In this paper, we study the existence and asymptotic behavior of nodal solutions to the following Kirchhoff problem−(a+b∫R3|∇u|2dx)Δu+V(|x|)u=f(|x|,u),inR3,u∈H1(R3), where V(x) is a smooth function, ...a,b are positive constants. Because the so-called nonlocal term (∫R3|∇u|2dx)Δu is involved in the equation, the variational functional of the equation has totally different properties from the case of b=0. Under suitable construction conditions, we prove that, for any positive integer k, the problem has a sign-changing solution ukb, which changes signs exactly k times. Moreover, the energy of ukb is strictly increasing in k, and for any sequence {bn}→0+(n→+∞), there is a subsequence {bns}, such that ukbns converges in H1(R3) to wk as s→∞, where wk also changes signs exactly k times and solves the following equation−aΔu+V(|x|)u=f(|x|,u),inR3,u∈H1(R3).
This paper is concerned with the following semilinear elliptic problem{−Δu=λm(x)f(u)inRN,u→0as|x|→+∞, where λ is a real parameter and m is a weight function which may be sign-changing. For the linear ...case, i.e., f(u)=u, we investigate the spectral structure. For the semilinear case, we study the existence and asymptotic behavior of one-sign and nodal solutions by bifurcation method.
This paper is devoted to the existence of nodal and multiple solutions of nonlinear problems involving the fractional Laplacian{(−Δ)su=f(x,u)in Ω,u=0on ∂Ω, where Ω⊂Rn (n⩾2) is a bounded smooth ...domain, s∈(0,1), (−Δ)s stands for the fractional Laplacian. When f is superlinear and subcritical, we prove the existence of a positive solution, a negative solution and a nodal solution. If f(x,u) is odd in u, we obtain an unbounded sequence of nodal solutions. In addition, the number of nodal domains of the nodal solutions are investigated.
In this article, we study the existence of localized nodal solutions for semiclassical quasilinear Choquard equations with subcritical growth $$-\varepsilon^p \Delta_{p} v +V(x)|v|^{p-2}v = ...\varepsilon^{\alpha-N} |v|^{q-2}v \int_{\mathbb{R}^N} \frac{|v(y)|^q}{|x-y|^{\alpha}}\,dy ,\quad x \in \mathbb{R}^N\,, $$ where \(N\geq 3\), \(1<p<N\), \(0<\alpha <\min\{2p,N-1\}\), \(p<q<p_\alpha^*\), \(p_\alpha^*= \frac{p(2N-\alpha)}{2(N-p)}\), \(V\) is a bounded function. By the perturbation method and the method of invariant sets of descending flow, for small \(\varepsilon\) we establish the existence of a sequence of localized nodal solutions concentrating near a given local minimum point of the potential function \(V\).
In this paper, we consider the following Kirchhoff equation in R3 with critical growth. (a+b∫R3|∇u|2dx)Δu−V(x)u+u5+μ|u|q−2u=0,inR3,u(x)→0,as|x|→∞,where V is the potential function, a,b,μ>0,5<q<6 are ...constants. We assume V is a radial function and is bounded from below by a positive constant. We prove that for any given positive integer k, the problem has a radial solution, having k nodal domains exactly.
We are concerned with the Neumann problem in some FLRW spacetimes(P){div(∇uf(u)f(u)2−|∇u|2)+f′(u)f(u)2−|∇u|2(N+|∇u|2f(u)2)=λNg(|x|,u)inB(R),|∇u|<f(u)inB(R),∂u∂ν=0on∂B(R), where ∂u∂ν denotes the ...outward normal derivative of u, B(R) is the Euclidean ball RN, N≥1, centered at 0 with radius R, f∈C2(I) is a positive function, I⊆R is an open interval containing 0, g:0,R×R→R is continuous, λ>0 is a parameter. We show that (P) has infinitely many radially symmetric sign-changing solutions under some appropriate conditions. The proof of our main result is based upon bifurcation techniques.
We study the nonlinearly coupled Choquard-type system(0.1){−Δu1+μ1u1=a1(Iα⁎|u1|p)|u1|p−2u1+β(Iα⁎|u2|p)|u1|p−2u1,x∈Ω,−Δu2+μ2u2=a2(Iα⁎|u2|p)|u2|p−2u2+β(Iα⁎|u1|p)|u2|p−2u2,x∈Ω,u1=u2=0on∂Ω, where Ω is a ...bounded smooth domain in RN with N≥3, p∈(N+αN,N+αN−2), α∈(0,N), Iα is the Riesz potential, and μ1,μ2,a1,a2,β are positive constants. For every k∈N, we prove that there exists βk>0 such that system (0.1) possesses k nodal solutions and k semi-nodal solutions for β∈(0,βk) and p>2. Additionally, the existence of least energy nodal solutions is also obtained.
In this paper, we are concerned with nodal solutions for a class of Kirchhoff-type problems
where
, a, b>0, f satisfies some asymptotically linear growth conditions. First of all, when N = 3, b>0 and
..., b>0 sufficiently small, we attain infinitely many nodal solutions by an equivalent transformation. Based on this, via a new and direct approach, the least energy sign-changing radial solutions can be obtained for the above problem. What's more, we also establish non-existence results of nodal solutions for
and b large enough.
We consider a non-local Shrödinger problem driven by the fractional Orlicz g-Laplace operator as follows(P)(−△g)αu+g(u)=K(x)f(x,u),inRd, where d≥3,(−△g)α is the fractional Orlicz g-Laplace operator, ...f:Rd×R→R is a measurable function and K is a positive continuous function. Employing the Nehari manifold method and without assuming the well-known Ambrosetti-Rabinowitz and differentiability conditions on the non-linear term f, we prove that the problem (P) has a ground state of fixed sign and a nodal (or sign-changing) solutions.
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