Infinitely many solutions for the superlinear Dirichlet problem with fixed inhomogeneous data in G-invariant domains: the full subcritical range

For a bounded smooth domain $\Omega\subset\mathbb{R}^N$ ($N\ge3$) invariant under a closed subgroup $G\subset O(N)$, consider $-\Delta u=|u|^{p-2}u+f$ in $\Omega$ with $u=u_0$ on $\partial\Omega$, where $(f,u_0)$ is a fixed $G$-invariant datum. When $f=u_0=0$ the problem is odd in $u$ and has infinitely many solutions for every $p\in(2,2^*)$, $2^*=2N/(N-2)$ being the critical Sobolev exponent; any nonzero $f$ or $u_0$ breaks the odd symmetry. For arbitrary fixed data, infinite multiplicity is known only below thresholds strictly smaller than $2^*$: Bahri–Lions reach $p<2(N-1)/(N-2)$, Bolle–Ghoussoub–Tehrani reach $p<2N/(N-1)$, and the symmetry-based line of Clapp–Hernández-Martínez (2006) and Ackermann–Cano–Hernández-Martínez (2017) raises these bounds in terms of the orbit dimension of the $G$-action, covering the full subcritical range only when all orbits are sufficiently large. A conjecture of Bahri–Lions (1988) asserts that the fixed-forcing multiplicity result extends to the full subcritical range; Molle–Passaseo (2022, 2023) confirmed it when $\Omega$ is a cube (for $f\in L^2$, $u_0=0$) by a nodal-partition method that does not use deformation from symmetry, leaving general smooth domains, nonhomogeneous boundary data, and any $G$-invariant formulation open. The open question is whether infinitely many $G$-invariant solutions exist for every $p\in(2,2^*)$ and every fixed $G$-invariant pair $(f,u_0)$ — including the trivial group on general smooth domains, where this is the classical problem open since 1980–88.

Background

The model problem is the superlinear semilinear Dirichlet problem

$$-\Delta u=|u|^{p-2}u+f\ \text{in }\Omega,\qquad u=u_0\ \text{on }\partial\Omega,$$

where $\Omega\subset\mathbb{R}^N$ ($N\ge3$) is bounded with smooth boundary and $2<p<2^*$, with $2^*:=\frac{2N}{N-2}$ the critical Sobolev exponent (the Sobolev embedding $H_0^1(\Omega)\hookrightarrow L^p(\Omega)$ is compact exactly for $p<2^*$). Writing $u_0$ also for its harmonic extension, weak solutions in $H_0^1(\Omega)$ — the closure of $C_c^\infty(\Omega)$ under the norm $\|u\|^2=\int_\Omega|\nabla u|^2$ — are critical points of the $C^1$ functional $J(u)=\frac12\|u\|^2-\frac1p\int_\Omega|u+u_0|^p-\int_\Omega fu$. When $f=u_0=0$ this functional is odd in $u$, and the symmetric mountain-pass theorem (minimizing the supremum of the functional over images of finite-dimensional subspaces under odd continuous maps) yields an unbounded sequence of critical values, hence infinitely many solutions, for every $p\in(2,2^*)$. Any nonzero forcing $f$, or any nonzero boundary datum $u_0$, destroys the oddness and with it every equivariant multiplicity argument.

Whether infinite multiplicity survives such fixed symmetry-breaking perturbations is a classical question. Struwe (1980), Bahri–Berestycki (1981), Bahri (1981) and Rabinowitz (1982) created the perturbation-from-symmetry program; Bahri proved for $u_0=0$ that the full subcritical range holds for a residual — that is, topologically generic — set of forcings $f$. For every fixed $f$, however, Bahri–Lions (1988, 1992), combining minimax topology with Morse-index estimates based on the Lieb–Cwikel–Rosenbljum inequality (the classical bound on the number of nonpositive eigenvalues of a Schrödinger operator $-\Delta+V$ on $\mathbb{R}^N$ by $\int V_-^{N/2}$), reached only $p<\frac{2(N-1)}{N-2}$; Tanaka (1989) obtained the related symmetric-mountain-pass Morse index bounds. In the nonhomogeneous boundary case, Candela–Salvatore (1998) and then Bolle–Ghoussoub–Tehrani (2000), using Bolle's continuation theorem (1999) — which follows critical levels along a path $t\mapsto I_t$ of functionals interpolating from the even problem at $t=0$ to the perturbed one at $t=1$, comparing the gaps between consecutive symmetric min-max levels $c_k$ with the drift of levels along the path — reached $p<\frac{2N}{N-1}$. Both bounds are strictly below $2^*$, and Bolle–Ghoussoub–Tehrani state explicitly that reaching the full range would require an estimate $|\partial_tI_t(v)|\le C_\epsilon(|I_t(v)|^\epsilon+1)$ at critical points for every $\epsilon>0$, which is beyond the existing technique.

Clapp–Hernández-Martínez (2006) brought domain symmetry into the program. Let $G\subset O(N)$ be a closed subgroup of the orthogonal group with $\Omega$ $G$-invariant. The natural space is the fixed-point subspace $H_0^1(\Omega)^G=\{u\in H_0^1(\Omega):u(gx)=u(x)\ \forall g\in G\}$; by the principle of symmetric criticality (Palais 1979), critical points of the restricted functional are genuine weak solutions. For $x\in\Omega$ the orbit $Gx=\{gx:g\in G\}$ is a compact submanifold, and its dimension $m$ measures the amount of symmetry: Brüning–Heintze (1978/79) proved that the Dirichlet eigenvalues restricted to the invariant subspace satisfy $\lambda_k^G\sim k^{2/(N-m)}$, so the invariant spectrum is sparser as if the effective dimension were $N-m$; correspondingly the $G$-invariant Sobolev embeddings improve, with a higher critical exponent $2^*_{N,m}=\frac{2(N-m)}{N-m-2}$ (infinite when $N-m\le2$), as in the classical radial case $G=O(N)$ on an annulus studied by Castro–Kurepa (1987). Faster growth of the $G$-equivariant min-max levels lets the Bolle-type continuation tolerate larger $p$: Clapp–Hernández-Martínez proved that for $G$-invariant $f\in C(\overline\Omega)$ and $u_0\in C^2(\partial\Omega)$ there are infinitely many $G$-invariant solutions for $p$ below explicit thresholds $p_{N,m}$ (if $u_0=0$) and $\widetilde p_{N,m}$ (if $u_0\ne0$) that reduce to the Bahri–Lions and Bolle–Ghoussoub–Tehrani bounds for trivial $G$, increase with the orbit dimension, but remain strictly below $2^*$; they also obtained polynomial-in-$k$ energy growth of the solution sequence with exponent $\gamma=\frac{2p}{(N-m)(p-2)}$. Ackermann–Cano–Hernández-Martínez (2017) re-derived the spectral input through $G$-adapted Lieb–Cwikel–Rosenbljum bounds (exponent $\frac{N-m}{2}$ in terms of the minimal orbit dimension $m$, plus weighted variants for cylindrical domains $\Omega_1\times B_r^\ell$ with rotations on the second factor) and extended the admissible ranges to $p<2+\frac{2}{N-m-2}$ for $u_0=0$ (no restriction at all when $N-m\le2$) and $p<2+\frac{2}{N-m-1}$ for $u_0\ne0$ (assuming Hölder-continuous $f$ and $C^2$ boundary data) — precisely the classical bounds with $N$ replaced by the effective dimension $N-m$. These cover the full subcritical range only when all $G$-orbits have dimension at least about $(N-2)/2$ (about $N/2$ in the nonhomogeneous case); for small orbit dimension — and a fortiori for the trivial group — the gap between the known thresholds and $2^*$ remains.

A recent parallel development clarifies what remains at stake. Bahri–Lions (1988) themselves posed as a conjecture that their fixed-forcing multiplicity result should extend to the full subcritical range; Ramos–Tavares–Zou (2009) refined the Bahri–Lions technique, but the exponent barrier persisted. The breakthrough is due to Molle–Passaseo (2022, 2023), who for $-\Delta u=|u|^{q-1}u+\psi$ in $\Omega$ with $u=0$ on $\partial\Omega$ (that is, $u_0=0$ with forcing $\psi$; their exponent range $1<q<\frac{N+2}{N-2}$ is exactly $p\in(2,2^*)$ with $p=q+1$ in the $|u|^{p-2}u$ convention) developed a minimization method on nodal partitions: solutions of Dirichlet problems on subdomains that are bilipschitz deformations of small cubes are pieced together, yielding infinitely many nodal solutions with prescribed nodal structure, without any deformation from symmetry. Their 2022 theorem requires a geometric condition (condition (2.53) in that paper) on $\Omega$, verified only in dimension one and in radial-type situations; their 2023 theorem proves the Bahri–Lions conjecture when $\Omega$ is a cube of $\mathbb{R}^N$: for every $\psi\in L^2(\Omega)$ and every $p\in(2,2^*)$ there are infinitely many solutions. General smooth domains, nonhomogeneous boundary data, and every $G$-invariant formulation remain outside this method's reach.

Problem Statement

Let $N\ge3$, let $G\subset O(N)$ be a closed subgroup, let $\Omega\subset\mathbb{R}^N$ be bounded with smooth boundary and $G$-invariant, and let $f\in C(\overline\Omega)$ and $u_0\in C^2(\partial\Omega)$ be $G$-invariant ($u_0$ extended harmonically into $\Omega$). For $2<p<2^*=\frac{2N}{N-2}$ consider

$$-\Delta u=|u|^{p-2}u+f\ \text{in }\Omega,\qquad u=u_0\ \text{on }\partial\Omega.\tag{$\wp$}$$

Primary question. Prove, or disprove, the following statement: for every such $N$, $G$, $\Omega$ and every fixed $G$-invariant pair $(f,u_0)$, problem $(\wp)$ has infinitely many $G$-invariant weak solutions with norms in $H_0^1(\Omega)^G$ (equivalently, energy values $J(u_k)$) tending to infinity, for every $p\in(2,2^*)$. The statement includes the trivial group $G=\{e\}$, where it becomes the classical problem with no symmetry assumption — the Bahri–Lions conjecture (1988), open in general even for a ball with fixed forcing, and solved to date only for cubes with homogeneous boundary data (Molle–Passaseo 2023). A disproof must exhibit one instance ($N$, $G$, $\Omega$, $p\in(2,2^*)$, $G$-invariant data) for which $(\wp)$ has only finitely many $G$-invariant solutions. Partial answers restricted to generic data, to sufficiently large orbit dimension, or to special geometries (annuli, balls, cylinders — and, for the trivial group with homogeneous boundary data, cubes, where Molle–Passaseo (2023) proves the full range) do not settle the question.

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