Sign-changing Solutions Research Articles

Generalized limit theorem and bifurcation for problems with Pucci's operator

We establish a new limiting result which extends the famous Whyburn's limit theorem. As applications, we study the existence and multiplicity of one-sign or sign-changing solutions with a prescribed number of simple zeros for the following problem \begin{equation*} \begin{cases} -\mathcal{M}_{\lambda,\Lambda}^+\left(D^2 u\right)=\mu f(u) &\text{in } \Omega, u=0&\text{on } \partial\Omega, \end{cases} \end{equation*} where $\mathcal{M}_{\lambda,\Lambda}^+$ denotes the Pucci extremal operator. Combining bifurcation approach with our generalized limit theorem, we determine the range of parameter $\mu$ in which the above problem has one or multiple one-sign or sign-changing solutions according to the behaviors of $f$ at $0$ and $\infty$, and whether $f$ satisfies the signum condition $f(s)s> 0$ for $s\neq0$.

Infinitely many sign-changing solutions for modified Kirchhoff-type equations in ℝ3

ABSTRACT Consider a class of modified Kirchhoff-type equations where the nonlinear term f is 4-superlinear at infinity. By using the method of invariant sets of descending flow, the existence of a sign-changing solution is obtained. When f is assumed to be odd, we prove that the above problem admits infinitely many sign-changing solutions. Moreover, when and the nonlinearity of power growth with 1<p, we establish some existence and non-existence results. For , the non-existence result relies on the deduction of some suitable Pohozaev identity. For , using the Nehari–Pohozaev manifold, we prove that the above problem admits a ground state solution.

Resonant Anisotropic (p,q)-Equations

We consider an anisotropic Dirichlet problem which is driven by the (p(z),q(z))-Laplacian (that is, the sum of a p(z)-Laplacian and a q(z)-Laplacian), The reaction (source) term, is a Carathéodory function which asymptotically as x±∞ can be resonant with respect to the principal eigenvalue of (−Δp(z),W01,p(z)(Ω)). First using truncation techniques and the direct method of the calculus of variations, we produce two smooth solutions of constant sign. In fact we show that there exist a smallest positive solution and a biggest negative solution. Then by combining variational tools, with suitable truncation techniques and the theory of critical groups, we show the existence of a nodal (sign changing) solution, located between the two extremal ones.

Sign-Changing Solutions for the One-Dimensional Non-Local sinh-Poisson Equation

Abstract We study the existence of sign-changing solutions for a non-local version of the sinh-Poisson equation on a bounded one-dimensional interval I, under Dirichlet conditions in the exterior of I. This model is strictly related to the mathematical description of galvanic corrosion phenomena for simple electrochemical systems. By means of the finite-dimensional Lyapunov–Schmidt reduction method, we construct bubbling families of solutions developing an arbitrarily prescribed number sign-alternating peaks. With a careful analysis of the limit profile of the solutions, we also show that the number of nodal regions coincides with the number of blow-up points.

Existence of least energy sign-changing solutions to Kirchhoff equation with critical growth

Existence of least energy sign-changing solutions to Kirchhoff equation with critical growth

Nonexistence of Sign-Changing Solutions for Some Elliptic Inequalities in Bounded Domains

Nonlinear elliptic inequalities in bounded domains and systems of such inequalities involving terms depending differently on the positive and negative parts of the sought function are considered. Sufficient conditions for the nonexistence of nontrivial solutions to the inequalities and systems under study in corresponding functional classes are obtained.

Sign-Changing Solutions for Chern–Simons–Schrödinger Equations with Asymptotically 5-Linear Nonlinearity

In this paper, we study the following Chern–Simons–Schrodinger equation $$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle -\Delta u+\omega u+\lambda \Big (\frac{h^{2}(|x|)}{|x|^{2}}+ \int _{|x|}^{+\infty }\frac{h(s)}{s}u^{2}(s)\hbox {d}s\Big )u=g(u) \quad \text{ in }\ {\mathbb {R}}^{2},\\ \displaystyle u\in H_r^1({\mathbb {R}}^{2}), \end{array}\right. } \end{aligned}$$ where $$\omega ,\lambda >0$$ and $$h(s)=\frac{1}{2}\int _{0}^{s}ru^{2}(r)\hbox {d}r$$ . Since the nonlinearity g is asymptotically 5-linear at infinity, there would be a competition between g and the nonlocal term. By constrained minimization arguments and the quantitative deformation lemma, we prove the existence of least energy sign-changing radial solution, which changes sign exactly once. Further, we study the concentration of the least energy sign-changing radial solutions as $$\lambda \rightarrow 0$$ .

Saddle solutions for the Choquard equation II

We study nodal solutions of saddle type for the Choquard equation −Δu+u=(Iα∗|u|p)|u|p−2uinRNwhere N≥3 and 2≤p<N+αN−2. The function Iα refers to the Riesz potential with order α∈(0,N). In a noncompact setting, we construct saddle solutions whose nodal domains are of conical shapes demonstrating polyhedral symmetry configurations in RN. More precisely, when N≥4, for given integers ℓ1,ℓ2≥2, this equation admits saddle solutions with its 4ℓ1ℓ2 nodal domains meeting at the origin. These among other saddle solutions not only reveal the nonlocal feature of the Choquard equation but also further complete the variational framework of constructing sign-changing solutions for the Choquard equation initiated in Ghimenti and Van Schaftingen (2016) and further developed in Xia and Wang (2019).

Existence of Positive and Sign-Changing Solutions to a Coupled Elliptic System with Mixed Nonlinearity Growth

In the present paper, we make a rigorous study of the solitary wave solutions to a coupled Schrodinger system with quadratic and cubic nonlinearity. This kind of system of Schrodinger equations arises from optics theory. First, the existence and nonexistence of nontrivial solutions, respectively, in focusing and defocusing cases are considered. Second, we prove the existence of multiple nontrivial solutions by using the Crandall–Rabinowitz local bifurcation theorems and calculate the exact Morse index of these solutions. Third, the continuous dependence on the parameter and asymptotic behavior of positive ground state solutions in the focusing case are also established. Particularly, from the mathematical point of view, we prove the behavior of positive solution coincides with the physical phenomena of Bang et al. (Opt Lett 22(22):1680–1682, 1997; Phys Rev E (3) 58(4):5057–5069, 1998). Finally, we prove the existence of sign-changing solutions.

Existence of nonradial positive and nodal solutions to a critical Neumann problem in a cone

We study the critical Neumann problem $ \begin{equation*} \begin{cases} -\Delta u = |u|^{2^*-2}u &amp;\text{in }\Sigma_\omega,\\ \quad\frac{\partial u}{\partial\nu} = 0 &amp;\text{on }\partial\Sigma_\omega, \end{cases} \end{equation*} $ in the unbounded cone $\Sigma_\omega: = \{tx:x\in\omega\text{ and }t&gt;0\}$, where $\omega$ is an open connected subset of the unit sphere $\mathbb{S}^{N-1}$ in $\mathbb{R}^N$ with smooth boundary, $N\geq 3$ and $2^*: = \frac{2N}{N-2}$. We assume that some local convexity condition at the boundary of the cone is satisfied. If $\omega$ is symmetric with respect to the north pole of $\mathbb{S}^{N-1}$, we establish the existence of a nonradial sign-changing solution. On the other hand, if the volume of the unitary bounded cone $\Sigma_\omega\cap B_1(0)$ is large enough (but possibly smaller than half the volume of the unit ball $B_1(0)$ in $\mathbb{R}^N$), we establish the existence of a positive nonradial solution.

On finite Morse index solutions of higher order fractional elliptic equations

<p style='text-indent:20px;'>We consider sign-changing solutions of the equation <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ (-{\Delta})^s u+\lambda u = |u|^{p-1}u \; \mbox{in}\; \mathbb R^n, $\end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ n\geq 1 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ \lambda&gt;0 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ p&gt;1 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ 1&lt;s\leq2 $\end{document}</tex-math></inline-formula>. The main goal of this work is to analyze the influence of the linear term <inline-formula><tex-math id="M5">\begin{document}$ \lambda u $\end{document}</tex-math></inline-formula>, in order to classify stable solutions possibly unbounded and sign-changing. We prove Liouville type theorems for stable solutions or solutions which are stable outside a compact set of <inline-formula><tex-math id="M6">\begin{document}$ \mathbb R^n $\end{document}</tex-math></inline-formula>. We first derive a monotonicity formula for our equation. After that, we provide integral estimate from stability which combined with Pohozaev-type identity to obtain nonexistence results in the subcritical case with the restrictive condition <inline-formula><tex-math id="M7">\begin{document}$ |u|_{L^{\infty}( \mathbb R^n)}^{p-1}&lt; \frac{\lambda (p+1) }{2} $\end{document}</tex-math></inline-formula>. The supercritical case needs more involved analysis, motivated by the monotonicity formula, we then reduce the nonexistence of nontrivial entire solutions which are stable outside a compact set of <inline-formula><tex-math id="M8">\begin{document}$ \mathbb R^n $\end{document}</tex-math></inline-formula>. Through this approach we give a complete classification of stable solutions <b>for all <inline-formula><tex-math id="M9">\begin{document}$ p&gt;1 $\end{document}</tex-math></inline-formula></b>. Moreover, for the case <inline-formula><tex-math id="M10">\begin{document}$ 0&lt;s\leq1 $\end{document}</tex-math></inline-formula>, finite Morse index solutions are classified in [<xref ref-type="bibr" rid="b19">19</xref>,<xref ref-type="bibr" rid="b25">25</xref>].

Existence of multiple and sign-changing solutions for a second-order nonlinear functional difference equation with periodic coefficients

In the present paper, we apply the method of invariant sets of descending flow to establish a series of criteria to ensure that a second-order nonlinear functional difference equation with periodic boundary conditions possesses at least one trivial solution and three nontrivial solutions. These nontrivial solutions consist of sign-changing solutions, positive solutions and negative solutions. Moreover, as an application of our theoretical results, an example is elaborated. Our results generalize and improve some existing ones.

Multiple solutions for fractional Schrödinger–Poisson system with critical or supercritical nonlinearity

In this paper, we study the following fractional Schrödinger–Poisson system (−Δ)su+V(x)u+ϕu=f(x,u)+λ|u|p−2u,inR3,(−Δ)tϕ=u2,inR3,where s∈(34,1), t∈(0,1), λ>0 is a real parameter, (−Δ)s is the fractional Laplace operator, p≥2s∗=63−2s. Under some suitable assumptions on V and f, by using Moser iterative method and truncation technique, we prove that the above equation has a ground state solution and a sign-changing solution.

Positive, negative and least energy nodal solutions for Kirchhoff equations in ℝ N

The paper deals with the following Kirchhoff equations: where , is real function satisfying quasicritical growth at infinity, and are positive and continuous functions. Combining Mountain Pass Theorem and compact embeddings in weighted Sobolev spaces, we establish the existence of at least a positive and a negative solution. Moreover, using a quantitative deformation lemma, we prove that the problem possesses one least energy sign-changing solution with two nodal domains. Finally, we show that the energy of is strictly larger than the ground state energy.

Multiplicity of sign-changing solutions for a supercritical nonlinear Schrödinger equation

In this paper, we consider the following supercritical nonlinear Schrödinger equation: (0.1)−Δu+λV(x)u=P(x)|u|p−2u+μ|u|q−2u,inRN,where μ>0, N≥3, 2<q<2∗≤p (2∗=2NN−2), V(x)≥0, P(x)≥0 is a bounded function. By using variational method and truncation method, we prove that (0.1) has at least k pairs of sign-changing solutions for large λ.

Existence of multiple solutions for nonlinear multi-point boundary value problems

In this paper, we study some nonlinear second order multi-point boundary value problems. We first give a lemma about the characteristic values of the corresponding linear operator. Then, by fixed point theorems in the recent existing literature, we obtain the existence of multiple solutions for these nonlinear second order multi-point boundary value problems, including two positive solutions, two negative solutions, and one sign-changing solution.

Infinitely many solutions for a new Kirchhoff-type equation with subcritical exponent

ABSTRACT In this article, we consider the following new Kirchhoff-type problem: where a and b are positive constants, is a bounded domain with boundary , with if , and if N = 1, 2. We show that the problem possesses infinitely many sign-changing solutions by using combination of invariant sets of descent flow and the Ljusternik–Schnirelman type minimax method. And an example for p = 2 is illustrated our results.

Sign-changing solutions of boundary value problems for semilinear $\Delta_{\gamma}$-Laplace equations

In this article, we study the multiplicity of weak solutions to the boundary value problem \begin{cases} -G_\alpha u= g(x,y,u) + f(x,y,u) &amp; \text{in } \Omega, u= 0 &amp; \text{on } \partial \Omega, \end{cases} where \Omega is a bounded domain with smooth boundary in \mathbb{R}^N \ (N \ge 2), \alpha \in \mathbb{N}, g(x,y,\xi), f(x,y,\xi) are Carathéodory functions and G_\alpha is the Grushin operator. We use the lower bounds of eigenvalues and an abstract theory on sign-changing solutions.

The Existence of the Sign-Changing Solutions for the Kirchhoff-Schrödinger-Poisson System in Bounded Domains

In this paper, we study a class of the Kirchhoff-Schrödinger-Poisson system. By using the quantitative deformation lemma and degree theory, the existence result of the least energy sign-changing solution u0 is obtained. Meanwhile, the energy doubling property is proved, that is, we prove that the energy of any sign-changing solution is strictly larger than twice that of the least energy. Moreover, we also get the convergence properties of u0 as the parameters b↘0 and λ↘0.

The Brézis–Nirenberg type problem for the p-Laplacian: infinitely many sign-changing solutions

In this paper we consider the quasilinear critical problem $$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} -\Delta _{p}u=\lambda \left| u\right| ^{q-2}u+\left| u\right| ^{p^{\star }-2}u &{} \mathrm {in}\;\Omega ,\\ u=0 &{} \mathrm {on}\;\partial \Omega , \end{array}\right. \end{aligned}$$where $$\Omega $$ is a bounded domain in $${\mathbb {R}}^{N}$$ with smooth boundary, $$-\Delta _{p}u:=\mathrm {div}(\left| \nabla u\right| ^{p-2}\nabla u)$$ is the p-Laplacian, $$N\ge 3,1 0$$ is a parameter. We investigate the multiplicity of sign-changing solutions to the problem and find the phenomenon depending on the positive solutions. Precisely, we show that the problem admits infinitely many pairs of sign-changing solutions when a positive solution exists. These results complete those obtained in Schechter and Zou (On the Brezis–Nirenberg problem, Arch Rational Mech Anal 197:337–356, 2010) for the cases $$p=q=2$$ and $$N\ge 7$$, and in Azorero and Peral (Existence and nonuniqueness for the p-Laplacian: nonlinear eigenvalues, Commun Partial Differ Equ 12:1389–1430, 1987) for the case of one positive solution. Our approach is based on variational methods combining upper-lower solutions and truncation techniques, and flow invariance arguments.