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Uniqueness of solutions for nonlinear Dirichlet problems with supercritical growth. (English) Zbl 1479.35429

Summary: We are concerned with Dirichlet problems of the form \[ \mathrm{div} (|Du|^{p-2} Du)+f(u)=0 \quad \text{in }\Omega,\qquad u=0\quad \text{on }\partial\Omega, \] where \(\Omega\) is a bounded domain of \(\mathbb{R}^n, n\geq 2, 1< p<n\) and \(f\) is a continuous function with supercritical growth from the viewpoint of the Sobolev embedding. In particular, if \(n=2\) and \(\gamma\colon [a,b]\to\mathbb{R}^2\) is a smooth curve such that \(\gamma (t_1)\neq\gamma (t_2)\) for \(t_1 \neq t_2\), we prove that, for \(\varepsilon >0\) small enough, there exists a unique solution of the Dirichlet problem in the domain \(\Omega =\Omega^{\Gamma}_{\varepsilon} =\{ (x_1, x_2)\in\mathbb{R}^2 :\mathrm{dist} \big( (x_1,x_2),\Gamma\big) < \varepsilon\}\), where \(\Gamma =\{\gamma (t):t\in [a,b]\}\). Moreover, we extend this uniqueness result to the case where \(n>2\) and \(\Omega\) is, for example, a domain of the type \[ \Omega =\widetilde{\Omega}^{\Gamma}_{\varepsilon,s} =\{ (x_1,x_2,y) : (x_1,x_2)\in\Omega^{\Gamma}_{\varepsilon}, y\in\mathbb{R}^{n-2}, |y|< s\}. \]

MSC:

35J62 Quasilinear elliptic equations
35J25 Boundary value problems for second-order elliptic equations
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
35J20 Variational methods for second-order elliptic equations
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References:

[1] A. Bahri and J.M. Coron, On a nonlinear elliptic equation involving the critical Sobolev exponent: the effect of the topology of the domain, Comm. Pure Appl. Math. 41 (1988), 253-294. · Zbl 0649.35033
[2] H. Brezis, Elliptic equations with limiting Sobolev exponents – the impact of topology, Comm. Pure Appl. Math. 39 (suppl.) (1986), S17-S39. · Zbl 0601.35043
[3] H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Comm. Pure Appl. Math. 36 (1983), no. 4, 437-477. · Zbl 0541.35029
[4] A. Carpio Rodriguez, M. Comte and R. Lewandowski, A nonexistence result for a nonlinear equation involving critical Sobolev exponent, Ann. Inst. H. Poincare Anal. Non Lineaire 9 (1992), no. 3, 243-261. · Zbl 0795.35032
[5] J.M. Coron, Topologie et cas limite des injections de Sobolev, C.R. Acad. Sci. Paris Ser. I Math. 299 (1984), no. 7, 209-212. · Zbl 0569.35032
[6] E.N. Dancer, A note on an equation with critical exponent, Bull. London Math. Soc. 20 (1988), no. 6, 600-602. · Zbl 0646.35027
[7] E.N. Dancer and K. Zhang, Uniqueness of solutions for some elliptic equations and systems in nearly star-shaped domains, Nonlinear Anal. 41 (2000), no. 5-6, Ser. A: Theory Methods, 745-761. · Zbl 0960.35035
[8] W.Y. Ding, Positive solutions of \(\Delta u + u^{(n+2)/(n−2)} = 0\) on contractible domains, J. Partial Differential Equations 2 (1989), no. 4, 83-88. · Zbl 0694.35067
[9] J. Kazdan and F.W. Warner, Remarks on some quasilinear elliptic equations, Comm. Pure Appl. Math. 28 (1975), no. 5, 567-597. · Zbl 0325.35038
[10] R. Molle and D. Passaseo, Positive solutions for slightly super-critical elliptic equations in contractible domains, C.R. Math. Acad. Sci. Paris 335 (2002), no. 5, 459-462. · Zbl 1010.35043
[11] R. Molle and D. Passaseo, Nonlinear elliptic equations with critical Sobolev exponent in nearly starshaped domains, C.R. Math. Acad. Sci. Paris 335 (2002), no. 12, 1029-1032. · Zbl 1032.35071
[12] R. Molle and D. Passaseo, Positive solutions of slightly supercritical elliptic equations in symmetric domains, Ann. Inst. H. Poincare Anal. Non Lineaire 21 (2004), no. 5, 639-656. · Zbl 1149.35353
[13] R. Molle and D. Passaseo, Nonlinear elliptic equations with large supercritical exponents, Calc. Var. Partial Differential Equations 26 (2006), no. 2, 201-225. · Zbl 1093.35022
[14] R. Molle and D. Passaseo, Multiple solutions of supercritical elliptic problems in perturbed domains, Ann. Inst. H. Poincare Anal. Non Lineaire 23 (2006), no. 3, 389-405. · Zbl 1172.35020
[15] R. Molle and D. Passaseo, Nonexistence of solutions for elliptic equations with supercritical nonlinearity in nearly nontrivial domains, Rend. Lincei Mat. Appl. 31 (2020), 121-130. · Zbl 1437.35311
[16] R. Molle and D. Passaseo, Nonexistence of solutions for Dirichlet problems with supercritical growth in tubular domains, Adv. Nonlinear Stud. 21 (2021), no. 1, 189-198. · Zbl 1487.35203
[17] L. Moschini, S.I. Pohozaev and A. Tesei, Existence and nonexistence of solutions of nonlinear Dirichlet problems with first order terms, J. Funct. Anal. 177 (2000), no. 2, 365-382. · Zbl 0976.35025
[18] D. Passaseo, Multiplicity of positive solutions of nonlinear elliptic equations with critical Sobolev exponent in some contractible domains, Manuscripta Math. 65 (1989), no. 2, 147-165. · Zbl 0701.35068
[19] D. Passaseo, On some sequences of positive solutions of elliptic problems with critical Sobolev exponent, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 3 (1992), no. 1, 15-21. · Zbl 0778.35036
[20] D. Passaseo, Existence and multiplicity of positive solutions for elliptic equations with supercritical nonlinearity in contractible domains, Rend. Accad. Naz. Sci. XL Mem. Mat. (5) 16 (1992), 77-98. · Zbl 0831.35064
[21] D. Passaseo, Nonexistence results for elliptic problems with supercritical nonlinearity in nontrivial domains, J. Funct. Anal. 114 (1993), no. 1, 97-105. · Zbl 0793.35039
[22] D. Passaseo, Multiplicity of positive solutions for the equation \(\Delta u + \lambda u + u^{2^* −1} = 0\) in noncontractible domains, Topol. Methods Nonlinear Anal. 2 (1993), no. 2, 343-366. · Zbl 0810.35029
[23] D. Passaseo, The effect of the domain shape on the existence of positive solutions of the equation \(\Delta u + u^{2^* −1} = 0\), Topol. Methods Nonlinear Anal. 3 (1994), no. 1, 27-54. · Zbl 0812.35032
[24] D. Passaseo, New nonexistence results for elliptic equations with supercritical nonlinearity, Differential Integral Equations 8 (1995), no. 3, 577-586. · Zbl 0821.35056
[25] D. Passaseo, Some concentration phenomena in degenerate semilinear elliptic problems, Nonlinear Anal. 24 (1995), no. 7, 1011-1025. · Zbl 0829.35043
[26] D. Passaseo, Multiplicity of nodal solutions for elliptic equations with supercritical exponent in contractible domains, Topol. Methods Nonlinear Anal. 8 (1996), no. 2, 245-262. · Zbl 0901.35034
[27] D. Passaseo, Nontrivial solutions of elliptic equations with supercritical exponent in contractible domains, Duke Math. J. 92 (1998), no. 2, 429-457. · Zbl 0943.35034
[28] S.I. Pohozaev, On the eigenfunctions of the equation \(\Delta u + \lambda f (u) = 0\), Soviet. Math. Dokl. 6 (1965), 1408-1411. · Zbl 0141.30202
[29] S.I. Pohozaev and A. Tesei, Existence and nonexistence of solutions of nonlinear Neumann problems, SIAM J. Math. Anal. 31 (1999), no. 1, 119-133. · Zbl 0944.35030
[30] O. Rey, The role of the Green’s function in a nonlinear elliptic equation involving the critical Sobolev exponent, J. Funct. Anal. 89 (1990), no. 1, 1-52. · Zbl 0786.35059
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