[Show abstract][Hide abstract] ABSTRACT: In this paper, we consider the existence of a non-trivial weak solution to a quasilinear elliptic equation with singular weights and multiple critical exponents in the whole space. Firstly, we get the existence of a local Palais–Smale sequence by verifying the geometric conditions of the Mountain Pass Lemma. Secondly, we study the concentration properties of the Palais–Smale sequence of a zero weak limit. Thirdly, we deduce by contradiction to eliminate the possibility of a zero weak limit case. Lastly, applying a monotonic inequality, we shall prove that the nontrivial weak limit of the Palais–Smale sequence is indeed a weak solution.
[Show abstract][Hide abstract] ABSTRACT: In this paper, we study the asymptotic behavior of radial extremal functions to an inequality involving Hardy potential and critical Sobolev exponent. Based on the asymptotic behavior at the origin and the infinity, we shall deduce a strict inequality between two best constants. Finally, as an application of this strict inequality, we consider the existence of a nontrivial solution of a quasilinear Brezis–Nirenberg type problem with Hardy potential and critical Sobolev exponent.
[Show abstract][Hide abstract] ABSTRACT: In this paper, we consider the existence of non-trivial solutions to semi-linear Brezis–Nirenberg type problems with Hardy potential and singular coefficients. First, we shall study the corresponding eigenvalue problem, and obtain some basic properties of eigenvalues and asymptotic estimates of the eigenfunctions and approximating eigenfunctions. Secondly, we consider the extremal functions of the best embedding constant, and get some crucial estimates for the cut-off function of the extremal functions. Thirdly, applying different variational theorems for distinct cases of those parameters appearing in the equation, we obtain two existence results for non-trivial solutions to semi-linear Brezis–Nirenberg type problems. Our existence results are divided into non-resonant and resonant cases.
[Show abstract][Hide abstract] ABSTRACT: In this paper, we study the asymptotic behavior of radial extremal functions to an inequality involving Hardy potential and critical Sobolev exponent. Based on the asymptotic behavior at the origin and the infinity, we shall deduce a strict inequality between two best constants. Finally, as an application of this strict inequality, we consider the existence of nontrivial solution of a quasilinear Brezis-Nirenberg type problem with Hardy potential and critical Sobolev exponent.
[Show abstract][Hide abstract] ABSTRACT: In this paper, we consider the existence and non-existence of non-trivial solutions to quasilinear Brezis–Nirenberg-type problems with singular weights. First, we shall obtain a compact imbedding theorem which is an extension of the classical Rellich–Kondrachov compact imbedding theorem, and consider the corresponding eigenvalue problem. Secondly, we deduce a Pohozaev-type identity and obtain a non-existence result. Thirdly, thanks to the generalized concentration compactness principle, we will give some abstract conditions when the functional satisfies the (PS)c condition. Finally, basing on the explicit form of the extremal function, we will obtain some existence results.
[Show abstract][Hide abstract] ABSTRACT: In this paper, using Mountain Pass Lemma and Linking Argument, we prove the existence of nontrivial weak solutions for the Dirichlet problem for the superlinear equation of Caffarelli-Kohn-Nirenberg type in the case where the parameter $\lambda\in (0, \lambda_2)$, $\lambda_2$ is the second positive eigenvalue of the quasilinear elliptic equation of Caffarelli-Kohn-Nirenberg type.
[Show abstract][Hide abstract] ABSTRACT: We study the existence of multiple solutions to a Caffarelli-Kohn-Nirenberg type equation with asymptotically linear term at infinity -div(|x| -αp |Du| p-2 Du)=|x| -(α+1)p+c f(u)inΩ,u=0on∂Ω, where Ω⊂ℝ n is a bounded regular domain such that 0∈Ω, 1<p<n, 0≤α<n-p p, c>0, and f∈C(ℝ) is such that f(t)=-f(-t) for all t∈ℝ and satisfies: (f1) lim t→0 f(t) |t| p-2 t=0, (f2) lim t→+∞ f(t) |t| p-2 t=l<+∞. In this case, the well-known Ambrosetti-Rabinowitz type condition doesn’t hold, hence it is difficult to verify the classical (PS) c condition. To overcome this difficulty, we use an equivalent version of Cerami’s condition, which allows the more general existence result.
[Show abstract][Hide abstract] ABSTRACT: In this paper, we consider the existence and non-existence of non-trivial solution to a Brezis-Nirenberg type problem with singular weights. First, we obtain a compact imbedding theorem which is an extension of the classical Rellich-Kondrachov compact imbedding theorem, and consider the corresponding eigenvalue problem. Secondly, we deduce a Pohozaev type identity and obtained a non-existence result. Thirdly, based on a generalized concentration compactness principle, we will give some abstract conditions when the functional satisfies the (PS)$_c$ condition. Finally, based on the explicit form of the extremal function, we will obtain some existence results to the problem.
[Show abstract][Hide abstract] ABSTRACT: In this article, using the Mountain Pass Lemma due to Ambrosetti and Rabinowitz, we obtain the existence of nontrivial stationary solutions of Generalized Kadomtsev–Petviashvili (GKP) equation in a bounded domain with smooth boundary and for superlinear nonlinear term f (u) which satisfies some growth condition. Based on a Pohozaev type variational identity for cylindrical symmetric solution, we obtain the nonexistence of the nontrivial cylindrical symmetric solution for super-critical nonlinearity, i.e. f(u)=|u| p −2 u where .
[Show abstract][Hide abstract] ABSTRACT: In this paper, we shall investigate the existence of multiple (positive) weak solutions for the Dirichlet problem of the p-Laplacian with singularity and cylindrical symmetry. The results depends heavily on parameters n,p,q,r,s and λ>0. By the technical decomposition of the associated Nehari manifold into three parts Λ+,Λ− and Λ0, and some compactness condition such as (PS) condition or local (PS) condition ((PS)c condition) at certain level of energy, we obtain two nonnegative minimizers of the energy functional on Λ+ and Λ−, respectively.
[Show abstract][Hide abstract] ABSTRACT: In this paper, using Mountain Pass Lemma and Linking Argument, we prove the existence of nontrivial weak solutions for the Dirichlet problem of the superlinear p-Laplacian equation – -Δ p =:-div(|∇u| p-2 ∇u)=λV(x)|u| p-2 u+f(x,u)inΩ,u=0on∂Ω,(P) where p>1, Ω is a bounded domain of ℝ N , V(x) is a changing sign function belonging to L s , for some s>N p, if 1<p<N, and s=1 if p>N and λ is a real parameter – in the case where the eigenvalue parameter λ∈(0,λ 2 ), λ 2 is the second positive eigenvalue of the p-Laplacian with indefinite weights.
[Show abstract][Hide abstract] ABSTRACT: In this paper, using the Mountain Pass Lemma without (PS) con-dition due to Ambrosetti and Rabinowitz, we obtain the existence of the non-trivial solitary waves of Generalized Kadomtsev-Petviashvili equation in multi-dimensional spaces and for superlinear nonlinear term f (u) which satisfies some growth condition. By the Pohozaev type variational identity, we obtain the nonexistence of the nontrivial solitary waves for power function nonlinear case, i.e. f (u) = up where p 2(2n - 1)/(2n - 3).
[Show abstract][Hide abstract] ABSTRACT: In this paper, we study the existence of multiple solutions to a Caffarelli-Kohn-Nirenberg type equation with asymptotically linear term at infinity. In this case, the well-known Ambrosetti-Rabinowtz type condition doesn't hold, hence it is difficult to verify the classical (PS)$_c$ condition. To overcome this difficulty, we use an equivalent version of Cerami's condition, which allows the more general existence result.
[Show abstract][Hide abstract] ABSTRACT: In this paper, we study the existence of multiple stationary solutions of Generalized Kadomtsev-Petviashvili (Abbr. GKP) equation in a bounded domain with smooth boundary and for superlinear nonlinear term f(u) = ‚jujp¡2u + jujq¡2u where 1 • p;q < 2⁄ = 2(2n¡1) 2n¡3 . Our methods are based on variational methods, and the results are divided into two cases according to the dierent values of the parameters p; q.