×

Local mountain passes for semilinear elliptic problems in unbounded domains. (English) Zbl 0844.35032

The authors consider the semilinear elliptic problem \[ \varepsilon^2 \Delta u- V(x) u+ f(u)= 0\quad\text{ in } \;\Omega,\quad u= 0\quad\text{on} \quad \partial\Omega,\quad u> 0,\tag{\(*\)} \] where \(\Omega\subset \mathbb R^n\) is a possibly unbounded domain. The function \(f\) is assumed to be of subcritical growth and \(f(\xi)/\xi\) is nondecreasing. The potential \(V\) is strictly positive and locally Hölder continuous. The main result of this paper states that there exists a positive solution of \((*)\) for sufficiently small \(\varepsilon> 0\), if \[ \inf_{G} V< \min_{\partial G} V \] holds for some domain \(G\) compactly contained in \(\Omega\). An asymptotic estimate for the solution is given, too. The proof of this result relies on a local mountain pass lemma. Since the energy functional associated to \((*)\) does not satisfy the usual Palais-Smale condition, the authors introduce a truncated functional, the critical points of which are also solutions of \((*)\) for small \(\varepsilon\).

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] M. del Pino, P. Felmer: Least energy solutions for elliptic equations in unbounded domains (To appear in Proc. Royal Soc. Edinburgh)
[2] W.Y Ding, W.M. Ni: ?On the existence of Positive Entire Solutions of a Semilinear Elliptic Equation?. Arch. Rat. Mech. Anal.91, 283-308 (1986) · Zbl 0616.35029 · doi:10.1007/BF00282336
[3] A. Floer, A. Weinstein: Nonspreading Wave Packets for the Cubic Schrödinger Equation with a Bounded Potential. J. Funct. Anal.69, 397-408 (1986) · Zbl 0613.35076 · doi:10.1016/0022-1236(86)90096-0
[4] B. Gidas, W.M. Ni, L. Nirenberg: Symmetry of positive solutions of nonlinear equations in ?N. Math. Anal. and Applications, Part A, Advances in Math. Suppl. Studies 7A, (ed. L Nachbin), Academic Press (1981), pp. 369-402
[5] W.C. Lien, S.Y Tzeng, H.C. Wang: Existence of solutions of semilinear elliptic problems in unbounded domains. Differential and Integral Equations6, 1281-1298 (1993) · Zbl 0837.35051
[6] Y.J. Oh: Existence of semi-classical bound states of nonlinear Schrödinger equations with potential on the class (V)a. Comm. Partial Differ. Eq.13, 1499-1519 (1988) · Zbl 0702.35228 · doi:10.1080/03605308808820585
[7] Y.J. Oh: Corrections to Existence of semi-classical bound states of nonlinear Schrödinger equations with potential on the class (V)a. Comm. Partial Differ. Eq.14, 833-834 (1989) · Zbl 0714.35078
[8] Y.J. Oh: On positive multi-lump bound states nonlinear Schrödinger equations under multiple well potential. Comm. Math. Phys.131, 223-253 (1990) · Zbl 0753.35097 · doi:10.1007/BF02161413
[9] P. Rabinowitz: On a class of nonlinear Schrödinger equations. Z. Angew. Math. Phys.43, 270-291 (1992) · Zbl 0763.35087 · doi:10.1007/BF00946631
[10] X. Wang: On concentration of positive bound states of nonlinear Schrödinger equations. Comm. Math. Phys.153, 229-244 (1993) · Zbl 0795.35118 · doi:10.1007/BF02096642
[11] M.C. Gui: Personal communication
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.