Sign-changing Solutions Research Articles

Ground state sign-changing solution for a logarithmic Kirchhoff-type equation in R 3

We investigate the following logarithmic Kirchhoff-type equation: ( a + b ∫ R 3 | ∇ u | 2 + V ( x ) u 2 d x ) [ − Δ u + V ( x ) u ] = | u | p − 2 u ln ⁡ | u | , x ∈ R 3 , where a , b > 0 are constants, 4 < p < 2 ∗ = 6 . Under some appropriate hypotheses on the potential function V , we prove the existence of a positive ground state solution, a ground state sign-changing solution and a sequence of solutions by using the constraint variational methods, topological degree theory, quantitative deformation lemma and symmetric mountain pass theorem. Our results complete those of Gao et al. [Appl. Math. Lett. 139(2023), 108539] with the case of 4 < p < 6 .

Existence and asymptotic behavior for ground state sign-changing solutions of fractional Schrödinger-Poisson system with steep potential well

<abstract><p>In this paper, we investigate the existence of ground state sign-changing solutions for the following fractional Schrödinger-Poisson system</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation} \begin{cases} (-\Delta)^s u+V_{\lambda} (x)u+\mu\phi u = f(u), & \; \mathrm{in}\; \; \mathbb{R}^3, \\ (-\Delta)^t \phi = u^2, & \; \mathrm{in}\; \; \mathbb{R}^3, \end{cases} \nonumber \end{equation} $\end{document} </tex-math></disp-formula></p> <p>where $ \mu > 0, s\in(\frac{3}{4}, 1), t\in(0, 1) $ and $ V_{\lambda}(x) $ = $ \lambda V(x)+1 $ with $ \lambda > 0 $. Under suitable conditions on $ f $ and $ V $, by using the constraint variational method and quantitative deformation lemma, if $ \lambda > 0 $ is large enough, we prove that the above problem has one least energy sign-changing solution. Moreover, for any $ \mu > 0 $, the least energy of the sign-changing solution is strictly more than twice of the energy of the ground state solution. In addition, we discuss the asymptotic behavior of ground state sign-changing solutions as $ \lambda\rightarrow \infty $ and $ \mu\rightarrow0 $.</p></abstract>

Existence of sign-changing solutions for the Hénon type parabolic equation with singular initial data

Existence of sign-changing solutions for the Hénon type parabolic equation with singular initial data

New sign-changing solutions for the 2D Lane-Emden problem with large exponents

New sign-changing solutions for the 2D Lane-Emden problem with large exponents

Interior Holder Regularity and Infinite Time Extinction to a doubly Nonlinear Degenerate Parabolic Equations of Signed Solutions

We study doubly nonlinear parabolic equation with sign changing solutions. We established the Holder regularity and the infinite time extinction of it within a parabolic domain. GANIT J. Bangladesh Math. Soc. 43.1 (2023) 75- 91

Ground State Sign-Changing Solutions for a Schrödinger–Poisson System with Steep Potential Well and Critical Growth

Ground State Sign-Changing Solutions for a Schrödinger–Poisson System with Steep Potential Well and Critical Growth

Interior regularity to the signed solution of the singular doubly nonlinear parabolic equations

We study doubly nonlinear parabolic equation with sign changing solutions. We establish the Hölder regularity of the singular parabolic equations within a parabolic domain. J. Bangladesh Acad. Sci. 47(2); 241-248: December 2023

Multiple solutions for the logarithmic fractional Kirchhoff equation with critical or supercritical nonlinearity

In this paper, we study the following logarithmic fractional Kirchhoff equation: ( a + b ∫ R 3 | ( − Δ ) s / 2 u | 2 d x ) ( − Δ ) s u + V ( x ) u = | u | p − 2 u log ⁡ u 2 + λ | u | q − 2 u , in R 3 , where a, b>0, ( − Δ ) s is the fractional Laplace operator with q ≥ 2 s ∗ = 6 3 − 2 s and V satisfies suitable conditions. Since the presence of supercritical exponents leads to a lack of compactness, we introduce a truncation function to reduce the exponents, allowing compactness to be recovered. We show that the above equation has a ground state solution and a sign-changing solution via the constraint variation method, the quantitative deformation lemma and Moser's iterative method.

Least energy sign-changing solutions for discrete Kirchhoff-type problems

In this paper, by variational technique coupled with quantitative lemma, we not only investigate the existence of least energy solutions, but also point out that the energy of the least energy sign-changing solutions is strictly larger than twice that of the ground state solutions.

Signed and Sign-Changing Solutions for a Kirchhoff-Type Problem Involving the Weighted N-Laplacian with Critical Double Exponential Growth

Signed and Sign-Changing Solutions for a Kirchhoff-Type Problem Involving the Weighted N-Laplacian with Critical Double Exponential Growth

Sign-changing solutions for a weighted Schrödinger–Kirchhoff equation with double exponential nonlinearities growth

This work is devoted to the existence of least energy nodal solutions for a nonlocal weighted Schrödinger–Kirchhoff problem, under boundary Dirichlet condition in the unit ball [Formula: see text] of [Formula: see text]. The nonlinearity of the equation is assumed to have double exponential growth in view of Trudinger–Moser type inequalities. By using the constrained minimization in Nehari set, the quantitative deformation Lemma and degree theory results, we prove our existence result.

Least energy sign-changing solutions for fractional critical Kirchhoff–Schrödinger–Poisson with steep potential well

Least energy sign-changing solutions for fractional critical Kirchhoff–Schrödinger–Poisson with steep potential well

Sign-changing solutions for Kirchhoff weighted equations under double exponential nonlinearities growth

Sign-changing solutions for Kirchhoff weighted equations under double exponential nonlinearities growth

Sign-changing solutions to critical Schrödinger equation with Hartree-type nonlinearity

Sign-changing solutions to critical Schrödinger equation with Hartree-type nonlinearity

Sign-changing solution for a generalized Kirchhoff problem in the fractional Orlicz-Sobolev space with nonsmooth nonlinearity

In this paper, we study a nonlocal generalized Kirchhoff problem driven by the fractional Orlicz g-Laplace operator and involving a nonsmooth nonlinearity. Although this problem has a double nonlocality, we prove the existence of a least-energy nodal (or sign-changing) solution. Note that the tools used are minimization argument and quantitative deformation lemma.

Existence of Signed and Sign-Changing Solutions for Weighted Kirchhoff Problems with Critical Exponential Growth

Existence of Signed and Sign-Changing Solutions for Weighted Kirchhoff Problems with Critical Exponential Growth

A Priori Estimates for Finite-Energy Sign-Changing Blowing-Up Solutions of Critical Elliptic Equations

Abstract We prove sharp pointwise blow-up estimates for finite-energy sign-changing solutions of critical equations of Schrödinger–Yamabe type on a closed Riemannian manifold $(M,g)$ of dimension $n \ge 3$. This is a generalisation of the so-called $C^{0}$-theory for positive solutions of Schrödinger–Yamabe-type equations. To deal with the sign-changing case, we develop a method of proof that combines an a priori bubble-tree analysis with a finite-dimensional reduction, and reduces the proof to obtaining sharp a priori blow-up estimates for a linear problem.

Sign-Changing Solutions for Discrete Schrödinger Equations with Asymptotically Linear Term

Sign-Changing Solutions for Discrete Schrödinger Equations with Asymptotically Linear Term

Sign-changing solution for an overdetermined elliptic problem on unbounded domain

Abstract We prove the existence of two smooth families of unbounded domains in ℝ N + 1 {\mathbb{R}^{N+1}} with N ≥ 1 {N\geq 1} such that { - Δ ⁢ u = λ ⁢ u in ⁢ Ω , u = 0 on ⁢ ∂ ⁡ Ω , ∂ ν ⁡ u = const on ⁢ ∂ ⁡ Ω , \left\{\begin{aligned} \displaystyle{}{-}\Delta u&\displaystyle=\lambda u&&% \displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle u&\displaystyle=0&&\displaystyle\phantom{}\text{on }\partial% \Omega,\\ \displaystyle\partial_{\nu}u&\displaystyle=\mathrm{const}&&\displaystyle% \phantom{}\text{on }\partial\Omega,\end{aligned}\right. admits a sign-changing solution. The domains bifurcate from the straight cylinder B 1 × ℝ {B_{1}\times\mathbb{R}} , where B 1 {B_{1}} is the unit ball in ℝ N {\mathbb{R}^{N}} . These results can be regarded as counterexamples to the Berenstein Conjecture on unbounded domain. Unlike most previous papers in this direction, a very delicate issue here is that there may be two-dimensional kernel space at some bifurcation point. Thus a Crandall–Rabinowitz-type bifurcation theorem from high-dimensional kernel space is also established to achieve the goal.

Sign changing solutions for critical double phase problems with variable exponent

In this paper, we deal with a double phase problem with variable exponent and a right-hand side consisting of a Carathéodory perturbation defined only locally and of a critical term. We stress that the presence of the critical term inhibits the possibility to apply results of the critical point theory to the corresponding energy functional. Instead, we use suitable cut-off functions and truncation techniques in order to work with a coercive functional. Then, using variational tools and an appropriate auxiliary coercive problem, we can produce a sequence of sign changing solutions to our main problem converging to 0 in L^{\infty} and in the Musielakk–Orlicz Sobolev space.