Multiple nodal solutions for the Schr\”odinger-Poisson system with an asymptotically cubic term

This paper deals with the following Schr\“odinger-Poisson system \begin{equation}\left\{\begin{aligned} &-\Delta u+u+ \lambda\phi u=f(u)\quad\mbox{in }\mathbb{R}^3,\\ &-\Delta \phi=u^{2}\quad\mbox{in }\mathbb{R}^3, \end{aligned}\right.\end{equation} where $\lambda>0$ and $f(u)$ is a nonlinear term asymptotically cubic at the infinity. Taking advantage of the Miranda theorem and deformation lemma, we combine some new analytic techniques to prove that for each positive integer $k,$ system \eqref{zhaiyaofc} admits a radial nodal solution $U_k^{\lambda}$, which has exactly $k+1$ nodal domains and the corresponding energy is strictly increasing in $k$. Moreover, for any sequence $\{\lambda_n\}\to 0_+$ as $n\to\infty,$ up to a subsequence, $U_k^{\lambda_n}$ converges to some $U_k^0\in H_r^1(\mathbb{R}^3)$, which is a radial nodal solution with exactly $k+1$ nodal domains of \eqref{zhaiyaofc} for $\lambda=0 $. These results give an affirmative answer to the open problem proposed in [Kim S, Seok J. Commun. Contemp. Math., 2012] for the Schr\”odinger-Poisson system with an asymptotically cubic term.

On solutions for a class of fractional Kirchhoff-type problems with Trudinger–Moser nonlinearity

In this paper, we prove the existence of at least three nontrivial solutions for the following class of fractional Kirchhoff-type problems: [Formula: see text] where [Formula: see text] is a constant, [Formula: see text] is a bounded open interval, [Formula: see text] is a continuous potential, the nonlinear term [Formula: see text] has exponential growth of Trudinger–Moser type, [Formula: see text] and [Formula: see text] denotes the standard Gagliardo seminorm of the fractional Sobolev space [Formula: see text]. More precisely, by exploring a minimization argument and the quantitative deformation lemma, we establish the existence of a nodal (or sign-changing) solution and by means of the Mountain Pass Theorem, we get one nonpositive and one nonnegative ground state solution. Moreover, we show that the energy of the nodal solution is strictly larger than twice the ground state level. When we regard [Formula: see text] as a positive parameter, we study the behavior of the nodal solutions as [Formula: see text].

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 > 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.

Ground state sign-changing solutions and infinitely many solutions for fractional logarithmic Schrödinger equations in bounded domains

We consider a class of fractional logarithmic Schrödinger equation in bounded domains. First, by means of the constraint variational method, quantitative deformation lemma and some new inequalities, the positive ground state solutions and ground state sign-changing solutions are obtained. These inequalities are derived from the special properties of fractional logarithmic equations and are critical for us to obtain our main results. Moreover, we show that the energy of any sign-changing solution is strictly larger than twice the ground state energy. Finally, we obtain that the equation has infinitely many nontrivial solutions. Our result complements the existing ones to fractional Schrödinger problems when the nonlinearity is sign-changing and satisfies neither the monotonicity condition nor Ambrosetti-Rabinowitz condition.

On a class of Kirchhoff equations involving an anisotropic operator and potential

In this work, we are concerned with a class of fractional equations of Kirchhoff type with potential. Using variational methods and a variant of quantitative deformation lemma, we prove the existence of a least energy sign-changing solution. Moreover, the existence of infinitely many solution is established.

Nodal solutions for double phase Kirchhoff problems with vanishing potentials

We consider the following ( p , q )-Laplacian Kirchhoff type problem − ( a + b ∫ R 3 | ∇ u | p d x ) Δ p u − ( c + d ∫ R 3 | ∇ u | q d x ) Δ q u + V ( x ) ( | u | p − 2 u + | u | q − 2 u ) = K ( x ) f ( u ) in R 3 , where a , b , c , d > 0 are constants, 3 2 < p < q < 3, V : R 3 → R and K : R 3 → R are positive continuous functions allowed for vanishing behavior at infinity, and f is a continuous function with quasicritical growth. Using a minimization argument and a quantitative deformation lemma we establish the existence of nodal 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 Existence Result for a Class of p-Kirchhoff-Type Problem with a Multilinear Growth Nonlinearity

In this paper, we firstly discuss the existence of the least energy sign-changing solutions for a class of p-Kirchhoff-type problems with a (2p-1)-linear growth nonlinearity. The quantitative deformation lemma and Non-Nehari manifold method are used in the paper to prove the main results. Remarkably, we use a new method to verify that Mb≠∅. The main results of our paper are the existence of the least energy sign-changing solution and its corresponding energy doubling property. Moreover, we also give the convergence property of the least energy sign-changing solution as the parameter b↘0.

Sign-Changing Solutions for a Class of Zero Mass Nonlocal Schrödinger Equations

We consider the following class of fractional Schrödinger equations: (-\Delta)^{\alpha}u+V(x)u=K(x)f(u)\quad\text{in }\mathbb{R}^{N}, where {\alpha\in(0,1)} , {N>2\alpha} , {(-\Delta)^{\alpha}} is the fractional Laplacian, V and K are positive continuous functions which vanish at infinity, and f is a continuous function. By using a minimization argument and a quantitative deformation lemma, we obtain the existence of a sign-changing solution. Furthermore, when f is odd, we prove that the above problem admits infinitely many nontrivial solutions. Our result extends to the fractional framework some well-known theorems proved for elliptic equations in the classical setting. With respect to these cases studied in the literature, the nonlocal one considered here presents some additional difficulties, such as the lack of decompositions involving positive and negative parts, and the non-differentiability of the Nehari Manifold, so that a careful analysis of the fractional spaces involved is necessary.

A Sign-Changing Solution for an Asymptotically Linear Schrödinger Equation

The aim of this paper is to present a sign-changing solution for a class of radially symmetric asymptotically linear Schrödinger equations. The proof is variational and the Ekeland variational principle is employed as well as a deformation lemma combined with Miranda’s theorem.