485 research outputs found

### On least Energy Solutions to A Semilinear Elliptic Equation in A Strip

We consider the following semilinear elliptic equation on a strip: $\left\{{array}{l} \Delta u-u + u^p=0 \ {in} \ \R^{N-1} \times (0, L), u>0,
\frac{\partial u}{\partial \nu}=0 \ {on} \ \partial (\R^{N-1} \times (0, L))
{array} \right.$ where $1< p\leq \frac{N+2}{N-2}$. When $1<p
0$ such that
for $L \leq L_{*}$, the least energy solution is trivial, i.e., doesn't depend
on $x_N$, and for $L >L_{*}$, the least energy solution is nontrivial. When $N
\geq 4, p=\frac{N+2}{N-2}$, it is shown that there are two numbers
$L_{*}<L_{**}$ such that the least energy solution is trivial when $L \leq
L_{*}$, the least energy solution is nontrivial when $L \in (L_{*}, L_{**}]$,
and the least energy solution does not exist when $L >L_{**}$. A connection
with Delaunay surfaces in CMC theory is also made.Comment: typos corrected and uniqueness adde

### Sign-changing solutions for Kirchhoff-type problems involving variable-order fractional Laplacian and critical exponents

In this paper, we are concerned with the Kirchhoff-type variable-order fractional Laplacian problem with critical variable exponent. By using constraint variational method and quantitative deformation lemma we show the existence of one least energy solution, which is strictly larger than twice of that of any ground state solution

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