Infinitely many solutions of the Dirichlet $p$-Laplacian with nonzero boundary data throughout the subcritical range

For the quasilinear elliptic Dirichlet problem $-\Delta_p u=|u|^{q-2}u+f(x)$ in a bounded $C^2$ domain, with boundary datum $u=\varphi$ on $\partial\Omega$, $1<p<q<p^*=Np/(N-p)$, $f\in C(\overline\Omega)$, $\varphi\in C^2(\overline\Omega)$, the known results (Bartolo 2013, via Bolle's perturbation method) establish infinitely many weak solutions only in a restricted explicit sub-range of the Sobolev subcritical range when the boundary datum is nonzero. The open problem asks whether multiplicity in fact holds for the full subcritical range $q\in(p,p^*)$ for arbitrary boundary data — including the semilinear case $p=2$ with nonzero datum up to the critical exponent $2^*=2N/(N-2)$, where the classical Bahri–Berestycki–Rabinowitz–Struwe theory covers only the zero-datum case.

Background

Let $\Omega\subset\mathbb{R}^N$ ($N>p$) be an open bounded domain with $C^2$ boundary. The $p$-Laplacian is the quasilinear operator $\Delta_p u=\operatorname{div}(|\nabla u|^{p-2}\nabla u)$, which degenerates where $\nabla u=0$ for $p>2$ and is singular for $p<2$; the model problem is

$$-\Delta_p u=|u|^{q-2}u+f(x)\ \text{in }\Omega,\qquad u=\varphi\ \text{on }\partial\Omega.\tag{$P_\varphi$}$$

Weak solutions live in the Sobolev class $\{u\in W^{1,p}(\Omega): u-\Phi\in W^{1,p}_0(\Omega)\}$ with $\Phi$ a $C^2$ extension of $\varphi$. The Sobolev embedding $W^{1,p}_0(\Omega)\hookrightarrow L^{p^*}(\Omega)$, with critical exponent $p^*=Np/(N-p)$, makes $q<p^*$ the natural subcritical regime in which the associated energy functional $J(u)=\frac1p\int_\Omega|\nabla u|^p-\frac1q\int_\Omega|u|^q-\int_\Omega fu$ satisfies the Palais–Smale compactness condition. When $f\equiv0$ and $\varphi\equiv0$, $J$ is even, and the symmetric mountain-pass theorem yields infinitely many weak solutions for every $q\in(p,p^*)$. A nonzero forcing term $f$ or a nonzero boundary datum $\varphi$ breaks the symmetry (evenness) of $J$; following the literature this is called a problem with broken symmetry. For $p=2$ with $\varphi\equiv0$, the perturbation theories of Struwe (1980), Bahri–Berestycki (1981) and Rabinowitz (1982) — often abbreviated BBRS — show that infinitely many solutions persist for the full subcritical range $q\in(2,2^*)$ despite the non-even perturbation $f$; the sharp growth estimates of the underlying symmetric min–max levels follow from the large Morse-index results of Bahri–Lions (1988). For $p\neq2$ the ambient space $W^{1,p}_0(\Omega)$ is Banach rather than Hilbert, Morse-index techniques are unavailable, and only restricted ranges are known: García Azorero and Peral Alonso (1991) treated the cube $\Omega=]0,1[^N$ with zero datum for $q$ below a root-defined exponent $\bar q(p)<p^*$, and Liu and Geng (2008) extended this to general smooth domains with an improved (still root-defined) range. Nonzero boundary data create an additional, more severe break of symmetry: writing $u=v+\Phi$, the shifted nonlinearity $|v+\Phi|^{q-2}(v+\Phi)$ contains cross terms growing like $|v|^{q-1}$, a superlinear perturbation outside the reach of the BBRS theory for $q$ near $p^*$. Bolle's perturbation method (Bolle 1999; Bolle, Ghoussoub and Tehrani 2000) splits the symmetric min–max levels along a one-parameter family and handles certain nonsymmetric problems, but again only below explicit exponent thresholds: Candela and Salvatore (1998) obtained infinitely many solutions of the semilinear problem with nonzero datum $g$ (and $f=0$) only for $q<2+2/N$, and Salvatore (2000) and Meng (2010) proved further restricted-range or resonant multiplicity results by the same method. Bartolo (2013) combined these lines: by Bolle's method she proved that $(P_\varphi)$ has infinitely many weak solutions on a general bounded $C^2$ domain with nonhomogeneous boundary data, but only for $q$ below explicit exponent bounds strictly smaller than $p^*$. Whether multiplicity holds throughout the full subcritical range — notably whether the semilinear case $p=2$ with nonzero datum admits infinitely many solutions for $q$ up to $2^*$, a range fully understood only in the zero-datum case — remains open.

Problem Statement

Let $\Omega\subset\mathbb{R}^N$ ($N>p>1$) be an open bounded domain with $C^2$ boundary, let $f\in C(\overline\Omega)$ and let $\varphi\in C^2(\overline\Omega)$, and let $p^*=Np/(N-p)$ denote the critical Sobolev exponent. Consider

$$-\Delta_p u=|u|^{q-2}u+f(x)\ \text{in }\Omega,\qquad u=\varphi\ \text{on }\partial\Omega.\tag{$P_\varphi$}$$

Determine whether $(P_\varphi)$ admits infinitely many weak solutions for every exponent $q$ in the full Sobolev subcritical range $q\in(p,p^*)$, for arbitrary boundary data — in particular for nonzero data $\varphi\not\equiv0$, where current results (Bartolo 2013, via Bolle's method) establish multiplicity only below explicit exponent thresholds strictly smaller than $p^*$. Two named sub-questions are part of the same objective: (i) the semilinear case $p=2$ with $\varphi\not\equiv0$ — do infinitely many solutions exist for all $q\in(2,2^*)$, $2^*=2N/(N-2)$, as they do for $\varphi\equiv0$ by the Bahri–Berestycki–Rabinowitz–Struwe theory? (ii) the quasilinear case — does multiplicity persist beyond the Bolle-type and root-type bounds up to $p^*$ (open even for $\varphi\equiv0$ when $p\neq2$)? The universal claim is confirmed by a proof that infinitely many weak solutions exist for the full range $q\in(p,p^*)$ in the stated settings; it is refuted by any single admissible $(f,\varphi)$ and exponent $q\in(p,p^*)$ for which $(P_\varphi)$ has only finitely many weak solutions. The case $\varphi\not\equiv0$ is of primary interest, and any admissible $(f,\varphi)$ meeting the stated hypotheses, including $\varphi\equiv0$ when $p\neq2$, refutes the universal claim.

References

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