In this paper, we consider a class of focusing nonlinear Schrödinger equations involving power-type nonlinearity with critical Sobolev ...exponent{(i∂∂t+Δ)u+|u|4N−2su=0,inR+×RN,u=u0(x)∈H1(RN)∩L2(RN,|x|2dx),fort=0,x∈RN, where 2N−2s=:p⋆ be the critical Sobolev exponent with s<N2. For dimension N≥1, the initial data u0 belongs to the energy space and |⋅|u0∈L2(RN) and the power index s satisfies 0,1∋s≡sc=N2−1p⋆, we prove that the problem is non-global existence in H1(RN) (here, finite-time blow-up occurs) with the energy of initial data Eu0 is negative. Moreover, we establish the stability for the solutions with the lower bound and the global a priori upper bound in dimension N≥2 related conservation laws. The motivation for this paper is inspired by the mass critical case with s=0 of the celebrated result of B. Dodson 9 and the work of R. Killip and M. Visan 18 represented with energy critical case for s=1. Our new results mention to nonlinear Schrödinger equation for interpolating between mass critical or mass super-critical (s≥0) and energy sub-critical or energy critical (s≤1).
We study the following critical Schrödinger system in R3$\mathbb {R}^3$:
...−Δu+λ1u=|u|4u+μ1|u|p−2u+αν|u|α−2u|v|β,−Δv+λ2v=|v|4v+μ2|v|p−2v+βν|u|α|v|β−2v,∫R3u2dx=a2and∫R3v2dx=b2,u,v∈H1(R3),$$\begin{equation*} {\begin{cases} -\Delta u+\lambda _1 u=|u|^{4}u+\mu _1|u|^{p-2}u+\alpha \nu|u|^{\alpha -2}u|v|^{\beta},\\ -\Delta v+\lambda _2 v=|v|^{4}v+\mu _2|v|^{p-2}v+\beta \nu|u|^{\alpha}|v|^{\beta -2}v,\\ \int _{\mathbb {R}^3} u^2dx=a^2\;\; \hbox{and}\;\; \int _{\mathbb {R}^3} v^2dx=b^2,\;\;u,v\in H^1(\mathbb {R}^3), \end{cases}} \end{equation*}$$where α,β>1,α+β=2∗=6$\alpha,\beta >1, \alpha +\beta =2^*=6$, p∈(2,6)$p\in (2,6)$, ν>0$\nu >0$, and μ1,μ2,λ1,λ2∈R$\mu _1, \mu _2, \lambda _1, \lambda _2\in \mathbb {R}$. Any (u,v)$(u,v)$ solving such system (for some λ1,λ2$\lambda _1,\lambda _2$) is called the normalized solution in the literature, where the normalization is settled in L2(R3)$L^2(\mathbb {R}^3)$. We show that this system has a positive ground state for p∈(2,103)$ p\in (2,\frac{10}{3})$ in the case of μ1,μ2>0$\mu _1,\mu _2>0$. For the case of 2<p<6$2<p<6$ and μ1,μ2<0$\mu _1,\mu _2<0$, we obtain the non‐existence results.
We consider the following prescribed scalar curvature equations in RN(0.1)−Δu=K(|y|)u2⁎−1,u>0inRN,u∈D1,2(RN), where K(r) is a positive function, 2⁎=2NN−2. We first prove a non-degeneracy result for ...the positive multi-bubbling solutions constructed in 26 by using the local Pohozaev identities. Then we use this non-degeneracy result to glue together bubbles with different concentration rate to obtain new solutions.
We consider a slightly subcritical Dirichlet problem with a non-power nonlinearity in a bounded smooth domain. For this problem, standard compact embeddings cannot be used to guarantee the existence ...of solutions as in the case of power-type nonlinearities. Instead, we use a Ljapunov-Schmidt reduction method to show that there is a positive solution which concentrates at a non-degenerate critical point of the Robin function. This is the first existence result for this type of generalized slightly subcritical problems.
In this note, we obtain a classification result for positive solutions to the critical
-Laplace equation in
with
≥ 4 and
>
for some number
such that
, which improves upon a similar result obtained by ...Ou (“On the classification of entire solutions to the critical
-Laplace equation,” 2022, arXiv:2210.05141) under the condition
In this paper we study the existence and multiplicity of positive solutions for the Schrödinger-Poisson system with critical growth:{−ε2Δu+V(x)u=f(u)+|u|3uϕ,x∈R3,−ε2Δϕ=|u|5,x∈R3,u∈H1(R3),u(x)>0,x∈R3, ...where ε>0 is a parameter, V:R3→R is a continuous function and f:R→R is a C1 function. Under a global condition for V we prove that the above problem has a ground state solution and relate the number of positive solutions with the topology of the set where V attains its minimum, by using variational methods.
This paper is concerned with the existence of positive solutions for a class of fractional Schrödinger-Poisson system with doubly critical exponents. In dealing with the case of critical nonlocal ...term, we need to develop new techniques in order to get the mountain pass solution.
In this paper, we study the multiplicity results of positive solutions for a class of Kirchhoff type problems involving critical growth terms. With the help of Nehari manifold and ...Ljusternik–Schnirelmann category theory, we investigate how the coefficient g(x) of the critical nonlinearity affects the number of positive solutions of that problem. Furthermore, we obtain a relationship between the number of positive solutions and the topology of the global maximum set of g.
This paper is concerned with the following fractional Schrödinger equations involving critical exponents: (−Δ)αu+V(x)u=k(x)f(u)+λ|u|2α∗−2uinRN, where (−Δ)α is the fractional Laplacian operator with ...α∈(0,1), N≥2, λ is a positive real parameter and 2α∗=2N/(N−2α) is the critical Sobolev exponent, V(x) and k(x) are positive and bounded functions satisfying some extra hypotheses. Based on the principle of concentration compactness in the fractional Sobolev space and the minimax arguments, we obtain the existence of a nontrivial radially symmetric weak solution for the above-mentioned equations without assuming the Ambrosetti–Rabinowitz condition on the subcritical nonlinearity.
In this paper we consider the quasilinear critical problem(Pλ){−Δpu=λuq−1+up⁎−1inΩu>0inΩu=0on∂Ω where Ω is a regular bounded domain in RN, N≥p2, 1<p<2, p≤q<p⁎, p⁎=Np/(N−p), λ>0 is a parameter. In ...spite of the lack of C2 regularity of the energy functional associated to (Pλ), we employ new Morse techniques to derive a multiplicity result of solutions. We show that there exists λ⁎>0 such that, for each λ∈(0,λ⁎), either (Pλ) has P1(Ω) distinct solutions or there exists a sequence of quasilinear problems approximating (Pλ), each of them having at least P1(Ω) distinct solutions. These results complete those obtained in 23 for the case p≥2.
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