Sign-changing solutions for a critical exponential problem with competing potentials
DOI:
https://doi.org/10.12775/TMNA.2025.025Keywords
Equation in divergent form, Del Pino and Felmer's penalization methodAbstract
We establish the existence, concentration, and exponential decay of a family of sign-changing solutions for a problem involving exponential critical growth, described by the equation: $$ \begin{cases} -\epsilon^{2}\mbox{div}\left(a(x)\nabla u\right)+V(x)u =K(x) f(u) & \mbox{in $\mathbb{R}^2,$}\\ u \in H^{1}\big(\mathbb{R}^{2}\big). \end{cases} \leqno{(\rom{P}_{\epsilon})} $$ To address the lack of compactness and the competition between the potentials, we employ variational methods alongside a suitable truncation of the nonlinearity.References
C.O. Alves, Existence and multiplicity of positive solutions for a class of equations, Adv. Nonlinear Stud. 5 (2005), 73–86.
C.O. Alves and G.M. Figueiredo, Multiplicity and concentration of positive solutions for a class of quasilinear problems, Adv. Nonlinear Stud. 11 (2011), 265–295.
C.O. Alves and D.S. Pereira, Multiplicity of multi-bump type nodal solutions for a class of elliptic problems with exponential critical growth in R2 , Proc. Edinb. Math. Soc. (2) 60 (2017), no. 2, 273–297.
C.O. Alves and S.H.M. Soares, On the location and profile of spike-layer nodal solutions to nonlinear Schrödinger equations, J. Math. Anal. Appl. 296 (2004), 563–577.
C.O. Alves and S.H.M. Soares, Nodal solutions for singularly perturbed equations with critical exponential growth, J. Differential Equations 234 (2007), no. 2, 464–484.
C.O. Alves and M. Souto, Existence of least energy nodal solution for a Schrödinger–Poisson system in bounded domains, Z. Angew. Math. Phys. (to appear).
S. Barile and G.M. Figueiredo, Existence of least energy positive, negative and nodal solutions for a class of p& q-problems with potentials vanishing at infinity, J. Math. Anal. Appl. 427 (2015), 1205–1233.
T. Bartsch and T. Weth, Three nodal solutions of singularly perturbed elliptic equations on domains without topology, Ann. Inst. H. Poincaré Anal. Non Linéaire 22 (2005), 259–281.
D.M. Cao, Nontrivial solution of semilinear elliptic equation with critical exponent in R2 , Comm. Partial Differential Equations 17 (1992), 407–435.
A. Castro, J. Cossio and J. Neuberger, A sign-changing solution for a super linear Dirichlet problem, Rocky Mountain J. Math. 27 (1997), no. 4, 1041–1053.
S. Cingolani, Semiclassical stationary states of Nonlinear Schrödinger equations with an external magnetic field, J. Differential Equations 188 (2003), 52–79.
S. Cingolani and M. Lazzo, Multiple positive solutions to nonlinear Schrödinger equations with competing potential functions, J. Differential Equations 160 (200), 118–138.
G.S.A. Costa and G.M. Figueiredo, Existence and concentration of nodal solutions for a subcritical p& q equation, Commun. Pure Appl. Anal. 19 (2020), no. 11, 5077–5095.
G.S.A. Costa and G.M. Figueiredo, On a critical exponential p& N equation type: existence and concentration of changing solutions, Bull. Braz. Math. Soc. (N.S.) 53 (2022), no. 1, 243–280.
M. Del Pino and P. Felmer, Local mountain passes for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differential Equations 4 (1996), 121–137.
E. Di Benedetto, C 1,α local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal. 7 (1985), 827–850.
J.M.B. do Ó, N -Laplacian equations in RN with critical growth, Abstr. Appl. Anal. 2 (1997), 301–315.
G.M. Figueiredo and M.F. Furtado, Positive solutions for some quasilinear equations with critical and supercritical growth, Nonlinear Anal. 66 (2007), 1600–1616.
G.M. Figueiredo and F.B.M. Nunes, Existence of a least energy nodal solution for a class of quasilinear elliptic equations with exponential growth, Funkc. Ekv. 64 (2021), 293–322.
G.M. Figueiredo and S.M.A. Salirrosas, Multiplicity of semiclassical states solutions for a weakly coupled Schrödinger system with critical growth in divergent form, Potential Anal. 59 (2023), 237–261.
J. Liu, Y. Wang and Z.-Q. Wang, Solutions for quasilinear equations via the Nehari method, Comm. Partial Differential Equations 29 (2004),no. 5–6, 879–901.
A. Lyberopoulos, Quasilinear scalar field equations with competing potentials, J. Differential Equations 251 (2011), no. 12, 3625–3657.
F. Mingwen and H. Yin, Existence and concentration of bound states of nonlinear Schrödinger equations with compactly supported and competing potentials, Pacific J. Math. 244 (2010), no. 2, 261–296
A. Pomponio, Schrödinger equation with critical Sobolev exponent, Differential Integral Equations 17 (2004), no. 5–6, 697–707.
A. Pomponio and S. Secchi, On a class of singularly perturbed elliptic equations in divergence form: existence and multiplicity results, J. Differential Equations 207 (2004), 229–266.
P.H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys. 43 (1992), 270–291.
S. Secchi and M. Squassina, On the location of concentration points for singularly perturbed elliptic equations, Adv. Differential Equations 9 (2004), no. 1–2, 221–239.
W. Wang, X. Yang and F. Zhao, Existence and concentration of ground states to a quasilinear problem with competing potentials, Nonlinear Anal. 102 (2014), 120–132.
X. Wang and B. Zeng, On concentration of positive bound states of nonlinear Schrödinger equations with competing potential functions, Siam J. Math. Anal. 28 (1997), no. 3, 633–655.
M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996.
Published
How to Cite
Issue
Section
Stats
Number of views and downloads: 0
Number of citations: 0
