In this paper, we concern with the following fourth order elliptic equations of Kirchhoff type {Δ^2u-（a＋bfR^3｜↓△u｜^2dx）△u＋V（x）u=f（x,u）,x∈R^3, u∈H^2（R3）,where a, b 〉 0 are constants and the primitive...In this paper, we concern with the following fourth order elliptic equations of Kirchhoff type {Δ^2u-（a＋bfR^3｜↓△u｜^2dx）△u＋V（x）u=f（x,u）,x∈R^3, u∈H^2（R3）,where a, b 〉 0 are constants and the primitive of the nonlinearity f is of superlinear growth near infinity in u and is also allowed to be sign-changing. By using variational methods, we establish the existence and multiplicity of solutions. Our conditions weaken the Ambrosetti- Rabinowitz type condition.展开更多
In this paper, we consider the following fourth-order equation of Kirchhoff type<br /> <p> <img src="Edit_bcc9844d-7cbc-494d-90c4-d75364de5658.bmp" alt="" /> </p> <p> ...In this paper, we consider the following fourth-order equation of Kirchhoff type<br /> <p> <img src="Edit_bcc9844d-7cbc-494d-90c4-d75364de5658.bmp" alt="" /> </p> <p> where <i>a</i>, <i>b</i> > 0 are constants, 3 < <i>p</i> < 5, <i>V</i> ∈ <i>C</i> (R<sup>3</sup>, R);Δ<sup>2</sup>: = Δ (Δ) is the biharmonic operator. By using Symmetric Mountain Pass Theorem and variational methods, we prove that the above equation admits infinitely many high energy solutions under some sufficient assumptions on <i>V</i> (<i>x</i>). We make some assumptions on the potential <i>V</i> (<i>x</i>) to solve the difficulty of lack of compactness of the Sobolev embedding. Our results improve some related results in the literature. </p>展开更多
This paper is devoted to studying the existence of solutions for the following logarithmic Schrödinger problem: −div(a(x)∇u)+V(x)u=ulogu2+k(x)| u |q1−2u+h(x)| u |q2−2u, x∈ℝN.(1)We first prove that the correspon...This paper is devoted to studying the existence of solutions for the following logarithmic Schrödinger problem: −div(a(x)∇u)+V(x)u=ulogu2+k(x)| u |q1−2u+h(x)| u |q2−2u, x∈ℝN.(1)We first prove that the corresponding functional I belongs to C1(HV1(ℝN),ℝ). Furthermore, by using the variational method, we prove the existence of a sigh-changing solution to problem (1).展开更多
In this paper,a class of Kirchhoff type equations in R^(N)(N≥3)with zero mass and a critical term is studied.Under some appropriate conditions,the existence of multiple solutions is obtained by using variational meth...In this paper,a class of Kirchhoff type equations in R^(N)(N≥3)with zero mass and a critical term is studied.Under some appropriate conditions,the existence of multiple solutions is obtained by using variational methods and a variant of Symmetric Mountain Pass theorem.The Second Concentration Compactness lemma is used to overcome the lack of compactness in critical problem.Compared to the usual Kirchhoff-type problems,we only require the nonlinearity to satisfy the classical superquadratic condition(Ambrosetti-Rabinowitz condition).展开更多
基金supported by Natural Science Foundation of China(11271372)Hunan Provincial Natural Science Foundation of China(12JJ2004)
文摘In this paper, we concern with the following fourth order elliptic equations of Kirchhoff type {Δ^2u-（a＋bfR^3｜↓△u｜^2dx）△u＋V（x）u=f（x,u）,x∈R^3, u∈H^2（R3）,where a, b 〉 0 are constants and the primitive of the nonlinearity f is of superlinear growth near infinity in u and is also allowed to be sign-changing. By using variational methods, we establish the existence and multiplicity of solutions. Our conditions weaken the Ambrosetti- Rabinowitz type condition.
文摘In this paper, we consider the following fourth-order equation of Kirchhoff type<br /> <p> <img src="Edit_bcc9844d-7cbc-494d-90c4-d75364de5658.bmp" alt="" /> </p> <p> where <i>a</i>, <i>b</i> > 0 are constants, 3 < <i>p</i> < 5, <i>V</i> ∈ <i>C</i> (R<sup>3</sup>, R);Δ<sup>2</sup>: = Δ (Δ) is the biharmonic operator. By using Symmetric Mountain Pass Theorem and variational methods, we prove that the above equation admits infinitely many high energy solutions under some sufficient assumptions on <i>V</i> (<i>x</i>). We make some assumptions on the potential <i>V</i> (<i>x</i>) to solve the difficulty of lack of compactness of the Sobolev embedding. Our results improve some related results in the literature. </p>
文摘This paper is devoted to studying the existence of solutions for the following logarithmic Schrödinger problem: −div(a(x)∇u)+V(x)u=ulogu2+k(x)| u |q1−2u+h(x)| u |q2−2u, x∈ℝN.(1)We first prove that the corresponding functional I belongs to C1(HV1(ℝN),ℝ). Furthermore, by using the variational method, we prove the existence of a sigh-changing solution to problem (1).
基金This work was supported by the National Natural Science Foundation of China(Grant Nos.11701346,11671239,11801338)the Natural Science Foundation of Shanxi Province(Grant No.201801D211001)+1 种基金the Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi(Grant No.2019L0024)the Research Project Supported by Shanxi Scholarship Council of China(Grant No.2020-005).
文摘In this paper,a class of Kirchhoff type equations in R^(N)(N≥3)with zero mass and a critical term is studied.Under some appropriate conditions,the existence of multiple solutions is obtained by using variational methods and a variant of Symmetric Mountain Pass theorem.The Second Concentration Compactness lemma is used to overcome the lack of compactness in critical problem.Compared to the usual Kirchhoff-type problems,we only require the nonlinearity to satisfy the classical superquadratic condition(Ambrosetti-Rabinowitz condition).
