In this paper we continue our investigation in [5, 7, 8] on multipeak solutions to the problem

formula here

where Δ = [sum ]Ni=1δ2/δx2i is the Laplace operator in ℝN, 2 < q < ∞ for N = 1, 2, 2 < q < 2N/(N−2) for N[ges ]3, and Q(x) is a bounded positive continuous function on ℝN satisfying the following conditions.

(Q1) Q has a strict local minimum at some point x0∈ℝN, that is, for some δ > 0

formula here

for all 0 < [mid ]x−x0[mid ] < δ.

(Q2) There are constants C, θ > 0 such that

formula here

for all [mid ]x−x0[mid ] [les ] δ, [mid ]y−y0[mid ] [les ] δ.

Our aim here is to show that corresponding to each strict local minimum point x0 of Q(x) in ℝN, and for each positive integer k, (1.1) has a positive solution with k-peaks concentrating near x0, provided ε is sufficiently small, that is, a solution with k-maximum points converging to x0, while vanishing as ε → 0 everywhere else in ℝN.

Solutions with peaks near the critical points of Q(x) are constructed for the problem

We establish the existence of 2k −1 positive solutions when Q(x) has k non-degenerate critical points in ℝN

Existence theorems and asymptotic properties will be obtained for boundary value problems of the form in an unbounded domain Ω⊆ RN(N ≥3) with smooth boundary, where Δ denotes the TV-dimensional Laplacian, τ — (N+ 2)/ (N — 2) is the critical Sobolev exponent, and is the completion of in the L2(Ω) norm of .

An existence theorem is obtained for a fourth-order semilinear elliptic problem in RN involving the critical Sobolev exponent (N + 4)/(N − 4), N>4. A preliminary result is that the best constant in the Sobolev embedding L2N/(N–4) (RN) is attained by all translations and dilations of (1 + ∣x∣2)(4-N)/2. The best constant is found to be

A class of weakly coupled systems of semilinear elliptic partial differential equations is considered in an exterior domain in ℝN, N > 3. Necessary and sufficient conditions are given for the existence of a positive solution (componentwise) with the asymptotic decay u(x) = O(|x|2-N ) as |x| —> ∞. Additional results concern the existence and structure of positive solutions u with finite energy in a neighbourhood of infinity.

Semilinear elliptic partial differential systems of second order with weak coupling are considered in exterior domains Ω ⊆ ℝN, N≧3. Conditions on the nonlinearities are given which guarantee the existence of solutions u with positive components in Ω such that u|∂Ω = 0 and u(x)→0 uniformly as |x|→∞. Asymptotic decay estimates for the solutions are established, including an exponential decay law under extra hypotheses.

Our main objective is to prove the existence of a pair of positive, exponentially decaying, classical solutions of the semilinear elliptic eigenvalue problem

1.1

in a smooth unbounded domain Ω ⊂ R N, N ≧ 2, where λ is a positive parameter and L is a uniformly elliptic operator in Ω defined by

The semilinear elliptic boundary value problem

1.1

will be considered in an exterior domain Ω ⊂ R n , n ≥ 2, with boundary ∂Ω ∊ C 2 + α , 0 < α < 1, where

1.2

D i = ∂/∂x i , i = 1, …, n. The coefficients aij , bi in (1.2) are assumed to be real-valued functions defined in Ω ∪ ∂Ω such that each , , and (aij (x)) is uniformly positive definite in every bounded domain in Ω. The Hölder exponent α is understood to be fixed throughout, 0 < α < 1 . The regularity hypotheses on f and g are stated as H 1 near the beginning of Section 2.