Search Results (5)

This paper studies the existence, regularity, and properties of normalized ground state solutions for the mixed fractional Schrödinger equations. For subcritical cases, we establish the boundedness and Sobolev regularity of solutions, derive Pohozaev identities, and prove the existence of radial, decreasing ground states, while showing nonexistence in the $L^2$-critical case. For $L^2$-supercritical exponents, we identify parameter regimes where ground states exist, characterized by a negative Lagrange multiplier. The analysis combines variational methods, scaling techniques, and the careful study of fibering maps to address challenges posed by competing nonlinearities and nonlocal interactions. Full article

In this paper, we study a class of nonlocal Schrödinger–Poisson–Slater equations: $-\Delta u+u+\lambda \left(I_{\alpha }\ast {\left|u\right|}^q\right){\left|u\right|}^{q-2}u={\left|u\right|}^{p-2}u$, where $q,p>1$, $\lambda >0$, and $I_{\alpha }$ is the Riesz potential. We obtain the existence, stability, and symmetry-breaking of solutions for both radial and nonradial cases. In the radial case, we use variational methods to establish the coercivity and weak lower semicontinuity of the energy functional, ensuring the existence of a positive solution when p is below a critical threshold $\overline{p}={\displaystyle \frac{4q+2\alpha }{2+\alpha }}$. In addition, we prove that the energy functional attains a minimum, guaranteeing the existence of a ground-state solution under specific conditions on the parameters. We also apply the Pohozaev identity to identify parameter regimes where only the trivial solution is possible. In the nonradial case, we use the Nehari manifold method to prove the existence of ground-state solutions, analyze symmetry-breaking by studying the behavior of the energy functional and identifying the parameter regimes in the nonradial case, and apply concentration-compactness methods to prove the global well-posedness of the Cauchy problem and demonstrate the orbital stability of the ground state. Our results demonstrate the stability of solutions in both radial and nonradial cases, identifying critical parameter regimes for stability and instability. This work enhances our understanding of the role of nonlocal interactions in symmetry-breaking and stability, while extending existing theories to multiparameter and higher-dimensional settings in the Schrödinger–Poisson–Slater model. Full article

We deal with the following singular perturbation Kirchhoff equation: $-\left({ϵ}^{2}a+ϵb{\int }_{{\mathbb{R}}^3}{|\nabla u|}^{2}dy\right)$$\Delta u+Q\left(y\right)u={\left|u\right|}^{p-1}u,u\in H^1\left({\mathbb{R}}^3\right),$ where constants $a,b,ϵ>0$ and $1<p<5$. In this paper, we prove the uniqueness of the concentrated solutions under some suitable assumptions on asymptotic behaviors of $Q\left(y\right)$ and its first derivatives by using a type of Pohozaev identity for a small enough $ϵ$. To some extent, our result exhibits a new phenomenon for a kind of $Q\left(x\right)$ which allows for different orders in different directions. Full article

In this paper, we study the nonlocal equation ${\left({\int }_{{\mathbb{R}}^N}{\left|u\left(x\right)\right|}^{2}dx\right)}^{\gamma }\Delta u=\lambda u+{\mu \left|u\right|}^{q-2}u+{\left|u\right|}^{p-2}u$, $x\mathrm{in}{\mathbb{R}}^N$ having a prescribed mass ${\int }_{{\mathbb{R}}^N}{\left|u\left(x\right)\right|}^{2}dx=c^2,$ where $N\ge 3$, $\mu$, $\gamma \in (0,+\infty )$, $q\in (2,2^{\ast })$, c is a positive constant, p, $q\in (2,2^{\ast })$ with $p\ne q$ and $2^{\ast }=\frac{2N}{N-2}$. This research is meaningful from a physical point of view. Using variational methods, we present some results on the nonexistence and existence of solutions under different cases p and q which improve upon the previous ones via topological theory. Full article

We prove the existence of a spherically symmetric solution for a Schrödinger equation with a nonlocal nonlinearity of Choquard type. This term is assumed to be subcritical and satisfy almost optimal assumptions. The mass of of the solution, described by its norm in the Lebesgue space, is prescribed in advance. The approach to this constrained problem relies on a Lagrange formulation and new deformation arguments. In addition, we prove that the obtained solution is also a ground state, which means that it realizes minimal energy among all the possible solutions to the problem. Full article