The purpose of this paper is to study a class of semilinear elliptic boundary value problems with degenerate boundary conditions which include as particular cases the Dirichlet and Robin problems. By making use of the Morse and Ljusternik-Schnirelman theories of critical points, we prove existence theorems of non-trivial solutions of our problem. The approach here is distinguished by the extensive use of the ideas and techniques characteristic of the recent developments in the theory of

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... ar elliptic boundary value problems with degenerate boundary conditions. The results here extend earlier theorems due to Ambrosetti-Lupo and Struwe to the degenerate case. 2010 Mathematics Subject Classification. Primary 35J65; Secondary 35J20, 47H10, 58E05. Key Words and Phrases. semilinear elliptic boundary value problem, degenerate boundary condition, multiple solution, Morse theory, Ljusternik-Schnirelman theory. 342 K. Taira (A) g ∈ C 1 (R) and g(0) = g (0) = 0. (B) The limits g (±∞) satisfy the conditions g (±∞) = lim s→±∞ g(s) s = +∞. Example 1.1. A simple example of the nonlinear term g(s) is given by the formula where p > 1 and q > 1. It is easy to verify that g(s) satisfies conditions (A) and (B). Since g(0) = 0, then u = 0 is a solution of the semilinear problem (1.3) for all λ. In this paper we establish existence theorems of non-trivial solutions (i.e., u = 0) of the semilinear problem (1.3). More precisely, our main purpose is to prove the following existence theorem, which is a generalization of Ambrosetti-Lupo [6, Theorem] to the degenerate case: Theorem 1.1. Assume that conditions (A) and (B) are satisfied. Then we have the following two assertions: ( i ) For each λ > λ 1 , the semilinear problem (1.3) has at least two non-trivial solutions u 1 , u 2 with u 1 > 0 in Ω and u 2 < 0 in Ω. ( ii ) For each λ > λ 2 , the semilinear problem (1.3) has at least a third non-trivial solution u 3 different from u 1 and u 2 . Rephrased, assertion (i) of Theorem 1.1 states that the semilinear problem (1.3) has at least two non-trivial solutions provided that the derivative f (s) = λ − g (s) of the function f (s) = λs − g(s) crosses the first eigenvalue λ 1 if |s| goes from 0 to ∞ (see Remark 1.1 below): Similarly, assertion (ii) of Theorem 1.1 states that the semilinear problem (1.3) has at least three non-trivial solutions provided that the derivative f (s) = λ − g (s) of f (s) crosses the two eigenvalues λ 1 and λ 2 if |s| goes from 0 to ∞: f (∞) = −∞ < λ 1 < λ 2 < λ = f (0). Remark 1.1. It is worth pointing out that the bifurcation solution curve (λ, u) of problem (1.3) is "formally" given by the formula