Qualitative analysis of single peaked solutions for the supercritical Hénon problem

Abstract

We consider the supercritical Hénon problem

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u= C(\alpha )|x|^{\alpha }u^{p_{\alpha }-\varepsilon } &{}{\text {in}~\Omega },\\ u>0 &{}{\text {in}~\Omega },\\ u=0 &{}{\text {on}~\partial \Omega }, \end{array}\right. } \end{aligned}$$

where $p_{\alpha }=\frac{N+2+2\alpha }{N-2}$, $\alpha >0$, $\Omega \subset {\mathbb {R}}^{N}$ is a smooth bounded domain containing the origin, $C(\alpha )=(N+\alpha )(N-2)$ and $N\ge 3$. The existence single peaked solutions has been established by Gladiali and Grossi in [J. Differential Equation 253, 2616–2645 (2012)]. In this paper, we investigate qualitative property of the single peaked solutions for small $|\varepsilon |$. More precisely, we obtain the non-degeneracy and uniqueness of the single peaked solutions for $\varepsilon >0$ small enough. Moreover, if $\varepsilon <0$ small, we show that above problem has no single peaked solutions. It is known that when $\alpha =0$, the non-degeneracy and uniqueness of the single peaked solutions will depend on the Robin function. However, if $\alpha >0$, the role of Robin function will disappear because of the presence of the weighted term $|x|^{\alpha }$. Also our results indicate that the Hénon critical exponent $p_{\alpha }=\frac{N+2+2\alpha }{N-2}$ plays an essential role in the Hénon equation as the critical exponent $\frac{N+2}{N-2}$ in the case $\alpha =0$.

Appendix A. Some known facts

In this section, we give some well-known facts used in this paper.

Lemma A.1

It holds

$$\begin{aligned}&\int _{{\mathbb {R}}^{N}}|x|^{\alpha }U_1^{p_{\alpha }}Z(x)dx=0, \end{aligned}$$

(A.1)

$$\begin{aligned}&\int _{{\mathbb {R}}^{N}}|x|^{\alpha }U^{p_{\alpha }}_{1}Z(x)\log (1+|x|^{2+\alpha })^{\frac{N-2}{2+\alpha }}dx=A(\alpha )<0, \end{aligned}$$

(A.2)

$$\begin{aligned}&\int _{{\mathbb {R}}^{N}}|x|^{\alpha }\frac{|x|^{2+\alpha }-1}{(1+|x|^{2+\alpha })^{\frac{N+4+3\alpha }{2+\alpha }}}dx=B(\alpha )<0. \end{aligned}$$

(A.3)

Proof

The proof of the above lemma can be found in [9, Lemma 3.7]. $\square$

Define

$$\begin{aligned} L_{\lambda }:=-\Delta -p_{\alpha }C(\alpha )|x|^{\alpha }\left( PU_{\lambda }\right) ^{p_{\alpha }-1}I. \end{aligned}$$

Then the linearized operator $L_{\lambda }$ is a map from $H^{1}_{0}(\Omega )$ to $L^{2}(\Omega )$.

Consider the linearized problem

$$\begin{aligned} {\left\{ \begin{array}{ll} L_{\lambda }\Phi =\xi \quad \text {in}~~\Omega ,\\ \Phi =0\quad \quad \text {on}~~ \partial \Omega ,\\ \displaystyle \int _{\Omega }\nabla \Phi \cdot \nabla V_{\lambda }dx=0, \end{array}\right. } \end{aligned}$$

(A.4)

where $V_{\lambda }:=\frac{\partial PU_{\lambda }}{\partial \lambda }$. We can show that $L_{\lambda }$ is invertible in the weighted $L^\infty$ space if $\lambda$ is large enough. More precisely, using the similar argument in [9, Proposition 4.2], we can obtain the following result.

Proposition A.2

There exists $\lambda _{0}>0$ and $C>0$, independent of $\lambda$, such that for all $\lambda \ge \lambda _{0}$ and all $\xi \in L^{\infty }(\Omega )$, problem (A.4) has a unique solution $\Phi \equiv L^{-1}_{\lambda }(\xi )$. Moreover,

$$\begin{aligned} \Vert L^{-1}_{\lambda }(\xi )\Vert _{*}\le C\Vert \xi \Vert _{**}. \end{aligned}$$