Non-radial Sign-changing Solutions Research Articles

Concentration on the Clifford torus for the critical problem in an annulus

In this paper, we are concerned with the existence of solutions for the following critical elliptic problem:(P){−Δu=|u|4N−2u, in Ω,u=0, on ∂Ω. where Ω:={x∈RN:a<|x|<b} is an annulus, N≥5. By using a finite reduction argument, we prove the existence of infinitely many non-radial sign-changing solutions concentrating on the Clifford torus.

Normalized solutions to p-Laplacian equations with combined nonlinearities* *Supported by National Natural Science Foundation of China (12031015, 11771428, 12026217, 11871302).

In this paper, we study the p-Laplacian equation with a L p -norm constraint: where N ⩾ 2, a > 0, , , and is a Lagrange multiplier, which appears due to the mass constraint ‖u‖ p = a. We assume that g is odd and L p -supercritical. When and μ > 0, we use Schwarz rearrangement and Ekeland variational principle to prove the existence of positive radial ground states for suitable μ. When and μ > 0 or and μ ⩽ 0, with an additional condition of g, we obtain a positive radial ground state if μ lies in a suitable range, by the Schwarz rearrangement and minimax theorems. Via a fountain theorem type argument, with suitable , we obtain infinitely many radial solutions for any N ⩾ 2 and establish the existence of infinitely many nonradial sign-changing solutions for N = 4 or N ⩾ 6. In any case mentioned above, the range of μ depends on the value of a: |μ| can be large if a > 0 is small.

Existence of sign-changing solutions to a Hamiltonian elliptic system in [formula omitted

In this paper, we consider the Hamiltonian elliptic system−ΔU+U=|V|p−1V,−ΔV+V=|U|q−1Uin RN, where N≥4 and the non-linearities p,q are superlinear and lies below the critical hyperbola. We shall prove the existence of nonradial sign-changing solutions. We shall work with the space of ϕ-equivariant functions where ϕ:Γ→{1,−1} and Γ⊂O(N), a closed subgroup of O(N). Depending on the size of the fixed point set Fix(Γ), we shall establish our result in two different ways.

Saddle solutions for the fractional Choquard equation

We study the saddle solutions for the fractional Choquard equation \begin{align*} (-\Delta)^{s}u+ u=(K_{\alpha}\ast|u|^{p})|u|^{p-2}u, \quad x\in \mathbb{R}^N \end{align*} where $s\in(0,1)$, $N\geq 3$ and $K_\alpha$ is the Riesz potential with order $\alpha\in (0,N)$. For every Coxeter group $G$ with rank $1\leq k\leq N$ and $p\in[2,\frac{N+\alpha}{N-2s})$, we construct a $G$-saddle solution with prescribed symmetric nodal configurations. This is a counterpart for the fractional Choquard equation of saddle solutions to the Choquard equation and further completes the existence of non-radial sign-changing solutions for this doubly nonlocal equation.

Infinitely many radial and non-radial sign-changing solutions for Schrödinger equations

Abstract In the present paper, a class of Schrödinger equations is investigated, which can be stated as − Δ u + V ( x ) u = f ( u ) , x ∈ ℝ N . - \Delta u + V(x)u = f(u),\;\;\;\;x \in {{\rm{\mathbb R}}^N}. If the external potential V is radial and coercive, then we give the local Ambrosetti-Rabinowitz super-linear condition on the nonlinearity term f ∈ C(ℝ, ℝ) which assures the problem has not only infinitely many radial sign-changing solutions, but also infinitely many non-radial sign-changing solutions.

A mass supercritical problem revisited

In any dimension $N\geq1$ and for given mass $m>0$, we revisit the nonlinear scalar field equation with an $L^2$ constraint: $$ -\Delta u=f(u)-\mu u, \quad u \in H^1(\mathbb{R}^N) \quad \text{with} \quad \|u\|^2_{L^2(\mathbb{R}^N)}=m. $$ where $\mu\in\mathbb{R}$ will arise as a Lagrange multiplier. Assuming only that the nonlinearity $f$ is continuous and satisfies weak mass supercritical conditions, we show the existence of ground states and reveal the basic behavior of the ground state energy $E_m$ as $m>0$ varies. In particular, to overcome the compactness issue when looking for ground states, we develop robust arguments which we believe will allow treating other $L^2$ constrained problems in general mass supercritical settings. Under the same assumptions, we also obtain infinitely many radial solutions for any $N\geq2$ and establish the existence and multiplicity of nonradial sign-changing solutions when $N\geq4$. Finally we propose two open problems.

Sign-changing solutions for some class of elliptic system

This paper is concerned with the existence of multiple non-radial sign-changing solutions for where 2<p<6, can be regarded as a parameter and is a positive continuous function. We proved that the above equation possesses a non-radial sign-changing solutions with exactly k maximum points and k minimum points for suitable range of α.

Variational analysis for nonlocal Yamabe-type systems

The paper is concerned with existence, multiplicity and asymptotic behavior of (weak) solutions for nonlocal systems involving critical nonlinearities. More precisely, we consider \begin{document}$\left\{ \begin{array}{*{35}{l}} \begin{align} &amp; M\left( [u]_{s}^{2}-\mu \int_{{{\mathbb{R}}^{3}}}{V}(x)|u{{|}^{2}}dx \right)\left[ {{(-\Delta )}^{s}}u-\mu V(x)u \right]-\phi |u{{|}^{2_{s,t}^{*}-2}}u \\ &amp; =\lambda h(x)|u{{|}^{p-2}}u+|u{{|}^{2_{s}^{*}-2}}u\quad ~~\text{in}~~~ {{\mathbb{R}}^{\text{3}}} \\ &amp; {{(-\Delta )}^{t}}\phi =|u{{|}^{2_{s,t}^{*}}}~~~ \text{in} ~~{{\mathbb{R}}^{3}}, \\ \end{align} &amp; \ &amp; \ &amp; {} \\\end{array} \right.$\end{document} where $ (-\Delta )^s $ is the fractional Lapalcian, $ [u]_{s} $ is the Gagliardo seminorm of $ u $, $ M:\mathbb{R}^+_0\rightarrow \mathbb{R}^+_0 $ is a continuous function satisfying certain assumptions, $ V(x) = {|x|^{-2s}} $ is the Hardy potential function, $ 2_{s, t}^* = {(3+2t)}/{(3-2s)} $, $ s, t\in (0, 1) $, $ \lambda, \mu $ are two positive parameters, $ 1&lt;p&lt;2_s^* = {6}/{(3-2s)} $ and $ h\in L^{{2_s^*}/{(2_s^*-p)}}(\mathbb{R}^3) $. By using topological methods and the Krasnoleskii's genus theory, we obtain the existence, multiplicity and asymptotic behaviour of solutions for above problem under suitable positive parameters $ \lambda $ and $ \mu $. Moreover, we also consider the existence of nonnegative radial solutions and non-radial sign-changing solutions. The main novelties are that our results involve the possibly degenerate Kirchhoff function and the upper critical exponent in the sense of Hardy–Littlehood–Sobolev inequality. We emphasize that some of the results contained in the paper are also valid for nonlocal Schrödinger–Maxwell systems on Cartan–Hadamard manifolds.

Nodal solutions for fractional Schrödinger-Poisson problems

In this paper, we consider the following fractional Schrodinger-Poisson problem, $$\begin{cases}\epsilon^{2s}(-\Delta)^{s}u+V(x)u+\phi{u}=\mid{u}\mid^{p-1}u, & x\in\mathbb{R}^N,\\(-\Delta)^t\phi=u^2, & x\in\mathbb{R}^N,\end{cases}$$ where e > 0 is a small parameter, N ⩾ 3 and V(x) is a potential function. We construct non-radial sign-changing solutions, whose components may have spikes clustering at the local minimum point of V(x).

Nondegeneracy of nonradial sign-changing solutions to the nonlinear Schrödinger equation

We prove that the non-radial sign-changing solutions to the nonlinear Schrodinger equation ∆u− u + |u|p−1u = 0 in R , u ∈ H(R ) constructed by Musso, Pacard and Wei [19] is non-degenerate. This provides the first example of non-degenerate sign-changing solution with finite energy to the above nonlinear Schrodinger equation.

Infinitely many non-radial solutions to a critical equation on annulus

In this paper, we build infinitely many non-radial sign-changing solutions to the critical problem:(P){−Δu=|u|4N−2u, in Ω,u=0, on ∂Ω, on the annulus Ω:={x∈RN:a<|x|<b}, N≥3. In particular, for any integer k large enough, we build a non-radial solution which look like the unique positive solution u0 to (P) crowned by k negative bubbles arranged on a regular polygon with radius r0 such that r0N−22u0(r0)=:maxa≤r≤b⁡rN−22u0(r).

Nonradial sign changing solutions to Lane–Emden problem in an annulus

In this paper we prove the existence of continua of nonradial solutions for the Lane–Emden equation in the annulus. In a first result we show that there are infinitely many global continua detaching from the curve of radial solutions with any prescribed number of nodal zones. Next, using the fixed point index in cone, we produce nonradial solutions with a new type of symmetry. This result also applies to solutions with fixed signed, showing that the set of solutions to the Lane–Emden problem has a very rich and complex structure.

Non-radial sign-changing solutions for the Schrödinger–Poisson problem in the semiclassical limit

We study a system of equations known as Schrodinger–Poisson problem proving that the set of sign-changing solutions has a rich structure in the semiclassical limit. Indeed we construct non-radial multi-peak solutions with an arbitrary large number of positive and negative peaks which are displaced in suitable symmetric configurations and which collapse to the same point as ϵ ⟶ 0. The proof is based on the Lyapunov–Schmidt reduction.

Large energy entire solutions for the Yamabe type problem of polyharmonic operator

In this paper, we consider the following Yamabe type problem of polyharmonic operator:(P){Dmu=|u|4mN−2muon SN,u∈Hm(SN), where N⩾2m+1, m∈N+, SN, is the unit sphere with the induced Riemannian metric g=gSN, and Dm is the elliptic differential operator of 2m order given byDm=∏k=1m(−Δg+14(N−2k)(N+2k−2)), where Δg is the Laplace–Beltrami operator on SN. We will show that the problem (P) has infinitely many non-radial sign-changing solutions.

A Biharmonic Equation in ℜ4 Involving Nonlinearities with Subcritical Exponential Growth

Abstract In this paper we consider a biharmonic equation of the form Δ2u+V(x)u = f (u) in the whole four-dimensional space ℜ4. Assuming that the potential V satisfies some symmetry conditions and is bounded away from zero and that the nonlinearity f is odd and has subcritical exponential growth (in the sense of an Adams’ type inequality), we prove a multiplicity result. More precisely we prove the existence of infinitely many nonradial sign-changing solutions and infinitely many radial solutions in H2(ℜ4). The main difficulty is the lack of compactness due to the unboundedness of the domain ℜ4 and in this respect the symmetries of the problem play an important role.

Infinitely many sign-changing solutions for a Schrödinger equation in RN

In this paper, we consider a Schrödinger equation − Δ u + ( λ a ( x ) + 1 ) u = f ( u ) . Applying Principle of Symmetric Criticality and the invariant set method, under some assumptions on a and f , we obtain an unbounded sequence of radial sign-changing solutions for the above equation in R N when λ > 0 large enough. As N = 4 or N ⩾ 6 , λ > 0 given, using Fountain Theorem and the Principle of Symmetric Criticality, we prove that there exists an unbounded sequence of non-radial sign-changing solutions for the above equation in R N .

Existence of many sign-changing nonradial solutions for semilinear elliptic problems on thin annuli

We study the existence of many nonradial sign-changing solutions of a superlinear Dirichlet boundary value problem in an annulus in $\mathbb R^N$. We use Nehari-type variational method and group invariance techniques to prove that the critical points of an action functional on some spaces of invariant functions in $H_{0}^{1,2}(\Omega_{\varepsilon})$, where $\Omega_{\varepsilon}$ is an annulus in $\mathbb R^N$ of width $\varepsilon$, are weak solutions (which in our case are also classical solutions) to our problem. Our result generalizes an earlier result of Castro et al. (See [A. Castro, J. Cossio and J. M. Neuberger, A minmax principle, index of the critical point, and existence of sign-changing solutions to elliptic boundary value problems , Electron. J. Differential Equations 2 (1998), 1–18]).