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Some remarks on elliptic equations with singular potentials and mixed boundary conditions. (English) Zbl 1237.35072
The authors study here the existence of nonnegative nontrivial solutions to the following degenerate singular semilinear elliptic problem with mixed boundary conditions in a smooth bounded domain \(\Omega\subset\mathbb R^n\) containing the origin:
\[ \begin{cases} -\text{div}(|x|^{-2\gamma}\nabla u)=\lambda |x|^{-2(\gamma+1)}u +\frac{u^p}{|x|^{2^*\gamma}}&\text{in }\Omega,\\ u\chi_{\Sigma_1}+|x|^{-2\gamma}\frac{\partial u}{\partial\nu} \chi_{\Sigma_2}=0&\text{on }\partial\Omega,\end{cases} \] where \(\chi_{\Sigma_i}\) is the characteristic function on \(\Sigma_i\); \(\Sigma_1,\Sigma_2\subset\partial\Omega\) are two smooth \((n-1)\)-dimensional manifolds, \(\Sigma_1\cap\Sigma_2=\emptyset\), \(\overline \Sigma_1\cup\overline\Sigma_2=\partial\Omega\) and \(\overline\Sigma_1 \cap\overline\Sigma_2\) is a smooth \((n-2)\)-dimensional manifold. Here \(\lambda\geq0\), \(-\infty<\gamma<\frac{n-2}{2}\) and \(n\geq3\).
Let \(\Gamma_{n,\gamma}(\Omega,\Sigma_1)\) denote the optimal Hardy-Sobolev constant under the above mixed boundary conditions; when \(\Sigma_1=\partial\Omega\) it reduces to the Hardy-Sobolev constant \(\Gamma_{n,\gamma}:=(\frac{n-2(\gamma+1)}{2})^2\) under Dirichlet boundary conditions, which is not attained (see [F. Catrina and Z. Q. Wang, Commun. Pure Appl. Math. 54, No. 2, 229–258 (2001; Zbl 1072.35506)] for details).
Under the above assumptions the authors prove that \(0<\Gamma_{n,\gamma}(\Omega,\Sigma_1)\leq\Gamma_{n,\gamma}\); moreover, they show that the strict inequality \(\Gamma_{n,\gamma}(\Omega,\Sigma_1)<\Gamma_{n,\gamma}\) holds if and only if \(\Gamma_{n,\gamma}(\Omega,\Sigma_1)\) is attained (this result in the case \(\gamma=0\) has been already obtained in [Z.-Q. Wang and M. Zhu, Electron. J. Differ. Equ. 2003, Paper No. 43, 8 p. (2003; Zbl 1109.35316)] and extended to the quasilinear setting in [the authors, J. Math. Anal. Appl. 332, No. 2, 1165–1188 (2007; Zbl 1166.35020)].
Some explicit examples of mixed boundary conditions for which the optimal Hardy-Sobolev constant is either attained or not achieved are given.
In the case \(\Gamma_{n,\gamma}(\Omega,\Sigma_1)=\Gamma_{n,\gamma}\) an improved Hardy-Sobolev inequality is given, modifying the proof used in [the authors, Calc. Var. Partial Differ. Equ. 23, No. 3, 327–345 (2005; Zbl 1207.35114)] under Dirichlet boundary conditions.
Taking advantage of this result, a uniform energy estimate in the space \(H_\gamma\) naturally associated to the improved Hardy-Sobolev inequality is obtained for the original semilinear problem. More precisely, the authors prove that for any nonlinearity \(0<p<2^*-1:=\frac{n+2}{n-2}\) the family of solutions \((\lambda,u)\in[0,\Gamma_{n,\gamma}(\Omega,\Sigma_1)]\times H_\gamma\) is a priori uniformly bounded in \(H_\gamma\).
Finally, the authors study the bifurcation behaviour of solutions for the semilinear problem.
If the optimal Hardy-Sobolev constant is attained, such a constant is shown to be the minimum of the spectrum and behaves as an eigenvalue. Precisely, if \(p<1\) the authors obtain bifurcation from infinity, while if \(p>1\) they get bifurcation from zero. Since the classical results by A. Ambrosetti and P. Hess [J. Math. Anal. Appl. 73, 411–422 (1980; Zbl 0433.35026)] and P. H. Rabinowitz [J. Funct. Anal. 7, 487–513 (1971; Zbl 0212.16504)] do not apply in that case, the authors do a direct study inspired by C. A. Stuart’s results [Prog. Nonlinear Differ. Equ. Appl. 27, 397–443 (1997; Zbl 0888.47045); Proc. Lond. Math. Soc., III. Ser. 57, No. 3, 511–541 (1988; Zbl 0673.35005)].
On the contrary, if \(\Gamma_{n,\gamma}(\Omega,\Sigma_1)=\Gamma_{n,\gamma}\) the authors show that there is no bifurcation in \(H_\gamma\), neither from infinity (\(p<1\)) nor from zero (\(p>1\)).

35J70 Degenerate elliptic equations
35J20 Variational methods for second-order elliptic equations
35J25 Boundary value problems for second-order elliptic equations
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
35J75 Singular elliptic equations
35J61 Semilinear elliptic equations
35B32 Bifurcations in context of PDEs
Full Text: DOI
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