Infinitely many solutions for the discrete Schrödinger equations with a nonlocal term

In the present paper, we consider the following discrete Schrödinger equations $$ - \biggl(a+b\sum_{k\in \mathbf{Z}} \vert \Delta u_{k-1} \vert ^{2} \biggr) \Delta ^{2} u_{k-1}+ V_{k}u_{k}=f_{k}(u_{k}) \quad k\in \mathbf{Z}, $$ − ( a + b ∑ k ∈ Z | Δ u k − 1 | 2 ) Δ 2 u k − 1 + V k u k = f k ( u k ) k ∈ Z , where a, b are two positive constants and $V=\{V_{k}\}$ V = { V k } is a positive potential. $\Delta u_{k-1}=u_{k}-u_{k-1}$ Δ u k − 1 = u k − u k − 1 and $\Delta ^{2}=\Delta (\Delta )$ Δ 2 = Δ ( Δ ) is the one-dimensional discrete Laplacian operator. Infinitely many high-energy solutions are obtained by the Symmetric Mountain Pass Theorem when the nonlinearities $\{f_{k}\}$ { f k } satisfy 4-superlinear growth conditions. Moreover, if the nonlinearities are sublinear at infinity, we obtain infinitely many small solutions by the new version of the Symmetric Mountain Pass Theorem of Kajikiya.

Ground states for critical fractional Schr\”{o}dinger-Poisson systems with vanishing potentials

This paper deals with a class of fractional Schr\”{o}dinger-Poisson system \[\begin{cases}\displaystyle (-\Delta )^{s}u+V(x)u-K(x)\phi |u|^{2^*_s-3}u=a (x)f(u), &x \in \R^{3}\\ (-\Delta )^{s}\phi=K(x)|u|^{2^*_s-1}, &x \in \R^{3}\end{cases} \]with a critical nonlocal term and multiple competing potentials, which may decay and vanish at infinity, where $s \in (\frac{3}{4},1)$, $ 2^*_s = \frac{6}{3-2s}$ is the fractional critical exponent. The problem is set on the whole space and compactness issues have to be tackled. By employing the mountain pass theorem, concentration-compactness principle and approximation method, the existence of a positive ground state solution is obtained under appropriate assumptions imposed on $V, K, a$ and $f$.

Multiplicity of positive solutions for a degenerate nonlocal problem with p-Laplacian

We consider a nonlinear boundary value problem with degenerate nonlocal term depending on the L q -norm of the solution and the p-Laplace operator. We prove the multiplicity of positive solutions for the problem, where the number of solutions doubles the number of “positive bumps” of the degenerate term. The solutions are also ordered according to their L q -norms.

Positive solutions for a degenerate Kirchhoff problem

By assuming that the Kirchhoff term has $K$ degeneracy points and other appropriated conditions, we have proved the existence of at least $K$ positive solutions other than those obtained in Santos Júnior and Siciliano [Positive solutions for a Kirchhoff problem with vanishing nonlocal term, J. Differ. Equ. 265 (2018), 2034–2043], which also have ordered $H_{0}^{1}(\Omega )$ -norms. A concentration phenomena of these solutions is verified as a parameter related to the area of a region under the graph of the reaction term goes to zero.

Weighted Choquard Equation Perturbed with Weighted Nonlocal Term

We investigate the following problem $$\begin{aligned} -\mathrm{div}(v(x)|\nabla u|^{m-2}\nabla u)+V(x)|u|^{m-2}u= \left( |x|^{-\theta }*\frac{|u|^{b}}{|x|^{\alpha }}\right) \frac{|u|^{b-2}}{|x|^{\alpha }}u+\lambda \left( |x|^{-\gamma }*\frac{|u|^{c}}{|x|^{\beta }}\right) \frac{|u|^{c-2}}{|x|^{\beta }}u \quad \text { in }{\mathbb {R}}^{N}, \end{aligned}$$ - div ( v ( x ) | ∇ u | m - 2 ∇ u ) + V ( x ) | u | m - 2 u = | x | - θ ∗ | u | b | x | α | u | b - 2 | x | α u + λ | x | - γ ∗ | u | c | x | β | u | c - 2 | x | β u in R N , where $$b, c, \alpha , \beta >0$$ b , c , α , β > 0 , $$\theta ,\gamma \in (0,N)$$ θ , γ ∈ ( 0 , N ) , $$N\ge 3$$ N ≥ 3 , $$2\le m< \infty$$ 2 ≤ m < ∞ and $$\lambda \in {\mathbb {R}}$$ λ ∈ R . Here, we are concerned with the existence of groundstate solutions and least energy sign-changing solutions and that will be done by using the minimization techniques on the associated Nehari manifold and the Nehari nodal set respectively.

The equilibrium measure for an anisotropic nonlocal energy

In this paper we characterise the minimisers of a one-parameter family of nonlocal and anisotropic energies $$I_\alpha $$ I α defined on probability measures in $${\mathbb {R}}^n$$ R n , with $$n\ge 3$$ n ≥ 3 . The energy $$I_\alpha $$ I α consists of a purely nonlocal term of convolution type, whose interaction kernel reduces to the Coulomb potential for $$\alpha =0$$ α = 0 and is anisotropic otherwise, and a quadratic confinement. The two-dimensional case arises in the study of defects in metals and has been solved by the authors by means of complex-analysis techniques. We prove that for $$\alpha \in (-1, n-2]$$ α ∈ ( - 1 , n - 2 ] , the minimiser of $$I_\alpha $$ I α is unique and is the (normalised) characteristic function of a spheroid. This result is a paradigmatic example of the role of the anisotropy of the kernel on the shape of minimisers. In particular, the phenomenon of loss of dimensionality, observed in dimension $$n=2$$ n = 2 , does not occur in higher dimension at the value $$\alpha =n-2$$ α = n - 2 corresponding to the sign change of the Fourier transform of the interaction potential.