## Topological Methods in Nonlinear Analysis

### The effect of the domain's configuration space on the number of nodal solutions of singularly perturbed elliptic equations

#### Abstract

We prove a new multiplicity result for nodal solutions of the Dirichlet problem for the singularly perturbed equation $-\varepsilon^2 \Delta u+u =f(u)$ for $\varepsilon> 0$ small on a bounded domain $\Omega\subset{\mathbb R}^N$. The nonlinearity $f$ grows superlinearly and subcritically. We relate the topology of the configuration space $C\Omega=\{(x,y)\in\Omega\times\Omega:x\not=y\}$ of ordered pairs in the domain to the number of solutions with exactly two nodal domains. More precisely, we show that there exist at least ${\rm cupl}(C\Omega)+2$ nodal solutions, where ${\rm cupl}$ denotes the cuplength of a topological space. We furthermore show that ${\rm cupl}(C\Omega)+1$ of these solutions have precisely two nodal domains, and the last one has at most three nodal domains.

#### Article information

Source
Topol. Methods Nonlinear Anal., Volume 26, Number 1 (2005), 109-133.

Dates
First available in Project Euclid: 23 June 2016

https://projecteuclid.org/euclid.tmna/1466705433

Mathematical Reviews number (MathSciNet)
MR2179353

Zentralblatt MATH identifier
1152.35039

#### Citation

Bartsch, Thomas; Weth, Tobias. The effect of the domain's configuration space on the number of nodal solutions of singularly perturbed elliptic equations. Topol. Methods Nonlinear Anal. 26 (2005), no. 1, 109--133. https://projecteuclid.org/euclid.tmna/1466705433

#### References

• A. Aftalion and F. Pacella, Qualitative properties of nodal solutions of semilinear elliptic equations in radially symmetric domains , preprint \ref\key 2
• A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications , J. Funct. Anal., 14 (1973), 349–381 \ref\key 3
• T. Bartsch, Critical point theory on partially ordered Hilbert spaces , J. Funct. Anal., 186 (2001), 117–152 \ref\key 4
• T. Bartsch, Z. L. Liu and T. Weth, Sign changing solutions to superlinear Schrödinger equations , Comm. Partial Differential Equations, 29(2004), 25–42 \ref\key 5
• T. Bartsch and Z.-Q. Wang, On the existence of sign changing solutions for semilinear Dirichlet problems , Topol. Methods Nonlinear Anal., 7 (1996), 115–131 \ref\key 6
• T. Bartsch and T. Weth, A note on additional properties of sign changing solutions to superlinear Schrödinger equations , Topol. Methods Nonlinear Anal., 22 (2003), 1–14 \ref\key 7 ––––, Three nodal solutions of singularly perturbed elliptic equations on domains without topology , to appear, Ann. Inst. H. Poincaré, Anal. Non Linéaire \ref\key 8
• T. Bartsch, T. Weth and M. Willem, Partial symmetry of least energy nodal solutions to some variational problems , to appear, J. Anal. Math. \ref\key 9
• V. Benci and G. Cerami, The effect of the domain topology on the number of positive solutions of nonlinear elliptic problems , Arch. Rational Mech. Anal., 114 (1991), 79–93 \ref\key 10 ––––, Multiple positive solutions of some elliptic problems via the Morse theory and the domain topology , Calc. Var. Partial Differential Equations, 2 (1994), 29–48 \ref\key 11
• A. Castro, J. Cossio and J. M. Neuberger, A sign-changing solution for a superlinear Dirichlet problem , Rocky Mountain J. Math., 27 (1997), 1041–1053 \ref\key 12 ––––, A minmax principle, index of the critical point, and existence of sign-changing solutions to elliptic BVPs , Electronic J. Differential Equations, 2 (1998), 18 \ref\key 13
• K. C. Chang, Infinite Dimensional Morse Theory and Multiple Solution Problems, Birkhäuser Boston (1993) \ref\key 14
• M. Clapp and D. Puppe, Critical point theory with symmetries , J. Reine Angew. Math., 418 (1991), 1–29 \ref\key 15
• M. Conti, S. Terracini and G. Verzini, Nehari's problem and competing species systems , Ann. Inst. H. Poincaré Anal. Non Linéaire, 19(2002), 871–888 \ref\key 16
• E. Dancer and Y. Du, Existence of sign-changing solutions for some semilinear problems with jumping nonlinearities , Proc. Roy. Soc. Edinburgh Sect. A, 124 (1994), 1165–1176 \ref\key 17 ––––, On sign changing solutions of certain semi-linear elliptic problems , Appl. Anal., 56(1995), 193–206 \ref\key 18
• E. Dancer and J. Wei, On the effect of domain topology in a singular perturbation problem , Topol. Methods Nonlinear Anal., 11(1998), 227–248 \ref\key 19
• E. N. Dancer and S. Yan, A singularly perturbed elliptic problem in bounded domains with nontrivial topology , Adv. Differential Equations, 4 (1999), 347–368 \ref\key 20
• K. Deimling, Ordinary differential equations in Banach spaces , Lecture Notes in Mathematics, 596 , Springer, Berlin (1978) \ref\key 21
• M. del Pino, P. L. Felmer and J. C. Wei, On the role of distance function in some singular perturbation problems , Comm. Partial Differential Equations, 25(2000), 155–177 \ref\key 22
• A. Dold, Algebraic Topology, Springer–Verlag, Berlin (1972) \ref\key 23
• G. Fournier and M. Willem, Relative category and the calculus of variations , Variational methods (Paris, 1988), vol. 4 of Progr. Nonlinear Differential Equations Appl., 95–104, Birkhäuser, Boston, MA (1990) \ref\key 24 ––––, Multiple solution of the forced double pendulum equation , Ann. Inst. H. Poincaré Anal. Non Linéaire, 6 (1989), 259–281 \ref\key 25
• B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle , Comm. Math. Phys., 68 (1979), 209–243 \ref\key 26
• S. J. Li and Z.-Q. Wang, Mountain pass theorems in order intervals and multiple solutions for semilinear elliptic Dirichlet problems , J. Anal. Math., 81 (2001), 373–396 \ref\key 27 ––––, Lusternik–Schnirelman theory in partially ordered Hilbert spaces , Trans. Amer. Math. Soc., 354 (2002), 3207–3227 \ref\key 28
• W. S. Massey, Homology and Cohomology Theory, Marcel Dekker, New York (1978) \ref\key 29
• E. Müller-Pfeiffer, On the number of nodal domains for eigenfunctions of elliptic differential operators , J. London Math. Soc. (2), 31 (1985), 91–100 \ref\key 30
• E. S. Noussair and J. C. Wei, On the effect of the domain geometry on the existence of nodal solutions in singular perturbations problems , Indiana Univ. Math. J., 46 (1996), 1255–1271 \ref\key 31
• W. M. Ni and J. C. Wei, On the location and profile of spike-layer solutions to singularly perturbed semilinear Dirichlet problems , Comm. Pure Appl. Math., 48 (1995), 731–768 \ref\key 32
• D. Smets M. and Willem, Partial symmetry and asymptotic behaviour for some elliptic variational problems , Calc. Var. Partial Differential Equations, 18 (2003), 57–75 \ref\key 33
• E. H. Spanier, Algebraic topology, McGraw Hill, New York (1966) \ref\key 34
• J. C. Wei and M. Winter, Symmetry of nodal solutions for singularly perturbed elliptic problems on a ball , to appear, Indiana Univ. Math. J. \ref\key 35
• M. Willem, Minimax Theorems, Birkhäuser, Boston (1996)