic equation involving critical exponent

In this paper, we study the following biharmonic equation where , K(1) > 0,K′(1) > 0, B₁(0) is the unit ball in (N≥6). We show that the aforementioned problem has infinitely many peak solutions, whose energy can be made arbitrarily large. Copyright © 2016 John Wiley & Sons, Ltd.     

Multiple solutions of an inhomogeneous Neumann problem for an elliptic system with critical Sobolev exponent

In this paper we prove the existence of two solutions for the inhomogeneous Neumann problem with critical Sobolev exponent.     

Entropy solutions for a doubly nonlinear elliptic problem with variable exponent

We study the boundary value problem in Ω, u=0 on ∂Ω, where Ω is a smooth bounded domain in RN (N?3) and is a p(x)-Laplace type operator with p(.):Ω→[1,+∞) a measurable function and b a continuous and nondecreasing function from R→R. We prove the existence and uniqueness of an entropy solution for L¹-data f.