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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 (\(p>1\)).

MSC:
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
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References:
[1] Ambrosetti, Positive solutions of asymptotically linear elliptic eigenvalue problems no, Math Anal Appl pp 73– (1989)
[2] Ambrosetti, ıa Multiplicity results for some non - linear elliptic equations, Funct Anal pp 137– (1996)
[3] Ambrosetti, Dual variational methods in critical point the - ory and applications, Funct Anal 14 pp 349– (1973) · Zbl 0273.49063 · doi:10.1016/0022-1236(73)90051-7
[4] Rabinowitz, Some global results for nonlinear eigenvalue problems, Funct Anal 7 pp 487– (1971) · Zbl 0212.16504 · doi:10.1016/0022-1236(71)90030-9
[5] Wang, Caffarelli Nirenberg inequalities with remainder terms no, Funct Anal pp 203– (2003) · Zbl 1037.26014
[6] Stampacchia, Problemi al contorno ellitici con dati discontinui dotati di soluzioni ho lderiane Pura no, Ann Mat Appl pp 51– (1960)
[7] Lions, The concentration - compactness principle in the Calculus of Varia - tions The limit case Parts I and II no no, Rev Mat pp 145– (1985)
[8] Stuart, Bifurcation in Lp for a semilinear elliptic equation London no, Proc Math Soc pp 57– (1988)
[9] Ambrosetti, Combined effects of concave and convex nonlinearities in some elliptic problems no, Funct Anal pp 122– (1994)
[10] Caffarelli, First order interpolation inequalities with weights Compositio no, Math pp 53– (1984)
[11] Colorado, Semilinear elliptic problems with mixed Dirichlet - Neumann boundary conditions no, Funct Anal pp 199– (2003)
[12] Denzler, Bounds for the heat diffusion through windows of given area no, Math Anal Appl pp 217– (1998) · Zbl 0898.35068
[13] Catrina, On the Caffarelli Nirenberg inequalities : sharp constants existence nonexistence ) and symmetry of extremal functions Pure no, Appl Math 54 pp 229– (2001) · Zbl 1072.35506
[14] Colorado, Eigenvalues and bifurcation for elliptic equations with mixed Dirichlet - Neumann boundary conditions related to Caffarelli Nirenberg inequalities Nonlinear no, Methods Anal 23 pp 239– (2004) · Zbl 1075.35014
[15] Stuart, Bifurcation from the essential spectrum Topological nonlinear anal - ysis II Nonlinear Differential Equations Vol Birkha user Boston MA, Appl 27 pp 397– (1995)
[16] Abdellaoui, Some results for quasilinear elliptic equations related to some Caffarelli Nirenberg inequalities Pure no, Appl Anal 2 pp 539– (2003) · Zbl 1148.35324
[17] Abdellaoui, Existence and nonexistence results for quasilinear elliptic equations involving the p - Laplacian with a critical potential Pura no, Mat Appl pp 182– (2003) · Zbl 1223.35151
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