Abstract We consider the following critical nonlocal Schrödinger problem with general nonlinearities − open="(" close=")"> ε <m:mrow class="MJX-TeXA...

In this paper, we investigate the following nonlinear magnetic Schrödinger equation with exponential growth: style='text-indent:20px;'> \begin{document}$ (-i\nabla+A(x))^2 u+V(x)u = f(x, |u|^2)u\ \mbox{in}\; \mathbb{R}^2 , $\end{document} style='text-indent:20px;'>where <tex-math i...

In the present paper, we study normalized solutions for following quasilinear Schr\"odinger equations: $$-\Delta u-u\Delta u^2+\lambda u=|u|^{p-2}u \quad \text{in}~\mathbb R^N,$$ with prescribed mass $$\int_{\mathbb R^N} u^2=a^2.$$ We first consider mass-supercritical case $p>4+\frac{4}{N}$, which has not been studied before. By using a perturbation method, succeed to prove existence of ground ...

For the problem (here u = u(x)) ∆u− u + αu + βu = 0, x ∈ R, lim |x|→∞ u(x) = 0 , with constants 1 ≤ p < q < r < n+2 n−2 , and α, β > 0, uniqueness of radial solution (called ground state solution) is not known. We present a procedure, which opens the way to produce computer assisted proofs of uniqueness for specific p, q, r, and n.