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Multiplicity of positive solutions for semilinear elliptic equations involving the critical Sobolev exponents. (English) Zbl 0879.35055

The authors are concerned with the multiplicity of positive solutions of \[ -\Delta u=Q(x)|u|^{p-1}u+\varepsilon|u|^{\sigma-1}u,\quad x\in\Omega,\quad u=0,\quad x\in\partial\Omega,\tag{1} \] where \(\Omega\) is a smooth and bounded domain in \(\mathbb{R}^N\), \(N\geq 3\), \(p=(N+2)/(N-2)\) so that \(p+1\) is the critical Sobolev exponent, \(\sigma\in[1,p)\), \(\varepsilon>0\) is a small number, an \(0\leq Q(x)\in C(\overline\Omega)\cap C^3(\Omega)\) satisfies some condition. They mainly prove the following:
1) Suppose that \(Q(x)\in C(\overline\Omega)\) and \(\sigma\in[1,p)\). Then the least energy solution \(u_\varepsilon\) of (1) satisfies \[ |\nabla u_\varepsilon|^2\to Q^{-(N-2)/2}_M S^{{N/2}}\delta_{x_0},\;|u_\varepsilon|^{p+1}\to Q^{-(N-2)}_M S^{{N/2}}\delta_{x_0}\text{ as }\varepsilon\to 0 \] in the sense of measure for some \(x_0\in\overline\Omega\) satisfying \(Q(x_0)= Q_M\).
2) Suppose that \(N\geq 4\), \(\sigma\in(1,p)\). Then there exists an \(\varepsilon_0>0\) such that for \(\varepsilon\in(0,\varepsilon_0)\), (1) has a solution \(u^j_\varepsilon\) satisfying \[ |\nabla u^j_\varepsilon|^2\to Q(a^j)^{-(N-2)/2}S^{{N/2}}\delta_{a^j},\;|u^j_\varepsilon|^{p+1}\to Q(a^j)^{-{N/2}}S^{{N/2}}\delta_{a^j}\text{ as }\varepsilon\to 0. \]

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
35B40 Asymptotic behavior of solutions to PDEs
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