Persistence of infinitely many solutions for the perturbed elliptic system $-\Delta u=v^{p-1}$, $-\Delta v=f(u)+h$ in the full subcritical range (Salvatore's open problems)

The nonhomogeneous Dirichlet system $-\Delta u=v^{p-1}$, $-\Delta v=f(u)+h(x)$ on a bounded smooth domain, with $1<p\le 2$ and $f$ an odd superlinear nonlinearity of subcritical growth, is known (Salvatore, 2010) to admit infinitely many weak solutions whenever the exponents and the integrability of $h$ satisfy an auxiliary quantitative condition tying $(p,q)$ to the integrability exponent of $h$; the same paper proves the unrestricted conclusion in the symmetric case $h=0$. The open questions are whether this artificial restriction can be removed so that infinite multiplicity persists for every exponent pair below the critical hyperbola $1/p+1/q=1-2/N$ (as the author conjectures at least under radial symmetry), whether an open dense (generic) set of perturbations $h$ yields infinitely many solutions, and whether the conclusion survives when the pure power $v^{p-1}$ is replaced by a general nonlinearity, so that the reduction to a single fourth-order quasilinear equation is no longer available. All three questions remain open in Salvatore (2010) and the surrounding perturbation-from-symmetry literature; the analogous exponent restriction is open even in the better-studied superquadratic regime.

Background

On a bounded smooth domain $\Omega\subset\mathbb{R}^N$ with Dirichlet boundary conditions, consider the coupled Lane–Emden system of Hamiltonian type

$$\begin{cases} -\Delta u=v^{p-1} & \text{in }\Omega,\\ -\Delta v=f(u)+h(x) & \text{in }\Omega,\\ u=v=0 & \text{on }\partial\Omega, \end{cases}$$

where $1<p\le 2$, $f$ is a continuous odd superlinear function, and $h$ is a given perturbation breaking the natural odd symmetry. Systems of this shape arise as stationary reaction–diffusion models and as model cases for Hamiltonian elliptic systems $-\Delta u=H_v$, $-\Delta v=H_u$, whose associated energy quadratic form has both infinitely many positive and infinitely many negative directions (a strongly indefinite structure). Because the first nonlinearity is exactly the pure power $v^{p-1}$, one can substitute $v=(-\Delta u)^{1/(p-1)}$ and equivalently study the single fourth-order quasilinear equation $-\Delta\bigl((-\Delta u)^{1/(p-1)}\bigr)=f(u)+h$ with Navier boundary conditions $u=\Delta u=0$ on $\partial\Omega$ (Navier conditions treat the fourth-order problem as a coupled pair of second-order Dirichlet problems). The natural energy space is the Banach space $E=W^{2,\frac{p}{p-1}}(\Omega)\cap W_0^{1,\frac{p}{p-1}}(\Omega)$ with norm $\|u\|_E=\bigl(\int_\Omega|\Delta u|^{\frac{p}{p-1}}dx\bigr)^{\frac{p-1}{p}}$, on which the energy functional $I_h(u)=\frac{p-1}{p}\int_\Omega|\Delta u|^{\frac{p}{p-1}}dx-\int_\Omega F(u)dx-\int_\Omega h u\,dx$, with $F(s)=\int_0^s f(t)dt$, is well defined and $C^1$ under suitable hypotheses; critical points of $I_h$ are weak solutions (and, for Hölder continuous $f$ and smooth $h$, classical $C^2$ solution pairs). For pure-power nonlinearities the compactness threshold for such systems is expressed by the critical hyperbola $1/p+1/q=1-2/N$: below this curve (i.e. $1/p+1/q>1-2/N$) the Sobolev embeddings of the natural energy space are compact and variational methods apply, while on or above it compactness is lost. The multiplicity theory for the unperturbed (odd, hence $\mathbb{Z}_2$-symmetric) problem $h=0$ is classical: assuming an Ambrosetti–Rabinowitz-type superlinearity condition (there exist $\mu>1+\frac{1}{p-1}$ and $s_0\ge 0$ with $0<\mu F(s)\le f(s)s$ for $|s|\ge s_0$), smallness of $f$ near zero, oddness of $f$, and, when the embedding $E\hookrightarrow L^\infty$ fails ($N\ge 4$, $\frac{N}{N-2}\le p\le 2$), a polynomial growth bound $|f(s)|\le a_1|s|^{q-1}+a_2$ with $\frac1p+\frac1q>1-\frac2N$, Salvatore proved that the symmetric system has infinitely many solutions via the symmetric mountain pass theorem (which produces an unbounded sequence of minimax levels for even functionals satisfying the Palais–Smale compactness condition, i.e. every sequence with bounded energy and vanishing gradient has a convergent subsequence). The nonhomogeneous case is far subtler, because a non-odd $h$ destroys the even symmetry on which the symmetric mountain pass rests. The scalar precedent is a classical program: Rabinowitz's perturbation method for $-\Delta u=|u|^{p-1}u+h$ (with precursors by Bahri–Berestycki and Struwe) yields infinitely many solutions only in a restricted exponent range, improved by Bahri–Lions and Tanaka via Morse-index estimates, while Bahri proved that, in the full natural subcritical range, an open dense set of perturbations $h\in H^{-1}(\Omega)$ gives infinitely many solutions — the scalar genericity phenomenon. For superquadratic Hamiltonian systems (both powers above 2), Tarsi and later Bonheure–Ramos extended these perturbation arguments (Bolle's method, a perturbative critical-point technique tracking minimax levels along a one-parameter family of functionals) to systems, again only in proper subregions of the subcritical range, with Bonheure–Ramos also proving open-dense/residual genericity statements; the 2014 survey by Bonheure, dos Santos and Tavares records that extending such theorems to the full superlinear range $pq>1$ was unknown except in dimension $N=1$. Within this landscape, dos Santos (2008) treated the scalar fourth-order quasilinear nonhomogeneous equation, and Salvatore (2010) treated the above coupled system with $1<p\le 2$: via the fourth-order reduction and Bolle's method she proved that the nonhomogeneous system has infinitely many solutions for every $h\in L^{\frac{\mu}{\mu-1}}(\Omega)$ whenever the parameters satisfy the additional restriction

$$\frac{\mu}{\mu-1}<\frac{2pq}{N\bigl(q(p-1)-p\bigr)}-1, \qquad\text{equivalently}\qquad \frac{1}{p}+\frac{1}{q}>1-\frac{2}{N}\cdot\frac{\mu-1}{2\mu-1},$$

which is strictly stronger than the critical-hyperbola condition $\frac1p+\frac1q>1-\frac2N$. Her paper closes by stating the three open problems considered below.

Problem Statement

Decide whether the persistence of infinite multiplicity under perturbation, established by Salvatore (2010) only under the auxiliary condition $\frac{\mu}{\mu-1}<\frac{2pq}{N(q(p-1)-p)}-1$, holds for the nonhomogeneous Dirichlet system

$$\begin{cases} -\Delta u=v^{p-1} & \text{in }\Omega,\\ -\Delta v=f(u)+h(x) & \text{in }\Omega,\\ u=v=0 & \text{on }\partial\Omega \end{cases}$$

on a bounded smooth domain $\Omega\subset\mathbb{R}^N$, in its natural full generality. The common objective is that the perturbation $h$ destroys the odd symmetry but not the infinitely many solutions; three linked questions, stated in the source, delimit the gap between what is proved and the natural statement:

(i) Full exponent range. For every $1<p\le 2$ and every growth exponent $q$ with $\frac1p+\frac1q>1-\frac2N$ (below the critical hyperbola), every odd continuous $f$ satisfying the superlinearity condition $0<\mu F(s)\le f(s)s$ for large $|s|$ (some $\mu>1+\frac{1}{p-1}$) and the growth bound $|f(s)|\le a_1|s|^{q-1}+a_2$, and every $h\in L^{\frac{\mu}{\mu-1}}(\Omega)$, does the system admit infinitely many distinct weak solutions (classical $C^2$ pairs when $f$ is Hölder continuous and $h$ smooth) — with no restriction linking $(p,q)$ and the integrability of $h$ beyond the critical hyperbola itself? The author conjectures an affirmative answer at least in the radially symmetric case: $\Omega$ a ball, $f$ and $h$ radial, infinitely many radial solutions.

(ii) Genericity in the perturbation. Does there exist an open dense (or at least residual) set of perturbations $h\in L^{\frac{\mu}{\mu-1}}(\Omega)$, topologized in that space (or, equivalently for the genericity question, in the dual space $E'$ of the energy space $E$), for which the system has infinitely many solutions, as Bahri proved for the scalar equation and Bonheure–Ramos for superquadratic systems?

(iii) Non-power first nonlinearity. If the pure power $v^{p-1}$ is replaced by a general nonlinearity $g(v)$ (not an exact pure power, even if it grows like a power asymptotically), so that the substitution $v=(-\Delta u)^{1/(p-1)}$ and the reduction to a single fourth-order Navier problem are unavailable, does the corresponding system $-\Delta u=g(v)$, $-\Delta v=f(u)+h$ still admit infinitely many solutions under the analogous superlinear/subcritical hypotheses?

If any of (i)–(iii) has a negative answer, the remaining question is which of the restrictions stated there is genuinely necessary, to be established by counterexample.

References

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