Solutions of biharmonic equations with steep potential wells

Sign-changing Multi-bump Solutions for Nonlinear Schrödinger Equations with Steep Potential Wells

We study the nonlinear Schrödinger equations: (Pλ) −∆u+(λa(x)+1)u = |u|p−1u, u ∈ H(R ), where p > 1 is a subcritical exponent, a(x) is a continuous function satisfying a(x) ≥ 0, 0 < lim inf |x|→∞ a(x) ≤ lim sup|x|→∞ a(x) < ∞ and a−1(0) consists of 2 connected bounded smooth components Ω1 and Ω2. We study the existence of solutions (uλ) of (Pλ) which converge to 0 in RN \ (Ω1 ∪Ω2) and to a presc...

The existence of nontrivial solution for a class of sublinear biharmonic equations with steep potential well

where 2u = ( u), N > 4, λ > 0, 1 < q < 2 and μ ∈ [0,μ0], 0 < μ0 <∞. The continuous function f verifies the assumptions: (f1) f (s) = o(|s|) as s→ 0; (f2) f (s) = o(|s|) as |s| →∞; (f3) F(u0) > 0 for some u0 > 0, where F(u) = ∫ u 0 f (t) dt. According to hypotheses (f1)–(f3), the number cf = max s =0 | f (s) s | > 0 is well defined (see [1]). The continuous functions α and K verify the assumptio...

Radial entire solutions for supercritical biharmonic equations ∗

We prove existence and uniqueness (up to rescaling) of positive radial entire solutions of supercritical semilinear biharmonic equations. The proof is performed with a shooting method which uses the value of the second derivative at the origin as a parameter. This method also enables us to find finite time blow up solutions. Finally, we study the convergence at infinity of regular solutions tow...

Existence and concentration of solutions for the nonlinear Kirchhoff type equations with steep well potential

Multiple Solutions for Biharmonic Equations with Asymptotically Linear Nonlinearities

Ruichang Pei1, 2 1 Center for Nonlinear Studies, Northwest University, Xi’an 710069, China 2 Department of Mathematics, Tianshui Normal University, Tianshui 741001, China Correspondence should be addressed to Ruichang Pei, [email protected] Received 26 February 2010; Revised 2 April 2010; Accepted 22 April 2010 Academic Editor: Kanishka Perera Copyright q 2010 Ruichang Pei. This is an open access ...