Search

In this article, we consider the multiplicity of positive solutions for a static Schrödinger-Poisson-Slater equation of the type −Δu+u2∗1∣4πx∣u=μf(x)∣u∣p−2u+g(x)∣u∣4uinR3,-\Delta u+\left({u}^{2}\ast \frac{1}{| 4\pi x| }\right)u=\mu f\left(x){| u| }^{p-2}u+g\left(x){| u| }^{4}u\hspace{1em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{3}, where μ>0\mu \gt 0, 10\mu \gt 0 small; while gg has kk strict local maximum points, we prove that the equation has at least k+1k+1 distinct positive solutions for μ>0\mu \gt 0 small by the Nehari manifold. Moreover, we show that one of the solutions is a ground state solution.

In this paper, the existence and multiplicity of positive solutions is established for Schrödinger‐Poisson system of the form −Δu+ϕu=u5+λ|x|βuγ,inΩ,−Δϕ=u2,inΩ,u>0,inΩ,u=ϕ=0,on∂Ω,where 0 ∈ Ω is a smooth bounded domain in R3, γ∈(0,1),0≤β<5+γ2, and λ > 0 is a real parameter. Combining with the variational method and Nehari manifold method, two positive solutions of the system are obtained. [ABSTRACT FROM AUTHOR]

In this article, we study the Brezis-Nirenberg result for Kirchhoff-type equation in dimension four. Using the variational method, the existence of positive solutions and the multiplicity of solutions are obtained. It is worth to point out that a global Palais-Smale compactness condition for the critical problem is obtained. Thanks to this global Palais-Smale compactness condition, k pairs of distinct solutions are obtained, where . [ABSTRACT FROM AUTHOR]

In this work, we are interested in considering the following nonlocal problem − a + b ∫ Ω | ∇ u | 2 d x Δ u = μ | u | 2 ∗ − 2 u + λ | u | q − 2 u , x ∈ Ω , u = 0 , x ∈ ∂ Ω , where Ω ⊂ R N ( N ≥ 4 ) is a smooth bounded domain, a ≥ 0 , b > 0 , 1 < q < 2 , μ , λ > 0 and 2 ∗ = 2 N N − 2 is the critical Sobolev exponent. By using the variational method and the critical point theorem, some existence and multiplicity results are obtained. [ABSTRACT FROM AUTHOR]

In this article, we consider a class of superlinear Kirchhoff-type equations with critical growth { − ( a + b ∫ R N ∣ ∇ u ∣ 2 d x ) Δ u = μ u 2 ∗ − 1 + λ k ( x ) u q − 1 , x ∈ R N , u ∈ D 1 , 2 ( R N ) , where λ , μ > 0 , N ≥ 4 , 2 ≤ q < 2 ∗ , 2 ∗ = 2 N N − 2 , a , b ≥ 0 and a + b > 0 , k satisfies some conditions. By using the variational method, the multiplicity of positive solutions and infinitely many pairs of solutions are obtained. [ABSTRACT FROM AUTHOR]

In the paper, by using a new condition, a class of asymptotically periodic Schrödinger equations with critical exponent are studied. Via the variational method, a positive ground state solution is obtained. [ABSTRACT FROM AUTHOR]

In this paper, we study the following nonlinear Kirchhoff-type equation (0.1) { − ( a + b ∫ R N | ∇ u | 2 d x ) △ u = | u | 2 ⁎ − 2 u + μ h ( x ) , x ∈ R N u ∈ D 1 , 2 ( R N ) , where a ≥ 0 , b > 0 , N ≥ 3 , 2 ⁎ = 2 N N − 2 , μ ≥ 0 and h ∈ L 2 ⁎ 2 ⁎ − 1 ( R N ) \ { 0 } is nonnegative. By using the variational method, we obtain the existence of positive solutions for Eq. (0.1) , under some assumptions on a , b , μ . [ABSTRACT FROM AUTHOR]

Abstract In this article, the following p -Laplacian problem with Kirchhoff term and critical exponent is studied − a + b ∫ R N | ∇ u | p d x Δ p u = u p ∗ − 1 + μ h (x) , x ∈ R N , u ∈ D 1 , p (R N) , where a , μ ≥ 0 , b > 0 , 1 < p < N , p ∗ = N p N − p , Δ p u = d i v ( | ∇ u | p − 2 ∇ u) , and h ∈ L p ∗ p ∗ − 1 (R N) is nonzero and nonnegative. When μ = 0 , with the help of the recent result of B. Sciunzi(Adv. Math. 291(2016) 12-23), infinitely many positive solutions are obtained by some analysis techniques. While μ > 0 , by the variational method, some results about positive solutions are obtained. [ABSTRACT FROM AUTHOR]