68 research outputs found
Normalized ground states for a biharmonic Choquard system in
In this paper, we study the existence of normalized ground state solutions
for the following biharmonic Choquard system \begin{align*}
\begin{split}
\left\{
\begin{array}{ll}
\Delta^2u=\lambda_1 u+(I_\mu*F(u,v))F_u (u,v),
\quad\mbox{in}\ \ \mathbb{R}^4,
\Delta^2v=\lambda_2 v+(I_\mu*F(u,v)) F_v(u,v),
\quad\mbox{in}\ \ \mathbb{R}^4,
\displaystyle\int_{\mathbb{R}^4}|u|^2dx=a^2,\quad
\displaystyle\int_{\mathbb{R}^4}|v|^2dx=b^2,\quad u,v\in H^2(\mathbb{R}^4),
\end{array}
\right.
\end{split}
\end{align*} where are prescribed, , with , are
partial derivatives of and have exponential subcritical or
critical growth in the sense of the Adams inequality. By using a minimax
principle and analyzing the behavior of the ground state energy with respect to
the prescribed mass, we obtain the existence of ground state solutions for the
above problem.Comment: arXiv admin note: text overlap with arXiv:2211.1370
Normalized solutions for a fractional -Laplacian Choquard equation with exponential critical nonlinearities
In this paper, we are concerned with the following fractional -Laplacian
Choquard equation
\begin{align*}
\begin{cases}
(-\Delta)^s_{N/s}u=\lambda |u|^{\frac{N}{s}-2}u +(I_\mu*F(u))f(u),\ \
\mbox{in}\ \mathbb{R}^N,
\displaystyle\int_{\mathbb{R}^N}|u|^{N/s} \mathrm{d}x=a^{N/s},
\end{cases}
\end{align*} where , is a
prescribed constant, ,
with , is the primitive function of , and is a
continuous function with exponential critical growth of Trudinger-Moser type.
Under some suitable assumptions on , we prove that the above problem admits
a ground state solution for any given , by using the constraint
variational method and minimax technique
Normalized ground states for a biharmonic Choquard equation with exponential critical growth
In this paper, we consider the normalized ground state solution for the
following biharmonic Choquard type problem \begin{align*}
\begin{split}
\left\{
\begin{array}{ll}
\Delta^2u-\beta\Delta u=\lambda u+(I_\mu*F(u))f(u),
\quad\mbox{in}\ \ \mathbb{R}^4,
\displaystyle\int_{\mathbb{R}^4}|u|^2dx=c^2,\quad u\in H^2(\mathbb{R}^4),
\end{array}
\right.
\end{split}
\end{align*} where , , ,
with , is the primitive function
of , and is a continuous function with exponential critical growth in
the sense of the Adams inequality. By using a minimax principle based on the
homotopy stable family, we obtain that the above problem admits at least one
ground state normalized solution.Comment: arXiv admin note: text overlap with arXiv:2210.0088
Normalized solutions for a fractional Choquard-type equation with exponential critical growth in
In this paper, we study the following fractional Choquard-type equation with
prescribed mass
\begin{align*}
\begin{cases}
(-\Delta)^{1/2}u=\lambda u +(I_\mu*F(u))f(u),\ \ \mbox{in}\ \mathbb{R},
\displaystyle\int_{\mathbb{R}}|u|^2 \mathrm{d}x=a^2,
\end{cases}
\end{align*} where denotes the -Laplacian operator,
, , with
, is the primitive function of , and is a
continuous function with exponential critical growth in the sense of the
Trudinger-Moser inequality. By using a minimax principle based on the homotopy
stable family, we obtain that there is at least one normalized ground state
solution to the above equation.Comment: arXiv admin note: text overlap with arXiv:2211.1370
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