Energy Sign-changing Solution Research Articles

Yamabe systems, optimal partitions and nodal solutions to the Yamabe equation

We give conditions for the existence of regular optimal partitions, with an arbitrary number \ell\geq 2 of components, for the Yamabe equation on a closed Riemannian manifold (M,g) . To this aim, we study a weakly coupled competitive elliptic system of \ell equations, related to the Yamabe equation. We show that this system has a least energy solution with nontrivial components if \dim M\geq 10 , (M,g) is not locally conformally flat, and satisfies an additional geometric assumption whenever \dim M=10 . Moreover, we show that the limit profiles of the components of the solution separate spatially as the competition parameter goes to -\infty , giving rise to an optimal partition. We show that this partition exhausts the whole manifold, and we prove the regularity of both the interfaces and the limit profiles, together with a free boundary condition. For \ell=2 the optimal partition obtained yields a least energy sign-changing solution to the Yamabe equation with precisely two nodal domains.

Least Energy Sign-Changing Solution for N-Kirchhoff Problems with Logarithmic and Exponential Nonlinearities

Least Energy Sign-Changing Solution for N-Kirchhoff Problems with Logarithmic and Exponential Nonlinearities

Sign-changing solutions for Kirchhoff-type variable-order fractional Laplacian problems

In this paper, we are concerned with the Kirchhoff-type variable-order fractional Laplacian problems involving critical exponents and logarithmic nonlinearity. By using the constraint variational method, we show the existence of one least energy sign-changing solution. Moreover, we show that this energy is strictly larger than twice the ground energy.

Existence and convergence of the least energy sign-changing solutions for nonlinear Kirchhoff equations on locally finite graphs

In this paper, we study the least energy sign-changing solutions to the following nonlinear Kirchhoff equation − ( a + b ∫ V | ∇ u | 2 d μ ) Δ u + c ( x ) u = f ( u ) on a locally finite graph G = ( V , E ), where a, b are positive constants. We use the constrained variational method to prove the existence of a least energy sign-changing solution u b of the above equation if c ( x ) and f satisfy certain assumptions, and to show the energy of u b is strictly larger than twice that of the least energy solutions. Moreover, if we regard b as a parameter, as b → 0 + , the solution u b converges to a least energy sign-changing solution of a local equation − a Δ u + c ( x ) u = f ( u ).

Convergence of least energy sign-changing solutions for logarithmic Schrödinger equations on locally finite graphs

In this paper, we study the following logarithmic Schrödinger equation −Δu+λa(x)u=ulogu2inVon a connected locally finite graph G=(V,E), where Δ denotes the graph Laplacian, λ>0 is a constant, and a(x)≥0 represents the potential. Using variational techniques in combination with the Nehari manifold method based on directional derivative, we can prove that, there exists a constant λ0>0 such that for all λ≥λ0, the above problem admits a least energy sign-changing solution uλ. Moreover, as λ→+∞, we prove that the solution uλ converges to a least energy sign-changing solution of the following Dirichlet problem −Δu=ulogu2inΩ,u(x)=0on∂Ω,where Ω={x∈V:a(x)=0} is the potential well.

Existence and multiplicity of sign-changing solutions for quasilinear Schrödinger equations with sub-cubic nonlinearity

In this paper, we consider the quasilinear Schrödinger equation−Δu+V(x)u−uΔ(u2)=g(u),x∈R3, where V and g are continuous functions. Without the coercive condition on V or the monotonicity condition on g, we show that the problem above has a least energy sign-changing solution and infinitely many sign-changing solutions. Our results especially solve the problem above in the case where g(u)=|u|p−2u (2<p<4) and complete some recent related works on sign-changing solutions, in the sense that, in the literature only the case g(u)=|u|p−2u (p≥4) was considered. The main results in the present paper are obtained by a new perturbation approach and the method of invariant sets of descending flow. In addition, in some cases where the functional merely satisfies the Cerami condition, a deformation lemma under the Cerami condition is developed.

1 / 2 -Laplacian problem with logarithmic and exponential nonlinearities

In this paper, based on a suitable fractional Trudinger–Moser inequality, we establish sufficient conditions for the existence result of least energy sign-changing solution for a class of one-dimensional nonlocal equations involving logarithmic and exponential nonlinearities. By using a main tool of constrained minimization in Nehari manifold and a quantitative deformation lemma, we consider both subcritical and critical exponential growths. This work can be regarded as the complement for some results of the literature.

The nodal solution for a problem involving the logarithmic and exponential nonlinearities

ABSTRACT In this work, we study the existence of the least energy sign-changing solution for the following degenerate Kirchhoff-type problem involving the fractional N/s-Laplacian with logarithmic and both subcritical and critical exponential nonlinearities: { M ( ∫ R 2 N | u ( x ) − u ( y ) | N s | x − y | 2 N d x d y ) ( − Δ ) N / s s u = | u | q − 2 uln ⁡ | u | 2 + μf ( u ) , in Ω , u = 0 , in R N ∖ Ω , where Ω ⊂ R N is a bounded domain. The proof is based on constrained variational method, fractional Trudinger–Moser inequality, quantitative deformation lemma and Brouwer's degree theory in Nehari sets. To be more precise, the least energy sign-changing solution is obtained by minimizing the energy functional on the sign-changing Nehari manifold.

Existence of least energy sign-changing solution for a class of fractional p&q-Laplacian problems with potentials vanishing at infinity

In this paper, we consider the fractional p &q -Laplacian equation: ( − Δ ) p s u + ( − Δ ) q s u + V ( x ) ( | u | p − 2 u + | u | q − 2 u ) = K ( x ) f ( u ) in R N , where s ∈ ( 0 , 1 ) , 1 < p < q < N s , and ( − Δ ) t s with t ∈ { p , q } is the fractional t-Laplacian operator, f is a C 1 real function and V, K are continuous, positive functions. By using constrained variational methods, a quantitative Deformation Lemma and Brouwer degree theory, we prove the existence of a least energy sign-changing solution.

Nodal Solutions for Quasilinear Schrödinger Equations with Asymptotically 3-Linear Nonlinearity

In this paper, we are concerned with the quasilinear Schr\"{o}dinger equation \begin{equation*} -\Delta u+V(x)u-u\Delta(u^2)=g(u),\ \ x\in \mathbb{R}^{N}, \end{equation*} where $N\geq3$, $V$ is radially symmetric and nonnegative, and $g$ is asymptotically 3-linear at infinity. In the case of $\inf_{\mathbb{R}^N}V>0$, we show the existence of a least energy sign-changing solution with exactly one node, and for any integer $k>0$, there are a pair of sign-changing solutions with $k$ nodes. Moreover, in the case of $\inf_{\mathbb{R}^N}V=0$, the problem above admits a least energy sign-changing solution with exactly one node. The proof is based on variational methods. In particular, some new tricks and the method of sign-changing Nehari manifold depending on a suitable restricted set are introduced to overcome the difficulty resulting from the appearance of asymptotically 3-linear nonlinearities.

Least energy sign-changing solutions for Schrödinger-Poisson systems with potential well

Abstract In this article, we investigate the existence of least energy sign-changing solutions for the following Schrödinger-Poisson system − Δ u + V ( x ) u + K ( x ) ϕ u = f ( u ) , x ∈ R 3 , − Δ ϕ = K ( x ) u 2 , x ∈ R 3 , \left\{\begin{array}{ll}-\Delta u+V\left(x)u+K\left(x)\phi u=f\left(u),\hspace{1.0em}&amp; x\in {{\mathbb{R}}}^{3},\\ -\Delta \phi =K\left(x){u}^{2},\hspace{1.0em}&amp; x\in {{\mathbb{R}}}^{3},\\ \hspace{1.0em}\end{array}\right. where the functions V ( x ) , K ( x ) V\left(x),K\left(x) have finite limits as ∣ x ∣ → ∞ | x| \to \infty satisfying some mild assumptions. By combining variational methods with the global compactness lemma, we obtain a least energy sign-changing solution with exactly two nodal domains, and its energy is strictly larger than twice that of least energy solutions.

Least energy sign-changing solutions for Kirchhoff-type problems with potential well

In this paper, we investigate the existence of least energy sign-changing solutions for the Kirchhoff-type problem −a+b∫R3|∇u|2dxΔu+V(x)u=f(u),x∈R3, where a, b &amp;gt; 0 are parameters, V∈C(R3,R), and f∈C(R,R). Under weaker assumptions on V and f, by using variational methods with the aid of a new version of global compactness lemma, we prove that this problem has a least energy sign-changing solution with exactly two nodal domains, and its energy is strictly larger than twice that of least energy solutions.

Least energy positive solutions of critical Schrödinger systems with mixed competition and cooperation terms: The higher dimensional case

Let Ω⊂RN be a smooth bounded domain. In this paper we investigate the existence of least energy positive solutions to the following Schrödinger system with d≥2 equations−Δui+λiui=|ui|p−2ui∑j=1dβij|uj|p in Ω,ui=0 on ∂Ω,i=1,...,d, in the case of a critical exponent 2p=2⁎=2NN−2 in high dimensions N≥5. We treat the focusing case (βii>0 for every i) in the variational setting βij=βji for every i≠j, dealing with a Brézis-Nirenberg type problem: −λ1(Ω)<λi<0, where λ1(Ω) is the first eigenvalue of (−Δ,H01(Ω)). We provide several sufficient conditions on the coefficients βij that ensure the existence of least energy positive solutions; these include the situations of pure cooperation (βij>0 for every i≠j), pure competition (βij≤0 for every i≠j) and coexistence of both cooperation and competition coefficients.In order to provide these results, we firstly establish the existence of nonnegative solutions with 1≤m≤d nontrivial components by comparing energy levels of the system with those of appropriate sub-systems. Moreover, based on new energy estimates, we obtain the precise asymptotic behaviors of the nonnegative solutions in some situations which include an analysis of a phase separation phenomena. Some proofs depend heavily on the fact that 1<p<2, revealing some different phenomena comparing to the special case N=4. Our results provide a rather complete picture in the particular situation where the components are divided in two groups. Besides, based on the results about the phase separation, we prove the existence of least energy sign-changing solution to the Brézis-Nirenberg problem−Δu+λu=μ|u|2⁎−2u,u∈H01(Ω), for μ>0, −λ1(Ω)<λ<0 for all N≥4, a result which is new in dimensions N=4,5.

On fractional logarithmic Schrödinger equations

Abstract We study the following fractional logarithmic Schrödinger equation: ( − Δ ) s u + V ( x ) u = u log u 2 , x ∈ R N , {\left(-\Delta )}^{s}u+V\left(x)u=u\log {u}^{2},\hspace{1em}x\in {{\mathbb{R}}}^{N}, where N ≥ 1 N\ge 1 , ( − Δ ) s {\left(-\Delta )}^{s} denotes the fractional Laplace operator, 0 &lt; s &lt; 1 0\lt s\lt 1 and V ( x ) ∈ C ( R N ) V\left(x)\in {\mathcal{C}}\left({{\mathbb{R}}}^{N}) . Under different assumptions on the potential V ( x ) V\left(x) , we prove the existence of positive ground state solution and least energy sign-changing solution for the equation. It is known that the corresponding variational functional is not well defined in H s ( R N ) {H}^{s}\left({{\mathbb{R}}}^{N}) , and inspired by Cazenave (Stable solutions of the logarithmic Schrödinger equation, Nonlinear Anal. 7 (1983), 1127–1140), we first prove that the variational functional is well defined in a subspace of H s ( R N ) {H}^{s}\left({{\mathbb{R}}}^{N}) . Then, by using minimization method and Lions’ concentration-compactness principle, we prove that the existence results.

Least energy sign-changing solutions of fractional Kirchhoff–Schrödinger–Poisson system with critical and logarithmic nonlinearity

In the present paper, we deal with the following fractional Kirchhoff–Schrödinger–Poisson system with logarithmic and critical nonlinearity: where and Ω is a bounded domain in with Lipschitz boundary. Combining constraint variational methods, topological degree theory and quantitative deformation arguments, we prove that the above problem has a least energy sign-changing solution . Moreover, we show that the energy of is strictly larger than two times the ground state energy. Finally, we regard b as a parameter and show the convergence property of as .

Least energy sign-changing solutions for a class of fractional Kirchhoff–Poisson system

Using the constraint variational method and a quantitative deformation lemma, we establish the existence of the least energy sign-changing solutions for the fractional Kirchhoff–Poisson system, a+b∫R3|(−Δ)s2u|2dx(−Δ)su+V(x)u+ϕ(x)u=f(x,u),(−Δ)tϕ=u2,x∈R3, where a &amp;gt; 0 is a constant, b∈R+ is a parameter, s, t ∈ (0, 1) and 4s + 2t &amp;gt; 3, (−Δ)s stands for the fractional Laplacian, V is a continuous, positive function, and f is nonlinear function satisfying suitable growth assumptions. Moreover, for any b &amp;gt; 0, we prove that the energy of the least energy sign-changing solution is strictly larger than twice the ground state energy. Furthermore, we show a convergence property of the least energy sign-changing solutions as the parameter b goes to zero. Our results complement an in-depth study of Wang, Radulescu, and Zhang [J. Math. Phys. 60, 011506 (2019)] in the sense that we are concerned with the nodal characteristics of the ground states.

Least energy sign-changing solution for N-Laplacian problem with logarithmic and exponential nonlinearities

In this paper, we establish sufficient conditions for the existence result of least energy sign-changing weak solution to the following quasilinear nonhomogeneous N-Laplacian equation with logarithmic and exponential nonlinearities:{−ΔNu=|u|p−2uln⁡|u|2+μf(u),inΩ,u=0,on∂Ω, where f(t) behaves like eα|t|N/(N−1). By using a main tool of constrained minimization in Nehari manifold, we consider both subcritical and critical exponential growths.

Least energy sign-changing solutions for a class of Schrödinger–Poisson system on bounded domains

This paper is concerned with the Schrödinger–Poisson system −Δu + ϕu = λu + μ|u|2u and −Δϕ = u2 setting on a bounded domain Ω⊂R3 with smooth boundary and λ,μ∈R being parameters. By using variational techniques in combination with the nodal Nehari manifold method, we show the existence of μ̄&amp;gt;0 such that for all (λ,μ)∈(−∞,λ1)×(μ̄,+∞), the above system has one least energy sign-changing solution, where λ1 &amp;gt; 0 is the first eigenvalue of −Δ,H01(Ω). The results of this paper are complementary to those in Alves and Souto [Z. Angew. Math. Phys. 65, 1153–1166 (2014)].

The Existence of Least Energy Sign-Changing Solution for Kirchhoff-Type Problem with Potential Vanishing at Infinity

In this paper, we study the Kirchhoff-type equation: − a + b ∫ ℝ 3 ∇ u 2 d x Δ u + V x u = Q x f u , in ℝ 3 , where a , b &gt; 0 , f ∈ C 1 ℝ 3 , ℝ , and V , Q ∈ C 1 ℝ 3 , ℝ + . V x and Q x are vanishing at infinity. With the aid of the quantitative deformation lemma and constraint variational method, we prove the existence of a sign-changing solution u to the above equation. Moreover, we obtain that the sign-changing solution u has exactly two nodal domains. Our results can be seen as an improvement of the previous literature.

Sign-changing solutions for Schr\xf6dinger\u2013Kirchhoff-type fourth-order equation with potential vanishing at infinity

The purpose of this paper is to study the existence of sign-changing solution to the following fourth-order equation: 0.1Δ2u−(a+b∫RN|∇u|2dx)Δu+V(x)u=K(x)f(u)in RN,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\Delta ^{2}u- \\biggl(a+ b \\int _{\\mathbb{R}^{N}} \\vert \\nabla u \\vert ^{2}\\,dx \\biggr) \\Delta u+V(x)u=K(x)f(u) \\quad\\text{in } \\mathbb{R}^{N}, $$\\end{document} where 5leq Nleq 7, Delta ^{2} denotes the biharmonic operator, K(x), V(x) are positive continuous functions which vanish at infinity, and f(u) is only a continuous function. We prove that the equation has a least energy sign-changing solution by the minimization argument on the sign-changing Nehari manifold. If, additionally, f is an odd function, we obtain that equation has infinitely many nontrivial solutions.