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Invariance of critical points under Kelvin transform and multiple solutions in exterior domains of \(\mathbb {R}^2\)

Abstract

Let \(\Omega ={\mathbb {R}}^2{\setminus }\overline{B(0,1)}\) be the exterior of the closed unit ball. We prove the existence of extremal constant-sign solutions as well as sign-changing solutions of the following boundary value problem

$$\begin{aligned} -\Delta u=a(x) f(u)\ \text{ in } \Omega ,\quad u=0\ \text{ on } \partial \Omega =\partial B(0,1), \end{aligned}$$

where the nonnegative coefficient a satisfies a certain integrability condition. We are looking for solutions in the space \(D^{1,2}_0(\Omega )\) which is the completion of \(C^\infty _c(\Omega )\) with respect to the \(\Vert \nabla \cdot \Vert _{2,\Omega }\)-norm. Unlike in the situation of \({\mathbb {R}}^N\) with \(N\ge 3\), the behavior of solutions in the borderline case \(N=2\) considered here is qualitatively significantly different, such as for example, constant-sign solutions in the borderline case are not decaying to zero at infinity, and instead are bounded away from zero. Our main tool in studying the above problem will be the Kelvin transform. We will first show that the Kelvin transform provides an isometric isomorphism between \(D^{1,2}_0(\Omega )\) and the Sobolev space \(H^1_0(B(0,1))\), which is order-preserving. This allows us to establish a one-to-one mapping between solutions of the problem above and solutions of an associated problem in the bounded domain B(0, 1) of the form:

$$\begin{aligned} -\Delta u=b(x) f(u)\ \text{ in } B(0,1),\quad u=0\ \text{ on } \partial B(0,1), \end{aligned}$$

where b satisfies an integrability condition in terms of the coefficient a. This duality approach given via the Kelvin transform allows us to handle nonlinearities under sub, super or asymptotically linear hypotheses.

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References

  1. Babuska, I.M., Vyborny, R.: Continuous dependence of eigenvalues on the domain. Czechoslov. Math. J. 15, 169–178 (1965)

    MathSciNet  MATH  Google Scholar 

  2. Carl, S., Perera, K.: Sign-changing and multiple solutions for the p-Laplacian. Abstr. Appl. Anal. 7(12), 613–625 (2002)

    MathSciNet  Article  Google Scholar 

  3. Carl, S., Motreanu, D.: Constant-sign and sign-changing solutions for nonlinear eigenvalue problems. Nonlinear Anal. 68, 2668–2676 (2008)

    MathSciNet  Article  Google Scholar 

  4. Carl, S., Costa, D.G., Tehrani, H.: Extremal and sign-changing solutions of supercritical logistic-type equations in \({\mathbb{R}}^N\). Calc. Var. Part. Differ. Equ. 54(4), 4143–4164 (2015)

    Article  Google Scholar 

  5. Carl, S., Costa, D.G., Tehrani, H.: \({\cal{D}}^{1,2}({\mathbb{R}}^N)\) versus \(C({\mathbb{R}}^N)\) local minimizer and Hopf-type maximum principle. J. Differ. Equ. 261(3), 2006–2025 (2016)

    Article  Google Scholar 

  6. Carl, S., Costa, D.G., Tehrani, H.: \(\cal{D}^{1,2}(\mathbb{R}^N)\) versus \(C(\mathbb{R}^N)\) local minimizer on manifolds and multiple solutions for zero-mass equations in \(\mathbb{R}^N\). Adv. Calc. Var. 11(3), 257–272 (2018)

    MathSciNet  Article  Google Scholar 

  7. Carl, S., Costa, D.G., Tehrani, H.: Extremal solutions of logistic-type equations in exterior domain in \(\mathbb{R}^2\). Nonlinear Anal. 176, 272–287 (2018)

    MathSciNet  Article  Google Scholar 

  8. Carl, S., Le, V.K., Motreanu, D.: Nonsmooth Variational Problems and Their Inequalities. Comparison Principles and Applications. Springer Monographs in Mathematics. Springer, New York (2007)

    Book  Google Scholar 

  9. Castro, A., Cossio, J., Neuberger, J.M.: A sign-changing solution for a superlinear Dirichlet problem. Rocky Mt. J. Math. 27(4), 1041–1053 (1997)

    MathSciNet  Article  Google Scholar 

  10. Di Benedetto, E.: \(C^{1,\alpha }\) Local regularity of weak solutions of degenerate elliptic equations. Nonlinear Anal. 7(8), 827–850 (1983)

    MathSciNet  Article  Google Scholar 

  11. Di Benedetto, E.: Partial Differential Equations. Birkhäuser, Boston (2010)

    Book  Google Scholar 

  12. Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (1983)

    Book  Google Scholar 

  13. Hörmander, L., Lions, J.L.: Sur la complétion par rapport a une intégrale de Dirichlet. Math. Scand. 4, 259–270 (1956)

    MathSciNet  Article  Google Scholar 

  14. Lieberman, G.: Boundary regularity for solutions of degenerate elliptic equations. Nonlinear Anal. 12(11), 1203–1219 (1988)

    MathSciNet  Article  Google Scholar 

  15. Motreanu, D., Motreanu, V.V., Papageorgiou, N.: Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems. Springer, New York (2014)

    Book  Google Scholar 

  16. Pucci, P., Serrin, J.: The Maximum Principle. Birkhäuser, Boston (2007)

    Book  Google Scholar 

  17. Simader, C.G., Sohr, H.: The Dirichlet Problem for the Laplacian in Bounded and Unbounded Domains. Pitman Research Notes in Mathematics Series. Addison Wesley Longman, Boston (1996)

    MATH  Google Scholar 

  18. Tehrani, H.: \(H^1\) versus \(C^1\) local minimizers on manifolds. Nonlinear Anal. 26(9), 1491–1509 (1996)

    MathSciNet  Article  Google Scholar 

  19. Tintarev, K., Fieseler, K.-H.: Concentration Compactness. Functional-Analytic Grounds and Applications. Imperial College Press, London (2007)

    Book  Google Scholar 

  20. Trudinger, N.S.: On Harnack type inequalities and their applications to quasilinear elliptic equations. Commun. Pure Appl. Math. 20, 721–747 (1967)

    MathSciNet  Article  Google Scholar 

  21. Wang, Z.-Q.: On a superlinear elliptic equation. Ann. Inst. Henri Poincaré 8, 43–57 (1991)

    MathSciNet  Article  Google Scholar 

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Acknowledgements

The authors are very grateful for the reviewer’s careful reading of the manuscript and useful comments that helped to improve the manuscript.

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Correspondence to Siegfried Carl.

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Communicated by A. Malchiodi.

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Carl, S., Costa, D.G., Fotouhi, M. et al. Invariance of critical points under Kelvin transform and multiple solutions in exterior domains of \(\mathbb {R}^2\). Calc. Var. 58, 65 (2019). https://doi.org/10.1007/s00526-019-1519-y

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  • DOI: https://doi.org/10.1007/s00526-019-1519-y

Mathematics Subject Classification

  • 35B38
  • 35B51
  • 35J20
  • 35J61