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In this paper, we are concerned with the ground states of the following planar Kirchhoff-type problem:\[ -\left(1+b\displaystyle\int_{\mathbb{R}^2}|\nabla u|^2\,{\rm d}x\right)\Delta u+\omega u=|u|^{p-2}u, \quad x\in\mathbb{R}^2. \]where $b,\, \omega >0$
are constants, $p>2$
. Based on variational methods, regularity theory and Schwarz symmetrization, the equivalence of ground state solutions for the above problem with the minimizers for some minimization problems is obtained. In particular, a new scale technique, together with Lagrange multipliers, is delicately employed to overcome some intrinsic difficulties.
$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}p$-SUPERLINEAR 
$p$-LAPLACIAN EQUATIONS
In this paper, we deduce new conditions for the existence of ground state solutions for the 
$p$-Laplacian equation which weaken the Ambrosetti–Rabinowitz type condition and the monotonicity condition for the function 
$$\begin{equation*} \left \{ \begin{array}{@{}ll} -\mathrm {div}(|\nabla u|^{p-2}\nabla u)+V(x)|u|^{p-2}u=f(x, u), \quad x\in {\mathbb {R}}^{N},\\[5pt] u\in W^{1, p}({\mathbb {R}}^{N}), \end{array} \right . \end{equation*}$$
$t\mapsto f(x, t)/|t|^{p-1}$. In particular, both 
$tf(x, t)$ and 
$tf(x, t)-pF(x, t)$ are allowed to be sign-changing in our assumptions.
In this paper, we consider the following Robin problem: where 
$$\begin{eqnarray*}\displaystyle \left\{ \begin{array}{ @{}ll@{}} \displaystyle - \Delta u= \mid x{\mathop{\mid }\nolimits }^{\alpha } {u}^{p} , \quad & \displaystyle x\in \Omega , \\ \displaystyle u\gt 0, \quad & \displaystyle x\in \Omega , \\ \displaystyle \displaystyle \frac{\partial u}{\partial \nu } + \beta u= 0, \quad & \displaystyle x\in \partial \Omega , \end{array} \right.&&\displaystyle\end{eqnarray*}$$
$\Omega $ is the unit ball in 
${ \mathbb{R} }^{N} $ centred at the origin, with 
$N\geq 3$, 
$p\gt 1$, 
$\alpha \gt 0$, 
$\beta \gt 0$, and 
$\nu $ is the unit outward vector normal to 
$\partial \Omega $. We prove that the above problem has no solution when 
$\beta $ is small enough. We also obtain existence results and we analyse the symmetry breaking of the ground state solutions.
We provide a detailed analysis of the minimizers of the functional $u \mapsto \int_{\Bbb R^n} |\nabla u|^2 + D \int_{\Bbb R^n} |u|^{\gamma}$ 
, $\gamma \in (0, 2)$ 
, subject to the constraint $\|u\|_{L^2} = 1$ 
. This problem, e.g., describes the long-time behavior of the parabolic Anderson in probability theory or ground state solutions of a nonlinear Schrödinger equation. While existence can be proved with standard methods, we show that the usual uniqueness results obtained with PDE-methods can be considerably simplified by additional variational arguments. In addition, we investigate qualitative properties of the minimizers and also study their behavior near the critical exponent 2.
