论文标题
具有L^2约束的非线性P拉普拉斯方程的归一化解决方案：质量超临界情况
Normalized solution to the nonlinear p-Laplacian equation with an L^2 constrain: mass supercritical case
论文作者
论文摘要
在本文中，我们研究了以下p-laplacian方程在某个方面的存在$ n \ geq3 $，其中$ l^2 $约束：\ begin {equation*} \ begin {case}}}-Δ__{p} p} \ Mathbb {r}^n，\\ {\ Vert U \ vert}^2_ {l^2（\ Mathbb {r}^n）} = m，\\ U \ in W^{1，p}（\ p} \ end {qore*}其中$-Δ_{p} u = div \ left（{{\ vert \ nabla u \ vert}^{p-2} {p-2} \ nabla u \ right）$，$ 2 \ leq p <n $，$ f \ in c（in c（in c） $μ\ in \ Mathbb {r} $将以Lagrange乘数出现，并且连续的非线性$ f $满足质量超临界条件。我们主要研究基态能源$ e_m $，$ m> 0 $在一定范围内更改，并旨在扩展非线性标量场方程时，当$ p = 2 $并减少非线性$ f $的约束条件。
In this paper, we study the existence of ground state solutions to the following p-Laplacian equation in some dimension $N\geq3$ with an $L^2$ constraint: \begin{equation*} \begin{cases} -Δ_{p}u+{\vert u\vert}^{p-2}u=f(u)-μu \quad \text{ in } \mathbb{R}^N,\\ {\Vert u\Vert}^2_{L^2(\mathbb{R}^N)}=m,\\ u\in W^{1,p}(\mathbb{R}^N)\cap L^2(\mathbb{R}^N), \end{cases} \end{equation*} where $-Δ_{p}u=div\left( {\vert\nabla u\vert}^{p-2}\nabla u \right)$, $2\leq p<N$, $f\in C(\mathbb{R},\mathbb{R})$, $m>0$, $μ\in\mathbb{R}$ will appear as a Lagrange multiplier and the continuous nonlinearity $f$ satisfies mass supercritical conditions. We mainly study the behavior of ground state energy $E_m$ with $m>0$ changing within a certain range and aim at extending nonlinear scalar field equation when $p=2$ and reducing the constraint condition of nonlinearity $f$.
