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Global boundedness and normalized solutions to a $p$-Laplacian equation

Raj Narayan Dhara, Matteo Rizzi

Abstract

In the paper, we prove the existence of radial solutions to \begin{equation}\notag%\label{main-eq-abstarct} %\begin{aligned} -Δ_p u+({\rm sgn}(p-s)+V(x))|u|^{p-2}u+λ|u|^{s-2}u=|u|^{q-2}u\qquad\text{in}\,\R^N \\ %\int_{\R^N}|u|^sdx&=ρ^s %\end{aligned} \end{equation} with prescribed $L^s(\R^N)$-norm, where $N\ge 3,\,p\in[2,N),\,s\in(1,p],\,q\in(p\frac{N+s}{N},\frac{Np}{N-p})$ and $V:\R^N\to\R$ is a suitable radial potential. We stress that $V$ is required to be radial but not necessarily bounded, and there are no assumptions about its sign. The case $V\equiv 0$ is also included. The proof is variational and relies on a min-max argument. A key-tool is the Pohozaev identity, which is shown to be true for any solution under quite weak assumptions about the potential $V$. This identity is proved with the aid of a new global boundedness result for subsolutions to a suitable $p$-Laplace equation.

Global boundedness and normalized solutions to a $p$-Laplacian equation

Abstract

In the paper, we prove the existence of radial solutions to \begin{equation}\notag%\label{main-eq-abstarct} %\begin{aligned} -Δ_p u+({\rm sgn}(p-s)+V(x))|u|^{p-2}u+λ|u|^{s-2}u=|u|^{q-2}u\qquad\text{in}\,\R^N \\ %\int_{\R^N}|u|^sdx&=ρ^s %\end{aligned} \end{equation} with prescribed -norm, where and is a suitable radial potential. We stress that is required to be radial but not necessarily bounded, and there are no assumptions about its sign. The case is also included. The proof is variational and relies on a min-max argument. A key-tool is the Pohozaev identity, which is shown to be true for any solution under quite weak assumptions about the potential . This identity is proved with the aid of a new global boundedness result for subsolutions to a suitable -Laplace equation.
Paper Structure (9 sections, 18 theorems, 156 equations)

This paper contains 9 sections, 18 theorems, 156 equations.

Table of Contents

  1. Introduction
  2. Global boundedness and the Pohozaev identity
  3. A deformation Theorem
  4. The right inverse of a non-linear operator
  5. The proof of Theorem \ref{['main-th-hom']}
  6. Symmetry
  7. Mountain pass geometry
  8. A bounded Palais-Smale sequence
  9. Compactness

Key Result

Theorem 1.1

Assume that $(V_1)$ holds and $\lim_{|x|\to\infty}V(x)=0$ if $\alpha\in\{N/p,\infty\}$. Then there exists $\rho_0>0$ such that, for any $\rho\in(0,\rho_0)$, there exists a constant $K_\rho>0$ and a solution $(\lambda_\rho,u_\rho)\in (0,\infty)\times X_r$ to problem main-eq such that $J_V(u_\rho)=c_{ Moreover, $u_\rho$ satisfies the Pohozaev identity $P_V(u)=0$ and $\rho^s\lambda_\rho\to\infty$ as

Theorems & Definitions (42)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Proposition 2.3
  • ...and 32 more