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, 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 concerned with the existence of nontrivial solutions for the following fourth-order equations of Kirchhoff type {Δ^2u-（a＋b∫R^N｜▽u｜^2dx）Δu＋λv（x）u=f（x,u）,x∈R^N,u∈H^2（R^N）,where a,b are p...This paper is concerned with the existence of nontrivial solutions for the following fourth-order equations of Kirchhoff type {Δ^2u-（a＋b∫R^N｜▽u｜^2dx）Δu＋λv（x）u=f（x,u）,x∈R^N,u∈H^2（R^N）,where a,b are positive constants, λ≥ 1 is a parameter, and the nonlinearity f is either superlinear or sublinear at infinity in u. With the help of the variational methods, we obtain the existence and multiplicity results in the working spaces.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
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>
基金This work was supported by the National Natural Science Foundation of China （Grant No. 11371282, 11571259）.
文摘This paper is concerned with the existence of nontrivial solutions for the following fourth-order equations of Kirchhoff type {Δ^2u-（a＋b∫R^N｜▽u｜^2dx）Δu＋λv（x）u=f（x,u）,x∈R^N,u∈H^2（R^N）,where a,b are positive constants, λ≥ 1 is a parameter, and the nonlinearity f is either superlinear or sublinear at infinity in u. With the help of the variational methods, we obtain the existence and multiplicity results in the working spaces.
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).
