Nonradial Solutions Research Articles

Radiation of the energy-critical wave equation with compact support

We prove exterior energy lower bounds for (nonradial) solutions to the energy-critical nonlinear wave equation in space dimensions 3≤d≤5, with compactly supported initial data. In particular, it is shown that nontrivial global solutions with compact spatial support must be radiative in a sharp sense. In space dimensions 3 and 4, a nontrivial soliton background is also considered. As an application, we obtain partial results on the rigidity conjecture concerning solutions with the compactness property, including a new proof for the global existence of such solutions.

Multiplicity and symmetry breaking for supercritical elliptic problems in exterior domains

Abstract We deal with the following semilinear equation in exterior domains − Δ u + u = a ( x ) | u | p − 2 u , u ∈ H 0 1 ( A R ) , where A R := { x ∈ R N : | x | > R } , N ⩾ 3 , R > 0. Assuming that the weight a is positive and satisfies some symmetry and monotonicity properties, we exhibit a positive solution having the same features as a, for values of p > 2 in a suitable range that includes exponents greater than the standard Sobolev critical one. In the special case of radial weight a, our existence result ensures multiplicity of nonradial solutions. We also provide an existence result for supercritical p in nonradial exterior domains.

Multiple non-radial solutions for coupled Schrödinger equations

The paper deals with the existence of non-radial solutions for an N-coupled nonlinear elliptic system. In the repulsive regime with some structure conditions on the coupling and for each symmetric subspace of rotation symmetry, we prove the existence of an infinite sequence of non-radial positive solutions and an infinite sequence of non-radial nodal solutions.

New type of solutions for Schrödinger equations with critical growth

We consider the following nonlinear Schrödinger equations with critical growth: −Δu+V(|y|)u=uN+2N−2,u>0inRN, where V(|y|) is a bounded positive radial function in C1, N ≥ 5. By using a finite reduction argument, we show that if r2V(r) has either an isolated local maximum or an isolated local minimum at r0 > 0 with V(r0) > 0, there exists infinitely many non-radial large energy solutions which are invariant under some sub-groups of O(3).

Non-radial ground state solutions for fractional Schrödinger–Poisson systems in $$\mathbb {R}^{2}$$

Non-radial ground state solutions for fractional Schrödinger–Poisson systems in $$\mathbb {R}^{2}$$

Long time asymptotics of small mass solutions for a chemotaxis-consumption system involving prescribed signal concentrations on the boundary

This paper investigates a parabolic–elliptic chemotaxis-consumption system with signal dependent sensitivity χ=χ(c) under no-flux/Dirichlet boundary conditions. For general χ which may allow singularities at c=0, the global existence and boundedness of radial large data solutions are established in dimensions d≥2. In particular, when χ(c)=1, we also find that the constructed solution converges asymptotically to a nonhomogeneous steady state if the initial mass is small. On the other hand, for the system with χ(c)=c, a Lyapunov-type inequality is derived. This inequality not only leads to a result on global existence of smooth solutions with non-radial large data in two dimensions but moreover, provides long-time asymptotics of non-radial (d=2) and radial (d≥2) solutions at suitably small mass levels.

Existence and symmetry breaking results for positive solutions of elliptic Hamiltonian systems

Abstract In this paper, we are interested in positive solutions of $$ \begin{align*}\left\{ \begin{array}{@{}ll} -\Delta u = a(x)v^{p-1}, \quad &\text{ in } \Omega,\\ -\Delta v = b(x)u^{q-1}, \quad &\text{ in } \Omega,\\ u,v>0, \quad &\text{ in } \Omega,\\ u=v=0, \quad &\text{ on } \partial\Omega, \end{array} \right. \end{align*} $$ where $\Omega $ is a bounded annular domain (not necessarily an annulus) in ${\mathbb {R}}^N (N \ge 3)$ and $ a(x), b(x)$ are positive continuous functions. We show the existence of a positive solution for a range of supercritical values of p and q when the problem enjoys certain mild symmetry and monotonicity conditions. We shall also address the symmetry breaking phenomena where the system is fully symmetric. Indeed, as a consequence of our results, we shall show that problem (1) has $\Bigl \lfloor \frac {N}{2} \Bigr \rfloor $ (the floor of $\frac {N}{2}$ ) positive non-radial solutions when $ a(x)=b(x)=1$ and $\Omega $ is an annulus with certain assumptions on the radii. In general, for the radial case where the domain is an annulus, we prove the existence of a non-radial solution provided $$ \begin{align*}(p-1)(q-1)> \Big(1+\frac{2N}{\lambda_H}\Big)^2\left(\frac{q}{p}\right),\end{align*} $$ where $\lambda _H$ is the best constant for the Hardy inequality on $\Omega .$ We remark that the best constant $\lambda _H$ for the Hardy inequality is just the characteristic of the domain, and is independent of the choices of p and $q.$ For this reason, the aforementioned inequality plays a major role to prove the existence and multiplicity of non-radial solutions when the problem is fully symmetric. Our proofs use a variational formulation on appropriate convex subsets for which the lack of compactness is recovered for the supercritical problem.

L2 Concentration of Blow-Up Solutions for the Nonlinear Schrödinger Equation with an Inhomogeneous Combined Non-Linearity

This article studies the Schrödinger equation with an inhomogeneous combined term i∂tu−(−Δ)su+λ1|x|−b|u|pu+λ2|u|qu=0, where s∈(12,1),λ1,λ2=±1,0<b<{2s,N} and p,q>0. We study the limit behaviour of the infinite blow-up solution at the blow-up time. When the parameters p,q,λ1 and λ2 have different values, we obtain the nonexistence of a strong limit for the non-radial solution and the L2 concentration for the radial solution. Interestingly, we find that the mass of the finite time blow-up solutions are concentrated in different ways for different parameters.

Infinitely many solutions for Hamiltonian system with critical growth

Abstract In this article, we consider the following elliptic system of Hamiltonian-type on a bounded domain: − Δ u = K 1 ( ∣ y ∣ ) ∣ v ∣ p − 1 v , in B 1 ( 0 ) , − Δ v = K 2 ( ∣ y ∣ ) ∣ u ∣ q − 1 u , in B 1 ( 0 ) , u = v = 0 on ∂ B 1 ( 0 ) , \left\{\begin{array}{ll}-\Delta u={K}_{1}\left(| y| ){| v| }^{p-1}v,\hspace{1.0em}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{B}_{1}\left(0),\\ -\Delta v={K}_{2}\left(| y| ){| u| }^{q-1}u,\hspace{1.0em}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{B}_{1}\left(0),\\ u=v=0\hspace{1.0em}& \hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}\partial {B}_{1}\left(0),\end{array}\right. where K 1 ( r ) {K}_{1}\left(r) and K 2 ( r ) {K}_{2}\left(r) are positive bounded functions defined in [ 0 , 1 ] \left[0,1] , B 1 ( 0 ) {B}_{1}\left(0) is the unit ball in R N {{\mathbb{R}}}^{N} , and ( p , q ) \left(p,q) is a pair of positive numbers lying on the critical hyperbola 1 p + 1 + 1 q + 1 = N − 2 N . \frac{1}{p+1}+\frac{1}{q+1}=\frac{N-2}{N}. Under some suitable further assumptions on the functions K 1 ( r ) {K}_{1}\left(r) and K 2 ( r ) {K}_{2}\left(r) , we prove the existence of infinitely many nonradial positive solutions whose energy can be made arbitrarily large. Our proof is based on the reduction method. The most ingredients of the article are using the Green representation and estimating the Green function and its regular part very carefully. For this purpose, some more extra ideas and techniques are needed. We believe that our method and techniques can be applied to other related problems.

An upper bound for the least energy of a sign-changing solution to a zero mass problem

Abstract We give an upper bound for the least possible energy of a sign-changing solution to the nonlinear scalar field equation − Δ u = f ( u ) , u ∈ D 1,2 ( R N ) , $-{\Delta}u=f\left(u\right), u\in {D}^{1,2}\left({\mathrm{R}}^{N}\right),$ where N ≥ 5 and the nonlinearity f is subcritical at infinity and supercritical near the origin. More precisely, we establish the existence of a nonradial sign-changing solution whose energy is smaller that 12c 0 if N = 5, 6 and smaller than 10c 0 if N ≥ 7, where c 0 is the ground state energy.

Segregated solutions for nonlinear Schrödinger systems with a large number of components

Abstract In this paper we are concerned with the existence of segregated non-radial solutions for nonlinear Schrödinger systems with a large number of components in a weak fully attractive or repulsive regime in presence of a suitable external radial potential.

Nonradial singular solutions for elliptic equations with exponential nonlinearity

<abstract><p>For any $ R > 0 $, infinitely many nonradial singular solutions can be constructed for the following equation:</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation} -\Delta u = e^u \;\;\; \mbox{in}\; B_R \backslash \{0\} , \;\;\;\;\;\;(0.1)\end{equation} $\end{document} </tex-math></disp-formula></p> <p>where $ B_R = \{x \in \mathbb{R}^N \; (N \geq 3): \; |x| < R\} $. To construct nonradial singular solutions, we need to consider asymptotic expansion at the isolated singular point $ x = 0 $ of a prescribed solution of (0.1). Then, nonradial singular solutions of (0.1) can be constructed by using the asymptotic expansion and introducing suitable weighted Hölder spaces.</p></abstract>

Nonradial solutions of quasilinear Schrödinger equations with general nonlinearity

Nonradial solutions of quasilinear Schrödinger equations with general nonlinearity

Nonradial solutions for coupled elliptic system with critical exponent in exterior domain

We consider the following coupled Schr$ \ddot{o} $dinger system with critical exponent in $ \mathbb{R}^3: $$ \left \{ \begin{aligned} &-\Delta u+\lambda V(|y|)u = K_1(|y|)u^5+u^2v^3,\qquad &\text{ in } \mathbb{R}^3\backslash B_\epsilon(0),\\ &-\Delta v+\lambda V(|y|)v = K_2(|y|)v^5+v^2u^3, \qquad &\text{ in } \mathbb{R}^3\backslash B_\epsilon(0),\\ &u >0, v>0, \quad &\text{ in } \mathbb{R}^3\backslash B_\epsilon(0), \\ & (u,v) = (0,0), \quad &\text{on } \partial B_\epsilon (0), \\ &u,v\in D^{1,2}(\mathbb{R}^3\backslash B_\epsilon(0))), \end{aligned} \right. $where $ V(|y|) $ is the potential function satifying $ 0<V(|y|)\leq C\frac{1}{(1+|y|)^4} $ in $ \mathbb{R}^3\backslash B_\epsilon(0), $ $ \lambda>0 $ is a constant. $ K_i ( i = 1,2 ) $ are smooth bounded functions satisfying some suitable assumptions. $ B_\epsilon(0) $ is the ball centered at the origin with radius $ \epsilon. $ By using Schmidt reduction arguments combine with the energy expansion and the critical point theory, we prove the existence of infinitely nonradial solutions for the system.

The Effects of Nonlinear Noise on the Fractional Schrödinger Equation

The aim of this work is to investigate the influence of nonlinear multiplicative noise on the Cauchy problem of the nonlinear fractional Schrödinger equation in the non-radial case. Local well-posedness follows from estimates related to the stochastic convolution and deterministic non-radial Strichartz estimates. Furthermore, the blow-up criterion is presented. Then, with the help of Itô’s lemma and stopping time arguments, the global solution is constructed almost surely. The main innovation is that the non-radial global solution is given under fractional-order derivatives and a nonlinear noise term.

Infinitely many nonradial positive solutions for multi-species nonlinear Schrödinger systems in [formula omitted

In this paper, we consider the multi-species nonlinear Schrödinger systems in RN:{−Δuj+Vj(x)uj=μjuj3+∑i=1;i≠jdβi,jui2ujin RN,uj(x)>0in RN,uj(x)→0as |x|→+∞,j=1,2,⋯,d, where N=2,3, μj>0 are constants, βi,j=βj,i≠0 are coupling parameters, d≥2 and Vj(x) are potentials. By Ljapunov-Schmidt reduction arguments, we construct infinitely many nonradial positive solutions of the above system under some mild assumptions on potentials Vj(x) and coupling parameters {βi,j}, without any symmetric assumptions on the limit case of the above system. Our result, giving a positive answer to the conjecture in Pistoia and Viara [50] and extending the results in [50,52], reveals new phenomenon in the case of N=2 and d=2 and is almost optimal for the coupling parameters {βi,j}.

Existence of non-radial solutions to semilinear elliptic systems on a unit ball in $${\mathbb {R}}^3$$

Existence of non-radial solutions to semilinear elliptic systems on a unit ball in $${\mathbb {R}}^3$$

Radial and nonradial solutions for nonautonomous Kirchhoff problems

In this paper, we study the following nonautonomous Kirchhoff problem: where , , , is a positive constant, is a parameter, and the potential functions belong to . The existence of radially symmetric and positive solution to the above problem is first established for all when are radially symmetric and , and the range of can be extended to with the aid of a coercive type assumption on . Moreover, we show the existence of infinitely many solutions with high energies via the fountain theorem under more general assumption on which allows it to be sign‐changing. When and , we show that the above problem possesses infinitely many solutions with negative critical values for small provided that the function belongs to a suitable space. In particular, by imposing a hypothesis on the potential controlling its growth at infinity, we obtain a nonradial solution via the mountain pass theorem and the principle of symmetric criticality.

A compact embedding result and its applications to a nonlocal Schrödinger equation

AbstractCompactness is a fundamental issue in nonlinear analysis. Assume and with , we consider a subspace of , which is the space of invariant functions corresponding to a subgroup G of , where is a kind of function spaces including fractional Sobolev spaces . We show that the embeddings are compact for provided Ω is compatible with G, where is the fractional Sobolev critical exponent. Moreover, the existence and multiplicity of radial and nonradial solutions were obtained of the following nonlocal Schrödinger equation: where is an integro‐differential operator has order 2s and can be given by and the kernel K satisfies the following properties: there is and such that for any ; , where .

Non-radial solutions for higher order Hénon-type equation with critical exponent

Non-radial solutions for higher order Hénon-type equation with critical exponent