[Paper Review] Infinite many Blow-up solutions for a Schrödinger quasilinear elliptic problem with a non-square diffusion term

Carlos Alberto Santos, Jiazheng Zhou|arXiv (Cornell University)|Mar 3, 2016

Nonlinear Partial Differential Equations7 references3 citations

TL;DR

This paper establishes the existence of infinitely many blow-up solutions for a quasilinear Schrödinger-type elliptic equation with a non-square diffusion term in $\mathbb{R}^N$, using a dual variational approach and careful analysis of the potential's oscillation. The key result is that under suitable decay and oscillation conditions on the coefficient $a(x)$, and for nonincreasing $g$, the problem admits uncountably many positive solutions blowing up at infinity.

ABSTRACT

In this paper, we consider existence of positive solutions for the Schrödinger quasilinear elliptic problem $$ \left\{ \begin{array}{l} Δ_pu+Δ_p(|u|^{2γ})|u|^{2γ-2}u = a(x)g(u)~ \mbox{on}~ \mathbb{R}^N,\\ u>0\ \mbox{in}~\mathbb{R}^N,\ u(x)\stackrel{\left|x ight| ightarrow \infty}{\longrightarrow} \infty, \end{array} ight. $$ where $a(x), ~x\in \mathbb{R}^N$ and $g(s)~s>0$ are a nonnegative and continuous functions with $g$ being nonincreasing as well, $γ>{1}/{2}$, and $N \geq 1$. By a dual approach we establish sufficient conditions for existence and multiplicity of solutions for this problem.

Motivation & Objective

• To establish existence and multiplicity of positive solutions for a quasilinear Schrödinger equation with non-square diffusion in $\mathbb{R}^N$.
• To address the lack of results on blow-up solutions for such quasilinear problems with non-square diffusion terms.
• To extend prior existence results for semilinear and radial cases to the quasilinear, non-radial setting with general nonincreasing nonlinearity $g$.
• To characterize the role of the oscillation of the potential $a(x)$, defined via $a_{\text{osc}}(r) = \overline{a}(r) - \underline{a}(r)$, in enabling infinite multiplicity of solutions.
• To develop a dual variational framework that handles the non-convex, non-standard diffusion structure arising from $\Delta_p(|u|^{2\gamma})|u|^{2\gamma-2}u$.

Proposed method

• Employ a dual variational approach by transforming the original problem into an equivalent dual formulation involving a new unknown function $w = f(u)$, where $f$ is a suitable diffeomorphism.
• Define $w_{\alpha}(r)$ as a radial solution to a transformed ODE, using the radial structure of the problem and the assumption of radial symmetry in the potential $a(x)$.
• Introduce the oscillation function $a_{\text{osc}}(r) = \overline{a}(r) - \underline{a}(r)$ to quantify deviations from radial symmetry and control the difference between upper and lower bounds of $a(x)$ on spheres.
• Use sub- and super-solution techniques in expanding balls $B_n(0)$ to construct a sequence of approximating solutions $w_n$, which converge to a global solution $w \in C^1(\mathbb{R}^N)$.
• Establish pointwise bounds on $w_{\alpha}(r)$ using integral estimates involving $\overline{a}(t)$ and the inverse of the function $\mathcal{G}(t) = \int_1^t \frac{ds}{g(s)^{1/(p-1)}}$, under the Keller-Osserman-type condition.
• Prove that for any $\alpha > \mathcal{A}$ and $\epsilon > 0$, there exists a solution $w$ with $w(x) \to \infty$ as $|x| \to \infty$, by showing that the solution branch extends to infinity using a contradiction argument involving the integral $\overline{H} = \int_0^\infty \mathcal{H}(s) ds$.

Experimental results

Research questions

• RQ1 Under what conditions on $a(x)$ and $g(u)$ does the quasilinear Schrödinger equation with non-square diffusion admit positive solutions that blow up at infinity?
• RQ2 How does the oscillation of the potential $a(x)$, measured by $a_{\text{osc}}(r)$, affect the existence and multiplicity of blow-up solutions?
• RQ3 Can the dual variational method be adapted to handle non-convex, non-standard diffusion terms of the form $\Delta_p(|u|^{2\gamma})|u|^{2\gamma-2}u$ with $\gamma > 1/2$?
• RQ4 Is it possible to construct uncountably many distinct positive solutions to this problem, even when $a(x)$ is not radially symmetric?
• RQ5 What is the role of the Keller-Osserman-type integral condition $\int_1^\infty \frac{dt}{g(t)^{1/(p-1)}} = \infty$ in ensuring the existence of blow-up solutions?

Key findings

• The problem admits infinitely many positive solutions that blow up at infinity, provided the potential $a(x)$ satisfies $\int_0^\infty r a_{\text{osc}}(r) \exp(\underline{A}(r)) \, dr < \infty$, where $\underline{A}(r) = \int_0^r s \underline{a}(s) \, ds$.
• For any $\alpha > \mathcal{A}$, there exists a solution $w_{\alpha}$ such that $w_{\alpha}(r) \leq \mathcal{G}^{-1}\left(r \left(\int_0^r \overline{a}(t) \, dt\right)^{1/p - 1}\right)$ for all sufficiently large $r$.
• The existence of a solution branch $w_{\alpha}$ with $\alpha > \mathcal{A}$ can be extended to infinity, implying the existence of a solution $w$ with $w(x) \to \infty$ as $|x| \to \infty$, under the condition $\int_0^\infty \mathcal{H}(s) \, ds \leq \overline{H} < \infty$.
• The construction yields uncountably many distinct solutions by varying the initial parameter $\alpha > \mathcal{A}$, leading to infinite multiplicity of blow-up solutions.
• The proof relies on a contradiction argument showing that if $S(\beta_0) < \infty$, then $\beta_0 \leq \alpha + \overline{H}$, which contradicts $\beta_0 > \alpha + \overline{H}$, thus forcing $S(\beta_0) = \infty$.
• The method successfully handles non-convex, non-square diffusion terms by transforming the problem into a dual setting where comparison principles and integral estimates apply.