Sharp thresholds for infinitely many solutions of quasilinear equations perturbed from symmetry

For quasilinear elliptic Dirichlet problems of modified nonlinear Schrödinger type, written in divergence form as $-\mathrm{div}(A(x,u)\nabla u)+\tfrac{1}{2}A_t(x,u)|\nabla u|^2=g(u)+f$ in $\Omega$ with $u=0$ on $\partial\Omega$ (model case $A(x,t)=1+t^2$, i.e. $\Delta u+\tfrac{1}{2}u\Delta u^2+|u|^{p-2}u+f=0$), existence of infinitely many weak solutions is known when a non-symmetric forcing $f$ breaks the symmetry of the energy functional, but only under sufficient conditions: Liu and Zhao (2013) proved such an existence theorem for fixed $f\in L^{2}(\Omega)$ in the exponent window $4<p<4N/(N-2)$ under structural hypotheses on the coefficient $A$, and every subsequent result again provides only sufficient conditions. The open problem flowing from this theorem is to determine the sharp range of the growth exponent for which infinitely many solutions persist under a fixed symmetry-breaking forcing: whether multiplicity extends to the critical exponent $4N/(N-2)$ and beyond, whether it extends down to the lower endpoint $p=4$ and below, and, conversely, whether there exist data $(A,g,f)$ in this class for which the problem has only finitely many solutions. All existing literature provides only sufficient conditions; no endpoint result, no determination of the sharp supercritical threshold, and no non-existence or finite-multiplicity result is known.

Background

A broad and classical theme in nonlinear variational methods is that symmetry forces multiplicity. By the symmetric mountain pass theorem of Ambrosetti and Rabinowitz (1973), an even (i.e. symmetric under $u\mapsto -u$), superlinear, coercive-at-zero energy functional on a Banach space has infinitely many critical points; for semilinear elliptic Dirichlet problems this yields infinitely many weak solutions when the nonlinearity is odd in $u$. The classical perturbation-from-symmetry program asks what survives when this symmetry is broken: if a non-odd term, such as a fixed forcing $f(x)$, is added, do infinitely many solutions persist? Foundational contributions by Bahri and Berestycki (1981) and Struwe (1980) produced infinitely many critical points for non-even functionals under restrictions, while Rabinowitz (1982) developed a perturbation method showing that a $\varepsilon$-small non-symmetric perturbation of an even functional still has as many critical points as prescribed. In the semilinear setting, the fixed-perturbation question was answered affirmatively in broad subcritical ranges by Schechter and Zou (2005) and Ramos, Tavares and Zou (2009), who established infinitely many — indeed sign-changing — solutions for fixed, arbitrarily large non-odd perturbations under explicit sufficient conditions on the growth exponents; the remaining benchmark open questions of the program concern the quasilinear class and the critical and endpoint exponents.

The quasilinear chapter of this program is the subject here. Quasilinear equations of modified nonlinear Schrödinger type, $\Delta u+\tfrac{1}{2}u\,\Delta u^{2}+|u|^{p-2}u+f=0\quad\text{in }\Omega,\qquad u=0\quad\text{on }\partial\Omega,$ arise as stationary modified nonlinear Schrödinger equations modelling, for instance, superfluid films and self-channeling of high-power laser beams; the term $\tfrac{1}{2}u\Delta u^{2}$ makes the equation quasilinear (the differential operator depends on $u$ itself, not only on $\nabla u$). This equation is the model case $A(x,t)=1+t^{2}$ of the general divergence-form problem $-\mathrm{div}\big(A(x,u)\nabla u\big)+\tfrac{1}{2}A_t(x,u)\,|\nabla u|^{2}=g(x,u)+f(x)\quad\text{in }\Omega,\qquad u=0\quad\text{on }\partial\Omega,$ with $A_t=\partial A/\partial t$ (the identity $\tfrac{1}{2}u\Delta u^{2}=u^{2}\Delta u+u|\nabla u|^{2}$ converts one form into the other; later work, e.g. Huang 2022, writes the general form with a coefficient matrix $a_{ij}(x,u)$). The natural weak solutions live in a Sobolev space adapted to the energy density $A(x,u)|\nabla u|^{2}$; the term $u^{2}|\nabla u|^{2}=\tfrac{1}{4}|\nabla(u^{2})|^{2}$ makes the Sobolev critical exponent for the superlinear term $|u|^{p-2}u$ equal to $2\cdot 2^{*}=4N/(N-2)$ for $N\ge 3$, where $2^{*}=2N/(N-2)$ is the usual critical exponent for $H_0^1(\Omega)$.

For this class, Liu and Zhao (2013) proved, by Bolle-type perturbation arguments (Bolle's method tracks how the critical values of a one-parameter family of functionals evolve with the parameter, extending Rabinowitz's perturbation idea to non-even limits), that infinitely many solutions exist when $f\in L^{2}(\Omega)$ breaks the symmetry and $4<p<4N/(N-2)$, under structural assumptions on the coefficients generalizing uniform ellipticity. Subsequent work produced further sufficient conditions: Candela, Palmieri and Salvatore (2018) treated the general divergence form with a fixed measurable forcing and even allowed suitable supercritical growth of $g$ via a weak Cerami–Palais–Smale compactness condition (a sequential compactness condition on Cerami sequences — sequences of approximate critical points along which the energy stays bounded and a norm-weighted gradient tends to zero — weaker than the classical Palais–Smale condition, and adapted to quasilinear functionals whose energy density depends on $u$); Gui and Ge (2021) removed the Ambrosetti–Rabinowitz superlinear condition (the standard assumption $\mu G(x,t)\le t\,g(x,t)$, $\mu>2$, forcing superlinear growth and a priori bounds); Zhang, Tang and Chen (2019) proved infinite multiplicity of small negative-energy solutions for sublinear modified Schrödinger equations; and a parallel 2022–2026 wave (Huang 2022; Jing, Liu and Liu 2025 in the semilinear case; Huang, Wang and Zhang 2026) handles $\varepsilon$-small perturbations of general quasilinear equations with prescribed numbers of solutions under very weak growth conditions, the key tool being nondegenerate critical values of the unperturbed functional (isolated critical values that are stable under small perturbations of the functional). Every one of these results is a sufficient condition. Nowhere in this literature is there a non-existence or finite-multiplicity result, or a treatment of the critical endpoint $p=4N/(N-2)$ or the lower endpoint $p=4$ for the fixed-forcing problem — so the question of the sharp multiplicity threshold remains open.

Problem Statement

For a bounded smooth domain $\Omega\subset\mathbb{R}^{N}$, $N\ge 3$, consider the quasilinear Dirichlet problem in divergence form $-\mathrm{div}\big(A(x,u)\nabla u\big)+\tfrac{1}{2}A_t(x,u)\,|\nabla u|^{2}=g(x,u)+f(x)\quad\text{in }\Omega,\qquad u=0\quad\text{on }\partial\Omega,$ with model case $A(x,t)=1+t^{2}$, $g(x,t)=|t|^{p-2}t$, i.e. $\Delta u+\tfrac{1}{2}u\,\Delta u^{2}+|u|^{p-2}u+f=0\quad\text{in }\Omega,\qquad u=0\quad\text{on }\partial\Omega,$ where the coefficients belong to the standard structural class for this divergence form — written $a_{ij}(x,s)$ in the general formulation ($A$ is the scalar model case), conditions (A1)–(A3) of Liu, Liu and Wang (Proc. Amer. Math. Soc. 141 (2013) 253–263): $a_{ij}\in C^{1}$, $a_{ij}=a_{ji}$, with $a_{0}(1+s^{2})|\xi|^{2}\le a_{ij}(x,s)\xi_{i}\xi_{j}\le a_{1}(1+s^{2})|\xi|^{2}$ for constants $a_{1}\ge a_{0}>0$, and $0\le D_{s}a_{ij}(x,s)\,s\,\xi_{i}\xi_{j}\le 2a_{ij}(x,s)\xi_{i}\xi_{j}$ (uniform ellipticity with prescribed quadratic growth in $t$) — and $f\in L^{2}(\Omega)$ is a fixed non-symmetric forcing term that breaks the odd symmetry of the energy functional. It is known that infinitely many weak solutions exist under sufficient conditions, notably in the superquadratic window $4<p<4N/(N-2)$ with Ambrosetti–Rabinowitz-type growth on $g$ and structural assumptions on $A$ (Liu–Zhao 2013). Determine the sharp range of the growth exponent for which infinitely many solutions persist under a fixed symmetry-breaking forcing. The sharp range comprises the exponent window on which infinitely many solutions persist for every fixed non-symmetric $f\in L^{2}(\Omega)$, together with the complementary data for which the problem has only finitely many solutions. The named regimes and the converse question below are the constituent cases of this range. Specifically: is the known exponent window optimal — do infinitely many solutions persist at the Sobolev critical exponent $p=4N/(N-2)$, in supercritical regimes, or down to the lower endpoint $p=4$ and below (the known sublinear results concern small negative-energy solutions under their own additional hypotheses, and do not address the endpoint $p=4$ or the full threshold for the fixed-forcing problem)? Conversely, does there exist data $(A,g,f)$ in this class for which the problem has only finitely many solutions?

References

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