Bound states for the Schr\"{o}dinger equation with mixed-type nonlinearites

We prove the existence results for the Schr\"odinger equation of the form $$-\Delta u + V(x) u = g(x,u), \quad x \in \mathbb{R}^N,$$ where $g$ is superlinear and subcritical in some periodic set $K$ and linear in $\mathbb{R}^N \setminus K$ for sufficiently large $|u|$. The periodic potential $V$ is such that $0$ lies in a spectral gap of $-\Delta+V$. We find a solution with the energy bounded by a certain min-max level, and infinitely many geometrically distinct solutions provided that $g$ is odd in $u$

Normalized ground states of the nonlinear Schr\"{o}dinger equation with at least mass critical growth

We propose a simple minimization method to show the existence of least energy solutions to the normalized problem \begin{cases} -\Delta u + \lambda u = g(u) \quad \mathrm{in} \ \mathbb{R}^N, \ N \geq 3, \\ u \in H^1(\mathbb{R}^N), \\ \int_{\mathbb{R}^N} |u|^2 \, dx = \rho > 0, \end{cases} where $\rho$ is prescribed and $(\lambda, u) \in \mathbb{R} \times H^1 (\mathbb{R}^N)$ is to be determined. The new approach based on the direct minimization of the energy functional on the linear combination of Nehari and Pohozaev constraints is demonstrated, which allows to provide general growth assumptions imposed on $g$. We cover the most known physical examples and nonlinearities with growth considered in the literature so far as well as we admit the mass critical growth at $0$

The semirelativistic Choquard equation with a local nonlinear term

We propose an existence result for the semirelativistic Choquard equation with a local nonlinearity in $\mathbb{R}^N$ \begin{equation*} \sqrt{\strut -\Delta + m^2} u - mu + V(x)u = \left( \int_{\mathbb{R}^N} \frac{|u(y)|^p}{|x-y|^{N-\alpha}} \, dy \right) |u|^{p-2}u - \Gamma (x) |u|^{q-2}u, \end{equation*} where $m > 0$ and the potential $V$ is decomposed as the sum of a $\mathbb{Z}^N$-periodic term and of a bounded term that decays at infinity. The result is proved by variational methods applied to an auxiliary problem in the half-space $\mathbb{R}_{+}^{N+1}$

Solutions to a nonlinear Maxwell equation with two competing nonlinearities in R3\mathbb{R}^3

We are interested in the nonlinear, time-harmonic Maxwell equation \nabla \times (\nabla \times \mathbf{E} ) + V(x) \mathbf{E} = h(x, \mathbf{E})\mbox{ in } \mathbb{R}^3 with sign-changing nonlinear term $h$, i.e. we assume that $h$ is of the form $$h(x, \alpha w) = f(x, \alpha) w - g(x, \alpha) w$$ for $w \in \mathbb{R}^3$, $|w|=1$ and $\alpha \in \mathbb{R}$. In particular, we can consider the nonlinearity consisting of two competing powers $h(x, \mathbf{E}) = |\mathbf{E}|^{p-2}\mathbf{E} - |\mathbf{E}|^{q-2}\mathbf{E}$ with $2 < q < p < 6$. Under appriopriate assumptions, we show that weak, cylindrically equivariant solutions of the special form are in one-to-one correspondence with weak solutions to a Schr\"odinger equation with a singular potential. Using this equivalence result we show the existence of the least energy solution among cylindrically equivariant solutions of the particular form to the Maxwell equation, as well as to the Schr\"odinger equation

Non-local to local transition for ground states of fractional Schr\"{o}dinger equations on RN\mathbb{R}^N

We consider the nonlinear fractional problem \begin{align*} (-\Delta)^{s} u + V(x) u = f(x,u) &\quad \hbox{in $\mathbb{R}^N$} \end{align*} We show that ground state solutions converge (along a subsequence) in $L^2_{\mathrm{loc}} (\mathbb{R}^N)$, under suitable conditions on $f$ and $V$, to a weak solution of the local problem as $s \to 1^-$.Comment: arXiv admin note: text overlap with arXiv:1907.1145